tag:blogger.com,1999:blog-53032460738241274712024-03-28T08:17:08.526-04:00Multiplication by Infinity(Indeterminate, like me. Think outside the box, but when you step outside the box ... try to keep one foot in)Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.comBlogger671125tag:blogger.com,1999:blog-5303246073824127471.post-50928191636511506622012-12-17T21:56:00.002-05:002012-12-18T12:11:37.987-05:00MoMath - Museum of Mathematics Opening Day<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmBj7IGxgKS6GzFZFhhppeXD5TgSY2a83GKmRyqslaZbADeAdxZydQH8cBQUHj9ZKYyUI6tlTcKemCI82NgEvGntiG2bqZoN7drbm9b-XRM3objKzQ7cdQgFrsnxEdEgzlqDe8sOz12Q/s1600/M0.jpg" imageanchor="1" style="margin-left: 1em;"><img border="0" height="304" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmBj7IGxgKS6GzFZFhhppeXD5TgSY2a83GKmRyqslaZbADeAdxZydQH8cBQUHj9ZKYyUI6tlTcKemCI82NgEvGntiG2bqZoN7drbm9b-XRM3objKzQ7cdQgFrsnxEdEgzlqDe8sOz12Q/s320/M0.jpg" width="320" /></a></div>
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I attended the Dec. 14 pre-opening and Dec. 15 opening day celebrations of America's one and only Museum of Mathematics in New York City. It was a great experience and a great museum. Glen Whitney, Director, and all his staff and contributors are to be congratulated on a job well done. The museum is located mid-block on 26th St. in Manhattan, between 5th and 6th Avenues.<br />
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Mathematics is THE ultimate language, but it needs work, and Mathematicians are working on it. From the youngest pre-School teacher to Alain Connes with his Non-Communicative Geometry project in France to The Langlands Program out of British Columbia and everyone in between, it is being worked.<br />
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Mathematics, taught VERY POORLY in America's Public Education school system, through no fault of the teachers themselves (they have the highest teacher drop-out rate) but rather the insane bureaucracy brought on by people in charge of Education who know NOTHING of Science, let alone Math (they are all political appointees), is THE deepest, THE widest, and THE tallest field of study.<br />
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It is a LANGUAGE, NOT a "Science." It transcends Science. It grew out of Logic. It is real, so real in fact, it would exist even if Reality itself never existed.<br />
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But forGET all the equations you ever learned by rote in a system maintained via TRADITION, of all the God-forsaken things, to make you HATE this MOST wonderful subject.<br />
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EXPERIENCE Mathematics, sans formulas, experience it hands on, interactively, at America's National Museum of Mathematics, in NYC. Bring the kids, including yourself, and be young again. A pleasant time is guaranteed for all.<br />
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Here are the names of the exhibits, not all but most of them, and prepare to THINK, not in a brain-stressing but rather in a fun and pleasant manner, THAT is what the museum is all about, and hopefully, each state and major city will repeat it's success, in time:<br />
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Light Grooves, Hyper Hyperboloid, Pattern Mesh, Structure Studio, Shapes of Space, Mathenaeum, Tracks of Galileo, Coaster Rollers, Twisted Thruway, String Product, Human Tree, Marble Multiplier, Math Square, Tessellation Station, 3D Doodle, Tile Factory, Sixth Sense, Monkey Around, Enigma Cafe, Rhythms of Life, Gallery of Innovation, Super Soma, Finding Fifteen, Feedback Fractals, Twist and Roll, and Marble Multiplier.<br />
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Several links before we show the tables.<br />
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<a href="http://momath.org/">The Official MoMath Website</a><br />
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<a href="https://www.facebook.com/MoMath1">The Official MoMath Facebook Page</a><br />
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<a href="http://www.scientificamerican.com/article.cfm?id=museum-puts-math-on-display">Scientific American magazine's article re MoMath's Opening</a><br />
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<a href="http://cosmiclog.nbcnews.com/_news/2011/06/13/6851255-exhibits-add-mirth-to-math"> Alan Boyle of Cosmic Log's ooriginal articl from a year ago descibing the museum to be</a><br />
From the article:<br />
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<span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px;">Although the museum is designed to appeal to all ages, the team is paying special attention to how well the exhibits go over with students in the fourth through the eighth grade.</span><br />
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"That's our sweet spot, for a very simple reason," Whitney said. "If you look at the trajectory of students going through the curriculum, things seem more or less fine up to the fourth grade. That period from the fourth to the eight grade is where we see a decline in the engagement of the students. Why are we opening a math museum in the first place? It's because we see cultural issues in this country."<br />
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International studies have shown that 15-year-old students in the U.S. perform well below the global average when it comes to math — specifically, 25th place out of 34 countries in 2009, when the Program for International Student Assessment's most recent test was conducted. EducationSecretary Arne Duncan said the results were "an absolute wakeup call for America."<br />
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Whitney has been awake and aware of this problem for a long time. He believes the standard sequence of math classes is way too limiting, and fails to engage students as much as they could be engaged. "Mathematics is actually much broader and richer than the list of topics that one reaches through the normal curriculum," he said.</div>
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<a href="https://www.facebook.com/media/set/?set=a.386563651429932.93509.100002289006504&type=3">My own MoMath Facebook Photo Album which will be expanded in time.</a><br />
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<a href="https://www.facebook.com/steve.colyer.1">My own Facebook page</a><br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuhy_gLefZubqKVE0yoOFHmCrbSEtzWOzrKCIf1lPsJrobB9rYm1zzjNyGJ4DThW2cTiEH-T9pL82sgvvg60cJj80AT7PbKJMs2X9vDHZF28FbWU1OofMC5910h8rGKrSfuC7JQQvTcA/s1600/M1.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuhy_gLefZubqKVE0yoOFHmCrbSEtzWOzrKCIf1lPsJrobB9rYm1zzjNyGJ4DThW2cTiEH-T9pL82sgvvg60cJj80AT7PbKJMs2X9vDHZF28FbWU1OofMC5910h8rGKrSfuC7JQQvTcA/s400/M1.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span aria-live="polite" class="fbPhotosPhotoCaption" id="fbPhotoSnowliftCaption" style="background-color: white; color: #333333; display: inline; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13px; line-height: 18px; outline: none; text-align: left; width: auto;" tabindex="0"><span class="hasCaption">Front entrance facing Madison Square Park, on 26th St. between 5th and 6th avenues.</span></span><span class="fbPhotoTagList" id="fbPhotoSnowliftTagList" style="background-color: white; color: #333333; display: inline; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13px; line-height: 18px; text-align: left;"><span class="fcg" style="color: grey;"> — at <span class="fbPhotoTagListTag withTagItem tagItem"><a class="taggee" data-hovercard-instant="1" data-hovercard="/ajax/hovercard/page.php?id=518890058124855" href="https://www.facebook.com/pages/Museum-of-Mathematics/518890058124855?ref=stream" style="color: #3b5998; cursor: pointer; text-decoration: initial;">Museum of Mathematics</a></span>.</span></span></td></tr>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyN7fS1N1xbi4A81iVYpLUq8BGXoMkPDL-rHnLmUUqvr_KHqtODYCmjqeLZyiNDNTH1fqm_7DuSsK5-CgPX_Kfe73sPbFWNUR4XfmrxoaPe4ls5HjcEEt4N5XRsmfKMlG1oKn3ejoTSQ/s1600/M2.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyN7fS1N1xbi4A81iVYpLUq8BGXoMkPDL-rHnLmUUqvr_KHqtODYCmjqeLZyiNDNTH1fqm_7DuSsK5-CgPX_Kfe73sPbFWNUR4XfmrxoaPe4ls5HjcEEt4N5XRsmfKMlG1oKn3ejoTSQ/s400/M2.jpg" width="300" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Main entrance to the museum.</td></tr>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjT_VZeA11UtmL_wL-p_qviXb97e_uCqwLxKTmnQzK5pE5DcsLlQceSWng95h8I5bYIiEXoqnr-nEMHMsZOaBjPRuk23YGfV-gMN-LzzRgZUgG_nn67f9aaDENUPaTD4jcj9H_zZjFd9w/s1600/M3.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjT_VZeA11UtmL_wL-p_qviXb97e_uCqwLxKTmnQzK5pE5DcsLlQceSWng95h8I5bYIiEXoqnr-nEMHMsZOaBjPRuk23YGfV-gMN-LzzRgZUgG_nn67f9aaDENUPaTD4jcj9H_zZjFd9w/s400/M3.jpg" width="300" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;">MoMath - Entrance view - pre-Opening night, Dec. 14, 2012</span><br />
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<span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;">The top Floor is Floor 0 and the bottom Floor (Entrance) is Floor -1</span></td></tr>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqONHfl4cgrKtr6gdhF_bJs02yT-wk07HryHhIW7Mk2Suf_ZhiHY2l-z565dtdqHgT2Idm8-FCU1fAgws9uI96UAsTIv0w51eUahPn5gGk_1C_sWqjfWVb-8tk_TiMSmr_zdfz34aJEA/s1600/M4.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqONHfl4cgrKtr6gdhF_bJs02yT-wk07HryHhIW7Mk2Suf_ZhiHY2l-z565dtdqHgT2Idm8-FCU1fAgws9uI96UAsTIv0w51eUahPn5gGk_1C_sWqjfWVb-8tk_TiMSmr_zdfz34aJEA/s640/M4.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;">Plaque of Founders</span></td></tr>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9fJ1dPy2NbHqN4OZifaY_qll-4qTiMhqmArAyQtPoJmTx07tz8gWcjAClmlVJEIX2w6KWd_gdfiWzlCJStk0KG5p_9O_u6Tx3-z9d8doXhhg_zi33aAzFXKOHztAuOk4-yJLZYM9sTQ/s1600/M5.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9fJ1dPy2NbHqN4OZifaY_qll-4qTiMhqmArAyQtPoJmTx07tz8gWcjAClmlVJEIX2w6KWd_gdfiWzlCJStk0KG5p_9O_u6Tx3-z9d8doXhhg_zi33aAzFXKOHztAuOk4-yJLZYM9sTQ/s640/M5.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">"Light Grooves", a holographic sculpture</td></tr>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQNRJ9lAsDq2nkWy7Z2jRuzWpPW_EgqEjR-QD2G3p4O_cDbLVeNppCqWY3eYikuuGK6SrPhlJjbvxxHUJybqaU7jPF8-baqIBXkN1VCztbPpRXlxNccWX49aKE1g3GaHxq2gUhcOTFtA/s1600/M6.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQNRJ9lAsDq2nkWy7Z2jRuzWpPW_EgqEjR-QD2G3p4O_cDbLVeNppCqWY3eYikuuGK6SrPhlJjbvxxHUJybqaU7jPF8-baqIBXkN1VCztbPpRXlxNccWX49aKE1g3GaHxq2gUhcOTFtA/s640/M6.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The same sculpture, but from an angle 90 degrees different. See if you can notice the difference.</td></tr>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvBYzK2yWPwg4j-8-S-IMJefnPKxsclxPs64G5M6XtuHfSYHAh9gvDzghqn0fFciQvJQuR5PIecA6hHK70SZoaBbpDwKhtTPxtnzQEoi1ZVz0nXg1lCVnlJvbmv7WlmLZU2vO6QDZaFA/s1600/M7.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvBYzK2yWPwg4j-8-S-IMJefnPKxsclxPs64G5M6XtuHfSYHAh9gvDzghqn0fFciQvJQuR5PIecA6hHK70SZoaBbpDwKhtTPxtnzQEoi1ZVz0nXg1lCVnlJvbmv7WlmLZU2vO6QDZaFA/s640/M7.jpg" width="480" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Mathematical pewter jewelry on sale in the Museum Shop.</td></tr>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbwrQOZhzI_z4AAilXuUl6aO0IvKIl82zjMDT26U-2GsEoJxyYs7xnKCXuIc3XZppTem7GkoAi2KDuYehLwl6X7lP5YzcCBJSzQ6INuw1aOsLbj9xH2Io2YRBNwCr-RLcBwPUBOECX1Q/s1600/M8.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbwrQOZhzI_z4AAilXuUl6aO0IvKIl82zjMDT26U-2GsEoJxyYs7xnKCXuIc3XZppTem7GkoAi2KDuYehLwl6X7lP5YzcCBJSzQ6INuw1aOsLbj9xH2Io2YRBNwCr-RLcBwPUBOECX1Q/s640/M8.jpg" width="480" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">More Mathematical jewelry</td></tr>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpv0zsX2ffGsjW212PX6zArW3aNMDxjBs9rBQiUgDBGaBFEmRsbjWbB2Yy1tA6BnwIE8ARoI9Vd-Iz43E3SJCOHwWk_U0TIn2AAlTCtWMgHAd1EgrN-X4EjORC2m6cbIXlADJYKSWN_w/s1600/M11.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpv0zsX2ffGsjW212PX6zArW3aNMDxjBs9rBQiUgDBGaBFEmRsbjWbB2Yy1tA6BnwIE8ARoI9Vd-Iz43E3SJCOHwWk_U0TIn2AAlTCtWMgHAd1EgrN-X4EjORC2m6cbIXlADJYKSWN_w/s640/M11.jpg" width="640" /></a></div>
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<tr><td class="tr-caption" style="text-align: center;">Structure Studio</td></tr>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgaF7FBIafHZShHPh_MkOrqTwbBAzc5FKDpPKOFTqSAbovjIdLqIp70RW1aWcbcEMq_8y26VnLpm_iH2FeDJnQFBwxssHt68BuGtwVFylbJsrKqCmeNCHy1K0Y3pHtdOC25YNX0O5ENA/s1600/M13.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgaF7FBIafHZShHPh_MkOrqTwbBAzc5FKDpPKOFTqSAbovjIdLqIp70RW1aWcbcEMq_8y26VnLpm_iH2FeDJnQFBwxssHt68BuGtwVFylbJsrKqCmeNCHy1K0Y3pHtdOC25YNX0O5ENA/s400/M13.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Shapes of Space</td></tr>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizzWu2Gl8PAZ5TnhAF8_NcdFxSev7JE8PFoSoWb2PU0srqC_QWUqBh0b1dnKlVdDLEw7nH2DYJjnw-772gcMLRLy-i4YKp5A-4pssvlA0CgVOAXoyevNgo-pGIPj-CxJxRF9NvwOAyWw/s1600/M14.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizzWu2Gl8PAZ5TnhAF8_NcdFxSev7JE8PFoSoWb2PU0srqC_QWUqBh0b1dnKlVdDLEw7nH2DYJjnw-772gcMLRLy-i4YKp5A-4pssvlA0CgVOAXoyevNgo-pGIPj-CxJxRF9NvwOAyWw/s640/M14.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The Mathenaeum kiosk and Tracks of Galileo</td></tr>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeoYImi_r5c4NfLudDgNvo4zpiiOGXI5gNxFvkjebNWlAUAElJ7h9iQLOJi0DHV56Q0pKUop5LDnL1oJ73xkTMvKt0PE-8D7wuOHAMBJyV_WsGqxL3LTtadlgd7TTuDu2wwqoOzBEMWQ/s1600/M15.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeoYImi_r5c4NfLudDgNvo4zpiiOGXI5gNxFvkjebNWlAUAElJ7h9iQLOJi0DHV56Q0pKUop5LDnL1oJ73xkTMvKt0PE-8D7wuOHAMBJyV_WsGqxL3LTtadlgd7TTuDu2wwqoOzBEMWQ/s640/M15.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Ride the square wheeled tricycles at the Twisted Thruway</td></tr>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKgT2MOjJxox9sJ7kycxJ86iQfwNMYmj1Lm4l3rW-9osy3_hfxpkrQOG30diZCtS2jYU77zWvh9EM_bNOEsrpgwfFmBTfL2C2Yy_biyJ7i482Lo6A8DFe56-WVa7WjJTg8Wl8R-j6DxA/s1600/M17.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKgT2MOjJxox9sJ7kycxJ86iQfwNMYmj1Lm4l3rW-9osy3_hfxpkrQOG30diZCtS2jYU77zWvh9EM_bNOEsrpgwfFmBTfL2C2Yy_biyJ7i482Lo6A8DFe56-WVa7WjJTg8Wl8R-j6DxA/s400/M17.jpg" width="300" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">String Product shows multiplication in 3-D, as well as being the center of the helical stairs to the next floor.</td></tr>
</tbody></table>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFc1FQR_DDJuCAMurVd3HD5zLpSLsa-eoUlPzJoBNVd9UVkS_g-PZr2bw-ud34kUUr5oXPU3j4gfYouhA4JaoCDjA7oznazguismDX6LAe2AWO9wqxECZ1orW3MM825xCDgKFeg85_pg/s1600/M18.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFc1FQR_DDJuCAMurVd3HD5zLpSLsa-eoUlPzJoBNVd9UVkS_g-PZr2bw-ud34kUUr5oXPU3j4gfYouhA4JaoCDjA7oznazguismDX6LAe2AWO9wqxECZ1orW3MM825xCDgKFeg85_pg/s640/M18.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Math Square is highly interactive and changes as you walk on it.</td></tr>
</tbody></table>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_F9yyFuXmJkyE4pu5Od0ztrdFrH_q29XKOCHEyAc02krulm1Gu8Qr4VzsjrkYz9rO0Bn0OHD2CvuthzWxldQplx7UNn6YdbS1CyqImOripzHukcVK1ivroWsUTpBVyWTTbl-3g4Q01Q/s1600/M19.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_F9yyFuXmJkyE4pu5Od0ztrdFrH_q29XKOCHEyAc02krulm1Gu8Qr4VzsjrkYz9rO0Bn0OHD2CvuthzWxldQplx7UNn6YdbS1CyqImOripzHukcVK1ivroWsUTpBVyWTTbl-3g4Q01Q/s640/M19.jpg" width="480" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The nodes on this sculpture provide sound as you touch them. </td></tr>
</tbody></table>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguwc_UPGFq5eFW9if3Oh3VDtHtRFB2LjuYgZ5lUgwao45TlttjRNbKOiQsGPC8qpNl1voYvTDNDF7EHBX71xW7e2oDUR48a4OpdLqLBqmgh43ng7wtccSwNVtduyJEV4CEjR10MGZIWA/s1600/M20.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguwc_UPGFq5eFW9if3Oh3VDtHtRFB2LjuYgZ5lUgwao45TlttjRNbKOiQsGPC8qpNl1voYvTDNDF7EHBX71xW7e2oDUR48a4OpdLqLBqmgh43ng7wtccSwNVtduyJEV4CEjR10MGZIWA/s640/M20.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Tessellation Station</td></tr>
</tbody></table>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvtBcqmuPslwUFYFteKA35B11E10RNCOZRdCLF7i0_wftXfxHxkkH_VnRIRdZHbbROZnVwundZTZ1zSmHIyUAn7vV18rzZwv1mGI3niXLxdbsvcbngRU0jvEyyRLGETEcIzKQFa-FMag/s1600/M21.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvtBcqmuPslwUFYFteKA35B11E10RNCOZRdCLF7i0_wftXfxHxkkH_VnRIRdZHbbROZnVwundZTZ1zSmHIyUAn7vV18rzZwv1mGI3niXLxdbsvcbngRU0jvEyyRLGETEcIzKQFa-FMag/s640/M21.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The Enigma Cafe has many Magic Puzzles at various tables to try out. </td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIfglEJw8RAPRWGIOf3ksRHnI5VAiqpDhkYX9gsZcfgsiKWdAd0TzLDAUqH96wOM7IHJOaTQ6dSAn3EoLyWiLQx5hUs7km9CDJvxWAjRqFa_dhZJC8Ydgk8hBysdinNpEtFGKR4lzyaw/s1600/M22.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIfglEJw8RAPRWGIOf3ksRHnI5VAiqpDhkYX9gsZcfgsiKWdAd0TzLDAUqH96wOM7IHJOaTQ6dSAn3EoLyWiLQx5hUs7km9CDJvxWAjRqFa_dhZJC8Ydgk8hBysdinNpEtFGKR4lzyaw/s640/M22.jpg" width="480" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Rhythm of Life</td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwN6mubHXiik5m3_BRQaxu08N4p824wxIYqBo6LXQE8CtPzzcsmX-wpwU6anj0g9wnPBUV9GpHhz2r3cQ64xnQxjnBU2egoLBEY0BR3Sq1n-bCRhMhS1l3YSPPG5u9GQNDnCnnCGDT0g/s1600/M23.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwN6mubHXiik5m3_BRQaxu08N4p824wxIYqBo6LXQE8CtPzzcsmX-wpwU6anj0g9wnPBUV9GpHhz2r3cQ64xnQxjnBU2egoLBEY0BR3Sq1n-bCRhMhS1l3YSPPG5u9GQNDnCnnCGDT0g/s640/M23.jpg" width="480" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Gallery of Innovation</td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwIEkKsBcDaTeuZjMMOHPwzs2kfsejYZeXcJyuuIm4V4hEkZjQ-4p6K783jt6rO6VWqk3zhn8qMCK2cZIRkgz7YX__HZYv9A60xwG7kuF5Ha0nX8R67beH1DhvTdUa3Z5XhtNjO1fNXQ/s1600/M24.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwIEkKsBcDaTeuZjMMOHPwzs2kfsejYZeXcJyuuIm4V4hEkZjQ-4p6K783jt6rO6VWqk3zhn8qMCK2cZIRkgz7YX__HZYv9A60xwG7kuF5Ha0nX8R67beH1DhvTdUa3Z5XhtNjO1fNXQ/s640/M24.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The Human Tree is a popular exhibit in which you move you arms in front of the camera, and the screen behind the camera projects your movements in fractal form.</td></tr>
</tbody></table>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhzRP_bwYWrbJy2dR_0TczdTZah1pObdih7c1gxr-BW7GjIyDRah7Eqh6R98Vn741wbsD3EehDqjA0frCZCL5ah9bNqntOvmdvLLLJvgSAeXA-ieh1IuEnvNfYGjSac1RU0Lh92cN-WQ/s1600/M25.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhzRP_bwYWrbJy2dR_0TczdTZah1pObdih7c1gxr-BW7GjIyDRah7Eqh6R98Vn741wbsD3EehDqjA0frCZCL5ah9bNqntOvmdvLLLJvgSAeXA-ieh1IuEnvNfYGjSac1RU0Lh92cN-WQ/s640/M25.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The Human Tree screen</td></tr>
</tbody></table>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBomDs9R0tcZj-38VzNvLwollUmuPPZZRb7y-UI0acov5ILJxMVRbrX2EjBOhVjaIuQH-0VhRwWY_txYfGDFbVth846Cge19AnxZBZOETc8VUxBaJk9MajYgiivpHaCvVWxom3q4iszw/s1600/M26.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBomDs9R0tcZj-38VzNvLwollUmuPPZZRb7y-UI0acov5ILJxMVRbrX2EjBOhVjaIuQH-0VhRwWY_txYfGDFbVth846Cge19AnxZBZOETc8VUxBaJk9MajYgiivpHaCvVWxom3q4iszw/s640/M26.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Twist and Roll challenges you to choose the right 3-D object and predict how it will roll.</td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyDhwgq0HD1PIt9w0M3nF87i-PLbMt3VrqGSLCKNPVa9z6Nn9sOAVu9OHXOAjzd0E8vmlOvnFqc8CbjjRDXiTpsN39y4YPMLmBlt2uGrC3ZG46Q7wQDrSAq8fQjjcMDwiuILOGFnX1xg/s1600/M27.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyDhwgq0HD1PIt9w0M3nF87i-PLbMt3VrqGSLCKNPVa9z6Nn9sOAVu9OHXOAjzd0E8vmlOvnFqc8CbjjRDXiTpsN39y4YPMLmBlt2uGrC3ZG46Q7wQDrSAq8fQjjcMDwiuILOGFnX1xg/s640/M27.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;">Some items I purchased last evening at pre-Opening. All are affordable approximately $12 each. The colored Moire coasters are more beautiful than seen in this lighting. Euler's Identity is considered by Mathematicians to be the most beautiful "equation" or "formula" in Math. It is composed of 2 operators (+, =) and 5 constants (pi, e, -1, 0, and i) only and shows how they interrelate:</span><br />
<br style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;" />
<span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;">e raised to the i(pi) +1 = 0 </span><br />
<br style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;" />
<span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;">or</span><br />
<br style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;" />
<span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;">e^i(pi) + 1 = 0</span></td></tr>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvQhKl2cbE19_R9M2EHohq6ytnfuqp8lByKJcjTkZ78lqv5_IEMdiV3PZGirfk0DowGFyVy58wZVqhDdPPCOPZeJgShWVpFqr0zQ4C3yPW2QnG-G6D0bWqvfbR5fi1Pa_dgqCRcaN-gA/s1600/M28.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvQhKl2cbE19_R9M2EHohq6ytnfuqp8lByKJcjTkZ78lqv5_IEMdiV3PZGirfk0DowGFyVy58wZVqhDdPPCOPZeJgShWVpFqr0zQ4C3yPW2QnG-G6D0bWqvfbR5fi1Pa_dgqCRcaN-gA/s640/M28.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">MoMath Museum Store</td></tr>
</tbody></table>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjG9PTt-PyNcNmGS-hwXR5qwgt8MFQmoLNMTddXX5_9Es-vHPvnYo6sSTHUU6MttejZrJqoNsiMOru3ws5Kktr0wm2OJd7mr7Q2VnWGgAxGwx7MDQ7ZRAp-Z-AELXN6LgVg2Md4zaSWQQ/s1600/M29.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjG9PTt-PyNcNmGS-hwXR5qwgt8MFQmoLNMTddXX5_9Es-vHPvnYo6sSTHUU6MttejZrJqoNsiMOru3ws5Kktr0wm2OJd7mr7Q2VnWGgAxGwx7MDQ7ZRAp-Z-AELXN6LgVg2Md4zaSWQQ/s640/M29.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">MoMath Musuem Store</td></tr>
</tbody></table>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1gZ-KnKFX07kXFPTlIQvJfukuqzMNeqwHWD8R0w3uDwjKNkaQ4GhCaQd3LlXNdLI2Gilgyu8ixs9_7k6fvKC-4i12UMDbUaoJ494srma7RVkz6MFNigU2xB4sCVqUJNJR42QscOSWCw/s1600/M30.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1gZ-KnKFX07kXFPTlIQvJfukuqzMNeqwHWD8R0w3uDwjKNkaQ4GhCaQd3LlXNdLI2Gilgyu8ixs9_7k6fvKC-4i12UMDbUaoJ494srma7RVkz6MFNigU2xB4sCVqUJNJR42QscOSWCw/s640/M30.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;">View from Madison Square Park across the street, after dusk. The museum store is on the left.</span></td></tr>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2ttmPdpeyy3BylbNG1_6QgKIe7c-uBCB05eRYGqzremfHgiAFsJDnzmm6k0lNsL4WadRhuIhBSmgaI92toH0g_eQFps7T6W7RNO_F6_hcnFfZku8MBUREMJj4I48ELh40tX43jfn3cQ/s1600/M31.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2ttmPdpeyy3BylbNG1_6QgKIe7c-uBCB05eRYGqzremfHgiAFsJDnzmm6k0lNsL4WadRhuIhBSmgaI92toH0g_eQFps7T6W7RNO_F6_hcnFfZku8MBUREMJj4I48ELh40tX43jfn3cQ/s640/M31.jpg" width="480" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;">A sculpture in Madison Sq. Park of the temporary (Through Feb. 13, 2013) sculpture: BUCKYBALL. Looking east in this picture, MoMath is to the left, or north.</span></td></tr>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEieTZc5gH952db7wT-aGhhzalVYmhAoYZJNJ5OSMnTJCzdEmG5UOAi7q9Oe1NAp5jNGluqimTPEJ51p717IIiQnkEnGEQfTh5I8-v_ajnC5HYl1Z-3iZ5EeOp5ahbGu-T3-izwT0v8HPA/s1600/M32.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEieTZc5gH952db7wT-aGhhzalVYmhAoYZJNJ5OSMnTJCzdEmG5UOAi7q9Oe1NAp5jNGluqimTPEJ51p717IIiQnkEnGEQfTh5I8-v_ajnC5HYl1Z-3iZ5EeOp5ahbGu-T3-izwT0v8HPA/s640/M32.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;">Looking North from the BUCKYBALL sculpture, MoMath is small and in the left center. Christmas Tree and the Empire State Building as well, which is 7 blocks north.</span></td></tr>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_DMaxbd5QnDX_QeEDt7Cu-QlqWWMTCNlwVbTca_f0IA3Cvp0CSHyvUl05S2k7QmNXfTurGItuPRTkLwn-5_kARz2TSPHjr7NXL0qjqSifW3tnWSwzYLM8M097cbxVzQ9AZKDb-Suw3w/s1600/M33.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_DMaxbd5QnDX_QeEDt7Cu-QlqWWMTCNlwVbTca_f0IA3Cvp0CSHyvUl05S2k7QmNXfTurGItuPRTkLwn-5_kARz2TSPHjr7NXL0qjqSifW3tnWSwzYLM8M097cbxVzQ9AZKDb-Suw3w/s640/M33.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;">From the Park, MoMath is in the lower left.</span></td></tr>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhErsqRwt7R2-CTBZVS4U_34Lu2091TMK6OFAOgDBKGfVdwlSAGDN-ypz8-vZvz3OEIfiynf42q_2HiZilvREusLGn25aj_kvZuwUlbTuQ2Ihn09ZoYAYDfmlpCMzSdw9MJJil_VuBC0w/s1600/M34.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhErsqRwt7R2-CTBZVS4U_34Lu2091TMK6OFAOgDBKGfVdwlSAGDN-ypz8-vZvz3OEIfiynf42q_2HiZilvREusLGn25aj_kvZuwUlbTuQ2Ihn09ZoYAYDfmlpCMzSdw9MJJil_VuBC0w/s320/M34.jpg" width="320" /></a></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkQ20kXSe8Y8znS_9uCfOxzS_89-fxcMqSRU7Hfwi7MrSzy8M5VuKcx-ypjgRj-F5bpkheJDzDBanHWY2STGksFVBdxAxP0Wd6HtvSOPEfbisvGRoOAfg2BHG97K-laQCzazS0_iTnDQ/s1600/M35.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkQ20kXSe8Y8znS_9uCfOxzS_89-fxcMqSRU7Hfwi7MrSzy8M5VuKcx-ypjgRj-F5bpkheJDzDBanHWY2STGksFVBdxAxP0Wd6HtvSOPEfbisvGRoOAfg2BHG97K-laQCzazS0_iTnDQ/s320/M35.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif; font-size: 13.333333969116211px; line-height: 17.981481552124023px; text-align: left;">With Glen Whitney, Director and Cindy Lawrence, Assistant Director, at MoMath pre-Opening night, Dec. 14, 2012</span></td></tr>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9Rw12dQghIF3XVtGqicB6dAN8ot6MmYM4v9eBnL2hi5agxlDXkLa9Nk53ByEp_JFl-2gdVyqYCfOUDLVvtgudtJQrR20jaUlRKItCybkORjRQ-znp4QoYBdmeFGyIoGIRVG4kX7k8jg/s1600/M36.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9Rw12dQghIF3XVtGqicB6dAN8ot6MmYM4v9eBnL2hi5agxlDXkLa9Nk53ByEp_JFl-2gdVyqYCfOUDLVvtgudtJQrR20jaUlRKItCybkORjRQ-znp4QoYBdmeFGyIoGIRVG4kX7k8jg/s640/M36.jpg" width="480" /></a></div>
<br />Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com10tag:blogger.com,1999:blog-5303246073824127471.post-80002478329782016812012-12-14T10:59:00.002-05:002012-12-14T22:27:23.164-05:00MoMATH: America's FIRST Math Museum Opens Tomorrow, and Today<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFqARWZi2ctC2kN19fOu8Jw1JA417DhFEgjQrg8lRPZGqL-oW8WupPnpAXrZ8Wqp1x8M5OYEXJCJon8UYCwYcfK4_vwH6_OPY1TGUNiDnatx9pX-2PcACxvwM2HZRQR1B1C9dq4tm1nw/s1600/MoMath.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="444" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFqARWZi2ctC2kN19fOu8Jw1JA417DhFEgjQrg8lRPZGqL-oW8WupPnpAXrZ8Wqp1x8M5OYEXJCJon8UYCwYcfK4_vwH6_OPY1TGUNiDnatx9pX-2PcACxvwM2HZRQR1B1C9dq4tm1nw/s640/MoMath.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">An artits conception of how MoMath would look from last year. The reality is a bit different and we will show you tomorrow when we return. However the sense of space in accurate in this wonderful two two floor museum dedicated to Mathematics alone.</td></tr>
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<a href="http://momath.org/" style="background-color: white; color: #6699cc; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14.545454025268555px; line-height: 18.18181800842285px; text-decoration: initial;">http://mom</a><a href="http://momath.org/" style="background-color: white; color: #6699cc; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14.545454025268555px; line-height: 18.18181800842285px; text-decoration: initial;">ath.org/</a><br /><br />I am happy to report that MoMath looks great and as the only museum of it's kind in America, one dedicated only to Mathematics, that this important step in making the public appreciate both the beauty and wide scope of a subject often taught in a boring and bland manner, is off to a eye-catching and enjoyable start.<br />
<span style="background-color: white; color: #333333; font-family: 'lucida grande', tahoma, verdana, arial, sans-serif;"><br /><br /><span style="font-size: x-small;"><span style="line-height: 18px;">Tomorrow, from 10-5 will be the first day The Museum of Mathematics opens to the public. I attended a members-only pre-opening this afternoon and bought a few items in the well-stocked shop with a separate street entrance and took a quick tour. It looks great, very clean and very entertaining for adults and children of all ages. There were many mathematics teachers and professors enjoying the exhibits as well as many children. My camera was buggy and I couldn't take many photos but I have worked out the kinks and will have more tomorrow.</span></span></span><br />
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Previously, I wrote:<br />
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<span style="background-color: white; color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14.545454025268555px; line-height: 18.18181800842285px;">The world's first Mathematics Museum is slated to open in New York City in 2012. Thanks to Alan Boyle of Cosmic Log for turning us on to this.</span><br />
<br style="background-color: white; color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14.545454025268555px; line-height: 18.18181800842285px;" />
<span style="background-color: white; color: #333333; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14.545454025268555px; line-height: 18.18181800842285px;">I sure wish there were more.</span><br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkS_7idlPpNIDpX1Vgt0C7j4nHfqa8PM3_jL-tW0PINvJF06aymHhYYG_Vrf0ECafI1mGJCkZTzUK7UfPFCnhcjeTjAI-3nzHmAH0b-lNxVHtsCc_m5AKzsDe0mZKh-SyIud_qiYp0iA/s1600/MoMath_exterior.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="208" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkS_7idlPpNIDpX1Vgt0C7j4nHfqa8PM3_jL-tW0PINvJF06aymHhYYG_Vrf0ECafI1mGJCkZTzUK7UfPFCnhcjeTjAI-3nzHmAH0b-lNxVHtsCc_m5AKzsDe0mZKh-SyIud_qiYp0iA/s320/MoMath_exterior.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">That's not me lol</td></tr>
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<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn6B7oe8qpNL1t4kainFl2JS4Yso-7VEhLKLrEOznNn-cuXWCHN7DIsn8UkrCg6KUvtCoEZMJp-OOMdzWeMD7n7HnlnWwFANvvnyGbGUQubn7mcFGZ-rzpopgo5XHauTfltpjEhSBVZQ/s1600/MusserAndMeNov282012NYASPrideDiscussion.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn6B7oe8qpNL1t4kainFl2JS4Yso-7VEhLKLrEOznNn-cuXWCHN7DIsn8UkrCg6KUvtCoEZMJp-OOMdzWeMD7n7HnlnWwFANvvnyGbGUQubn7mcFGZ-rzpopgo5XHauTfltpjEhSBVZQ/s320/MusserAndMeNov282012NYASPrideDiscussion.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 13.333333015441895px;">George Musser Jr. and me Nov. 28, 2012 at the New York Academy of Sciences , WTC7 NYC, panel discussion on Pride and Science: Where are all the Flying Cars? (A: they exist, you just better have a lot of money), more on this later as well as my participation in the NYC roundtable on Time the next week.<br />
Photograph by Robert Ricci, copyright 2012<br />
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<a href="http://cosmiclog.msnbc.msn.com/_news/2011/06/13/6851255-exhibits-add-mirth-to-math" style="background-color: white; color: #6699cc; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 14.545454025268555px; line-height: 18.18181800842285px; text-decoration: initial;">Click here to see Cosmic log's article on same.</a>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com4tag:blogger.com,1999:blog-5303246073824127471.post-91867190529311176092012-12-14T10:53:00.002-05:002012-12-14T10:53:40.115-05:00Happy Holidays MATH PHYSICS Shopping! <br />
Clifford Pickover has a followup to his excellent The Math Book, titled The Physics Book. Union College professor and laser cooling specialist Chad Orzel reviews it <a href="http://ciples/2011/12/the_physics_book_by_clifford_p.php?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+scienceblogs%2Funcertainprinciples+%28Uncertain+Principles%29">here</a> and Pickover's own page describes it <a href="http://sprott.physics.wisc.edu/pickover/physics-book.html">here.</a><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2-2S77pHF9pW9GqZza567MZEnem-S_RKSIsi7NoT9ulEdVtQuh1qj0MFZjdOWFIkhYzB6thsJnv7pik5XZOM8jxzX7P4LMbBXPK969paKm7-sJOdFt-5eZgHCz9k2mRLIvTpcASECIg/s1600/physicsbook-cover.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2-2S77pHF9pW9GqZza567MZEnem-S_RKSIsi7NoT9ulEdVtQuh1qj0MFZjdOWFIkhYzB6thsJnv7pik5XZOM8jxzX7P4LMbBXPK969paKm7-sJOdFt-5eZgHCz9k2mRLIvTpcASECIg/s400/physicsbook-cover.jpg" width="400" /></a></div>
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Pickover's Math Book is one of five I strongly recommend for the budding genius in your family be they 8-80 or beyond:<br />
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These are IMO the five best introductory books to Mathematics that prove that the field IS ANYTHING BUT BORING, but is indeed a beautiful and exciting Field of Study.<br />
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State regulations in America's States, exceedingly boring in themselves, hamper our Teachers in making the students understand this VERY important subject. Math is overly tested here in the USA, and at too early an age, to the point of impressing our young and oh so important citizens that the subject seems positively evil, and useless.<br />
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"What is Math good for?" the childrens cry! This mantra is far too common. We. Must. Debunk.<br />
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Well, here are five books that will hopefully dispel that faulty thinking.<br />
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I list them from simplest to deepest, so this is the order in which I would recommend them to be read, with links to Amazon:<br />
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1) <a href="http://www.amazon.com/Mathematics-Ideas-Really-Need-ideas/dp/1847241476/ref=sr_1_1?ie=UTF8&s=books&qid=1302355159&sr=8-1">50 Mathematical Ideas You Really Need To Know</a> by Tony Crilly<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEin4zumRWB4e7cMrxKYthQ9O1KRhhnsh5coLvm9sl9FrLtJFYMd-j_FzQQRfvOuNviUhFvl3ZCytc912vwhOqe_PhKweMz1CC_bIFxy97aihRApkGT_TpbDlV12Eqrh7XLhr8dQ5pv8CQ/s1600/Fifty.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEin4zumRWB4e7cMrxKYthQ9O1KRhhnsh5coLvm9sl9FrLtJFYMd-j_FzQQRfvOuNviUhFvl3ZCytc912vwhOqe_PhKweMz1CC_bIFxy97aihRApkGT_TpbDlV12Eqrh7XLhr8dQ5pv8CQ/s400/Fifty.jpg" width="300" /></a></div>
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Back on Feb. 8, 2010, I did a positive review of this book, outlining it even, twice, which you can call up by clicking <a href="http://tetrahedral.blogspot.com/2010/02/50-mathematical-ideas-you-really-need.html">here</a>.<br />
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There are wonderful little timelines at the bottom of each of the 50 four-page chapters. I spent a considerable amount of time typing them in into a gross History of Mathematics at the very beginning of this year 2011, and you can call that up by clicking <a href="http://tetrahedral.blogspot.com/2011/01/mathematics-timeline-part-3-and.html">here</a>.<br />
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This book is wonderfully cheap, and oddly, actually costs less at a local bookstore than at Amazon. I don't recommend this book for everyone. Only those aged eight to eighty. :-)<br />
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Mathematics starts here. This is your launching pad.<br />
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2) <a href="http://www.amazon.com/Math-Book-Pythagoras-Milestones-Mathematics/dp/1402757964/ref=sr_1_1?s=books&ie=UTF8&qid=1302355248&sr=1-1">The MαTH βOOK: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics </a>by Clifford A. Pickover<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzZv82KguqtrSc0qHL6pkpFkfCqDjb0F5e0C7CR32yr1kxfhsj4n0Ei6BIsaVPXBj_em7J3t0BTcS7w0Pq6_kvXFPoEWpXOhkSPi_vdjbMz0fEgr6_Y2ScyNBdypyoCTXpO2t00BFVEQ/s1600/MathBook.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzZv82KguqtrSc0qHL6pkpFkfCqDjb0F5e0C7CR32yr1kxfhsj4n0Ei6BIsaVPXBj_em7J3t0BTcS7w0Pq6_kvXFPoEWpXOhkSPi_vdjbMz0fEgr6_Y2ScyNBdypyoCTXpO2t00BFVEQ/s400/MathBook.jpg" width="300" /></a></div>
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Anyone who doesn't think that Math is beautiful, hasn't read this book. Heck, just skim through the pages. I would be surprised if you didn't buy it, just to own it. Beautiful pictures, suitable for framing. Beautiful prose.<br />
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3) <a href="http://www.amazon.com/Mathematics-1001-Absolutely-Everything-Explanations/dp/1554077192/ref=sr_1_1?s=books&ie=UTF8&qid=1302355357&sr=1-1">Mathematics 1001: Absolutely Everything That Matters About Mathematics in 1001 Bite-Sized Explanations</a> by Richard Elwes<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5L9PyIVlsko4rvqA5yVWhNqXNWAiq58QuBYs3MEgM_ieV1EYOJPWt0VSCQHtLIDH71pXYAX-V8uf3xaSyBLmbyer4gKYnY1Wn0iB4pRdfaZ6Nm7e9zTsih5pFKu7Wv59URx8fZ1JWYg/s1600/Math1001.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5L9PyIVlsko4rvqA5yVWhNqXNWAiq58QuBYs3MEgM_ieV1EYOJPWt0VSCQHtLIDH71pXYAX-V8uf3xaSyBLmbyer4gKYnY1Wn0iB4pRdfaZ6Nm7e9zTsih5pFKu7Wv59URx8fZ1JWYg/s1600/Math1001.jpg" /></a></div>
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You've seen 50 Mathematical ideas, then 250 milestones. Time to ratchet up the Knowledge Quotient. Try 1001. Currently my favorite read and my launching pad for ideas when I'm bored. And I hate being bored. I've got a fever, and the only prescription during those times is this book. Or more cowbell. Both'll work.<br />
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Dr. Elwes has a webpage for this book, including the very small amount of errata, which can be found <a href="http://richardelwes.co.uk/mathematics-1001/">here</a>.<br />
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John Baez has a nice recent review of the book: <a href="http://johncarlosbaez.wordpress.com/2011/12/06/maths-1001/">here</a>.<br />
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4) <a href="http://www.amazon.com/Eulers-Gem-Polyhedron-Formula-Topology/dp/0691126771/ref=sr_1_1?ie=UTF8&s=books&qid=1302355489&sr=1-1">Euler's Gem: The Polyhedron Formula and the Birth of Topology</a> by Dave Richeson<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiMQL9ZqHI3s8blqs1jry1b54z8h3is1F8aQTrLlF5pABDjN-z7IIYgSIaTGwShbdQXIL5eMx9b-pmUYjAOUnPXOupJ6ucIW4yCNRRcPFRlS7Nl6FQAVTlX1LQ3dnRcZmAPR9RG7vhyphenhypheng/s1600/Euler%2527s+Gem.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiMQL9ZqHI3s8blqs1jry1b54z8h3is1F8aQTrLlF5pABDjN-z7IIYgSIaTGwShbdQXIL5eMx9b-pmUYjAOUnPXOupJ6ucIW4yCNRRcPFRlS7Nl6FQAVTlX1LQ3dnRcZmAPR9RG7vhyphenhypheng/s1600/Euler%2527s+Gem.jpg" /></a></div>
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There's more to Topology than mathematicians being unable to distinguish a coffee cup from a donut, unlike policemen, who don't care; they enjoy both. This is the first of these books that have actual EQUATIONS in them, but don't freak out. They're straightforward, and Dave expositates beautifully.<br />
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It's all about Leonhard Euler in oh so many ways. I can't recommend this book strongly enough.<br />
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5) <a href="http://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809/ref=sr_1_1?s=books&ie=UTF8&qid=1302355579&sr=1-1">The Princeton Companion to Mathematics</a> edited by Timothy Gowers, et. al.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgP8brsIraxnpcwJdzfx4AIyNotJ5sfiOr0koaVJ-OsKjrTa2Hk-f8Pl7OKfoBbORJYLo3iZgsfgtsdooEU2cUrlZLwJouola_hIIDSU5UOOSuVEvEhLy4xmMgwqZoVJ5r6rtcl2AJ1Sg/s1600/Princeton+Companion.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgP8brsIraxnpcwJdzfx4AIyNotJ5sfiOr0koaVJ-OsKjrTa2Hk-f8Pl7OKfoBbORJYLo3iZgsfgtsdooEU2cUrlZLwJouola_hIIDSU5UOOSuVEvEhLy4xmMgwqZoVJ5r6rtcl2AJ1Sg/s320/Princeton+Companion.jpeg" width="256" /></a></div>
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And now it's time for Grad School. Not every Mathematician knows what their fellows are up to. It's been said by a famous Mathematician, that if they were stranded on a desert island and could have only one book, it would be the Princeton Companion. Read it and you'll see why. It is superb.<br />
Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com6tag:blogger.com,1999:blog-5303246073824127471.post-32115460862105760462012-12-14T10:52:00.000-05:002012-12-14T10:52:05.642-05:0014 Math Holidays Every Math Major Should Know<br />
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Pi Day, e Day, Square Root Day, Odd Day (isn't that every day?), Powers of Ten Day, and 9 other Math Holidays. It's all here, baby!</div>
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<span class="Apple-style-span" style="color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: x-small;"><span class="Apple-style-span" style="line-height: 19px;">DISCLAIMER: I did not write this but liked it enough to do my copy'n'paste thing (my specialty!) and put it on my weblog. Plus, I'm a sucker for lists! I got it from a random e-mail in my Inbox from one Jasmine Hall, <a href="http://www.onlineclasses.org/2011/03/23/14-holidays-every-math-major-must-know/">here</a>.</span></span></div>
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<span class="Apple-style-span" style="color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: x-small;"><span class="Apple-style-span" style="line-height: 19px;">At first I thought it was spam and it may very well end up being so, and unless and/or until I explore Jasmine's website I wish to express that I am not endorsing this school nor am I saying you should not explore it. I assume my readership is intelligent to make up their own minds one way or another.</span></span></div>
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<span class="Apple-style-span" style="color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: x-small;"><span class="Apple-style-span" style="line-height: 19px;">In any event, the mini-essay is good Marketing (another specialty), and as said I enjoyed it so here it is:</span></span></div>
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<span class="Apple-style-span" style="color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: x-small;"><a href="http://www.onlineclasses.org/2011/03/23/14-holidays-every-math-major-must-know/" style="color: #cf480d; outline: none; text-decoration: initial;">14 Holidays Every Math Major Must Know</a></span></h1>
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Math, however unfairly, has a reputation for being a bit dull. Yet math nerds know that the subject can be just as fascinating and fun as any other <a href="http://www.onlineclasses.org/blog" style="color: #e58734; outline: none;">college major</a> out there. Of course, convincing others who aren’t mathematically inclined of this fact can be difficult. Luckily, there are some fun holidays out there that can get even the most resistant of individuals to enjoy celebrating some of the fundamentals of mathematics. Here are just a few of the ones well worth celebrating.</div>
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<li style="margin: 0px 0px 0px 30px; padding-bottom: 2px; padding-left: 0px !important; padding-right: 0px !important; padding-top: 2px;"><strong><a href="http://hubpages.com/hub/Pi_Day_A_Holiday_for_Math_Geeks" style="color: #e58734; outline: none;">Pi Day</a>: </strong>Celebrated on March the 14th in the US, this holiday recognizes the mathematical constant of Pi, which is often abbreviated to 3.14– hence the date of the holiday. Math geeks can celebrate by enjoying the wonders of Pi through math, watching the movie <em>Pi, </em>eating actual pie or some Pi-inspired art.</li>
<li style="margin: 0px 0px 0px 30px; padding-bottom: 2px; padding-left: 0px !important; padding-right: 0px !important; padding-top: 2px;"><strong><a href="http://boingboing.net/2009/03/03/march-is-math-holida.html" style="color: #e58734; outline: none;">Square Root Day</a>: </strong>The date of Square Root Day changes depending on the year. For instance, square root day could be 3/3/09 or 4/4/16, meaning this holiday only comes around once in a great while, so you should party it up while you can. Some ideas for enjoying square root day include cooking up some delicious root veggies, square dancing or anything else punny involving squares or roots.</li>
<li style="margin: 0px 0px 0px 30px; padding-bottom: 2px; padding-left: 0px !important; padding-right: 0px !important; padding-top: 2px;"><strong><a href="http://www.awm-math.org/newsletter/199909/Skhs.html" style="color: #e58734; outline: none;">Sonia Kovalevsky Mathematics Days</a>: </strong>Women in math will love this event. Mostly celebrated at middle and high schools, this holiday isn’t set on a fixed date, but usually takes place in the spring. It is meant to encourage young women to pursue a career in a math or science field, inspired by Sonia Kovalevsky, an important Russian mathematician. Math geeks can attend lectures on this day or participate in workshops.</li>
<li style="margin: 0px 0px 0px 30px; padding-bottom: 2px; padding-left: 0px !important; padding-right: 0px !important; padding-top: 2px;"><strong><a href="http://www.squarecirclez.com/blog/today-is-e-day/4133" style="color: #e58734; outline: none;">e Day</a>: </strong>While not as well-known as Pi, e is also an irrational number that occurs naturally in the grand scheme of mathematics. Discovered by a number of mathematicians, it’s useful in helping puzzle out exponential and logarithmic functions. The rough numerical equivalent of e is 2.7, making the logical day to celebrate it February 7th. As to how you celebrate e Day, well, that’s up to you. You can only eat foods that start with e, read the poetry of ee cummings, watch the E! Network or just do some fun math related to e.</li>
<li style="margin: 0px 0px 0px 30px; padding-bottom: 2px; padding-left: 0px !important; padding-right: 0px !important; padding-top: 2px;"><strong><a href="http://mathfuture.wikispaces.com/events" style="color: #e58734; outline: none;">Math 2.0 Day</a>: </strong>Use this holiday to celebrate the intersection of math and technology. Only July 8th, spend your day using math programs, attending tech lectures and appreciating the subject on the web.</li>
<li style="margin: 0px 0px 0px 30px; padding-bottom: 2px; padding-left: 0px !important; padding-right: 0px !important; padding-top: 2px;"><strong><a href="http://mathematicianspictures.com/PI/CASUAL_PI.htm" style="color: #e58734; outline: none;">Pi Approximation Day</a>: </strong>Some prefer to celebrate Pi not on the decimal equivalent to Pi, but instead on the fraction that represents it: 22/7. Twenty two divided by seven gives you the approximate value of Pi, hence the name of the holiday. Celebrations of this day are pretty much the same as those on 3/14, so why not celebrate twice a year with twice the pie?</li>
<li style="margin: 0px 0px 0px 30px; padding-bottom: 2px; padding-left: 0px !important; padding-right: 0px !important; padding-top: 2px;"><strong><a href="http://www.oddday.net/" style="color: #e58734; outline: none;">Odd Day</a>: </strong>Odd day is a day that singles out those wonderful, wacky odd numbers. It occurs when three consecutive odd numbers make up a date– something that happens only six times a century. The last Odd Day was 5/7/09 and the next will be on 7/9/11. Enjoy Odd Day by, well, being odd.</li>
<li style="margin: 0px 0px 0px 30px; padding-bottom: 2px; padding-left: 0px !important; padding-right: 0px !important; padding-top: 2px;"><strong><a href="http://unreasonablefaith.com/2010/10/09/powers-of-ten-day" style="color: #e58734; outline: none;">Powers of Ten Day</a>: </strong>This holiday is all about seeing the world in a different light, though different magnitudes of 10 to be more precise. It was celebrated on 10/10/10 and isn’t due to come around again for quite some time, so if you missed your chance to celebrate in 2010, you likely won’t live to see this holiday come round again.</li>
<li style="margin: 0px 0px 0px 30px; padding-bottom: 2px; padding-left: 0px !important; padding-right: 0px !important; padding-top: 2px;"><strong><a href="http://www.worldmathsday.com/" style="color: #e58734; outline: none;">World Maths Day</a>: </strong>This is the day when math finally gets its due. Celebrated internationally on March 1st, the holiday recognizes all things mathematical, focusing special attention on getting kids enthused about a career in math or doing equations. You can celebrate World Maths (or Math if you’re not a fan of the British spelling) Day any way you like, so long as it involves the subject.</li>
<li style="margin: 0px 0px 0px 30px; padding-bottom: 2px; padding-left: 0px !important; padding-right: 0px !important; padding-top: 2px;"><strong><a href="http://www.holidayinsights.com/moreholidays/October/nationalmoleday.htm" style="color: #e58734; outline: none;">Mole Day</a>: </strong>Know the math behind chemistry? Then you’ve likely heard of Avagadro’s number (6.02×10^23) that’s used as a basic unit of measure in chemistry, more commonly referred to as a Mole. It’s observed on October 23rd from 6:02 am to 6:02 pm, and can include enjoying anything mole related from mole sauce to Whack-a-Mole. The punnier, the better.</li>
<li style="margin: 0px 0px 0px 30px; padding-bottom: 2px; padding-left: 0px !important; padding-right: 0px !important; padding-top: 2px;"><strong><a href="http://en.wikipedia.org/wiki/Pythagorean_theorem" style="color: #e58734; outline: none;">Pythagorean Theorem Day</a>:</strong> Pythagoras’ theorem states that the length of sides of a right triangle will always fit the equation a squared + b squared = c squared. Thus, this holiday is celebrated on dates which meet this criteria. For example, 6/8/10 would be one such date. Enjoy this holiday by playing the triangle, doing some geometry and eating Greek food.</li>
<li style="margin: 0px 0px 0px 30px; padding-bottom: 2px; padding-left: 0px !important; padding-right: 0px !important; padding-top: 2px;"><strong><a href="http://letsplaymath.net/2010/09/25/math-storytelling-day" style="color: #e58734; outline: none;">Math Storytelling Day</a>: </strong>On Math Storytelling Day, those who love math can have fun making up and sharing math-related stories. They can involve puzzles, logic, human relationships, just about anything so long as there’s math in there somewhere. This holiday is observed on September 25th and can be a lot of fun for kids and adults alike.</li>
<li style="margin: 0px 0px 0px 30px; padding-bottom: 2px; padding-left: 0px !important; padding-right: 0px !important; padding-top: 2px;"><strong><a href="http://www.g4g-com.org/" style="color: #e58734; outline: none;">Celebration of Mind</a>: </strong>Held in honor or Martin Gardner’s birthday, this holiday held on October 21st encourages a fun and playful approach to mathematics and logic puzzles. Celebrants can mark the day by doing fun math puzzles, performing magic tricks, or even sharing math stories.</li>
<li style="margin: 0px 0px 0px 30px; padding-bottom: 2px; padding-left: 0px !important; padding-right: 0px !important; padding-top: 2px;"><strong><a href="http://oakford.wordpress.com/2008/11/24/fibonacci-day" style="color: #e58734; outline: none;">Fibonacci Day</a>: </strong>If you’re a math nerd, you’ve more than likely heard of Fibonacci’s sequence. This sequence, made famous by the Italian mathematician, creates a spiral and begins with the numbers 1, 1, 2, 3, so the holiday is celebrated on November 23rd of each year. There are no set guidelines for celebration, so those who want to mark the occasion can do anything from delve into the sequence to enjoy Italian food.</li>
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Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com23tag:blogger.com,1999:blog-5303246073824127471.post-16238551303732238812012-12-14T10:49:00.001-05:002012-12-14T10:49:34.200-05:00FUN with MATH !!<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJWNCtg8XFRnvzpI_0hGQkWb8RPrt2w8NiPhGxhjv8MDA2VoCylCfa0cMFLDutsDlBfDwDg2RrEnRaffOW26VW6_RCdqDOhz62kvM-90MJiWVPVBSNax55KXOOJe1Yv7QGZ1ur_4Y2uQ/s1600/Pi+Pie.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJWNCtg8XFRnvzpI_0hGQkWb8RPrt2w8NiPhGxhjv8MDA2VoCylCfa0cMFLDutsDlBfDwDg2RrEnRaffOW26VW6_RCdqDOhz62kvM-90MJiWVPVBSNax55KXOOJe1Yv7QGZ1ur_4Y2uQ/s1600/Pi+Pie.jpg" /></a></div>
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Thanks to Ulla Mattfolk of Finland for this:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilW8_QIiaKrXxJwFx9O7cJNfbYItK7bBPoZcVxV7zlzwWCB4V7utho8UVuzPnxDAMZZG13VlPdcLez721U8Y2AB8ZOdMMcAE6NJFi5aJP1UlsIt3nmbCL4i5u5LgxWJ5N30aW4ltnL0A/s1600/MathAmazing.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="218" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilW8_QIiaKrXxJwFx9O7cJNfbYItK7bBPoZcVxV7zlzwWCB4V7utho8UVuzPnxDAMZZG13VlPdcLez721U8Y2AB8ZOdMMcAE6NJFi5aJP1UlsIt3nmbCL4i5u5LgxWJ5N30aW4ltnL0A/s400/MathAmazing.jpg" width="400" /></a></div>
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Math professor Dave Richeson's Double Torus Clothesline Trick:<br />
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Lady has problem with basic clock arithmetic:<br />
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Integration Joke<br />
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Right Brain Math<br />
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Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com12tag:blogger.com,1999:blog-5303246073824127471.post-81806325837955304452012-12-14T10:40:00.001-05:002012-12-14T10:40:23.555-05:00The MATHEMATICS Timeline<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyjBfMK5EDSXO1bIXjIvNxbHWxEa7N47vGVtOtZTn7meJYR5GIm4aSYL-Jkwoj-c-ZaL4XDOB-BomEv4KsYYgNWfI6LoHfQGHYjf5s7VFp4IAfttA1fatdomakgpuq770PjOLQFbKFng/s1600/Crilly%2527s+Book.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyjBfMK5EDSXO1bIXjIvNxbHWxEa7N47vGVtOtZTn7meJYR5GIm4aSYL-Jkwoj-c-ZaL4XDOB-BomEv4KsYYgNWfI6LoHfQGHYjf5s7VFp4IAfttA1fatdomakgpuq770PjOLQFbKFng/s1600/Crilly%2527s+Book.jpg" /></a></div>
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I reviewed UK Mathematics Historian Tony Crilly's <b>50 Mathematical Ideas You Really Need to Know</b> in February ==> <a href="http://tetrahedral.blogspot.com/2010/02/50-mathematical-ideas-you-really-need.html">HERE</a>. That is the first link and very important as it lists the 50 chapters, 4 pages each, of this wonderfully concise book.<br />
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In <a href="http://tetrahedral.blogspot.com/2011/01/math-timeline-up-to-and-including.html">Part 1</a>, I combined the little timelines in each chapter up though Chapter 14, up to and including Algebra. <b>Numbers in parentheses are Chapter numbers</b>, which is why I asked you to click on that first link, first. I also have introductory notes there.<br />
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In <a href="http://tetrahedral.blogspot.com/2011/01/math-timeline-part-2-of-3-post-algebra.html">Part 2</a>, I added Chapters 15-19, up to and including Calculus.<br />
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In this Part 3, all 50 chapter timelines are shown. The new items in Chapters 20-50 are in bold.<br />
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This list seems long, and it is (prints out at 13+ pages), but it's useful, important, yet ... finite! :-) I shall be referring to this timeline in the coming months.<br />
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Part 4 will be analysis of this list.<br />
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Enjoy, and Go, Go Euclid !<br />
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<b>THE MATHEMATICS TIMELINE</b><br />
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30,000 BC - Paleolithic peoples in Europe make number marks on bones (2)<br />
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<b>3000 BC - The Babylonians use a sexagesimal number system for financial dealings (44)</b><br />
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<b>2800 BC - The legend of the Lo Shu square is born (42)</b><br />
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2000 BC - The Babylonians use symbols for numbers (2)<br />
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2000 BC - The Babylonians observe pi is roughly 3 (5)<br />
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1950 BC - The Babylonians work with quadratic equations (14)<br />
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<b>1850 BC - The Babylonians know "Pythagoras's Theorem" (21)</b><br />
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1800 BC - Fractions are used in Babylonian cultures (3)<br />
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<b>c. 1800 BC - The Rhind papyrus is written in Egypt (41)</b><br />
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1750 BC - The Babylonians compile tables of square roots (4)<br />
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1650 BC - The Egyptians make use of unit fractions (3)<br />
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700 BC - The Babylonians use zero as a placeholder in their number system (1)<br />
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525 BC - The Pythagoreans study geometrically arranged square numbers (4)<br />
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525 BC - The Pythagoreans are associated with both perfect and abundant numbers (10)<br />
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c. 500 BC - Fragmentary evidence exists for Pascal's triangle in Sanskrit (13)<br />
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c. 450 BC - Zeno ridicules infinitesimals with a paradox (19)<br />
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<b>450 BC - Anaxogoras attempts to square the circle while in prison (20)</b><br />
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350 BC - Aristotle rejects an actual infinite (7)<br />
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c. 335 BC - Aristotle formalizes the logic of the the syllogism (16)<br />
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c. 300 BC - The theory of the irrational numbers by Eudoxus is published in Book 5 of Euclid's <i>Elements</i> (4)<br />
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300 BC - Euclid's <i>Elements</i> gives a proof that there are infinitely many prime numbers (9)<br />
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300 BC - Book 9 of Euclid's <i>Elements</i> discusses perfect numbers (10)<br />
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c. 300 BC - The extreme and mean ratio is published in Euclid's <i>Elements</i> (12)<br />
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c. 300 BC - Euclid's <i>Elements</i> provides the model for mathematical proof (17)<br />
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c. 300 BC - Euclid's algorithm is published in Book 7 of <i>Elements</i> (15)<br />
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<b>c. 300 BC - Euclid defines the conic sections (22)</b><br />
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<b>c. 300 BC - Euclid shows there are five regular polyhedra (23)</b><br />
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<b>c. 300 BC - Euclid describes a three-dimensional world (24)</b><br />
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<b>c. 300 BC - Euclid includes the parallel postulate in his <i>Elements</i> (27)</b><br />
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250 BC - Archimedes gives the close approximation of pi of 22/7 (5)<br />
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<b>c. 250 BC - Archimedes investigates spirals (22)</b><br />
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<b>c. 250 BC - Archimedes investigates truncated polyhedra (23)</b><br />
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230 BC - Eratosthenes of Cyrene describes a method for sieving out prime numbers from the whole numbers (9)<br />
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<b>c. 225 BC - Apollonius of Perga publishes <i>Conics</i> (22)</b><br />
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<b>200 BC - Chinese mathematicians use arrays of numbers (39)</b><br />
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<b>55 BC - Julius Caesar invades Britain and uses codes to communicate with his generals (40)</b><br />
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100 - The Chinese devise a system for calculating with fractions (3)<br />
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100 - Nicomachus of Gerasa gives a classification of numbers based on perfect numbers (10)<br />
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250 - Diophantus of Alexandria publishes Arithmetica (14)<br />
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300 - Sun Tzu discovers the Chinese Algorithm (15)<br />
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600 - The forerunner of our modern decimal notation is used in India (2)<br />
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628 - Brahmagupta uses zero and states rules for its use with other numerals (1)<br />
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630 - Brahmagupta gives methods for computing square roots (4)<br />
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810 - Al-Khwarizmi gives the word "algorithm" to mathematics (15)<br />
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825 - Derived from "al-jabr" Al-Khwarizmi gives the word "algebra" to mathematics (14)<br />
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830 - Mahavira has ideas on how zero interacts with other numerals (1)<br />
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c. 1070 - Omar Khayyam discovers Pascal's triangle, which in some countries is named after him (13)<br />
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1100 - Bhaskara uses zero as a symbol in algebra and attempts to show how it is manipulated (1)<br />
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<b>c. 1100 - Bhaskara deals with permutations and combinations (41)</b><br />
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1200 - The Hindu-Arabic system of writing numerals 1,...,9, and a zero, spreads (2)<br />
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1202 - Fibonacci uses the extra symbol 0 added to the Hindu-Arabic system of numerals 1,...,9 but not as a number on par with them (1)<br />
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1202 - Fibonacci publishes the Liber Abaci and Fibonacci numbers (11)<br />
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1202 - Fibonacci popularizes the bar notation of fractions (3)<br />
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1202 - Fibonacci publishes work on congruences in Liber Abaci (15)<br />
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1303 - Zhu Shijie defines Pascal's triangle and shows how to sum certain sequences (13)<br />
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<b>1335 - Richard of Wallingford writes a groundbreaking treatise on Trigonometry (21)</b><br />
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<b>1494 - Luca Pacioli publishes financial tables and an account of double-entry bookkeeping (44)</b><br />
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1509 - Paciola publishes The Divine Proportion (12)<br />
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1550 - The square root symbol is introduced (4)<br />
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<b>1571 - Francois Viete publishes a book on trigonometry and trigonometric tables (21)</b><br />
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1572 - Rafael Bombelli calculates with imaginary numbers (8)<br />
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1585 - Simon Stevin sets out a theory of decimal fractions (3)<br />
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1591 - Francois Viete writes a mathematical text in terms of letters for knowns and unknowns (14)<br />
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1600 - The symbols of the decimal system take their recognizable modern forms (2)<br />
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1603 - Pietro Cataldi finds the 6th and 7th perfect numbers, 2^16(2^17 -1) = 8,589,869,056 and 2^18(2^19 - 1) = 137,438,691,328 (10)<br />
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1618 - John Napier encounters a constant, e, in connection with logarithms (6)<br />
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<b>1631 - Galileo gives "Galilean transformations" for falling bodies (48)</b><br />
<br />
1639 - Girard Desargues introduces the concept of infinity into geometry (7)<br />
<br />
<b>1639 - Pascal discovers his theorem while only 16 years old (28)</b><br />
<br />
<b>1654 - Blaise Pascal lays the foundations of probability theory (21)(33)</b><br />
<br />
1655 - John Wallis is credited with being the first to use the "love knot" symbol for infinity (7)<br />
<br />
<b>1657 - Christiaan Huygens writes the first published work on probability (31)(33)</b><br />
<br />
1664 - Pascal's paper on the properties of Pascal's triangle is published posthumously (13)<br />
<br />
<b>1665 - Fermat dies, leaving no record of his "wonderful proof" (49)</b><br />
<br />
1660's-1670's - Newton and Leibniz take the first steps in Calculus (19)<br />
<br />
<b>1672 - Mohr shows that all Euclidean constructions can be carried out with compasses alone (20)</b><br />
<br />
<b>1676 - Romer calculates the speed of light from observations of the moons of Jupiter (48)</b><br />
<br />
1687 - Descartes promotes mathematical rigour as a model in his <i>Discourse on Method</i> (17)<br />
<br />
<b>1687 - Newton's <i>Principia</i> describes the classical laws of motion (48)</b><br />
<br />
<b>1690 - de la Loubere produces a Siamese method for constructing magic squares (42)</b><br />
<br />
<b>1693 - Bernard Frenicle de Bessy lists all the 880 possible 4 x 4 magic squares (42)</b><br />
<br />
1700 - The fractional line "-" is in general use (as is %) (3)<br />
<br />
<b>1704 - Newton classifies the cubic curves (22)</b><br />
<br />
1706 - William Jones introduces the pi symbol (5)<br />
<br />
<b>1713 - Waldegrave gives the first mathematical solution of a two-player game (47)</b><br />
<br />
1714 - Leibniz discusses the harmonic triangle (13)<br />
<br />
<b>1718 - Abraham de Moivre publishes <i>The Doctrine of Chance</i>, with expanded editions following in 1738 and 1756 (33)(37)</b><br />
<br />
<b>1718 - Abraham de Moivre investigates morality statistics and the foundation of the theory of annuities (44)</b><br />
<br />
1724 - Daniel Bernoulli expresses the numbers of the Fibonacci sequence in terms of the golden ratio (11)<br />
<br />
1727 - Euler uses the notation e in connection with the theory of logarithms; it is sometimes called Euler's number (6)<br />
<br />
<b>1733 - De Moivre publishes work on the normal curve as an approximation to the binomial distribution (35)</b><br />
<br />
1734 - Berkeley draws attention to foundational weaknesses in Calculus (19)<br />
<br />
<b>1735 - Euler solves the problem of the bridges of Konigsberg (29)</b><br />
<br />
1742 - Goldbach speculates that every even number (more than 2) is a sum of two primes (9)<br />
<br />
1748 - Euler calculates e to 23 digits; he is given the credit for the discovery of the famous formula e^i(pi) + 1 = 0 around this time (6)<br />
<br />
<b>1750 - Euler's theorem lays the foundations for public key cryptography (40)</b><br />
<br />
<b>1752 - Euler gives his formula for the number of vertices, edges and faces in a polyhedron (23)</b><br />
<br />
<b>1753 - Euler proves Fermat's last theorem for the case n=3 (49)</b><br />
<br />
<b>1756 - James Dobson publishes <i>First Lectures on Insurances</i> (44)</b><br />
<br />
1761 - Lambert proves that pi is irrational (5)<br />
<br />
<b>1763 - Bayes's essay on probability is published (32)</b><br />
<br />
<b>1770 - Euler produces a squared (42)</b><br />
<br />
1777 - Euler first uses the symbol i to represent the square root of -1 (8)<br />
<br />
<b>1779 - Euler explores the theory of Latin squares (43)</b><br />
<br />
<b>1785 - Condorcet applies probability to the analysis of juries and electoral systems (31)</b><br />
<br />
<b>1801 - Gauss publishes <i>Discourses on Arithmetic</i> including a section on the construction of a regular 17-gon by ruler and compasses (20)</b><br />
<br />
1806 - Argand's diagrammatic representation leads to the name "Argand diagram" (8)<br />
<br />
<b>1806 - Brianchon discovers the dual theorem of Pascal's theorem (28)</b><br />
<br />
<b>1806 - Adrien-Marie Legendre fits data by least squares (36)</b><br />
<br />
<b>1809 - Carl Friedrich Gauss uses the least-squares method in Astronomical problems (36)</b><br />
<br />
<b>1810 - Charles Babbage mentions the travelling salesperson problem as an interesting one (46)</b><br />
<br />
1811 - Carl Friedrich Gauss works with functions of complex number variables (8)<br />
<br />
<b>1812 - Laplace publishes his essay on a deterministic world (26)</b><br />
<br />
<b>1812 - Laplace publishes his two volume <i>Analytical Theory of Probabilities </i>(31)</b><br />
<br />
1820 - Cauchy formalizes calculus in a rigourous way (19)<br />
<br />
<b>1820 - Gauss uses the normal distribution (as the Gaussian) in astronomy as a law of error (35)</b><br />
<br />
<b>1822 - Karl Feuerbach describes the nine point circle of a triangle (21)</b><br />
<br />
<b>1825 - Legendre and Dirichlet independently prove Fermat's Last Theorem for the case n=5 (49)</b><br />
<br />
<b>1826 - Fourier anticipates linear programming; Gauss solves linear equations by Gaussian elimination (45)</b><br />
<br />
<b>1829-31 - Lobachevsky and Bolyai publish their work on hyperbolic geometry (27)</b><br />
<br />
<b>1831 - The travelling salesperson problem appears as a practical problem (46)</b><br />
<br />
<b>1832 - Galois proposes the idea of groups of permutations (38)</b><br />
<br />
<b>1835 - Quetelet uses the normal curve to measure divergence from the average man (35)</b><br />
<br />
1837 - William R. Hamilton treats complex numbers as ordered pairs of real numbers (8)<br />
<br />
<b>1837 - Wantzel proves that the classical problems of duplicating a cube and trisecting an angle cannot be solved with ruler and compass (20)</b><br />
<br />
<b>1837 - Simeon-Denis Poisson describes the distribution named after him (34)</b><br />
<br />
1838 - De Morgan introduces the term "Mathematical Induction" (17)<br />
<br />
<b>1839 - Lame proves Fermat's Last Theorem for the case n=7 (49)</b><br />
<br />
<b>1843 - Kummer claims he has proved Fermat's Last Theorem, but Dirichlet exposes flaw (49)</b><br />
<br />
<b>1844 - Morse transmits the first message using his code (4)</b><br />
<br />
<b>1846 - Kirkman anticipates the discovery of Steiner triple systems (28)</b><br />
<br />
1847 - Boole publishes The Mathematical Analysis of Logic (16)<br />
<br />
<b>1848 - The Institute of Actuaries is founded in London (44)</b><br />
<br />
<b>1850 - J.J. Sylvester introduces the term "matrix" (39)</b><br />
<br />
<b>1850 - Kirkman poses the 15 schoolgirls problem (41)</b><br />
<br />
<b>1852 - Guthrie, De Morgan's student, puts the 4-colour problem to him (30)</b><br />
<br />
1854 - Riemann introduces the Riemann integral (19)<br />
<br />
<b>1854 - Riemann lectures on the foundation of geometry (27)</b><br />
<br />
<b>1854 - Cayley attempts to generalize the concept of a group (38)</b><br />
<br />
<b>1854 - Riemann begins his work on the zeta function (50)</b><br />
<br />
<b>1858 - Mobius and Listing introduce the Mobius strip (23)</b><br />
<br />
<b>1858 - Cayley publishes <i>Memoir on the Theory of Matrices</i> (39)</b><br />
<br />
<b>1859 - Riemann proves key solutions to The Riemann Hypothesis lie in a critical strip and puts forward his conjecture (50)</b><br />
<br />
<b>1865 - Mendel proposes the existence of genes and laws of inheritance (37)</b><br />
<br />
<b>1870 - The distribution acquires the name "normal" (35)</b><br />
<br />
1872 - Richard Dedekind sets out a theory of irrational numbers (4)<br />
<br />
1872 - Cantor takes a tentative step in the creation of set theory (18)<br />
<br />
1872 - Klein unifies geometry via group theory (27)<br />
<br />
<b>1872 - Felix Klein begins a programme for classifying geometry using groups (38)</b><br />
<br />
1873 - Hermite proves e is a transcendental (6)<br />
<br />
<b>1873 - Brocard produces his exhaustive work on the triangle (21)</b><br />
<br />
1874 - Cantor treats the notion of infinity rigorously, specifying different orders of infinity (7)<br />
<br />
<b>1874 - Carl Schorlemmer links chemistry with "trees" (29)</b><br />
<br />
1876 - Fechner writes on psychological experiments to determine the proportions of the most "aesthetic" rectangle (12)<br />
<br />
<b>1877 - Cantor is surprised by his controversial discoveries in dimension theory (24)</b><br />
<br />
<b>1878 - Georg Frobenius proves some of the key results of matrix algebra (39)</b><br />
<br />
<b>1879 - Cayley works on a precursor of modern fractals (25)</b><br />
<br />
<b>1879 - Kempe is believed to solve the 4-color problem. He hasn't. (30)</b><br />
<br />
1881 - Venn produces "Venn Diagrams" for sets (18)<br />
<br />
<b>1881 - Newcomb discovers what becomes known as Benford's law (34)</b><br />
<br />
<b>1881 - Michelson measures the speed of light with great accuracy (48)</b><br />
<br />
1882 - Lindemann proves that pi is transcendental (5)<br />
<br />
<b>1882 - Lindemann proves the circle cannot be squared (20)</b><br />
<br />
<b>1885-8 - Galton introduces regression and correlation (36)</b><br />
<br />
<b>1887 - The Lorenz transformations are first written down (48)</b><br />
<br />
<b>1889 - Poincare encounters chaos in his work on the three-body problem for which he is awarded a prize by King Oscar of Sweden (26)</b><br />
<br />
<b>1890 - Peano proves a solid square is a curve (the space-filling curve) (22)</b><br />
<br />
<b>1890 - Heawood exposes errors in Kempe's 4-colour proof and proves a 5-colour theorem (30)</b><br />
<br />
<b>1891 - Evgraf Fedorov and Arthur Schonflies independently classify the 230 crystallographic groups (38)</b><br />
<br />
<b>1892 - Fano discovers the Fano plane, the simplest example of a projective geometry (28)</b><br />
<br />
1896 - The prime number theorem on the distribution of primes is proved (9)<br />
<br />
<b>1896 - Pearson publishes contributions to correlation and regression (36)</b><br />
<br />
<b>1896 - De la Vallee-Poussin and Hadamard show all important zeros lie <i>within</i> Riemann's critical strip (50)</b><br />
<br />
<b>1898 - Bortkiewicz analyses the deaths of Prussian cavalrymen (34)</b><br />
<br />
<b>1899 - Pick publishes his theorem on the area of polygons (28)</b><br />
<br />
<b>1900 - Tarry shows there are no orthogonal Latin squares of order 6 (43)</b><br />
<br />
<b>1900 - Hilbert places Reimann's Hypothesis in his list of key problems for mathematicians to solve. It is Hilbert's personally favorite problem, and still unsolved as of 2011 (50)</b><br />
<br />
<b>1901 - Aleksandr Lyapunov proves the Central Limit theorem rigourously using characteristic functions (35)</b><br />
<br />
1902 - Lebesgue sets out the theory of the Lebesgue integral (19)<br />
<br />
<b>1902 - Farkas gives a solution of inequality systems (45)</b><br />
<br />
<b>1904 - von Koch creates his snowflake curve (25)</b><br />
<br />
<b>1904 - Spearman uses rank correlation as a tool for psychological studies (36)</b><br />
<br />
<b>1905 - Einstein publishes <i>On the electrodynamics of moving bodies</i> , the paper that describes special relativity (48)</b><br />
<br />
<b>1907 - von Lindemann claims a proof of Fermat's Last Theorem, but is shown to be wrong (49)</b><br />
<br />
<b>1908 - Hardy and Weinberg show why dominant genes do not supplant recessive genes (37)</b><br />
<br />
<b>1908 - Wolfskehl offers a prize for solutions of Fermat's Last Theorem within the next 100 years (49)</b><br />
<br />
<b>1909 - Brouwer's work changes our notion of dimension (24)</b><br />
<br />
1910 - Russell and Whitehead attempt to reduce mathematics to logic (16)<br />
<br />
<b>1912 - Keynes publishes his <i>Treatise on Probability</i> which influences his theories of economics and statistics (31)</b><br />
<br />
<b>1914 - Hardy proves there are infinitely many solutions along Riemann's line (50)</b><br />
<br />
<b>1915 - Einstein publishes <i>The field equations for gravitation</i>, his paper that describes the theory of general relativity, based on Riemannian geometry (27)(48)</b><br />
<br />
<b>1918 - Hausdorff introduces his concept of fractional dimension (25)</b><br />
<br />
<b>1918 - Fisher reconciles Darwin's theory with the Mendelian theory of heredity (37)</b><br />
<br />
<b>1919 - Hausdorff introduces the notion of the fractional "Hausdorff dimension" (24)</b><br />
<br />
<b>1919 - Julia and Fatou investigate fractal structures in the complex plane (25)</b><br />
<br />
1920's - Emmy Noether publishes papers in the development of modern abstract algebra (14)<br />
<br />
<b>1920's - Menger and Urysohn define curves as part of topology (22)</b><br />
<br />
<b>1920's - Bose considers Einstein's theory of light as an occupancy problem (33)</b><br />
<br />
<b>1920's - The Enigma machine is developed (40)</b><br />
<br />
1923 - Bartok composes his "Dance Suite", believed to be inspired by the Fibonacci numbers (11)<br />
<br />
<b>1925 - Heisenberg uses matrix mechanics in quantum theory (39)</b><br />
<br />
<b>1925 - Fisher suggests using Latin squares to design statistical experiments (43)</b><br />
<br />
<b>1926 - Boruvka introduces the greedy algorithm (46)</b><br />
<br />
1930 - Bartel van der Waerden publishes his famous <i>Moderne Algebra</i> (14)<br />
<br />
<b>1930 - Kuratowski proves his planar graphs theorem (29)</b><br />
<br />
<b>1930 - Frank Ramsey works in combinatorics (41)</b><br />
<br />
1931 - Godel proves that any formal axiomatic mathematical system contains undecidable statements (18)<br />
<br />
<b>1933 - Kolmogorov presents probability in an axiomatic way (31)</b><br />
<br />
<b>1935 - George Polya develops counting techniques for graphs as algebra (29)</b><br />
<br />
<b>1937 - De Finetti champions subjective probability as an alternative to the frequency theory (32)</b><br />
<br />
<b>1939 - Benford restates the law of distribution of first digits (34)</b><br />
<br />
1939 - The pseudonym Bourbaki is first used by French mathematicians (18)<br />
<br />
<b>1939 - Richard von Mises proposes the birthday problem (33)</b><br />
<br />
<b>1944 - von Neumann and Morgenstern publish <i>Theory of Games and Economic Behavior</i> (47)</b><br />
<br />
<b>1945 - Stigler solves the diet problem by a heuristic method (45)</b><br />
<br />
<b>1947 - Dantzig formulates the simplex method and solves the diet problem by linear programming (45)</b><br />
<br />
<b>1950 - Jimmy Savage and Dennis Lindley spearhead the modern Bayesian movement (32)</b><br />
<br />
<b>1950's - The term "Bayesian" comes into use for the first time (32)</b><br />
<br />
<b>1950 - Zipf derives a formula relating word use to vocabulary (34)</b><br />
<br />
<b>1950 - Richard Hamming publishes a key paper on error-detecting and error-correcting codes (4)</b><br />
<br />
<b>1950 - Tucker proposes the prisoner's dilemma and nash proposes the nash equilibrium (47)</b><br />
<br />
<b>1953 - The double helix structure of DNA is discovered (37)</b><br />
<br />
<b>1954 - Dantzig and Dijkstra propose methods for attacking the travelling salesperson problem (46)</b><br />
<br />
1960s - Abraham Robinson devises a non-standard arithmetic based on the notion of the infinitesimal (7)<br />
<br />
<b>1960 - Euler's conjecture about the non-existence of certain pairs of Latin squares is disproved by Bose, Parker and Shrikhande (43)</b><br />
<br />
<b>1961 - Stephen Smale proves the Poincare conjecture in dimensions greater than 4 (23)</b><br />
<br />
<b>1961 - Lorenz observes the butterfly effect. (26)</b><br />
<br />
1963 - The Fibonacci Quarterly, a journal devoted to the number theory of the Fibonacci sequence, is founded (11)<br />
<br />
1964 - Cohen proves the independence of the continuum hypothesis (18)<br />
<br />
1965 - Lofti Zadeh develops fuzzy logic (16)<br />
<br />
1966 - Chen Jingrun almost confirms the Goldbach conjecture (9)<br />
<br />
1967 - Bishop proves results exclusively by constructive methods (17)<br />
<br />
1970's - The Chinese remainder theorem is applied to message encryption (15)<br />
<br />
<b>1970's - Public key cryptography is developed (40)</b><br />
<br />
<b>1970 - String theory conceives of our universe having 10, 11 or 26 dimensions (24)</b><br />
<br />
<b>1971 - Robert May investigates chaos in the population model (26)</b><br />
<br />
<b>1971 - Ray-Chaudhuri and Wilson prove the existence of Kirkman's systems (41)</b><br />
<br />
<b>1971 - Cook formulates the P versus NP concept for algorithms (46)</b><br />
<br />
1975 - The International Organization for Standardization (ISO) defines the A paper size (12)<br />
<br />
<b>1975 - Mandelbrot introduces the term fractal (25)</b><br />
<br />
1976 - Imre Lakatos publishes the influential <i>Proofs and Refutations</i> (17)<br />
<br />
<b>1976 - Appel and Haken give a computer-based proof for the general result of the 4-colour problem (30)</b><br />
<br />
<b>1979 - Sudoku-like games are invented in New York (43)</b><br />
<br />
<b>1982 - Michael Freedman proves the Poincare conjecture in dimension equal to 4 (23)</b><br />
<br />
<b>1982 - Maynard Smith publishes <i>Evolution and the Theory of Games</i> (47)</b><br />
<br />
<b>1983 - The classification of finite simple groups is completed and the enormous theorem proved (38)</b><br />
<br />
<b>1984 - Karmarker at Bell Labs derives a new algorithm for solving linear programming problems (45)</b><br />
<br />
1987 - The underground train system in Japan is based on fuzzy logic (16)<br />
<br />
<b>1986 - Sallows creates his letter-based square (42)</b><br />
<br />
<b>1992 - The International Society for Bayesian Analysis is founded (32)</b><br />
<br />
<b>1994 - The computer proof of the 4-colour problem is simplified by remains a computer-based proof (30)</b><br />
<br />
<b>1994 - Nash is awarded the Nobel Prize in Economics for his work on game theory (47)</b><br />
<br />
<b>1994 - Wiles finally proves Fermat's Last Theorem (49)</b><br />
<br />
<b>1999 - Eric Rains and Neil Sloane extend tree counting (29)</b><br />
<br />
<b>2002 - Perelman proves the Poincare conjecture for dimension 3 (23)</b><br />
<br />
<b>2003 - The Poisson distribution is used in the analysis of fish stocks in the North Atlantic (34)</b><br />
<br />
<b>2004 - Chaos theory reaches popular culture in the film <i>The Butterfly Effect</i> (26)</b><br />
<br />
<b>2004 - David Applegate solves the travelling salesperson problem for all 24,978 cities in Sweden (46)</b><br />
<br />
<b>2004 - The first 10 trillion zeros of Riemann's Hypothesis are verified to be on the critical line. (50)</b><br />
<br />
2006 - The great prime search project finds the 44th Mersenne prime (with almost ten million digits) and yet another new perfect number can be generated (10)<br />
<br />
2007 - e is calculated to 10^11 digits (6)<br />
<br />
2007 - Sculptor Peter Randall-Page creates the 70 tonne sculpture "Seed" based on the Fibonacci sequence for the Eden Project in Cornwall, UK (11)<br />
<br />
<a href="http://www.amazon.com/Mathematics-Ideas-Really-Need-ideas/dp/1847241476/ref=sr_1_1?ie=UTF8&qid=1294507308&sr=8-1">Source material, here. Thank you, Tony Crilly, great book!</a><br />
Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com0tag:blogger.com,1999:blog-5303246073824127471.post-13109684631202741692012-12-14T10:32:00.004-05:002012-12-14T10:38:32.019-05:00Mathematics Jokes<br />
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No matter what I serve my guests, they seem to like my Math jokes best.</div>
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*********</div>
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A mathematician and an engineer are sitting at a table drinking when a very beautiful woman walks in and sits down at the bar.</div>
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The mathematician sighs. "I'd like to talk to her, but first I have to cover half the distance between where we are and where she is, then half of the distance that remains, then half of that distance, and so on. The series is infinite. There'll always be some finite distance between us."</div>
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The engineer gets up and starts walking. "Ah, well, I figure I can get close enough for all practical purposes."</div>
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A mathematician is a device for turning coffee into theorems</div>
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... <b>Alfréd Rényi</b></div>
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A topologist is a mathematician who can't tell the difference between a doughnut and a coffee mug.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTUFyLns6h4UGWoFfBf7TPaBoM0EANe16IU14Km94vBXd5CyEzcQKnCKKtedjoV77B4O_17AhItYxI4CJyFc9-ToVFj7QypBML_bM2SGEt2u8qJvw_d_SCuVJEFyDImyTXQhC9FcbFew/s1600/Math-jokes.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5530863715222695218" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTUFyLns6h4UGWoFfBf7TPaBoM0EANe16IU14Km94vBXd5CyEzcQKnCKKtedjoV77B4O_17AhItYxI4CJyFc9-ToVFj7QypBML_bM2SGEt2u8qJvw_d_SCuVJEFyDImyTXQhC9FcbFew/s400/Math-jokes.jpg" style="cursor: pointer; display: block; height: 400px; margin: 0px auto 10px; text-align: center; width: 327px;" /></a></div>
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Did you know that all numbers are interesting? What’s that? You don’t believe me? Well I have a proof. Suppose not every number is interesting. Then let n be the smallest uninteresting number. That’s a rather interesting property isn’t it?</div>
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... Ron Graham</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs6Pa3Ohwvd47zPIwhzdTdHHQFKpT5psLdS1b4xADHTc0Uj5MnF_Mh2FSGao4Kvw6MWl7tTMlao6mCh_48BMdq6JNFTaYHStdY1PRA2oGYZeSzyKZ7VV_tek5pmqUR2MlCrsskPm9weA/s1600/Derive.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5530865524137209810" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs6Pa3Ohwvd47zPIwhzdTdHHQFKpT5psLdS1b4xADHTc0Uj5MnF_Mh2FSGao4Kvw6MWl7tTMlao6mCh_48BMdq6JNFTaYHStdY1PRA2oGYZeSzyKZ7VV_tek5pmqUR2MlCrsskPm9weA/s400/Derive.jpg" style="cursor: pointer; display: block; height: 250px; margin: 0px auto 10px; text-align: center; width: 250px;" /></a></div>
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Q: What is the difference between a mathematician and a philosopher?</div>
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A: The mathematician only needs paper, pencil, and a trash bin for his work - the philosopher can do without the trash bin...</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTM6XT8YqGSNglT7mNDsznEyv38vsL5cRNdiVSDZSVVPemrar2lyNXyyII3NGM9V7fBs5VNIjH2X8EW95AM588DG3Gzh4-_zjjR7e3kqRUblWAJpEG7dBKJsbscjYbegOYVkT5O7zsUQ/s1600/Math_comic3.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5530866997107825122" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTM6XT8YqGSNglT7mNDsznEyv38vsL5cRNdiVSDZSVVPemrar2lyNXyyII3NGM9V7fBs5VNIjH2X8EW95AM588DG3Gzh4-_zjjR7e3kqRUblWAJpEG7dBKJsbscjYbegOYVkT5O7zsUQ/s400/Math_comic3.jpg" style="cursor: pointer; display: block; height: 330px; margin: 0px auto 10px; text-align: center; width: 270px;" /></a></div>
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Q: What is the difference between a Ph.D. in mathematics and a large pizza?</div>
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A: A large pizza can feed a family of four.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheW_Eg3L3yEH_HJi1yH6-1WBlcpDCTwIgJxQk8gS274Jq1ctIyji00cLC3cQqa4hdy0YTMKdQQAZCde4Rkv5wlt5AWrm9MikX2CJMLBNjwb-iQDRNhynEam-SzLmKYK7DKWlTzze3RVg/s1600/math1.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5530893771026785890" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheW_Eg3L3yEH_HJi1yH6-1WBlcpDCTwIgJxQk8gS274Jq1ctIyji00cLC3cQqa4hdy0YTMKdQQAZCde4Rkv5wlt5AWrm9MikX2CJMLBNjwb-iQDRNhynEam-SzLmKYK7DKWlTzze3RVg/s400/math1.gif" style="cursor: pointer; display: block; height: 284px; margin: 0px auto 10px; text-align: center; width: 378px;" /></a></div>
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When the math professor's wife returns home from work, she finds an envelope on the living room table. She opens it and finds a letter from her husband:</div>
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<i>My dearest wife,<br /><br />We have been married for nearly thirty years, and I still love you as much as on the day I proposed. You must realize, however, that you are now 54 years old and no longer able to satisfy certain needs I still have. I very much hope that you are not hurt to learn that, while you're reading this, I'm in a hotel room with an 18-year-old freshman girl from my calculus class. I'll be home before midnight.<br /><br />Your husband, who will never stop loving you.</i></div>
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When the professor returns from the hotel shortly before midnight, he also finds an envelope in the living room. He opens it and reads:</div>
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<i>My beloved husband,<br /><br />You may recall that you, too, are 54 years old and no longer able to satisfy certain needs I still have. I thus hope that you are not hurt to learn that, while you're reading this, I am in a hotel room with the 18-year-old pool boy.<br /><br />Your loving wife.<br /><br />P.S. As a mathematician, you are certainly aware of the fact that 18 goes into 54 many more times than 54 goes into 18. Therefore, don't stay up and wait for me.</i></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhR9y0duRUM3NB-l366UErE9zkgaoZoYccKlmhULXu-G_lfUxhOUHHvHU4neNs4fbDtEFR46LKakq1hTKW5G6qJtTXYfzOuSh7F-fQo0Lq92KbA4aZykJk5ClkXZnUl3N3CE58o4K4G2Q/s1600/math-jokes-find-x-here-it-is.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5530894807079577778" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhR9y0duRUM3NB-l366UErE9zkgaoZoYccKlmhULXu-G_lfUxhOUHHvHU4neNs4fbDtEFR46LKakq1hTKW5G6qJtTXYfzOuSh7F-fQo0Lq92KbA4aZykJk5ClkXZnUl3N3CE58o4K4G2Q/s400/math-jokes-find-x-here-it-is.jpg" style="cursor: pointer; display: block; height: 310px; margin: 0px auto 10px; text-align: center; width: 394px;" /></a></div>
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Q. Why do mathematicians like national parks?</div>
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A. Because of the natural logs.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0PMaQRyMxiQLjVGDqT852rBUb9-sk2Pxh5kmMZdIIoSdp2gNlfGaE1Joxm1anEg8bDjKIlDRucfupeq_U4iH8yI60h1-4juYJ70FPvlruDA6Ho6U-TEfGfgsyIKkUzSA7wEVRQLE0fw/s1600/happy.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5530895797928454722" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0PMaQRyMxiQLjVGDqT852rBUb9-sk2Pxh5kmMZdIIoSdp2gNlfGaE1Joxm1anEg8bDjKIlDRucfupeq_U4iH8yI60h1-4juYJ70FPvlruDA6Ho6U-TEfGfgsyIKkUzSA7wEVRQLE0fw/s400/happy.gif" style="cursor: pointer; display: block; height: 400px; margin: 0px auto 10px; text-align: center; width: 394px;" /></a></div>
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Q: Why didn’t Newton discover group theory?</div>
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A: Because he wasn’t Abel.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgV2xu5jBN0c-5ICBE6dwjmQXo1t7LiDvvDpCpKLzHR6bGYY-997ld0C1aY8Sxh8zgYuIcD-lhmNL6LX5YhLmowyAX1Zd4V7qln9Oim4meZGQMPkR-jB9nw_Vc3q3lbyJ5yTrMJc_lBrA/s1600/blonde_equation2.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5531015591992108018" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgV2xu5jBN0c-5ICBE6dwjmQXo1t7LiDvvDpCpKLzHR6bGYY-997ld0C1aY8Sxh8zgYuIcD-lhmNL6LX5YhLmowyAX1Zd4V7qln9Oim4meZGQMPkR-jB9nw_Vc3q3lbyJ5yTrMJc_lBrA/s400/blonde_equation2.jpg" style="cursor: pointer; display: block; height: 284px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a></div>
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The integral of e raised to the power of x equals the function of u raised to the power of n.</div>
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(Write it out in notation to see the joke)</div>
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Did you really write it out? You didn't do that in your head? ;-)</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxc9Aa7gAY3RtFtEmEpC8JFSVXmbigvkgEl9m7YLUx2BZMeEkkcDs_HJUYBEC0GQj4CGzw5_t10QskHprTzPLHmlwlvc6nBpDM7PBSHk-bY6ATSqG1lYdUikedOTVoDmu6wZ-PRAMe3w/s1600/MathChem.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5531020979247886162" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxc9Aa7gAY3RtFtEmEpC8JFSVXmbigvkgEl9m7YLUx2BZMeEkkcDs_HJUYBEC0GQj4CGzw5_t10QskHprTzPLHmlwlvc6nBpDM7PBSHk-bY6ATSqG1lYdUikedOTVoDmu6wZ-PRAMe3w/s400/MathChem.gif" style="cursor: pointer; display: block; height: 300px; margin: 0px auto 10px; text-align: center; width: 328px;" /></a></div>
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True story:</div>
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A student walked into his discrete math class late and in order not to interrupt he put his late slip on the teacher's desk furtively without the teacher noticing. The teacher noticed the slip on his desk afterwards. He commented "I see you put this slip on my desk without me noticing. I guess that's why they call this class discrete mathematics."</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAy7b3k7damT8wu_eU7mNuw2vwX9jVSnodOu7R0F_TNsY-eaQ2St0rhiZEB6NhJsEezLezxLHMOzDrzVlTVStW5sFNI1gA11BYzf9FW5hpnfrTj_PjtB-ZHmvZ0W6N9dO8Fu4-7HPUFA/s1600/Strings.png" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5531021429198113346" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAy7b3k7damT8wu_eU7mNuw2vwX9jVSnodOu7R0F_TNsY-eaQ2St0rhiZEB6NhJsEezLezxLHMOzDrzVlTVStW5sFNI1gA11BYzf9FW5hpnfrTj_PjtB-ZHmvZ0W6N9dO8Fu4-7HPUFA/s400/Strings.png" style="cursor: pointer; display: block; height: 386px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a></div>
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There is a shipwreck, and the only three survivors are a Doctor, a Lawyer, and a Mathematician, in a rowboat.</div>
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After some time drifting about the seas, eventually they get get to talking and get to know each other. One day the doctor asks, "Is it better to have a wife or a girlfriend? I would say it's better to have a wife. I work long hard and emotional hours, and it's really great to have a caring wife who cooks great meals, cleans my clothes, and expertly manages our home and children."</div>
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The lawyer says, "I think it's better to have a girlfriend. I'm a Divorce Lawyer and the cost to the man in Divorce is so extreme I don't see where having a wife is worth the risk."</div>
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The mathematician says, "I think it's better to have both."</div>
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"What !?" say the doctor and lawyer. "Why?"</div>
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"Because," the mathematician says, "You can tell your wife you're working late, and your girlfriend you need to spend time with your family, which gives you more time to work on proving the Riemann Hypothesis !"<br />
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Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com3tag:blogger.com,1999:blog-5303246073824127471.post-10521733117955857112012-06-22T19:50:00.000-04:002012-06-22T19:51:16.387-04:00CERN Announces July 4 Press Conference re Higgs Boson<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;">Today CERN announced there will be an announcement regarding the progress in verifying the existence of The Higgs Boson on July 4, 2012. "Born on the 4th of July" is how one scientist describes it. Well perhaps, we shall see. But it looks good. ;-)<br />
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip2o7m8RcH5-Gsv71DVRA_ZEdtVmCUKvbVu5H8q611DkVXDaKpDtVL5NtbGOnVTcOzth0ORp1YRCBNytl6hTLoyAN7w7gRymBUgvWHp9PcIQkuOba94bsSF4zvSZMAqy5f1Nar6WSzYQ/s1600/Standard+Model.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="283" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip2o7m8RcH5-Gsv71DVRA_ZEdtVmCUKvbVu5H8q611DkVXDaKpDtVL5NtbGOnVTcOzth0ORp1YRCBNytl6hTLoyAN7w7gRymBUgvWHp9PcIQkuOba94bsSF4zvSZMAqy5f1Nar6WSzYQ/s400/Standard+Model.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 12px; line-height: 20px; text-align: -webkit-auto;">A diagram summarizing the tree-level interactions between elementary particles described in the Standard Model. Vertices (darkened circles) represent types of particles, and edges (blue arcs) connecting them represent interactions that can take place. The organization of the diagram is as follows: the top row of vertices (leptons and quarks) are the matter particles; the second row of vertices (photon, W/Z, gluons) are the force mediating particles; and the bottom row is the Higgs boson.</span>
</td></tr>
</tbody></table>
<span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;">The</span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;"> </span><b style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;">Standard Model</b><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;"> </span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;">of</span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;"> </span><a href="http://en.wikipedia.org/wiki/Particle_physics" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 1.5em; text-decoration: none;" title="Particle physics">particle physics</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;"> </span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;">is a theory concerning the</span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;"> </span><a href="http://en.wikipedia.org/wiki/Electromagnetism" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 1.5em; text-decoration: none;" title="Electromagnetism">electromagnetic</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;">,</span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;"> </span><a href="http://en.wikipedia.org/wiki/Weak_interaction" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 1.5em; text-decoration: none;" title="Weak interaction">weak</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;">, and</span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;"> </span><a href="http://en.wikipedia.org/wiki/Strong_interaction" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 1.5em; text-decoration: none;" title="Strong interaction">strong</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;"> </span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;">nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon experimental confirmation of the existence of</span><a href="http://en.wikipedia.org/wiki/Quark" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 1.5em; text-decoration: none;" title="Quark">quarks</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;">. Since then, discoveries of the</span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;"> </span><a href="http://en.wikipedia.org/wiki/Bottom_quark" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 1.5em; text-decoration: none;" title="Bottom quark">bottom quark</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;"> </span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;">(1977), the</span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;"> </span><a href="http://en.wikipedia.org/wiki/Top_quark" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 1.5em; text-decoration: none;" title="Top quark">top quark</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;"> </span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;">(1995) and the</span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;"> </span><a href="http://en.wikipedia.org/wiki/Tau_neutrino" style="background-color: white; background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 1.5em; text-decoration: none;" title="Tau neutrino">tau neutrino</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;"> </span><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 1.5em;">(2000) have given further credence to the Standard Model. Because of its success in explaining a wide variety of experimental results, the Standard Model is sometimes regarded as a "theory of almost everything".</span><br />
<div class="mw-content-ltr" dir="ltr" id="mw-content-text" lang="en" style="direction: ltr; font-family: sans-serif; font-size: 13px; line-height: 20px;">
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The Standard Model falls short of being a <a href="http://en.wikipedia.org/wiki/Theory_of_everything" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theory of everything">complete theory of fundamental interactions</a> because it does not incorporate the physics of <a href="http://en.wikipedia.org/wiki/Dark_energy" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dark energy">dark energy</a> nor of the full theory of <a href="http://en.wikipedia.org/wiki/Gravitation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gravitation">gravitation</a> as described by <a href="http://en.wikipedia.org/wiki/General_relativity" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="General relativity">general relativity</a>. The theory does not contain any viable <a href="http://en.wikipedia.org/wiki/Dark_matter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dark matter">dark matter</a> particle that possesses all of the required properties deduced from observational <a href="http://en.wikipedia.org/wiki/Cosmology" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cosmology">cosmology</a>. It also does not correctly account for <a href="http://en.wikipedia.org/wiki/Neutrino_oscillation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Neutrino oscillation">neutrino oscillations</a> (and their non-zero masses). Although the Standard Model is believed to be theoretically self-consistent, it has several apparently unnatural properties giving rise to puzzles like the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Strong_CP_problem" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Strong CP problem">strong CP problem</a> and the <a href="http://en.wikipedia.org/wiki/Hierarchy_problem" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hierarchy problem">hierarchy problem</a>.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Nevertheless, the Standard Model is important to <a href="http://en.wikipedia.org/wiki/Theoretical_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theoretical physics">theoretical</a> and <a href="http://en.wikipedia.org/wiki/Experimental_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Experimental physics">experimental</a> particle physicists alike. For theorists, the Standard Model is a paradigmatic example of a <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">quantum field theory</a>, which exhibits a wide range of physics including <a href="http://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spontaneous symmetry breaking">spontaneous symmetry breaking</a>, <a href="http://en.wikipedia.org/wiki/Anomaly_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Anomaly (physics)">anomalies</a>, non-perturbative behavior, etc. It is used as a basis for building more <a href="http://en.wikipedia.org/wiki/Physics_beyond_the_Standard_Model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physics beyond the Standard Model">exotic models</a> that incorporate <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Hypothetical_particle" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hypothetical particle">hypothetical particles</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Extra_dimensions_(disambiguation)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Extra dimensions (disambiguation)">extra dimensions</a>, and elaborate symmetries (such as <a href="http://en.wikipedia.org/wiki/Supersymmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Supersymmetry">supersymmetry</a>) in an attempt to explain experimental results at variance with the Standard Model, such as the existence of dark matter and neutrino oscillations. In turn, experimenters have incorporated the Standard Model into simulators to help search for new physics <a href="http://en.wikipedia.org/wiki/Physics_beyond_the_Standard_Model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physics beyond the Standard Model">beyond the Standard Model</a>.</div>
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Recently, the Standard Model has found applications in fields besides particle physics, such as <a href="http://en.wikipedia.org/wiki/Astrophysics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Astrophysics">astrophysics</a>, cosmology, and <a href="http://en.wikipedia.org/wiki/Nuclear_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nuclear physics">nuclear physics</a>.</div>
<table class="toc" id="toc" style="background-color: #f9f9f9; border: 1px solid rgb(170, 170, 170); font-size: 12px; padding: 5px;"><tbody>
<tr><td><div id="toctitle" style="direction: ltr; text-align: center;">
<h2 style="background-image: none; border: none; display: inline; font-size: 12px; margin: 0px 0px 0.6em; overflow: hidden; padding: 0px;">
Contents</h2>
<span class="toctoggle" style="-webkit-user-select: none;"> [<a class="internal" href="http://en.wikipedia.org/wiki/Standard_model#" id="togglelink" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">hide</a>] </span></div>
<ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin: 0.3em 0px; padding: 0px; text-align: left;">
<li class="toclevel-1 tocsection-1" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Historical_background" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1</span> <span class="toctext">Historical background</span></a></li>
<li class="toclevel-1 tocsection-2" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Overview" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2</span> <span class="toctext">Overview</span></a></li>
<li class="toclevel-1 tocsection-3" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Particle_content" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3</span> <span class="toctext">Particle content</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin: 0px 0px 0px 2em; padding: 0px;">
<li class="toclevel-2 tocsection-4" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Fermions" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.1</span> <span class="toctext">Fermions</span></a></li>
<li class="toclevel-2 tocsection-5" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Gauge_bosons" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.2</span> <span class="toctext">Gauge bosons</span></a></li>
<li class="toclevel-2 tocsection-6" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Higgs_boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.3</span> <span class="toctext">Higgs boson</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-7" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Field_content" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4</span> <span class="toctext">Field content</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin: 0px 0px 0px 2em; padding: 0px;">
<li class="toclevel-2 tocsection-8" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Spin_1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.1</span> <span class="toctext">Spin 1</span></a></li>
<li class="toclevel-2 tocsection-9" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Spin_1.E2.81.842" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.2</span> <span class="toctext">Spin <sup style="line-height: 1em;">1</sup>⁄<sub style="line-height: 1em;">2</sub></span></a></li>
<li class="toclevel-2 tocsection-10" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Spin_0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.3</span> <span class="toctext">Spin 0</span></a></li>
<li class="toclevel-2 tocsection-11" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Lagrangian" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.4</span> <span class="toctext">Lagrangian</span></a></li>
<li class="toclevel-2 tocsection-12" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Higgs_mechanism" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.5</span> <span class="toctext">Higgs mechanism</span></a></li>
<li class="toclevel-2 tocsection-13" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Masses_and_CKM_matrix" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.6</span> <span class="toctext">Masses and CKM matrix</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-14" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Theoretical_aspects" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5</span> <span class="toctext">Theoretical aspects</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin: 0px 0px 0px 2em; padding: 0px;">
<li class="toclevel-2 tocsection-15" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Construction_of_the_Standard_Model_Lagrangian" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5.1</span> <span class="toctext">Construction of the Standard Model Lagrangian</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin: 0px 0px 0px 2em; padding: 0px;">
<li class="toclevel-3 tocsection-16" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Quantum_chromodynamics_sector" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5.1.1</span> <span class="toctext">Quantum chromodynamics sector</span></a></li>
<li class="toclevel-3 tocsection-17" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Electroweak_sector" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5.1.2</span> <span class="toctext">Electroweak sector</span></a></li>
<li class="toclevel-3 tocsection-18" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Higgs_sector" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5.1.3</span> <span class="toctext">Higgs sector</span></a></li>
</ul>
</li>
<li class="toclevel-2 tocsection-19" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Additional_symmetries_of_the_Standard_Model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5.2</span> <span class="toctext">Additional symmetries of the Standard Model</span></a></li>
<li class="toclevel-2 tocsection-20" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#List_of_Standard_Model_fermions" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5.3</span> <span class="toctext">List of Standard Model fermions</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-21" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Tests_and_predictions" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6</span> <span class="toctext">Tests and predictions</span></a></li>
<li class="toclevel-1 tocsection-22" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Challenges" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">7</span> <span class="toctext">Challenges</span></a></li>
<li class="toclevel-1 tocsection-23" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#See_also" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">8</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-24" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Notes_and_references" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">9</span> <span class="toctext">Notes and references</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin: 0px 0px 0px 2em; padding: 0px;">
<li class="toclevel-2 tocsection-25" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Notes" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">9.1</span> <span class="toctext">Notes</span></a></li>
<li class="toclevel-2 tocsection-26" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#References" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">9.2</span> <span class="toctext">References</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-27" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#Further_reading" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">10</span> <span class="toctext">Further reading</span></a></li>
<li class="toclevel-1 tocsection-28" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#External_links" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">11</span> <span class="toctext">External links</span></a></li>
</ul>
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</tbody></table>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Historical background">edit</a>]</span><span class="mw-headline" id="Historical_background">Historical background</span></h2>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The first step towards the Standard Model was <a href="http://en.wikipedia.org/wiki/Sheldon_Lee_Glashow" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sheldon Lee Glashow">Sheldon Glashow</a>'s discovery in 1960 of a way to combine the <a href="http://en.wikipedia.org/wiki/Electromagnetism" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electromagnetism">electromagnetic</a> and <a href="http://en.wikipedia.org/wiki/Weak_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak interaction">weak interactions</a>.<sup class="reference" id="cite_ref-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[1]</a></sup> In 1967 <a href="http://en.wikipedia.org/wiki/Steven_Weinberg" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Steven Weinberg">Steven Weinberg</a><sup class="reference" id="cite_ref-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup> and <a href="http://en.wikipedia.org/wiki/Abdus_Salam" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Abdus Salam">Abdus Salam</a><sup class="reference" id="cite_ref-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup> incorporated the <a href="http://en.wikipedia.org/wiki/Higgs_mechanism" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Higgs mechanism">Higgs mechanism</a><sup class="reference" id="cite_ref-3" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[4]</a></sup><sup class="reference" id="cite_ref-4" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-4" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[5]</a></sup><sup class="reference" id="cite_ref-5" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-5" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[6]</a></sup> into Glashow's <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Electroweak_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electroweak theory">electroweak theory</a>, giving it its modern form.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The Higgs mechanism is believed to give rise to the <a href="http://en.wikipedia.org/wiki/Mass" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mass">masses</a> of all the <a href="http://en.wikipedia.org/wiki/Elementary_particle" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Elementary particle">elementary particles</a> in the Standard Model. This includes the masses of the <a href="http://en.wikipedia.org/wiki/W_and_Z_bosons" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="W and Z bosons">W and Z bosons</a>, and the masses of the <a href="http://en.wikipedia.org/wiki/Fermion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermion">fermions</a>, i.e. the <a href="http://en.wikipedia.org/wiki/Quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark">quarks</a> and <a href="http://en.wikipedia.org/wiki/Lepton" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lepton">leptons</a>.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
After the <a href="http://en.wikipedia.org/wiki/Neutral_current" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Neutral current">neutral weak currents</a> caused by <span class="unicode;" style="white-space: nowrap;">Z</span> boson exchange <a href="http://en.wikipedia.org/wiki/Gargamelle" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gargamelle">were discovered</a> at <a href="http://en.wikipedia.org/wiki/CERN" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="CERN">CERN</a> in 1973,<sup class="reference" id="cite_ref-6" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-6" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[7]</a></sup><sup class="reference" id="cite_ref-7" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-7" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[8]</a></sup><sup class="reference" id="cite_ref-8" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-8" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[9]</a></sup><sup class="reference" id="cite_ref-9" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-9" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[10]</a></sup> the electroweak theory became widely accepted and Glashow, Salam, and Weinberg shared the 1979 <a href="http://en.wikipedia.org/wiki/Nobel_Prize_in_Physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nobel Prize in Physics">Nobel Prize in Physics</a> for discovering it. The W and Z bosons were discovered experimentally in 1981, and their masses were found to be as the Standard Model predicted.</div>
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The theory of the <a href="http://en.wikipedia.org/wiki/Strong_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Strong interaction">strong interaction</a>, to which many contributed, acquired its modern form around 1973–74, when experiments confirmed that the <a href="http://en.wikipedia.org/wiki/Hadron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hadron">hadrons</a> were composed of fractionally charged quarks.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Overview">edit</a>]</span><span class="mw-headline" id="Overview">Overview</span></h2>
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At present, <a href="http://en.wikipedia.org/wiki/Matter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Matter">matter</a> and <a href="http://en.wikipedia.org/wiki/Energy" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Energy">energy</a> are best understood in terms of the <a href="http://en.wikipedia.org/wiki/Kinematics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Kinematics">kinematics</a> and <a href="http://en.wikipedia.org/wiki/Fundamental_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fundamental interaction">interactions</a> of <a href="http://en.wikipedia.org/wiki/Elementary_particle" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Elementary particle">elementary particles</a>. To date, physics has reduced the <a href="http://en.wikipedia.org/wiki/Scientific_law" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Scientific law">laws</a> governing the behavior and interaction of all known forms of matter and energy to a small set of fundamental laws and theories. A major goal of physics is to find the "common ground" that would unite all of these theories into one integrated <a href="http://en.wikipedia.org/wiki/Theory_of_everything" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theory of everything">theory of everything</a>, of which all the other known laws would be special cases, and from which the behavior of all matter and energy could be derived (at least in principle).<sup class="reference" id="cite_ref-10" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-10" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[11]</a></sup></div>
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The Standard Model groups two major extant theories—<a href="http://en.wikipedia.org/wiki/Electroweak_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electroweak interaction">quantum electroweak</a> and <a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum chromodynamics">quantum chromodynamics</a>—into an internally consistent theory that describes the interactions between all known particles in terms of <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">quantum field theory</a>. For a technical description of the fields and their interactions, see <a href="http://en.wikipedia.org/wiki/Standard_Model_(mathematical_formulation)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Standard Model (mathematical formulation)">Standard Model (mathematical formulation)</a>.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Particle content">edit</a>]</span><span class="mw-headline" id="Particle_content">Particle content</span></h2>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=4" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Fermions">edit</a>]</span><span class="mw-headline" id="Fermions">Fermions</span></h3>
<table class="wikitable" style="background-color: #f9f9f9; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; float: right; font-size: 13px; margin: 0px 0px 1em 1em;"><caption style="font-weight: bold;">Organization of <a href="http://en.wikipedia.org/wiki/Fermion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermion">Fermions</a></caption><tbody>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"></th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Electric_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electric charge">Charge</a></th><th colspan="2" style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">First <a href="http://en.wikipedia.org/wiki/Generation_(particle_physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Generation (particle physics)">generation</a></th><th colspan="2" style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Second generation</th><th colspan="2" style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Third generation</th></tr>
<tr><th rowspan="2" style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark">Quarks</a></th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">+2/3</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Up_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Up quark">Up</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center; vertical-align: middle;"><span class="unicode;" style="white-space: nowrap;">u</span></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Charm_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Charm quark">Charm</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center; vertical-align: middle;"><span class="unicode;" style="white-space: nowrap;">c</span></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Top_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Top quark">Top</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center; vertical-align: middle;"><span class="unicode;" style="white-space: nowrap;">t</span></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">−1/3</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Down_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Down quark">Down</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center; vertical-align: middle;"><span class="unicode;" style="white-space: nowrap;">d</span></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Strange_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Strange quark">Strange</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center; vertical-align: middle;"><span class="unicode;" style="white-space: nowrap;">s</span></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Bottom_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bottom quark">Bottom</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center; vertical-align: middle;"><span class="unicode;" style="white-space: nowrap;">b</span></td></tr>
<tr><th rowspan="2" style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Lepton" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lepton">Leptons</a></th><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">−1</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Electron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron">Electron</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center; vertical-align: middle;"><span class="unicode;" style="white-space: nowrap;">e<span style="display: inline-block; font-size: 11px; line-height: 1.2em; margin-bottom: -0.3em; text-align: left; vertical-align: 0.8em;">−</span></span></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Muon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Muon">Muon</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center; vertical-align: middle;"><span class="unicode;" style="white-space: nowrap;">μ<span style="display: inline-block; font-size: 11px; line-height: 1.2em; margin-bottom: -0.3em; text-align: left; vertical-align: 0.8em;">−</span></span></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Tau_(particle)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tau (particle)">Tau</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center; vertical-align: middle;"><span class="unicode;" style="white-space: nowrap;">τ<span style="display: inline-block; font-size: 11px; line-height: 1.2em; margin-bottom: -0.3em; text-align: left; vertical-align: 0.8em;">−</span></span></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">0</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Electron_neutrino" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron neutrino">Electron neutrino</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center; vertical-align: middle;"><span class="unicode;" style="white-space: nowrap;">ν<span style="display: inline-block; font-size: 11px; line-height: 1.2em; margin-bottom: -0.3em; text-align: left; vertical-align: -0.4em;"><br />e</span></span></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Muon_neutrino" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Muon neutrino">Muon neutrino</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center; vertical-align: middle;"><span class="unicode;" style="white-space: nowrap;">ν<span style="display: inline-block; font-size: 11px; line-height: 1.2em; margin-bottom: -0.3em; text-align: left; vertical-align: -0.4em;"><br />μ</span></span></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Tau_neutrino" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tau neutrino">Tau neutrino</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center; vertical-align: middle;"><span class="unicode;" style="white-space: nowrap;">ν<span style="display: inline-block; font-size: 11px; line-height: 1.2em; margin-bottom: -0.3em; text-align: left; vertical-align: -0.4em;"><br />τ</span></span></td></tr>
</tbody></table>
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The Standard Model includes 12 <a href="http://en.wikipedia.org/wiki/Elementary_particle" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Elementary particle">elementary particles</a> of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Spin-1/2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin-1/2">spin <img alt="{}_\frac{1}{2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/c/b/6cb7bc6b1ab08027622389e01eced3ee.png" style="border: none; margin: 0px; vertical-align: middle;" /></a> known as <a href="http://en.wikipedia.org/wiki/Fermion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermion">fermions</a>. According to the <a href="http://en.wikipedia.org/wiki/Spin-statistics_theorem" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin-statistics theorem">spin-statistics theorem</a>, fermions respect the <a href="http://en.wikipedia.org/wiki/Pauli_exclusion_principle" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pauli exclusion principle">Pauli exclusion principle</a>. Each fermion has a corresponding <a href="http://en.wikipedia.org/wiki/Antiparticle" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Antiparticle">antiparticle</a>.</div>
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The fermions of the Standard Model are classified according to how they interact (or equivalently, by what <a href="http://en.wikipedia.org/wiki/Charge_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Charge (physics)">charges</a> they carry). There are six <a href="http://en.wikipedia.org/wiki/Quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark">quarks</a> (<a href="http://en.wikipedia.org/wiki/Up_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Up quark">up</a>, <a href="http://en.wikipedia.org/wiki/Down_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Down quark">down</a>, <a href="http://en.wikipedia.org/wiki/Charm_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Charm quark">charm</a>, <a href="http://en.wikipedia.org/wiki/Strange_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Strange quark">strange</a>,<a href="http://en.wikipedia.org/wiki/Top_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Top quark">top</a>, <a href="http://en.wikipedia.org/wiki/Bottom_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bottom quark">bottom</a>), and six <a href="http://en.wikipedia.org/wiki/Lepton" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lepton">leptons</a> (<a href="http://en.wikipedia.org/wiki/Electron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron">electron</a>, <a href="http://en.wikipedia.org/wiki/Electron_neutrino" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron neutrino">electron neutrino</a>, <a href="http://en.wikipedia.org/wiki/Muon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Muon">muon</a>, <a href="http://en.wikipedia.org/wiki/Muon_neutrino" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Muon neutrino">muon neutrino</a>, <a href="http://en.wikipedia.org/wiki/Tau_(particle)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tau (particle)">tau</a>, <a href="http://en.wikipedia.org/wiki/Tau_neutrino" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tau neutrino">tau neutrino</a>). Pairs from each classification are grouped together to form a <a href="http://en.wikipedia.org/wiki/Generation_(particle_physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Generation (particle physics)">generation</a>, with corresponding particles exhibiting similar physical behavior (see table).</div>
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The defining property of the quarks is that they carry <a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge">color charge</a>, and hence, interact via the <a href="http://en.wikipedia.org/wiki/Strong_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Strong interaction">strong interaction</a>. A phenomenon called <a href="http://en.wikipedia.org/wiki/Color_confinement" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color confinement">color confinement</a> results in quarks being perpetually (or at least since very soon after the start of the <a href="http://en.wikipedia.org/wiki/Big_Bang" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Big Bang">Big Bang</a>) bound to one another, forming color-neutral composite particles (<a href="http://en.wikipedia.org/wiki/Hadron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hadron">hadrons</a>) containing either a quark and an antiquark (<a href="http://en.wikipedia.org/wiki/Meson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Meson">mesons</a>) or three quarks (<a href="http://en.wikipedia.org/wiki/Baryon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Baryon">baryons</a>). The familiar <a href="http://en.wikipedia.org/wiki/Proton" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Proton">proton</a> and the <a href="http://en.wikipedia.org/wiki/Neutron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Neutron">neutron</a> are the two baryons having the smallest mass. Quarks also carry <a href="http://en.wikipedia.org/wiki/Electric_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electric charge">electric charge</a> and <a href="http://en.wikipedia.org/wiki/Weak_isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak isospin">weak isospin</a>. Hence they interact with other fermions both <a href="http://en.wikipedia.org/wiki/Electromagnetism" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electromagnetism">electromagnetically</a> and via the <a href="http://en.wikipedia.org/wiki/Weak_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak interaction">weak interaction</a>.</div>
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The remaining six fermions do not carry colour charge and are called leptons. The three <a href="http://en.wikipedia.org/wiki/Neutrino" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Neutrino">neutrinos</a> do not carry electric charge either, so their motion is directly influenced only by the <a href="http://en.wikipedia.org/wiki/Weak_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak interaction">weak nuclear force</a>, which makes them notoriously difficult to detect. However, by virtue of carrying an electric charge, the electron, muon, and tau all interact electromagnetically.</div>
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Each member of a generation has greater mass than the corresponding particles of lower generations. The first generation charged particles do not decay; hence all ordinary (baryonic) matter is made of such particles. Specifically, all atoms consist of electrons orbiting <a href="http://en.wikipedia.org/wiki/Atomic_nucleus" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Atomic nucleus">atomic nuclei</a> ultimately constituted of up and down quarks. Second and third generations charged particles, on the other hand, decay with very short half lives, and are observed only in very high-energy environments. Neutrinos of all generations also do not decay, and pervade the universe, but rarely interact with baryonic matter.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=5" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Gauge bosons">edit</a>]</span><span class="mw-headline" id="Gauge_bosons">Gauge bosons</span></h3>
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Summary of interactions between particles described by the Standard Model.</div>
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The above interactions form the basis of the standard model. Feynman diagrams in the standard model are built from these vertices. Modifications involving Higgs boson interactions and neutrino oscillations are commonly added. The charge of the W bosons are dictated by the fermions they interact with.</div>
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In the Standard Model, <a href="http://en.wikipedia.org/wiki/Gauge_boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge boson">gauge bosons</a> are defined as <a href="http://en.wikipedia.org/wiki/Force_carrier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Force carrier">force carriers</a> that mediate the strong, weak, and electromagnetic <a href="http://en.wikipedia.org/wiki/Fundamental_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fundamental interaction">fundamental interactions</a>.</div>
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Interactions in physics are the ways that particles influence other particles. At a <a href="http://en.wikipedia.org/wiki/Macroscopic_scale" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Macroscopic scale">macroscopic level</a>, electromagnetism allows particles to interact with one another via <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Electric" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electric">electric</a> and <a href="http://en.wikipedia.org/wiki/Magnetic_field" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Magnetic field">magnetic</a> fields, and gravitation allows particles with mass to attract one another in accordance with Einstein's theory of<a href="http://en.wikipedia.org/wiki/General_relativity" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="General relativity">general relativity</a>. The Standard Model explains such forces as resulting from matter particles<a href="http://en.wikipedia.org/wiki/Static_forces_and_virtual-particle_exchange" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Static forces and virtual-particle exchange">exchanging other particles</a>, known as <i>force mediating particles</i> (strictly speaking, this is only so if interpreting literally what is actually an <i>approximation method</i> known as <a href="http://en.wikipedia.org/wiki/Perturbation_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Perturbation theory">perturbation theory</a>)<sup class="Template-Fact" style="line-height: 1em; white-space: nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources from August 2010">citation needed</span></a></i>]</sup>. When a force mediating particle is exchanged, at a macroscopic level the effect is equivalent to a force influencing both of them, and the particle is therefore said to have<i>mediated</i> (i.e., been the agent of) that force. The <a href="http://en.wikipedia.org/wiki/Feynman_diagram" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Feynman diagram">Feynman diagram</a> calculations, which are a graphical representation of the perturbation theory approximation, invoke "force mediating particles", and when applied to analyze <a href="http://en.wikipedia.org/wiki/Particle_accelerator" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Particle accelerator">high-energy scattering experiments</a> are in reasonable agreement with the data. However, perturbation theory (and with it the concept of a "force-mediating particle") fails in other situations. These include low-energy <a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum chromodynamics">quantum chromodynamics</a>, <a href="http://en.wikipedia.org/wiki/Bound_state" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bound state">bound states</a>, and <a href="http://en.wikipedia.org/wiki/Soliton" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Soliton">solitons</a>.</div>
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The gauge bosons of the Standard Model all have <a href="http://en.wikipedia.org/wiki/Spin_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin (physics)">spin</a> (as do matter particles). The value of the spin is 1, making them <a href="http://en.wikipedia.org/wiki/Boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Boson">bosons</a>. As a result, they do not follow the <a href="http://en.wikipedia.org/wiki/Pauli_exclusion_principle" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pauli exclusion principle">Pauli exclusion principle</a> that constrains <a href="http://en.wikipedia.org/wiki/Fermion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermion">fermions</a>: thus bosons (e.g. photons) do not have a theoretical limit on their spatial density (number per volume). The different types of gauge bosons are described below.</div>
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<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Photon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Photon">Photons</a> mediate the electromagnetic force between electrically charged particles. The photon is massless and is well-described by the theory of <a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum electrodynamics">quantum electrodynamics</a>.</li>
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<li style="margin-bottom: 0.1em;">The <a href="http://en.wikipedia.org/wiki/W_and_Z_bosons" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="W and Z bosons"><span class="unicode;" style="white-space: nowrap;">W<span style="display: inline-block; font-size: 11px; line-height: 1.2em; margin-bottom: -0.3em; text-align: left; vertical-align: 0.8em;">+</span></span>, <span class="unicode;" style="white-space: nowrap;">W<span style="display: inline-block; font-size: 11px; line-height: 1.2em; margin-bottom: -0.3em; text-align: left; vertical-align: 0.8em;">−</span></span>, and <span class="unicode;" style="white-space: nowrap;">Z</span></a> gauge bosons mediate the <a href="http://en.wikipedia.org/wiki/Weak_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak interaction">weak interactions</a> between particles of different flavors (all <a href="http://en.wikipedia.org/wiki/Quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark">quarks</a> and leptons). They are massive, with the <span class="unicode;" style="white-space: nowrap;">Z</span> being more massive than the <span class="unicode;" style="white-space: nowrap;">W<span style="display: inline-block; font-size: 11px; line-height: 1.2em; margin-bottom: -0.3em; text-align: left; vertical-align: 0.8em;">±</span></span>. The weak interactions involving the <span class="unicode;" style="white-space: nowrap;">W<span style="display: inline-block; font-size: 11px; line-height: 1.2em; margin-bottom: -0.3em; text-align: left; vertical-align: 0.8em;">±</span></span> exclusively act on <i>left-handed</i> particles and <i>right-handed</i> antiparticles only. Furthermore, the <span class="unicode;" style="white-space: nowrap;">W<span style="display: inline-block; font-size: 11px; line-height: 1.2em; margin-bottom: -0.3em; text-align: left; vertical-align: 0.8em;">±</span></span> carries an electric charge of +1 and −1 and couples to the electromagnetic interaction. The electrically neutral <span class="unicode;" style="white-space: nowrap;">Z</span> boson interacts with both left-handed particles and antiparticles. These three gauge bosons along with the photons are grouped together, as collectively mediating the<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Electroweak" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electroweak">electroweak</a> interaction.</li>
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<li style="margin-bottom: 0.1em;">The eight <a href="http://en.wikipedia.org/wiki/Gluon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gluon">gluons</a> mediate the <a href="http://en.wikipedia.org/wiki/Strong_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Strong interaction">strong interactions</a> between <a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge">color charged</a> particles (the quarks). Gluons are massless. The eightfold multiplicity of gluons is labeled by a combination of color and anticolor charge (e.g. red–antigreen).<sup class="reference" id="cite_ref-11" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-11" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[nb 1]</a></sup> Because the gluon has an effective color charge, they can also interact among themselves. The gluons and their interactions are described by the theory of quantum chromodynamics.</li>
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The interactions between all the particles described by the Standard Model are summarized by the diagrams on the right of this section.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=6" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Higgs boson">edit</a>]</span><span class="mw-headline" id="Higgs_boson">Higgs boson</span></h3>
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Main article: <a href="http://en.wikipedia.org/wiki/Higgs_boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Higgs boson">Higgs boson</a></div>
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The Higgs particle is a hypothetical massive <a href="http://en.wikipedia.org/wiki/Scalar_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Scalar field theory">scalar</a> <a href="http://en.wikipedia.org/wiki/Elementary_particle" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Elementary particle">elementary particle</a> theorized by <a href="http://en.wikipedia.org/wiki/Robert_Brout" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Robert Brout">Robert Brout</a>, <a href="http://en.wikipedia.org/wiki/Fran%C3%A7ois_Englert" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="François Englert">François Englert</a>, <a href="http://en.wikipedia.org/wiki/Peter_Higgs" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Peter Higgs">Peter Higgs</a>, <a href="http://en.wikipedia.org/wiki/Gerald_Guralnik" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gerald Guralnik">Gerald Guralnik</a>, <a href="http://en.wikipedia.org/wiki/C._R._Hagen" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="C. R. Hagen">C. R. Hagen</a>, and <a href="http://en.wikipedia.org/wiki/Tom_W._B._Kibble" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tom W. B. Kibble">Tom Kibble</a> in 1964 (see <a href="http://en.wikipedia.org/wiki/1964_PRL_symmetry_breaking_papers" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="1964 PRL symmetry breaking papers">1964 PRL symmetry breaking papers</a>) and is a key building block in the Standard Model.<sup class="reference" id="cite_ref-12" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-12" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[12]</a></sup><sup class="reference" id="cite_ref-Peter_W._Higgs_1964_508-509_13-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-Peter_W._Higgs_1964_508-509-13" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[13]</a></sup><sup class="reference" id="cite_ref-14" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-14" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[14]</a></sup><sup class="reference" id="cite_ref-15" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-15" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[15]</a></sup> It has no intrinsic <a href="http://en.wikipedia.org/wiki/Spin_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin (physics)">spin</a>, and for that reason is classified as a <a href="http://en.wikipedia.org/wiki/Boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Boson">boson</a> (like the gauge bosons, which have <a href="http://en.wikipedia.org/wiki/Integer" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Integer">integer</a> spin). Because an exceptionally large amount of energy and beam luminosity are theoretically required to observe a Higgs boson in high energy colliders, it is the only fundamental particle predicted by the Standard Model that has yet to be observed.</div>
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The Higgs boson plays a unique role in the Standard Model, by explaining why the other elementary particles, except the<a href="http://en.wikipedia.org/wiki/Photon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Photon">photon</a> and <a href="http://en.wikipedia.org/wiki/Gluon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gluon">gluon</a>, are massive. In particular, the Higgs boson would explain why the photon has no mass, while the <a href="http://en.wikipedia.org/wiki/W_and_Z_bosons" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="W and Z bosons">W and Z bosons</a> are very heavy. Elementary particle masses, and the differences between <a href="http://en.wikipedia.org/wiki/Electromagnetism" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electromagnetism">electromagnetism</a> (mediated by the photon) and the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Weak_force" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak force">weak force</a> (mediated by the W and Z bosons), are critical to many aspects of the structure of microscopic (and hence macroscopic) matter. In <a href="http://en.wikipedia.org/wiki/Electroweak_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electroweak interaction">electroweak theory</a>, the Higgs boson generates the masses of the leptons (electron, muon, and tau) and quarks.</div>
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As yet, no experiment has conclusively detected the existence of the Higgs boson. It is hoped that the <a href="http://en.wikipedia.org/wiki/Large_Hadron_Collider" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Large Hadron Collider">Large Hadron Collider</a> at <a href="http://en.wikipedia.org/wiki/CERN" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="CERN">CERN</a> will confirm the existence of this particle. As of August 2011, a significant portion of the possible masses for the Higgs have been excluded at 95% confidence level: <a href="http://en.wikipedia.org/wiki/Compact_Muon_Solenoid" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Compact Muon Solenoid">CMS</a> has excluded the mass ranges <span class="nowrap" style="white-space: nowrap;">145–216 GeV</span>,<span class="nowrap" style="white-space: nowrap;">226–288 GeV</span> and <span class="nowrap" style="white-space: nowrap;">310–400 GeV</span>,<sup class="reference" id="cite_ref-16" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-16" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[16]</a></sup> while the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/ATLAS" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ATLAS">ATLAS</a> experiment has excluded <span class="nowrap" style="white-space: nowrap;">146–232 GeV</span>, <span class="nowrap" style="white-space: nowrap;">256–282 GeV</span> and <span class="nowrap" style="white-space: nowrap;">296–466 GeV</span>.<sup class="reference" id="cite_ref-17" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-17" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[17]</a></sup> Note that these exclusions apply only to the Standard Model Higgs, and that more complex Higgs sectors which are possible in <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Beyond_the_Standard_Model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Beyond the Standard Model">Beyond the Standard Model</a> scenarios may be significantly more difficult to characterize. CERN director general <a href="http://en.wikipedia.org/wiki/Rolf-Dieter_Heuer" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Rolf-Dieter Heuer">Rolf Heuer</a> has predicted that by the end of 2012 either the Standard Model Higgs boson will be observed, or excluded in all mass ranges, implying that the Standard Model is not the whole story.<sup class="reference" id="cite_ref-18" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-18" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[18]</a></sup></div>
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On December 13, 2011 CERN announced that both ATLAS and CMS experiments had detected 'hints' of the Higgs boson in at approximately <span class="nowrap" style="white-space: nowrap;">124 GeV</span>. These results were not sufficiently strong to announce that the Higgs boson had been found (ATLAS showed a 2.3 <a href="http://en.wikipedia.org/wiki/Standard_deviation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Standard deviation">sigma</a> level of certainty for an excess at <span class="nowrap" style="white-space: nowrap;">126 GeV</span>, while CMS showed a 1.9 sigma level excess at <span class="nowrap" style="white-space: nowrap;">124 GeV</span>) but the fact that two separate experiments show excesses in the same energy range has led to much excitement in the particle physics world.<sup class="reference" id="cite_ref-19" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-19" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[19]</a></sup></div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=7" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Field content">edit</a>]</span><span class="mw-headline" id="Field_content">Field content</span></h2>
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The Standard Model has the following fields:</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=8" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Spin 1">edit</a>]</span><span class="mw-headline" id="Spin_1">Spin 1</span></h3>
<ol style="line-height: 1.5em; list-style-image: none; margin: 0.3em 0px 0px 3.2em; padding: 0px;">
<li style="margin-bottom: 0.1em;">A <a class="mw-redirect" href="http://en.wikipedia.org/wiki/U(1)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="U(1)">U(1)</a> <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gauge_field" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge field">gauge field</a> <i>B</i><sub style="line-height: 1em;"><i>μν</i></sub> with coupling <i>g</i>′ (weak U(1), or <a href="http://en.wikipedia.org/wiki/Weak_hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak hypercharge">weak hypercharge</a>)</li>
<li style="margin-bottom: 0.1em;">An <a class="mw-redirect" href="http://en.wikipedia.org/wiki/SU(2)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="SU(2)">SU(2)</a> gauge field <i>W</i><sub style="line-height: 1em;"><i>μν</i></sub> with coupling <i>g</i> (weak SU(2), or <a href="http://en.wikipedia.org/wiki/Weak_isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak isospin">weak isospin</a>)</li>
<li style="margin-bottom: 0.1em;">An <a class="mw-redirect" href="http://en.wikipedia.org/wiki/SU(3)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="SU(3)">SU(3)</a> gauge field <i>G</i><sub style="line-height: 1em;"><i>μν</i></sub> with coupling <i>g</i><sub style="line-height: 1em;">s</sub> (strong SU(3), or <a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge">color charge</a>)</li>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=9" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Spin 1⁄2">edit</a>]</span><span class="mw-headline" id="Spin_1.E2.81.842">Spin <span class="frac nowrap" style="white-space: nowrap;"><sup style="line-height: 1em;">1</sup>⁄<sub style="line-height: 1em;">2</sub></span></span></h3>
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The spin <img alt="{}_{\frac{1}{2}}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/8/7/e874acf0840bfb5b0ffaa6ca131e28f5.png" style="border: none; margin: 0px; vertical-align: middle;" /> particles are in <a href="http://en.wikipedia.org/wiki/Representation_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Representation theory">representations</a> of the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gauge_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge group">gauge groups</a>. For the U(1) group, we list the value of the <a href="http://en.wikipedia.org/wiki/Weak_hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak hypercharge">weak hypercharge</a> instead. The left-handed fermionic fields are:</div>
<ol style="line-height: 1.5em; list-style-image: none; margin: 0.3em 0px 0px 3.2em; padding: 0px;">
<li style="margin-bottom: 0.1em;">An SU(3) triplet, SU(2) doublet, with U(1) weak hypercharge <img alt="{}_{\frac{1}{3}}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/f/a/9fafb9c32756b54852cc224b83723699.png" style="border: none; vertical-align: middle;" /> (left-handed quarks)</li>
<li style="margin-bottom: 0.1em;">An SU(3) triplet, SU(2) singlet, with U(1) weak hypercharge <img alt="{}_{\frac{2}{3}}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/f/a/dfa02945107ead2105a0e1990c9be0c6.png" style="border: none; vertical-align: middle;" /> (left-handed down-type antiquark)</li>
<li style="margin-bottom: 0.1em;">An SU(3) singlet, SU(2) doublet with U(1) weak hypercharge −1 (left-handed lepton)</li>
<li style="margin-bottom: 0.1em;">An SU(3) triplet, SU(2) singlet, with U(1) weak hypercharge <img alt="{}_{-\frac{4}{3}}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/f/0/ff0e3de7e43cfedf5e96d207a1a16b2a.png" style="border: none; vertical-align: middle;" /> (left-handed up-type antiquark)</li>
<li style="margin-bottom: 0.1em;">An SU(3) singlet, SU(2) singlet with U(1) weak hypercharge 2 (left-handed antilepton)</li>
</ol>
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By CPT symmetry, there is a set of right-handed fermions with the opposite quantum numbers.</div>
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This describes one <i>generation</i> of leptons and quarks, and there are three generations, so there are three copies of each field. Note that there are twice as many left-handed lepton field components as left-handed antilepton field components in each generation, but an equal number of left-handed quark and antiquark fields.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=10" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Spin 0">edit</a>]</span><span class="mw-headline" id="Spin_0">Spin 0</span></h3>
<ol style="line-height: 1.5em; list-style-image: none; margin: 0.3em 0px 0px 3.2em; padding: 0px;">
<li style="margin-bottom: 0.1em;">An SU(2) doublet H with U(1) hyper-charge +1 (Higgs field)</li>
</ol>
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Note that <img alt="{}_{\left|H\right|^2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/6/1/e61c0116c9f1a9b9f188b3840a8cc9e3.png" style="border: none; margin: 0px; vertical-align: middle;" />, summed over the two SU(2) components, is invariant under both SU(2) and under U(1), and so it can appear as a <a href="http://en.wikipedia.org/wiki/Renormalization" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Renormalization">renormalizable</a> term in the <a href="http://en.wikipedia.org/wiki/Lagrangian" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lagrangian">Lagrangian</a>, as can its square.<sup class="noprint Inline-Template" style="line-height: 1em; white-space: nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Please_clarify" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:Please clarify"><span title="The text in the vicinity of this tag needs clarification or removal of jargon from July 2009">clarification needed</span></a></i>]</sup></div>
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This field acquires a <a href="http://en.wikipedia.org/wiki/Vacuum_expectation_value" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vacuum expectation value">vacuum expectation value</a>, leaving a combination of the <a href="http://en.wikipedia.org/wiki/Weak_isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak isospin">weak isospin</a>, <img alt="{}_{I^3}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/2/8/a28fbdc7ac339252acc0b5d1b49cb871.png" style="border: none; margin: 0px; vertical-align: middle;" />, and weak hypercharge unbroken. This is the electromagnetic gauge group, and the photon remains massless. The standard formula for the electric charge (which defines the normalization of the <a href="http://en.wikipedia.org/wiki/Weak_hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak hypercharge">weak hypercharge</a>, <img alt="{}_Y" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/0/a/10af5c0fca77dce40875f3d9d71ad4bf.png" style="border: none; margin: 0px; vertical-align: middle;" />, which would otherwise be somewhat arbitrary) is:<sup class="reference" id="cite_ref-20" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-20" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[nb 2]</a></sup></div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="Q = I_\mathrm{3} + \frac{Y}{2}." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/1/4/514d186ec08ca8c546ef997b5dca71bc.png" style="border: none; vertical-align: middle;" /></dd></dl>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=11" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Lagrangian">edit</a>]</span><span class="mw-headline" id="Lagrangian">Lagrangian</span></h3>
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The <a href="http://en.wikipedia.org/wiki/Lagrangian" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lagrangian">Lagrangian</a> for the spin 1 and spin <span class="frac nowrap" style="white-space: nowrap;"><sup style="line-height: 1em;">1</sup>⁄<sub style="line-height: 1em;">2</sub></span> fields is the most general renormalizable gauge field Lagrangian with no fine tunings:</div>
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;">Spin 1:</li>
</ul>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\int - {1\over 4} B_{\mu\nu} B^{\mu\nu} - {1\over 4}\mathrm{tr} W_{\mu\nu}W^{\mu\nu} - {1\over 4} \mathrm{tr}G_{\mu\nu} G^{\mu\nu}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/c/f/7cffbb68e03bfbfc3870195de9a5886f.png" style="border: none; vertical-align: middle;" /></dd></dl>
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where the traces are over the SU(2) and SU(3) indices hidden in <i>W</i> and <i>G</i> respectively. The two-index objects are the field strengths derived from <i>W</i> and <i>G</i> the vector fields. There are also two extra hidden parameters: the theta angles for SU(2) and SU(3).</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The spin-<span class="frac nowrap" style="white-space: nowrap;"><sup style="line-height: 1em;">1</sup>⁄<sub style="line-height: 1em;">2</sub></span> particles can have no mass terms because there is no right/left helicity pair with the same SU(2) and SU(3) representation and the same weak hypercharge. This means that if the gauge charges were conserved in the vacuum, none of the spin <span class="frac nowrap" style="white-space: nowrap;"><sup style="line-height: 1em;">1</sup>⁄<sub style="line-height: 1em;">2</sub></span> particles could ever swap helicity, and they would all be massless.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
For a neutral fermion, for example a hypothetical right-handed lepton <i>N</i> (or <i>N</i><sup style="line-height: 1em;"><i>α</i></sup> in relativistic two-spinor notation), with no SU(3), SU(2) representation and zero charge, it is possible to add the term:<sup class="noprint Inline-Template" style="line-height: 1em; white-space: nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Please_clarify" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:Please clarify"><span title="The text in the vicinity of this tag needs clarification or removal of jargon from July 2009">clarification needed</span></a></i>]</sup></div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\int M N^\alpha N^\beta \epsilon_{\alpha\beta} + \bar{N_\dot{\alpha}}\bar{N_\dot{\beta}}\epsilon^{\dot\alpha\dot\beta}." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/4/4/044f41a968b9292ddd597e1a35dc66e9.png" style="border: none; vertical-align: middle;" /></dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
This term gives the neutral fermion a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Majorana_mass" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Majorana mass">Majorana mass</a>. Since the generic value for <i>M</i> will be of order 1, such a particle would generically be unacceptably heavy. The interactions are completely determined by the theory – the leptons introduce no extra parameters.</div>
<h3 style="background-image: none; border-bottom-style: none; font-size: 18px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=12" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Higgs mechanism">edit</a>]</span><span class="mw-headline" id="Higgs_mechanism">Higgs mechanism</span></h3>
<div class="rellink relarticle mainarticle" style="font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em;">
Main article: <a href="http://en.wikipedia.org/wiki/Higgs_mechanism" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Higgs mechanism">Higgs mechanism</a></div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The Lagrangian for the Higgs includes the most general renormalizable self interaction:</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="S_{\mathrm{Higgs}} = \int d^4x\left[(D_\mu H)^*(D^\mu H) + \lambda(|H|^2 - v^2)^2\right]." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/d/e/2de54b61d3e966e1c090b4243be63a03.png" style="border: none; vertical-align: middle;" /></dd></dl>
</dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The parameter <i>v</i><sup style="line-height: 1em;">2</sup> has dimensions of mass squared, and it gives the location where the classical Lagrangian is at a minimum. In order for the Higgs mechanism to work, <i>v</i><sup style="line-height: 1em;">2</sup> must be a positive number. <i>v</i> has units of mass, and it is the only parameter in the Standard Model which is not dimensionless. It is also much smaller than the Planck scale; it is approximately equal to the Higgs mass, and sets the scale for the mass of everything else. This is the only real fine-tuning to a small nonzero value in the Standard Model, and it is called the <a href="http://en.wikipedia.org/wiki/Hierarchy_problem" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hierarchy problem">Hierarchy problem</a>.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
It is traditional to choose the SU(2) gauge so that the Higgs doublet in the vacuum has expectation value (<i>v</i>,0).</div>
<h3 style="background-image: none; border-bottom-style: none; font-size: 18px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=13" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Masses and CKM matrix">edit</a>]</span><span class="mw-headline" id="Masses_and_CKM_matrix">Masses and CKM matrix</span></h3>
<div class="rellink relarticle mainarticle" style="font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em;">
Main article: <a href="http://en.wikipedia.org/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cabibbo–Kobayashi–Maskawa matrix">Cabibbo–Kobayashi–Maskawa matrix</a></div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The rest of the interactions are the most general spin-0 spin-<span class="frac nowrap" style="white-space: nowrap;"><sup style="line-height: 1em;">1</sup>⁄<sub style="line-height: 1em;">2</sub></span> <a href="http://en.wikipedia.org/wiki/Yukawa_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Yukawa interaction">Yukawa interactions</a>, and there are many of these. These constitute most of the free parameters in the model. The Yukawa couplings generate the masses and mixings once the Higgs gets its vacuum expectation value.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The terms <i>L</i><sup style="line-height: 1em;">*</sup><i>HR</i><sup class="noprint Inline-Template" style="line-height: 1em; white-space: nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Please_clarify" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:Please clarify"><span title="The text in the vicinity of this tag needs clarification or removal of jargon from July 2009">clarification needed</span></a></i>]</sup> generate a mass term for each of the three generations of leptons. There are 9 of these terms, but by relabeling L and R, the matrix can be diagonalized. Since only the upper component of <i>H</i> is nonzero, the upper SU(2) component of <i>L</i> mixes with <i>R</i> to make the electron, the muon, and the tau, leaving over a lower massless component, the neutrino. Note: <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Neutrino_oscillations" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Neutrino oscillations">Neutrino oscillations</a> show neutrinos have mass.<sup class="reference" id="cite_ref-21" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-21" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[20]</a></sup> See also: <a href="http://en.wikipedia.org/wiki/Pontecorvo%E2%80%93Maki%E2%80%93Nakagawa%E2%80%93Sakata_matrix" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pontecorvo–Maki–Nakagawa–Sakata matrix">Pontecorvo–Maki–Nakagawa–Sakata matrix</a>.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The terms QHU<sup class="noprint Inline-Template" style="line-height: 1em; white-space: nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Please_clarify" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:Please clarify"><span title="The text in the vicinity of this tag needs clarification or removal of jargon from July 2009">clarification needed</span></a></i>]</sup> generate up masses, while QHD<sup class="noprint Inline-Template" style="line-height: 1em; white-space: nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Please_clarify" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:Please clarify"><span title="The text in the vicinity of this tag needs clarification or removal of jargon from July 2009">clarification needed</span></a></i>]</sup> generate down masses. But since there is more than one right-handed singlet in each generation, it is not possible to diagonalize both with a good basis for the fields, and there is an extra CKM matrix.</div>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=14" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Theoretical aspects">edit</a>]</span><span class="mw-headline" id="Theoretical_aspects">Theoretical aspects</span></h2>
<h3 style="background-image: none; border-bottom-style: none; font-size: 18px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=15" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Construction of the Standard Model Lagrangian">edit</a>]</span><span class="mw-headline" id="Construction_of_the_Standard_Model_Lagrangian">Construction of the Standard Model Lagrangian</span></h3>
<table class="wikitable" style="background-color: #f9f9f9; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; float: right; font-size: 13px; margin: 0px 0px 1em 1em;"><caption style="font-weight: bold;">Parameters of the Standard Model</caption><tbody>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Symbol</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Description</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Renormalization<br />
scheme (point)</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Value</th></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>m</i><sub style="line-height: 1em;">e</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Electron mass</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">511 keV</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>m</i><sub style="line-height: 1em;">μ</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Muon mass</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">105.7 MeV</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>m</i><sub style="line-height: 1em;">τ</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Tau mass</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">1.78 GeV</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>m</i><sub style="line-height: 1em;">u</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Up quark mass</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>μ</i><sub style="line-height: 1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/MSbar_scheme" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="MSbar scheme"><span style="text-decoration: overline;">MS</span></a></sub> = 2 GeV</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">1.9 MeV</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>m</i><sub style="line-height: 1em;">d</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Down quark mass</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>μ</i><sub style="line-height: 1em;"><span style="text-decoration: overline;">MS</span></sub> = 2 GeV</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">4.4 MeV</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>m</i><sub style="line-height: 1em;">s</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Strange quark mass</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>μ</i><sub style="line-height: 1em;"><span style="text-decoration: overline;">MS</span></sub> = 2 GeV</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">87 MeV</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>m</i><sub style="line-height: 1em;">c</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Charm quark mass</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>μ</i><sub style="line-height: 1em;"><span style="text-decoration: overline;">MS</span></sub> = <i>m</i><sub style="line-height: 1em;">c</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">1.32 GeV</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>m</i><sub style="line-height: 1em;">b</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Bottom quark mass</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>μ</i><sub style="line-height: 1em;"><span style="text-decoration: overline;">MS</span></sub> = <i>m</i><sub style="line-height: 1em;">b</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">4.24 GeV</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>m</i><sub style="line-height: 1em;">t</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Top quark mass</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="new" href="http://en.wikipedia.org/w/index.php?title=On-shell_scheme&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="On-shell scheme (page does not exist)">On-shell scheme</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">172.7 GeV</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>θ</i><sub style="line-height: 1em;">12</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">CKM 12-mixing angle</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">13.1°</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>θ</i><sub style="line-height: 1em;">23</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">CKM 23-mixing angle</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">2.4°</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>θ</i><sub style="line-height: 1em;">13</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">CKM 13-mixing angle</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">0.2°</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>δ</i></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">CKM <a href="http://en.wikipedia.org/wiki/CP_violation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="CP violation">CP-violating</a> Phase</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">0.995</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>g</i><sub style="line-height: 1em;">1</sub> or <i>g'</i></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">U(1) gauge coupling</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>μ</i><sub style="line-height: 1em;"><span style="text-decoration: overline;">MS</span></sub> = <i>m</i><sub style="line-height: 1em;">Z</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">0.357</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>g</i><sub style="line-height: 1em;">2</sub> or <i>g</i></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">SU(2) gauge coupling</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>μ</i><sub style="line-height: 1em;"><span style="text-decoration: overline;">MS</span></sub> = <i>m</i><sub style="line-height: 1em;">Z</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">0.652</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>g</i><sub style="line-height: 1em;">3</sub> or <i>g</i><sub style="line-height: 1em;">s</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">SU(3) gauge coupling</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>μ</i><sub style="line-height: 1em;"><span style="text-decoration: overline;">MS</span></sub> = <i>m</i><sub style="line-height: 1em;">Z</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">1.221</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>θ</i><sub style="line-height: 1em;">QCD</sub></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">QCD <a href="http://en.wikipedia.org/wiki/Vacuum_angle" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vacuum angle">vacuum angle</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">~0</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>μ</i></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Higgs quadratic coupling</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Unknown</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>λ</i></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Higgs self-coupling strength</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Unknown</td></tr>
</tbody></table>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Technically, <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">quantum field theory</a> provides the mathematical framework for the Standard Model, in which a <a href="http://en.wikipedia.org/wiki/Lagrangian" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lagrangian">Lagrangian</a> controls the dynamics and kinematics of the theory. Each kind of particle is described in terms of a dynamical <a href="http://en.wikipedia.org/wiki/Field_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Field (physics)">field</a> that pervades <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Space-time" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Space-time">space-time</a>. The construction of the Standard Model proceeds following the modern method of constructing most field theories: by first postulating a set of symmetries of the system, and then by writing down the most general <a href="http://en.wikipedia.org/wiki/Renormalization" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Renormalization">renormalizable</a> Lagrangian from its particle (field) content that observes these symmetries.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The <a href="http://en.wikipedia.org/wiki/Global_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Global symmetry">global</a> <a href="http://en.wikipedia.org/wiki/Poincar%C3%A9_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Poincaré group">Poincaré symmetry</a> is postulated for all relativistic quantum field theories. It consists of the familiar <a href="http://en.wikipedia.org/wiki/Translational_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Translational symmetry">translational symmetry</a>, <a href="http://en.wikipedia.org/wiki/Rotational_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Rotational symmetry">rotational symmetry</a> and the inertial reference frame invariance central to the theory of <a href="http://en.wikipedia.org/wiki/Special_relativity" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special relativity">special relativity</a>. The <a href="http://en.wikipedia.org/wiki/Local_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Local symmetry">local</a> SU(3)×SU(2)×U(1) gauge symmetry is an <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Internal_symmetries" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Internal symmetries">internal symmetry</a> that essentially defines the Standard Model. Roughly, the three factors of the gauge symmetry give rise to the three fundamental interactions. The fields fall into different <a href="http://en.wikipedia.org/wiki/Representation_of_a_Lie_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Representation of a Lie group">representations</a> of the various symmetry groups of the Standard Model (see table). Upon writing the most general Lagrangian, one finds that the dynamics depend on 19 parameters, whose numerical values are established by experiment. The parameters are summarized in the table at right.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=16" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Quantum chromodynamics sector">edit</a>]</span><span class="mw-headline" id="Quantum_chromodynamics_sector">Quantum chromodynamics sector</span></h4>
<div class="rellink relarticle mainarticle" style="font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em;">
Main article: <a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum chromodynamics">Quantum chromodynamics</a></div>
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The quantum chromodynamics (QCD) sector defines the interactions between quarks and gluons, with SU(3) symmetry, generated by T<sup style="line-height: 1em;">a</sup>. Since leptons do not interact with gluons, they are not affected by this sector. The Dirac Lagrangian of the quarks coupled to the gluon fields is given by</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\mathcal{L}_{QCD} = i\overline U (\partial_\mu-ig_sG_\mu^a T^a)\gamma^\mu U + i\overline D (\partial_\mu-i g_s G_\mu^a T^a)\gamma^\mu D." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/1/0/010b341a7f5b5b271765e05e35906aa3.png" style="border: none; vertical-align: middle;" /></dd></dl>
</dd></dl>
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<img alt="G_\mu^a" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/4/0/740f065882ef87090a8245f80adf2d9c.png" style="border: none; margin: 0px; vertical-align: middle;" /> is the SU(3) gauge field containing the gluons, <img alt="\gamma^\mu" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/6/0/d60917d1e76e82865bc6b29078c627c5.png" style="border: none; margin: 0px; vertical-align: middle;" /> are the Dirac matrices, D and U are the Dirac spinors associated with up- and down-type <a href="http://en.wikipedia.org/wiki/Quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark">quarks</a>, and g<sub style="line-height: 1em;">s</sub> is the strong coupling constant.</div>
<h4 style="background-image: none; border-bottom-style: none; font-size: 15px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=17" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Electroweak sector">edit</a>]</span><span class="mw-headline" id="Electroweak_sector">Electroweak sector</span></h4>
<div class="rellink relarticle mainarticle" style="font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em;">
Main article: <a href="http://en.wikipedia.org/wiki/Electroweak_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electroweak interaction">Electroweak interaction</a></div>
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The electroweak sector is a <a href="http://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Yang–Mills theory">Yang–Mills gauge theory</a> with the symmetry group U(1)×SU(2)<sub style="line-height: 1em;">L</sub>,</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="
\mathcal{L}_\mathrm{EW} =
\sum_\psi\bar\psi\gamma^\mu
\left(i\partial_\mu-g^\prime{1\over2}Y_\mathrm{W}B_\mu-g{1\over2}\vec\tau_\mathrm{L}\vec W_\mu\right)\psi" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/7/7/0773d59e0d5b8875beaf7bfd9a28ce40.png" style="border: none; vertical-align: middle;" /></dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
where <i>B</i><sub style="line-height: 1em;"><i>μ</i></sub> is the U(1) gauge field; <i>Y</i><sub style="line-height: 1em;">W</sub> is the <a href="http://en.wikipedia.org/wiki/Weak_hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak hypercharge">weak hypercharge</a>—the generator of the U(1) group; <img alt="\vec{W}_\mu" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/f/f/0fff3ddd9e672bbf786c278b1dee1eda.png" style="border: none; margin: 0px; vertical-align: middle;" /> is the three-component SU(2) gauge field; <img alt="\vec{\tau}_\mathrm{L}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/7/e/c7e8cdd9046acbdffde617074832a162.png" style="border: none; margin: 0px; vertical-align: middle;" /> are the <a href="http://en.wikipedia.org/wiki/Pauli_matrices" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pauli matrices">Pauli matrices</a>—infinitesimal generators of the SU(2) group. The subscript L indicates that they only act on left fermions; <i>g</i>′ and <i>g</i> are coupling constants.</div>
<h4 style="background-image: none; border-bottom-style: none; font-size: 15px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=18" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Higgs sector">edit</a>]</span><span class="mw-headline" id="Higgs_sector">Higgs sector</span></h4>
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In the Standard Model, the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Higgs_field" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Higgs field">Higgs field</a> is a complex <a href="http://en.wikipedia.org/wiki/Spinor" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spinor">spinor</a> of the group <a class="mw-redirect" href="http://en.wikipedia.org/wiki/SU(2)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="SU(2)">SU(2)</a><sub style="line-height: 1em;">L</sub>:</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="
\varphi={1\over\sqrt{2}}
\left(
\begin{array}{c}
\varphi^+ \\ \varphi^0
\end{array}
\right)\;,
" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/c/7/ac75d9a8adae2f22ea4a7c5cf9f1fba8.png" style="border: none; vertical-align: middle;" /></dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
where the indexes + and 0 indicate the electric charge (<i>Q</i>) of the components. The weak isospin (<i>Y</i><sub style="line-height: 1em;">W</sub>) of both components is 1.</div>
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Before symmetry breaking, the Higgs Lagrangian is:</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\mathcal{L}_\mathrm{H} = \varphi^\dagger
\left({\partial^\mu}-
{i\over2} \left( g'Y_\mathrm{W}B^\mu + g\vec\tau\vec W^\mu \right)\right)
\left(\partial_\mu + {i\over2} \left( g'Y_\mathrm{W}B_\mu
+g\vec\tau\vec W_\mu \right)\right)\varphi \ - \ {\lambda^2\over4}\left(\varphi^\dagger\varphi-v^2\right)^2\;," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/8/3/183b8fa6b5d552a86de8c59492a93e94.png" style="border: none; vertical-align: middle;" /></dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
which can also be written as:</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\mathcal{L}_\mathrm{H} = \left|
\left(\partial_\mu + {i\over2} \left( g'Y_\mathrm{W}B_\mu
+g\vec\tau\vec W_\mu \right)\right)\varphi\right|^2 \ - \ {\lambda^2\over4}\left(\varphi^\dagger\varphi-v^2\right)^2\;." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/e/7/0e78ca9aad901d4520cdcfef8a2a7845.png" style="border: none; vertical-align: middle;" /></dd></dl>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=19" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Additional symmetries of the Standard Model">edit</a>]</span><span class="mw-headline" id="Additional_symmetries_of_the_Standard_Model">Additional symmetries of the Standard Model</span></h3>
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From the theoretical point of view, the Standard Model exhibits four additional global symmetries, not postulated at the outset of its construction, collectively denoted<b>accidental symmetries</b>, which are continuous <a class="mw-redirect" href="http://en.wikipedia.org/wiki/U(1)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="U(1)">U(1)</a> <a href="http://en.wikipedia.org/wiki/Global_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Global symmetry">global symmetries</a>. The transformations leaving the Lagrangian invariant are:</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\psi_\text{q}(x)\rightarrow e^{i\alpha/3}\psi_\text{q}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/e/7/ee7842092eb7ca4dfb838919b683f7f2.png" style="border: none; vertical-align: middle;" /></dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="E_L\rightarrow e^{i\beta}E_L\text{ and }(e_R)^c\rightarrow e^{i\beta}(e_R)^c" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/2/0/320e3ca192ec3b2844a07bd9cabdece4.png" style="border: none; vertical-align: middle;" /></dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="M_L\rightarrow e^{i\beta}M_L\text{ and }(\mu_R)^c\rightarrow e^{i\beta}(\mu_R)^c" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/d/3/4d3ecd3d2c8d0adf36e0d0b8b372ecf0.png" style="border: none; vertical-align: middle;" /></dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="T_L\rightarrow e^{i\beta}T_L\text{ and }(\tau_R)^c\rightarrow e^{i\beta}(\tau_R)^c." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/b/a/7ba3172756e5b1922eb116bdfb3c9b7f.png" style="border: none; vertical-align: middle;" /></dd></dl>
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The first transformation rule is shorthand meaning that all quark fields for all generations must be rotated by an identical phase simultaneously. The fields <img alt="M_L" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/6/5/465a03898e7ee28c033a2c52b4fd4e91.png" style="border: none; margin: 0px; vertical-align: middle;" />, <img alt="T_L" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/5/e/45eba687a3a95f0c00c3aebb70cb4de1.png" style="border: none; margin: 0px; vertical-align: middle;" /> and <img alt="(\mu_R)^c" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/1/6/416427a2bb10a85376cd7155ac69f492.png" style="border: none; margin: 0px; vertical-align: middle;" />, <img alt="(\tau_R)^c" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/9/6/196c63164434db6ff3febde03d3725a2.png" style="border: none; margin: 0px; vertical-align: middle;" /> are the 2nd (muon) and 3rd (tau) generation analogs of <img alt="E_L" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/6/9/869e0885def75ae6da81f5cc07e071d9.png" style="border: none; margin: 0px; vertical-align: middle;" /> and <img alt="(e_R)^c" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/7/5/d75a23a87aa7c70b9c6742d6e2b8ed2a.png" style="border: none; margin: 0px; vertical-align: middle;" /> fields.</div>
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By <a href="http://en.wikipedia.org/wiki/Noether%27s_theorem" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Noether's theorem">Noether's theorem</a>, each symmetry above has an associated <a href="http://en.wikipedia.org/wiki/Conservation_law" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Conservation law">conservation law</a>: the conservation of <a href="http://en.wikipedia.org/wiki/Baryon_number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Baryon number">baryon number</a>, <a href="http://en.wikipedia.org/wiki/Lepton_number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lepton number">electron number</a>, <a href="http://en.wikipedia.org/wiki/Lepton_number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lepton number">muon number</a>, and <a href="http://en.wikipedia.org/wiki/Lepton_number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lepton number">tau number</a>. Each quark is assigned a baryon number of <img alt="{}_{\frac{1}{3}}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/f/a/9fafb9c32756b54852cc224b83723699.png" style="border: none; margin: 0px; vertical-align: middle;" />, while each antiquark is assigned a baryon number of <img alt="{}_{-\frac{1}{3}}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/a/d/1adec07238e3c2e98929096cc2c8ca28.png" style="border: none; margin: 0px; vertical-align: middle;" />. Conservation of baryon number implies that the number of quarks minus the number of antiquarks is a constant. Within experimental limits, no violation of this conservation law has been found.</div>
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Similarly, each electron and its associated neutrino is assigned an electron number of +1, while the <a href="http://en.wikipedia.org/wiki/Positron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Positron">anti-electron</a> and the associated anti-neutrino carry a −1 electron number. Similarly, the muons and their neutrinos are assigned a muon number of +1 and the tau leptons are assigned a tau lepton number of +1. The Standard Model predicts that each of these three numbers should be conserved separately in a manner similar to the way baryon number is conserved. These numbers are collectively known as <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Lepton_family_number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lepton family number">lepton family numbers</a> (LF).</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Symmetry works differently for quarks than for leptons, mainly because the Standard Model predicts (incorrectly) that <a href="http://en.wikipedia.org/wiki/Neutrino" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Neutrino">neutrinos</a> are massless. However, in 2002 it was discovered that neutrinos have mass (now established to be not greater than 0.28 electron volts), and as neutrinos <a href="http://en.wikipedia.org/wiki/Neutrino_oscillation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Neutrino oscillation">oscillate</a> between flavors (muon neutrinos have been observed changing to tau neutrinos) the discovery of neutrino mass indicates that the conservation of lepton family number is violated.<sup class="reference" id="cite_ref-22" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-22" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[21]</a></sup></div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
In addition to the accidental (but exact) symmetries described above, the Standard Model exhibits several <b>approximate symmetries</b>. These are the "SU(2) custodial symmetry" and the "SU(2) or SU(3) quark flavor symmetry."</div>
<table class="wikitable" style="background-color: #f9f9f9; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; font-size: 13px; margin: 0px 0px 1em 1em;"><caption style="font-weight: bold;">Symmetries of the Standard Model and Associated Conservation Laws</caption><tbody>
<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Symmetry_in_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Symmetry in physics">Symmetry</a></th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Lie_Group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lie Group">Lie Group</a></th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Symmetry Type</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Conservation_law" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Conservation law">Conservation Law</a></th></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Poincar%C3%A9_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Poincaré group">Poincaré</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Translational_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Translational symmetry">Translations</a><a href="http://en.wikipedia.org/wiki/Semidirect_product" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Semidirect product">×</a><a href="http://en.wikipedia.org/wiki/Lorentz_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lorentz group">SO(3,1)</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Global_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Global symmetry">Global symmetry</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Energy" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Energy">Energy</a>, <a href="http://en.wikipedia.org/wiki/Momentum" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Momentum">Momentum</a>, <a href="http://en.wikipedia.org/wiki/Angular_momentum" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Angular momentum">Angular momentum</a></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gauge_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge group">Gauge</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/SU(3)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="SU(3)">SU(3)</a>×<a class="mw-redirect" href="http://en.wikipedia.org/wiki/SU(2)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="SU(2)">SU(2)</a>×<a class="mw-redirect" href="http://en.wikipedia.org/wiki/U(1)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="U(1)">U(1)</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Local_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Local symmetry">Local symmetry</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge">Color charge</a>, <a href="http://en.wikipedia.org/wiki/Weak_isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak isospin">Weak isospin</a>, <a href="http://en.wikipedia.org/wiki/Electric_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electric charge">Electric charge</a>, <a href="http://en.wikipedia.org/wiki/Weak_hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak hypercharge">Weak hypercharge</a></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Baryon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Baryon">Baryon</a> phase</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/U(1)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="U(1)">U(1)</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Accidental <a href="http://en.wikipedia.org/wiki/Global_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Global symmetry">Global symmetry</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Baryon_number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Baryon number">Baryon number</a></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Electron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron">Electron</a> phase</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/U(1)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="U(1)">U(1)</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Accidental <a href="http://en.wikipedia.org/wiki/Global_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Global symmetry">Global symmetry</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Lepton_number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lepton number">Electron number</a></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Muon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Muon">Muon</a> phase</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/U(1)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="U(1)">U(1)</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Accidental <a href="http://en.wikipedia.org/wiki/Global_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Global symmetry">Global symmetry</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Lepton_number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lepton number">Muon number</a></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Tau_(particle)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tau (particle)">Tau</a> phase</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/U(1)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="U(1)">U(1)</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Accidental <a href="http://en.wikipedia.org/wiki/Global_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Global symmetry">Global symmetry</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Lepton_number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lepton number">Tau number</a></td></tr>
</tbody></table>
<table class="wikitable" style="background-color: #f9f9f9; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; font-size: 13px; margin: 0px 0px 1em 1em;"><caption style="font-weight: bold;">Field content of the Standard Model</caption><tbody>
<tr><th colspan="2" style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Field_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Field (physics)">Field</a><br />
(1st generation)</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Spin_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin (physics)">Spin</a></th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gauge_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge group">Gauge group</a><br />
Representation</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Baryon_number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Baryon number">Baryon</a><br />
Number</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Lepton_number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lepton number">Electron</a><br />
Number</th></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Left-handed <a href="http://en.wikipedia.org/wiki/Quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark">quark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="Q_\text{L}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/d/1/fd10f804f240daa590a114e1bd165d82.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Spinor" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spinor"><img alt="\frac{1}{2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/d/c/3dcde285c447d48b6bc42bb636112d6c.png" style="border: none; vertical-align: middle;" /></a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">(<a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge"><b>3</b></a>, <a href="http://en.wikipedia.org/wiki/Weak_isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak isospin"><b>2</b></a>, <a href="http://en.wikipedia.org/wiki/Weak_hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak hypercharge">+<span class="frac nowrap" style="white-space: nowrap;"><sup style="line-height: 1em;">1</sup>⁄<sub style="line-height: 1em;">3</sub></span></a>)</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="\frac{1}{3}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/3/c/e3c84416f9085bb72f7db037d93dce15.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Left-handed <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Up_antiquark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Up antiquark">up antiquark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bar u_\text{L} \equiv (u_\text{R})^c\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/7/0/67030e07a3d5e9ea67519544b6c13cfd.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Spinor" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spinor"><img alt="\frac{1}{2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/d/c/3dcde285c447d48b6bc42bb636112d6c.png" style="border: none; vertical-align: middle;" /></a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="\left(\bar{\mathbf{3}}, \mathbf{1}, -\frac{4}{3}\right)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/4/7/947080c2542fc57535198b6113055d87.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="-\frac{1}{3}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/9/c/59c23d1355e203750fa3102b93da1ee6.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Left-handed <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Down_antiquark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Down antiquark">down antiquark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bar d_\text{L} \equiv (d_\text{R})^c\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/9/5/b950f8a06fa2f0cde711ca0b29bd4927.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Spinor" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spinor"><img alt="\frac{1}{2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/d/c/3dcde285c447d48b6bc42bb636112d6c.png" style="border: none; vertical-align: middle;" /></a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="\left(\bar{\mathbf{3}}, \mathbf{1}, -\frac{2}{3}\right)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/a/1/da16e93b65c1a94242df6025e5aa60c1.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="-\frac{1}{3}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/9/c/59c23d1355e203750fa3102b93da1ee6.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Left-handed lepton</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="L_\text{L}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/a/9/ba96ca1914cf23587223b25ea7597b78.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Spinor" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spinor"><img alt="\frac{1}{2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/d/c/3dcde285c447d48b6bc42bb636112d6c.png" style="border: none; vertical-align: middle;" /></a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">(<a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge"><img alt="\mathbf{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></a>, <a href="http://en.wikipedia.org/wiki/Weak_isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak isospin"><img alt="\mathbf{2}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/f/f/8ff20f47faf7aaa53587e2376e3ed128.png" style="border: none; vertical-align: middle;" /></a>, <a href="http://en.wikipedia.org/wiki/Weak_hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak hypercharge"><img alt="-1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/c/f3c1d3a00c1ad4b653ef1ce33e47e721.png" style="border: none; vertical-align: middle;" /></a>)</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/0/6/d06c48671eacd7f1e2afde7289e483d5.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Left-handed <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Antielectron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Antielectron">antielectron</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bar e_\text{L} \equiv (e_\text{R})^c\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/a/a/aaa163d91c0c241a358d02806e8b67b3.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Spinor" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spinor"><img alt="\frac{1}{2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/d/c/3dcde285c447d48b6bc42bb636112d6c.png" style="border: none; vertical-align: middle;" /></a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">(<a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge"><img alt="\mathbf{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></a>, <a href="http://en.wikipedia.org/wiki/Weak_isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak isospin"><img alt="\mathbf{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></a>, <a href="http://en.wikipedia.org/wiki/Weak_hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak hypercharge"><img alt="+2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/e/b/beb954bb17ff5ba73db1bfe6cc90fa33.png" style="border: none; vertical-align: middle;" /></a>)</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="-1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/c/f3c1d3a00c1ad4b653ef1ce33e47e721.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hypercharge">Hypercharge</a> <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gauge_field" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge field">gauge field</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="B_\mu\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/f/f/7ffe907a4af2cff8732f15ffa8b6edc0.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Four-vector" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Four-vector"><img alt="1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/0/6/d06c48671eacd7f1e2afde7289e483d5.png" style="border: none; vertical-align: middle;" /></a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">(<a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge"><img alt="\mathbf{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></a>, <a href="http://en.wikipedia.org/wiki/Weak_isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak isospin"><img alt="\mathbf{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></a>, <a href="http://en.wikipedia.org/wiki/Weak_hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak hypercharge"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></a>)</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Isospin">Isospin</a> <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gauge_field" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge field">gauge field</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="W_\mu\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/c/4/cc408f7e78bd8391271da7478f8e6c19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Four-vector" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Four-vector"><img alt="1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/0/6/d06c48671eacd7f1e2afde7289e483d5.png" style="border: none; vertical-align: middle;" /></a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">(<a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge"><img alt="\mathbf{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></a>, <a href="http://en.wikipedia.org/wiki/Weak_isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak isospin"><img alt="\mathbf{3}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/1/b/21b18e410898272ca1a05200db7f5142.png" style="border: none; vertical-align: middle;" /></a>, <a href="http://en.wikipedia.org/wiki/Weak_hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak hypercharge"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></a>)</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Gluon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gluon">Gluon</a> field</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="G_\mu\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/5/0/e50fe99bdd70932c7772639d17a3a428.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Four-vector" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Four-vector"><img alt="1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/0/6/d06c48671eacd7f1e2afde7289e483d5.png" style="border: none; vertical-align: middle;" /></a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">(<a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge"><img alt="\mathbf{8}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/a/8/ba89adcae706c554b6c7e35256ba8d0d.png" style="border: none; vertical-align: middle;" /></a>, <a href="http://en.wikipedia.org/wiki/Weak_isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak isospin"><img alt="\mathbf{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></a>, <a href="http://en.wikipedia.org/wiki/Weak_hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak hypercharge"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></a>)</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Higgs_field" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Higgs field">Higgs field</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="H\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/c/f3c93a0a7b6ff16f891205bee4b04736.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Scalar_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Scalar (physics)"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">(<a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge"><img alt="\mathbf{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></a>, <a href="http://en.wikipedia.org/wiki/Weak_isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak isospin"><img alt="\mathbf{2}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/f/f/8ff20f47faf7aaa53587e2376e3ed128.png" style="border: none; vertical-align: middle;" /></a>, <a href="http://en.wikipedia.org/wiki/Weak_hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak hypercharge"><img alt="+1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/8/7/88772a82403b94411223fbb8f69e4bde.png" style="border: none; vertical-align: middle;" /></a>)</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td></tr>
</tbody></table>
<h3 style="background-image: none; border-bottom-style: none; font-size: 18px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=20" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: List of Standard Model fermions">edit</a>]</span><span class="mw-headline" id="List_of_Standard_Model_fermions">List of Standard Model fermions</span></h3>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
This table is based in part on data gathered by the <a href="http://en.wikipedia.org/wiki/Particle_Data_Group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Particle Data Group">Particle Data Group</a>.<sup class="reference" id="cite_ref-23" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-23" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[22]</a></sup></div>
<table class="wikitable" style="background-color: #f9f9f9; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; font-size: 13px; margin: auto;"><caption style="font-weight: bold;"><b>Left-handed fermions in the Standard Model</b></caption><tbody>
<tr><th colspan="8" style="background-color: navajowhite; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Generation 1</th></tr>
<tr style="background-color: #ffdddd; background-position: initial initial; background-repeat: initial initial;"><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Fermion<br />
(left-handed)</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Symbol</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Electric_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electric charge">Electric<br />charge</a></th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Weak_isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak isospin">Weak<br />isospin</a></th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Weak_hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak hypercharge">Weak<br />hypercharge</a></th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge">Color<br />charge</a> <sup class="reference" id="cite_ref-s1_24-0" style="font-weight: normal; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s1-24" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 1]</a></sup></th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"><a href="http://en.wikipedia.org/wiki/Mass" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mass">Mass</a><sup class="reference" id="cite_ref-s2_25-0" style="font-weight: normal; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s2-25" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 2]</a></sup></th></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Electron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron">Electron</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="e^-\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/a/6/7a60726b092e9ce66a4c9bfbe3b34180.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/c/f3c1d3a00c1ad4b653ef1ce33e47e721.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1/2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/c/7/dc74155f9123c0ea21e0fcdd48c3f8f0.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/c/f3c1d3a00c1ad4b653ef1ce33e47e721.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">511 keV</td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Positron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Positron">Positron</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="e^+\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/6/2/6628d2982b6403260b2659908582f407.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/8/7/88772a82403b94411223fbb8f69e4bde.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/e/b/beb954bb17ff5ba73db1bfe6cc90fa33.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">511 keV</td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Electron_neutrino" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron neutrino">Electron neutrino</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\nu_e\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/1/e/e1e423137f11587f90586c2ebfcbc1d8.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/5/c/05cab4a49dfc9b7b1327cc15eab681d7.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/c/f3c1d3a00c1ad4b653ef1ce33e47e721.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">< 0.28 eV<sup class="reference" id="cite_ref-s4_26-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s4-26" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 3]</a></sup><sup class="reference" id="cite_ref-s4b_27-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s4b-27" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 4]</a></sup></td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Electron_antineutrino" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron antineutrino">Electron antineutrino</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bar\nu_e\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/2/6/3264a95d06d23280bce1a4b4c6721753.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">< 0.28 eV<sup class="reference" id="cite_ref-s4_26-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s4-26" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 3]</a></sup><sup class="reference" id="cite_ref-s4b_27-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s4b-27" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 4]</a></sup></td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Up_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Up quark">Up quark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="u\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/1/e/61efad693efe8e0ffd7d7bc042b427ef.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+2/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/9/4/1942ed13d5af7f23bac1650e523d7eb3.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/5/c/05cab4a49dfc9b7b1327cc15eab681d7.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/8/e/08e98b019ed284066bba9882f94d20d1.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{3}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/1/b/21b18e410898272ca1a05200db7f5142.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">~ 3 MeV<sup class="reference" id="cite_ref-s3_28-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s3-28" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 5]</a></sup></td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Up_antiquark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Up antiquark">Up antiquark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bar{u}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/b/8/bb800c335257fca67d8da7d62d09a820.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-2/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/2/3/c23f5e67ea75ff03bde3bde54ff37537.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-4/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/c/d/ccd7d1c965c4c8f798086d866d97a359.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{\bar{3}}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/3/1/e31ea28169d02a7bec19ad3936b4628d.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">~ 3 MeV<sup class="reference" id="cite_ref-s3_28-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s3-28" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 5]</a></sup></td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Down_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Down quark">Down quark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="d\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/d/6/3d6de401d007ea0cffc99610ad623239.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/9/8/598ed7491136e7eab1a7d3b30b44f6c9.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1/2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/c/7/dc74155f9123c0ea21e0fcdd48c3f8f0.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/8/e/08e98b019ed284066bba9882f94d20d1.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{3}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/1/b/21b18e410898272ca1a05200db7f5142.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">~ 6 MeV<sup class="reference" id="cite_ref-s3_28-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s3-28" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 5]</a></sup></td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Down_antiquark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Down antiquark">Down antiquark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bar{d}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/1/c/e1cff07384389db9cd28f79fa163e4a7.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/8/e/08e98b019ed284066bba9882f94d20d1.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+2/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/9/4/1942ed13d5af7f23bac1650e523d7eb3.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{\bar{3}}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/3/1/e31ea28169d02a7bec19ad3936b4628d.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">~ 6 MeV<sup class="reference" id="cite_ref-s3_28-3" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s3-28" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 5]</a></sup></td></tr>
<tr><th colspan="8" style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"></th></tr>
<tr><th colspan="8" style="background-color: navajowhite; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Generation 2</th></tr>
<tr style="background-color: #ffdddd; background-position: initial initial; background-repeat: initial initial;"><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Fermion<br />
(left-handed)</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Symbol</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Electric<br />
charge</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Weak<br />
isospin</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Weak<br />
hypercharge</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Color<br />
charge <sup class="reference" id="cite_ref-s1_24-1" style="font-weight: normal; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s1-24" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 1]</a></sup></th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Mass <sup class="reference" id="cite_ref-s2_25-1" style="font-weight: normal; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s2-25" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 2]</a></sup></th></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Muon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Muon">Muon</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\mu^-\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/a/b/cab83cf8ff1cfbd72b25872440671b8c.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/c/f3c1d3a00c1ad4b653ef1ce33e47e721.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1/2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/c/7/dc74155f9123c0ea21e0fcdd48c3f8f0.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/c/f3c1d3a00c1ad4b653ef1ce33e47e721.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">106 MeV</td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Antimuon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Antimuon">Antimuon</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\mu^+\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/b/3/cb3a75f0f30a19bf92a4be4190ad0dde.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/8/7/88772a82403b94411223fbb8f69e4bde.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/e/b/beb954bb17ff5ba73db1bfe6cc90fa33.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">106 MeV</td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Muon_neutrino" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Muon neutrino">Muon neutrino</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\nu_\mu\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/f/6/cf612bc30e225bebb52109fa7610f6be.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/5/c/05cab4a49dfc9b7b1327cc15eab681d7.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/c/f3c1d3a00c1ad4b653ef1ce33e47e721.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">< 0.28 eV<sup class="reference" id="cite_ref-s4_26-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s4-26" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 3]</a></sup><sup class="reference" id="cite_ref-s4b_27-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s4b-27" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 4]</a></sup></td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Muon_antineutrino" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Muon antineutrino">Muon antineutrino</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bar\nu_\mu\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/3/1/731213e759c48c800e96119670365a71.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">< 0.28 eV<sup class="reference" id="cite_ref-s4_26-3" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s4-26" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 3]</a></sup><sup class="reference" id="cite_ref-s4b_27-3" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s4b-27" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 4]</a></sup></td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Charm_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Charm quark">Charm quark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="c\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/8/1/08163b03d3a58471d7f88fc4e581a282.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+2/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/9/4/1942ed13d5af7f23bac1650e523d7eb3.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/5/c/05cab4a49dfc9b7b1327cc15eab681d7.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/8/e/08e98b019ed284066bba9882f94d20d1.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{3}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/1/b/21b18e410898272ca1a05200db7f5142.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">~ 1.337 GeV</td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Charm_antiquark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Charm antiquark">Charm antiquark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bar{c}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/e/b/6eb414580d84b62703780b39ab0e7b5b.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-2/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/2/3/c23f5e67ea75ff03bde3bde54ff37537.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-4/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/c/d/ccd7d1c965c4c8f798086d866d97a359.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{\bar{3}}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/3/1/e31ea28169d02a7bec19ad3936b4628d.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">~ 1.3 GeV</td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Strange_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Strange quark">Strange quark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="s\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/0/4/d0438646c1f482faffdd1bac9a841799.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/9/8/598ed7491136e7eab1a7d3b30b44f6c9.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1/2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/c/7/dc74155f9123c0ea21e0fcdd48c3f8f0.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/8/e/08e98b019ed284066bba9882f94d20d1.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{3}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/1/b/21b18e410898272ca1a05200db7f5142.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">~ 100 MeV</td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Strange_antiquark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Strange antiquark">Strange antiquark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bar{s}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/9/3/59395df6869af03183e7870c6aae5483.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/8/e/08e98b019ed284066bba9882f94d20d1.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+2/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/9/4/1942ed13d5af7f23bac1650e523d7eb3.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{\bar{3}}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/3/1/e31ea28169d02a7bec19ad3936b4628d.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">~ 100 MeV</td></tr>
<tr><th colspan="8" style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;"></th></tr>
<tr><th colspan="8" style="background-color: navajowhite; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Generation 3</th></tr>
<tr style="background-color: #ffdddd; background-position: initial initial; background-repeat: initial initial;"><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Fermion<br />
(left-handed)</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Symbol</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Electric<br />
charge</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Weak<br />
isospin</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Weak<br />
hypercharge</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Color<br />
charge<sup class="reference" id="cite_ref-s1_24-2" style="font-weight: normal; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s1-24" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 1]</a></sup></th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Mass<sup class="reference" id="cite_ref-s2_25-2" style="font-weight: normal; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s2-25" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 2]</a></sup></th></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Tau_(particle)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tau (particle)">Tau</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\tau^-\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/0/c/c0cc55c3741f9dcea64265798bcce1fa.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/c/f3c1d3a00c1ad4b653ef1ce33e47e721.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1/2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/c/7/dc74155f9123c0ea21e0fcdd48c3f8f0.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/c/f3c1d3a00c1ad4b653ef1ce33e47e721.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">1.78 GeV</td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Antitau" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Antitau">Antitau</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\tau^+\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/2/2/e2223788f70e4f40918bf325a3fd1c6d.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/8/7/88772a82403b94411223fbb8f69e4bde.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/e/b/beb954bb17ff5ba73db1bfe6cc90fa33.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">1.78 GeV</td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Tau_neutrino" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tau neutrino">Tau neutrino</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\nu_\tau\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/8/5/885aa4b15f8f9480e82bd2c5f84e6ec2.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/5/c/05cab4a49dfc9b7b1327cc15eab681d7.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/c/f3c1d3a00c1ad4b653ef1ce33e47e721.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">< 0.28 eV<sup class="reference" id="cite_ref-s4_26-4" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s4-26" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 3]</a></sup><sup class="reference" id="cite_ref-s4b_27-4" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s4b-27" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 4]</a></sup></td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Tau_antineutrino" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tau antineutrino">Tau antineutrino</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bar\nu_\tau\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/a/7/9a70a6f7e092d456132f29e020ef25da.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{1}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/2/1/1215bb986441f5182489dd4279d1b0bb.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">< 0.28 eV<sup class="reference" id="cite_ref-s4_26-5" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s4-26" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 3]</a></sup><sup class="reference" id="cite_ref-s4b_27-5" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-s4b-27" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[lhf 4]</a></sup></td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Top_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Top quark">Top quark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="t\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/c/6/0c68620ee2ea4f1286fcd672a47ea080.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+2/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/9/4/1942ed13d5af7f23bac1650e523d7eb3.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/5/c/05cab4a49dfc9b7b1327cc15eab681d7.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/8/e/08e98b019ed284066bba9882f94d20d1.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{3}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/1/b/21b18e410898272ca1a05200db7f5142.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">171 GeV</td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Top_antiquark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Top antiquark">Top antiquark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bar{t}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/a/6/ea6d4736827652cb2d68459145d7a9e8.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-2/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/2/3/c23f5e67ea75ff03bde3bde54ff37537.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-4/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/c/d/ccd7d1c965c4c8f798086d866d97a359.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{\bar{3}}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/3/1/e31ea28169d02a7bec19ad3936b4628d.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">171 GeV</td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a href="http://en.wikipedia.org/wiki/Bottom_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bottom quark">Bottom quark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="b\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/f/c/5fce6fb65e297d5b7e9a07717b52fc59.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/9/8/598ed7491136e7eab1a7d3b30b44f6c9.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="-1/2\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/c/7/dc74155f9123c0ea21e0fcdd48c3f8f0.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/8/e/08e98b019ed284066bba9882f94d20d1.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{3}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/1/b/21b18e410898272ca1a05200db7f5142.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">~ 4.2 GeV</td></tr>
<tr><td style="background-color: #efefef; background-position: initial initial; background-repeat: initial initial; border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Bottom_antiquark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bottom antiquark">Bottom antiquark</a></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bar{b}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/4/6/34659bf3c3335941f95907cf4ca3142d.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+1/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/8/e/08e98b019ed284066bba9882f94d20d1.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/2/b7277a3bac777f05d8acc80a07ed0e19.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="+2/3\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/9/4/1942ed13d5af7f23bac1650e523d7eb3.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><img alt="\bold{\bar{3}}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/3/1/e31ea28169d02a7bec19ad3936b4628d.png" style="border: none; vertical-align: middle;" /></td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">~ 4.2 GeV</td></tr>
<tr><td colspan="8" style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><ol class="references" style="font-size: 12px; line-height: 1.5em; list-style-image: none; margin: 0.3em 0px 0.5em 3.2em; padding: 0px; text-align: left;">
<li id="cite_note-s1-24" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink">^ <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s1_24-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s1_24-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s1_24-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>c</b></i></sup></a></span> <span class="reference-text">These are not ordinary <a href="http://en.wikipedia.org/wiki/Abelian_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Abelian group">abelian</a> <a href="http://en.wikipedia.org/wiki/Electric_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electric charge">charges</a>, which can be added together, but are labels of <a href="http://en.wikipedia.org/wiki/Group_representation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Group representation">group representations</a> of <a href="http://en.wikipedia.org/wiki/Lie_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lie group">Lie groups</a>.</span></li>
<li id="cite_note-s2-25" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink">^ <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s2_25-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s2_25-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s2_25-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>c</b></i></sup></a></span> <span class="reference-text">Mass is really a coupling between a left-handed fermion and a right-handed fermion. For example, the mass of an electron is really a coupling between a left-handed electron and a right-handed electron, which is the <a href="http://en.wikipedia.org/wiki/Antiparticle" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Antiparticle">antiparticle</a> of a left-handed <a href="http://en.wikipedia.org/wiki/Positron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Positron">positron</a>. Also neutrinos show large mixings in their mass coupling, so it's not accurate to talk about neutrino masses in the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Flavor_(particle_physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Flavor (particle physics)">flavor</a> basis or to suggest a left-handed electron antineutrino.</span></li>
<li id="cite_note-s4-26" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink">^ <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s4_26-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s4_26-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s4_26-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>c</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s4_26-3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>d</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s4_26-4" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>e</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s4_26-5" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>f</b></i></sup></a></span> <span class="reference-text">The Standard Model assumes that neutrinos are massless. However, several contemporary experiments prove that <a href="http://en.wikipedia.org/wiki/Neutrino_oscillation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Neutrino oscillation">neutrinos oscillate</a> between their <a href="http://en.wikipedia.org/wiki/Flavour_(particle_physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Flavour (particle physics)">flavour</a> states, which could not happen if all were massless. It is straightforward to extend the model to fit these data but there are many possibilities, so the mass <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Eigenstate" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eigenstate">eigenstates</a> are still <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Open_question" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Open question">open</a>. See <a href="http://en.wikipedia.org/wiki/Neutrino#Mass" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Neutrino">neutrino mass</a>.</span></li>
<li id="cite_note-s4b-27" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink">^ <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s4b_27-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s4b_27-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s4b_27-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>c</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s4b_27-3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>d</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s4b_27-4" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>e</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s4b_27-5" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>f</b></i></sup></a></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">W.-M. Yao <i>et al</i>. (<a href="http://en.wikipedia.org/wiki/Particle_Data_Group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Particle Data Group">Particle Data Group</a>) (2006). <a class="external text" href="http://pdg.lbl.gov/2007/reviews/numixrpp.pdf" rel="nofollow" style="background-image: url(http://upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif); background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 18px; text-decoration: none;">"Review of Particle Physics: Neutrino mass, mixing, and flavor change"</a>. <i><a href="http://en.wikipedia.org/wiki/Journal_of_Physics_G" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Journal of Physics G">Journal of Physics G</a></i> <b>33</b>: 1. <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/astro-ph/0601168" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">astro-ph/0601168</a>. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/2006JPhG...33....1Y" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">2006JPhG...33....1Y</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1088%2F0954-3899%2F33%2F1%2F001" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1088/0954-3899/33/1/001</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Review+of+Particle+Physics%3A++Neutrino+mass%2C+mixing%2C+and+flavor+change&rft.jtitle=%5B%5BJournal+of+Physics+G%5D%5D&rft.aulast=W.-M.+Yao+%27%27et+al%27%27.+%28%5B%5BParticle+Data+Group%5D%5D%29&rft.au=W.-M.+Yao+%27%27et+al%27%27.+%28%5B%5BParticle+Data+Group%5D%5D%29&rft.date=2006&rft.volume=33&rft.pages=1&rft_id=info:arxiv/astro-ph%2F0601168&rft_id=info:bibcode/2006JPhG...33....1Y&rft_id=info:doi/10.1088%2F0954-3899%2F33%2F1%2F001&rft_id=http%3A%2F%2Fpdg.lbl.gov%2F2007%2Freviews%2Fnumixrpp.pdf&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-s3-28" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink">^ <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s3_28-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s3_28-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s3_28-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>c</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-s3_28-3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>d</b></i></sup></a></span> <span class="reference-text">The <a href="http://en.wikipedia.org/wiki/Mass" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mass">masses</a> of <a href="http://en.wikipedia.org/wiki/Baryon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Baryon">baryons</a> and <a href="http://en.wikipedia.org/wiki/Hadron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hadron">hadrons</a> and various cross-sections are the experimentally measured quantities. Since quarks can't be isolated because of <a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum chromodynamics">QCD</a> <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Colour_confinement" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Colour confinement">confinement</a>, the quantity here is supposed to be the mass of the quark at the <a href="http://en.wikipedia.org/wiki/Renormalization" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Renormalization">renormalization</a> scale of the QCD scale.</span></li>
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Log plot of masses in the Standard Model.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=21" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Tests and predictions">edit</a>]</span><span class="mw-headline" id="Tests_and_predictions">Tests and predictions</span></h2>
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<img alt="" height="39" src="http://upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" style="border: none; vertical-align: middle;" width="50" /></div>
</td><td class="mbox-text" style="border: none; padding: 0.25em 0.5em; width: 387px;"><span class="mbox-text-span">This section <b>needs additional <a href="http://en.wikipedia.org/wiki/Wikipedia:Citing_sources#Inline_citations" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; color: #0b0080; padding: 0px !important; text-decoration: none;" title="Wikipedia:Citing sources">citations</a> for <a href="http://en.wikipedia.org/wiki/Wikipedia:Verifiability" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; color: #0b0080; padding: 0px !important; text-decoration: none;" title="Wikipedia:Verifiability">verification</a></b>. Please help <a class="external text" href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit" style="background-image: url(data:image/png; background-position: 0% 0%; background-repeat: no-repeat no-repeat; color: #663366; padding-bottom: 0px !important; padding-left: 0px !important; padding-right: 13px; padding-top: 0px !important; text-decoration: none;">improve this article</a> by adding citations to <a href="http://en.wikipedia.org/wiki/Wikipedia:Identifying_reliable_sources" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; color: #0b0080; padding: 0px !important; text-decoration: none;" title="Wikipedia:Identifying reliable sources">reliable sources</a>. Unsourced material may be <a href="http://en.wikipedia.org/wiki/Template:Citation_needed" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; color: #0b0080; padding: 0px !important; text-decoration: none;" title="Template:Citation needed">challenged</a> and <a href="http://en.wikipedia.org/wiki/Wikipedia:Verifiability#Burden_of_evidence" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; color: #0b0080; padding: 0px !important; text-decoration: none;" title="Wikipedia:Verifiability">removed</a>. <small><i>(April 2008)</i></small></span></td></tr>
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The Standard Model (SM) predicted the existence of the <a href="http://en.wikipedia.org/wiki/W_and_Z_bosons" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="W and Z bosons">W and Z bosons</a>, <a href="http://en.wikipedia.org/wiki/Gluon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gluon">gluon</a>, and the <a href="http://en.wikipedia.org/wiki/Top_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Top quark">top</a> and<a href="http://en.wikipedia.org/wiki/Charm_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Charm quark">charm quarks</a> before these particles were observed. Their predicted properties were experimentally confirmed with good precision. To give an idea of the success of the SM, the following table compares the measured masses of the W and Z bosons with the masses predicted by the SM:</div>
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<tr><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Quantity</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">Measured (GeV)</th><th style="background-color: #f2f2f2; border: 1px solid rgb(170, 170, 170); padding: 0.2em; text-align: center;">SM prediction (GeV)</th></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Mass of W boson</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">80.387 ± 0.019</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">80.390 ± 0.018</td></tr>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">Mass of Z boson</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">91.1876 ± 0.0021</td><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;">91.1874 ± 0.0021</td></tr>
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The SM also makes several predictions about the decay of Z bosons, which have been experimentally confirmed by the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Large_Electron-Positron_Collider" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Large Electron-Positron Collider">Large Electron-Positron Collider</a> at <a href="http://en.wikipedia.org/wiki/CERN" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="CERN">CERN</a>.</div>
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In May 2012 <a href="http://en.wikipedia.org/wiki/BaBar_experiment" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="BaBar experiment">BaBar Collaboration</a> reported that their recently analyzed data may suggest possible flaws in the Standard Model of particle physics.<sup class="reference" id="cite_ref-29" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-29" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[23]</a></sup><sup class="reference" id="cite_ref-30" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-30" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[24]</a></sup> These data show that a particular type of particle decay called "B to D-star-tau-nu" happens more often than the Standard Model says it should. In this type of decay, a particle called the B-bar meson decays into a D meson, an antineutrino and a tau-lepton. While the level of certainty of the excess (3.4 sigma) is not enough to claim a break from the Standard Model, the results are a potential sign of something amiss and are likely to impact existing theories, including those attempting to deduce the properties of <a href="http://en.wikipedia.org/wiki/Higgs_boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Higgs boson">Higgs bosons</a>.<sup class="reference" id="cite_ref-31" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-31" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[25]</a></sup></div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=22" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Challenges">edit</a>]</span><span class="mw-headline" id="Challenges">Challenges</span></h2>
<div class="rellink boilerplate seealso" style="font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em;">
See also: <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Beyond_the_Standard_Model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Beyond the Standard Model">Beyond the Standard Model</a></div>
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<tr><td class="mbox-image" style="border: none; padding: 2px 0px 2px 0.5em; text-align: center;"><a class="image" href="http://en.wikipedia.org/wiki/File:Question_book-new.svg" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; color: #0b0080; padding: 0px !important; text-decoration: none;"><img alt="Question book-new.svg" height="39" src="http://upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" style="border: none; vertical-align: middle;" width="50" /></a></td><td class="mbox-text" style="border: none; padding: 0.25em 0.5em; width: 160px;"><span class="mbox-text-span">This unreferenced section requires <a href="http://en.wikipedia.org/wiki/Wikipedia:Citing_sources" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; color: #0b0080; padding: 0px !important; text-decoration: none;" title="Wikipedia:Citing sources">citations</a> to ensure<a href="http://en.wikipedia.org/wiki/Wikipedia:Verifiability" style="background-image: none !important; background-position: initial initial !important; background-repeat: initial initial !important; color: #0b0080; padding: 0px !important; text-decoration: none;" title="Wikipedia:Verifiability">verifiability</a>.</span></td></tr>
</tbody></table>
<table class="wikitable" style="background-color: #f9f9f9; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; float: right; font-size: 13px; margin: 0em 1em 1em; width: 300px;"><caption style="font-weight: bold;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Unsolved_problems_in_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Unsolved problems in physics">Unsolved problems in physics</a></caption><tbody>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px; text-align: left;">
<li style="margin-bottom: 0.1em;">What gives rise to the Standard Model of particle physics?</li>
<li style="margin-bottom: 0.1em;">Why do particle masses and <a href="http://en.wikipedia.org/wiki/Coupling_constant" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Coupling constant">coupling constants</a>have the values that we measure?</li>
<li style="margin-bottom: 0.1em;">Does the <a href="http://en.wikipedia.org/wiki/Higgs_boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Higgs boson">Higgs boson</a> really exist?</li>
<li style="margin-bottom: 0.1em;">Why are there three <a href="http://en.wikipedia.org/wiki/Generation_(particle_physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Generation (particle physics)">generations</a> of particles?</li>
<li style="margin-bottom: 0.1em;">Why is there more matter than <a href="http://en.wikipedia.org/wiki/Antimatter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Antimatter">antimatter</a> in the universe?</li>
<li style="margin-bottom: 0.1em;">Where does <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Dark_Matter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dark Matter">Dark Matter</a> fit into the model? Is it even a new particle?</li>
</ul>
</td></tr>
</tbody></table>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Self-consistency of the Standard Model has not been mathematically proven. While computational approximations (for example using <a href="http://en.wikipedia.org/wiki/Lattice_gauge_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lattice gauge theory">lattice gauge theory</a>) exist, it is not known whether they converge in the limit. A key question related to the consistency is the <a href="http://en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Yang–Mills existence and mass gap">Yang–Mills existence and mass gap</a> problem.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
There is some experimental evidence consistent with <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Neutrinos" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Neutrinos">neutrinos</a> having <a href="http://en.wikipedia.org/wiki/Mass" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mass">mass</a>, which the Standard Model does not allow.<sup class="reference" id="cite_ref-32" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Standard_model#cite_note-32" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[26]</a></sup> To accommodate such findings, the Standard Model can be modified by adding a non-renormalizable interaction of lepton fields with the square of the Higgs field. This is natural in certain <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Grand_unified_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Grand unified theory">grand unified theories</a>, and if new physics appears at about 10<sup style="line-height: 1em;">16</sup> <a href="http://en.wikipedia.org/wiki/Electronvolt" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electronvolt">GeV</a>, the neutrino masses are of the right order of magnitude.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Currently, there is one elementary particle predicted by the Standard Model that has yet to be observed: the <a href="http://en.wikipedia.org/wiki/Higgs_boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Higgs boson">Higgs boson</a>. A major reason for building the <a href="http://en.wikipedia.org/wiki/Large_Hadron_Collider" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Large Hadron Collider">Large Hadron Collider</a> is that the high energies of which it is capable are expected to make the Higgs boson observable. However, as of January 2012, there is only indirect empirical evidence for the existence of the Higgs boson, so that its discovery cannot be claimed. Moreover, some theoretical concerns have been raised positing that elementary scalar Higgs particles cannot exist (see <a href="http://en.wikipedia.org/wiki/Quantum_triviality" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum triviality">Quantum triviality</a>).</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Theoretical and experimental <a href="http://en.wikipedia.org/wiki/Research" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Research">research</a> has attempted to extend the Standard Model into a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Unified_Field_Theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Unified Field Theory">Unified Field Theory</a> or a <a href="http://en.wikipedia.org/wiki/Theory_of_everything" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theory of everything">Theory of everything</a>, a complete theory explaining all physical phenomena including constants. Inadequacies of the Standard Model that motivate such research include:</div>
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;">It does not attempt to explain <a href="http://en.wikipedia.org/wiki/Gravitation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gravitation">gravitation</a>, although a theoretical particle known as a <a href="http://en.wikipedia.org/wiki/Graviton" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Graviton">graviton</a> would help explain it, and unlike for the strong and electroweak interactions of the Standard Model, there is no known way of describing <a href="http://en.wikipedia.org/wiki/General_relativity" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="General relativity">general relativity</a>, the canonical theory of gravitation, consistently in terms of <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">quantum field theory</a>. The reason for this is, among other things, that quantum field theories of gravity generally break down before reaching the <a href="http://en.wikipedia.org/wiki/Planck_scale" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Planck scale">Planck scale</a>. As a consequence, we have no reliable theory for the very early universe;</li>
<li style="margin-bottom: 0.1em;">Some consider it to be <i>ad-hoc</i> and inelegant, requiring 19 numerical constants whose values are unrelated and arbitrary. Although the Standard Model, as it now stands, can explain why neutrinos have masses, the specifics of neutrino mass are still unclear. It is believed that explaining neutrino mass will require an additional 7 or 8 constants, which are also arbitrary parameters;</li>
<li style="margin-bottom: 0.1em;">The Higgs mechanism gives rise to the <a href="http://en.wikipedia.org/wiki/Hierarchy_problem" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hierarchy problem">hierarchy problem</a> if any new physics (such as quantum gravity) is present at high energy scales. In order for the weak scale to be much smaller than the <a href="http://en.wikipedia.org/wiki/Planck_scale" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Planck scale">Planck scale</a>, severe fine tuning of Standard Model parameters is required;</li>
<li style="margin-bottom: 0.1em;">It should be modified so as to be consistent with the emerging "Standard Model of <a href="http://en.wikipedia.org/wiki/Cosmology" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cosmology">cosmology</a>." In particular, the Standard Model cannot explain the observed amount of<a href="http://en.wikipedia.org/wiki/Cold_dark_matter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cold dark matter">cold dark matter</a> (CDM) and gives contributions to <a href="http://en.wikipedia.org/wiki/Dark_energy" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dark energy">dark energy</a> which are far too large. It is also difficult to accommodate the observed predominance of matter over antimatter (<a href="http://en.wikipedia.org/wiki/Matter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Matter">matter</a>/<a href="http://en.wikipedia.org/wiki/Antimatter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Antimatter">antimatter</a> <a href="http://en.wikipedia.org/wiki/Baryon_asymmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Baryon asymmetry">asymmetry</a>). The <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Isotropic" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Isotropic">isotropy</a> and <a href="http://en.wikipedia.org/wiki/Homogeneity_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Homogeneity (physics)">homogeneity</a> of the visible universe over large distances seems to require a mechanism like cosmic <a href="http://en.wikipedia.org/wiki/Inflation_(cosmology)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Inflation (cosmology)">inflation</a>, which would also constitute an extension of the Standard Model.</li>
</ul>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Currently no proposed <a href="http://en.wikipedia.org/wiki/Theory_of_everything" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theory of everything">Theory of everything</a> has been conclusively verified.</div>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=23" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: See also">edit</a>]</span><span class="mw-headline" id="See_also">See also</span></h2>
<div class="noprint tright portal" style="border: 1px solid rgb(170, 170, 170); clear: right; float: right; margin: 0.5em 0px 0.5em 0.5em;">
<table style="background-color: #f9f9f9; background-position: initial initial; background-repeat: initial initial; font-size: 11px; line-height: 12px; max-width: 175px;"><tbody>
<tr><td style="text-align: center;"><a class="image" href="http://en.wikipedia.org/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><img alt="Portal icon" height="28" src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" style="border: none; vertical-align: middle;" width="28" /></a></td><td style="font-style: italic; font-weight: bold; padding: 0px 0.2em; vertical-align: middle;"><a href="http://en.wikipedia.org/wiki/Portal:Mathematics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Portal:Mathematics">Mathematics portal</a></td></tr>
</tbody></table>
</div>
<table class="metadata mbox-small" style="background-color: #f9f9f9; border: 1px solid rgb(170, 170, 170); clear: right; float: right; font-size: 11px; line-height: 1.25em; margin: 4px 0px 4px 1em; width: 238px;"><tbody>
<tr><td class="mbox-image" style="border: none; padding: 2px 0px 2px 0.9em; text-align: center;"><img alt="Book icon" height="30" src="http://upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Office-book.svg/30px-Office-book.svg.png" style="border: none; vertical-align: middle;" width="30" /></td><td class="mbox-text" style="border: none; padding: 0.25em 0.9em; width: 170px;"><i><b><a href="http://en.wikipedia.org/wiki/Book:Particles_of_the_Standard_Model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Book:Particles of the Standard Model">Book: Particles of the Standard Model</a></b></i></td></tr>
<tr><td class="mbox-text" colspan="2" style="border: none; padding: 0.25em 0.9em; width: 212px;"><small><a href="http://en.wikipedia.org/wiki/Wikipedia:Books" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:Books">Wikipedia books</a> are collections of articles that can be downloaded or ordered in print.</small></td></tr>
</tbody></table>
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/1964_PRL_symmetry_breaking_papers" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="1964 PRL symmetry breaking papers">1964 PRL symmetry breaking papers</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/C._R._Hagen" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="C. R. Hagen">C. R. Hagen</a></li>
<li style="margin-bottom: 0.1em;">Diagrams: <a href="http://en.wikipedia.org/wiki/Feynman_diagram" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Feynman diagram">Feynman diagram</a> - <a href="http://en.wikipedia.org/wiki/Penguin_diagram" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Penguin diagram">Penguin diagram</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Elementary_particle" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Elementary particle">Elementary particle</a>: <a href="http://en.wikipedia.org/wiki/Boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Boson">Boson</a>, <a href="http://en.wikipedia.org/wiki/Fermion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermion">Fermion</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Flavour_(particle_physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Flavour (particle physics)">Flavour</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Fundamental_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fundamental interaction">Fundamental interaction</a>:<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum electrodynamics">Quantum electrodynamics</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Strong_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Strong interaction">Strong interaction</a>: <a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge">Color charge</a>, <a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum chromodynamics">Quantum chromodynamics</a>, <a href="http://en.wikipedia.org/wiki/Quark_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark model">Quark model</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Weak_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak interaction">Weak interaction</a>: <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Electroweak_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electroweak theory">Electroweak theory</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Fermi_theory_of_beta_decay" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermi theory of beta decay">Fermi theory of beta decay</a>, <a href="http://en.wikipedia.org/wiki/Weak_hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak hypercharge">Weak hypercharge</a>, <a href="http://en.wikipedia.org/wiki/Weak_isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak isospin">Weak isospin</a></li>
</ul>
</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Gauge_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge theory">Gauge theory</a>: <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Nontechnical_introduction_to_gauge_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nontechnical introduction to gauge theory">Nontechnical introduction to gauge theory</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Generation_(particle_physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Generation (particle physics)">Generation</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Higgs_mechanism" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Higgs mechanism">Higgs mechanism</a>: <a href="http://en.wikipedia.org/wiki/Higgs_boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Higgs boson">Higgs boson</a>, <a href="http://en.wikipedia.org/wiki/Higgsless_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Higgsless model">Higgsless model</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/John_Clive_Ward" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John Clive Ward">J. C. Ward</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Sakurai_Prize" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sakurai Prize">J. J. Sakurai Prize for Theoretical Particle Physics</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Lagrangian" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lagrangian">Lagrangian</a></li>
<li style="margin-bottom: 0.1em;"><a class="new" href="http://en.wikipedia.org/w/index.php?title=Noncommutative_Standard_Model&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Noncommutative Standard Model (page does not exist)">Noncommutative Standard Model</a></li>
<li style="margin-bottom: 0.1em;">Open questions: <a href="http://en.wikipedia.org/wiki/BTeV_experiment" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="BTeV experiment">BTeV experiment</a>, <a href="http://en.wikipedia.org/wiki/CP_violation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="CP violation">CP violation</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Neutrino_mass" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Neutrino mass">Neutrino masses</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Quark_matter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark matter">Quark matter</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">Quantum field theory</a></li>
<li style="margin-bottom: 0.1em;">Standard Model: <a href="http://en.wikipedia.org/wiki/Standard_Model_(mathematical_formulation)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Standard Model (mathematical formulation)">Mathematical formulation of</a>, <a href="http://en.wikipedia.org/wiki/Physics_beyond_the_Standard_Model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physics beyond the Standard Model">Physics beyond the Standard Model</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Unparticle_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Unparticle physics">Unparticle physics</a></li>
</ul>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=24" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Notes and references">edit</a>]</span><span class="mw-headline" id="Notes_and_references">Notes and references</span></h2>
<h3 style="background-image: none; border-bottom-style: none; font-size: 18px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=25" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Notes">edit</a>]</span><span class="mw-headline" id="Notes">Notes</span></h3>
<div class="reflist" style="font-size: 12px; list-style-type: decimal; margin-bottom: 0.5em;">
<ol class="references" style="line-height: 1.5em; list-style-image: none; list-style-type: inherit; margin: 0.3em 0px 0.5em 3.2em; padding: 0px;">
<li id="cite_note-11" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-11" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">Technically, there are nine such color–anticolor combinations. However, there is one color-symmetric combination that can be constructed out of a linear superposition of the nine combinations, reducing the count to eight.</span></li>
<li id="cite_note-20" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-20" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">The normalization <img alt="{}_{Q=I^3+Y}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/2/f/72f37d83bddf21cc94cf93c3842cc648.png" style="border: none; vertical-align: middle;" /> is sometimes used instead.</span></li>
</ol>
</div>
<h3 style="background-image: none; border-bottom-style: none; font-size: 18px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=26" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: References">edit</a>]</span><span class="mw-headline" id="References">References</span></h3>
<div class="reflist references-column-width" style="-webkit-column-width: 35em; font-size: 12px; list-style-type: decimal; margin-bottom: 0.5em;">
<ol class="references" style="line-height: 1.5em; list-style-image: none; list-style-type: inherit; margin: 0.3em 0px 0.5em 3.2em; padding: 0px;">
<li id="cite_note-0" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">S.L. Glashow (1961). "Partial-symmetries of weak interactions". <i><a href="http://en.wikipedia.org/wiki/Nuclear_Physics_(journal)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nuclear Physics (journal)">Nuclear Physics</a></i> <b>22</b>(4): 579–588. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1961NucPh..22..579G" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1961NucPh..22..579G</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1016%2F0029-5582%2861%2990469-2" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1016/0029-5582(61)90469-2</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Partial-symmetries+of+weak+interactions&rft.jtitle=%5B%5BNuclear+Physics+%28journal%29%7CNuclear+Physics%5D%5D&rft.aulast=S.L.+Glashow&rft.au=S.L.+Glashow&rft.date=1961&rft.volume=22&rft.issue=4&rft.pages=579%E2%80%93588&rft_id=info:bibcode/1961NucPh..22..579G&rft_id=info:doi/10.1016%2F0029-5582%2861%2990469-2&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-1" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">S. Weinberg (1967). "A Model of Leptons". <i><a href="http://en.wikipedia.org/wiki/Physical_Review_Letters" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review Letters">Physical Review Letters</a></i> <b>19</b> (21): 1264–1266. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1967PhRvL..19.1264W" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1967PhRvL..19.1264W</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRevLett.19.1264" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRevLett.19.1264</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+Model+of+Leptons&rft.jtitle=%5B%5BPhysical+Review+Letters%5D%5D&rft.aulast=S.+Weinberg&rft.au=S.+Weinberg&rft.date=1967&rft.volume=19&rft.issue=21&rft.pages=1264%E2%80%931266&rft_id=info:bibcode/1967PhRvL..19.1264W&rft_id=info:doi/10.1103%2FPhysRevLett.19.1264&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-2" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;">A. Salam (1968). N. Svartholm. ed. <i>Elementary Particle Physics: Relativistic Groups and Analyticity</i>. <a class="new" href="http://en.wikipedia.org/w/index.php?title=Nobel_Symposium&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Nobel Symposium (page does not exist)">Eighth Nobel Symposium</a>. Stockholm: <a class="new" href="http://en.wikipedia.org/w/index.php?title=Almquvist_and_Wiksell&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Almquvist and Wiksell (page does not exist)">Almquvist and Wiksell</a>. pp. 367.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Particle+Physics%3A+Relativistic+Groups+and+Analyticity&rft.aulast=A.+Salam&rft.au=A.+Salam&rft.date=1968&rft.pages=pp.%26nbsp%3B367&rft.place=Stockholm&rft.pub=%5B%5BAlmquvist+and+Wiksell%5D%5D&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-3" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">F. Englert, R. Brout (1964). "Broken Symmetry and the Mass of Gauge Vector Mesons". <i><a href="http://en.wikipedia.org/wiki/Physical_Review_Letters" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review Letters">Physical Review Letters</a></i> <b>13</b> (9): 321–323. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1964PhRvL..13..321E" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1964PhRvL..13..321E</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRevLett.13.321" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRevLett.13.321</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Broken+Symmetry+and+the+Mass+of+Gauge+Vector+Mesons&rft.jtitle=%5B%5BPhysical+Review+Letters%5D%5D&rft.aulast=F.+Englert%2C+R.+Brout&rft.au=F.+Englert%2C+R.+Brout&rft.date=1964&rft.volume=13&rft.issue=9&rft.pages=321%E2%80%93323&rft_id=info:bibcode/1964PhRvL..13..321E&rft_id=info:doi/10.1103%2FPhysRevLett.13.321&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-4" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-4" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">P.W. Higgs (1964). "Broken Symmetries and the Masses of Gauge Bosons".<i><a href="http://en.wikipedia.org/wiki/Physical_Review_Letters" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review Letters">Physical Review Letters</a></i> <b>13</b> (16): 508–509. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1964PhRvL..13..508H" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1964PhRvL..13..508H</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRevLett.13.508" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRevLett.13.508</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Broken+Symmetries+and+the+Masses+of+Gauge+Bosons&rft.jtitle=%5B%5BPhysical+Review+Letters%5D%5D&rft.aulast=P.W.+Higgs&rft.au=P.W.+Higgs&rft.date=1964&rft.volume=13&rft.issue=16&rft.pages=508%E2%80%93509&rft_id=info:bibcode/1964PhRvL..13..508H&rft_id=info:doi/10.1103%2FPhysRevLett.13.508&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-5" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-5" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">G.S. Guralnik, C.R. Hagen, T.W.B. Kibble (1964). "Global Conservation Laws and Massless Particles". <i><a href="http://en.wikipedia.org/wiki/Physical_Review_Letters" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review Letters">Physical Review Letters</a></i> <b>13</b> (20): 585–587. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/1964PhRvL..13..585G" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1964PhRvL..13..585G</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRevLett.13.585" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRevLett.13.585</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Global+Conservation+Laws+and+Massless+Particles&rft.jtitle=%5B%5BPhysical+Review+Letters%5D%5D&rft.aulast=G.S.+Guralnik%2C+C.R.+Hagen%2C+T.W.B.+Kibble&rft.au=G.S.+Guralnik%2C+C.R.+Hagen%2C+T.W.B.+Kibble&rft.date=1964&rft.volume=13&rft.issue=20&rft.pages=585%E2%80%93587&rft_id=info:bibcode/1964PhRvL..13..585G&rft_id=info:doi/10.1103%2FPhysRevLett.13.585&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-6" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-6" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">F.J. Hasert <i>et al.</i> (1973). "Search for elastic muon-neutrino electron scattering".<i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Physics_Letters_B" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physics Letters B">Physics Letters B</a></i> <b>46</b>: 121. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1973PhLB...46..121H" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1973PhLB...46..121H</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1016%2F0370-2693%2873%2990494-2" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1016/0370-2693(73)90494-2</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Search+for+elastic+muon-neutrino+electron+scattering&rft.jtitle=%5B%5BPhysics+Letters+B%5D%5D&rft.aulast=F.J.+Hasert+%27%27et+al.%27%27&rft.au=F.J.+Hasert+%27%27et+al.%27%27&rft.date=1973&rft.volume=46&rft.pages=121&rft_id=info:bibcode/1973PhLB...46..121H&rft_id=info:doi/10.1016%2F0370-2693%2873%2990494-2&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-7" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-7" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">F.J. Hasert <i>et al.</i> (1973). "Observation of neutrino-like interactions without muon or electron in the gargamelle neutrino experiment". <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Physics_Letters_B" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physics Letters B">Physics Letters B</a></i> <b>46</b>: 138. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/1973PhLB...46..138H" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1973PhLB...46..138H</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1016%2F0370-2693%2873%2990499-1" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1016/0370-2693(73)90499-1</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Observation+of+neutrino-like+interactions+without+muon+or+electron+in+the+gargamelle+neutrino+experiment&rft.jtitle=%5B%5BPhysics+Letters+B%5D%5D&rft.aulast=F.J.+Hasert+%27%27et+al.%27%27&rft.au=F.J.+Hasert+%27%27et+al.%27%27&rft.date=1973&rft.volume=46&rft.pages=138&rft_id=info:bibcode/1973PhLB...46..138H&rft_id=info:doi/10.1016%2F0370-2693%2873%2990499-1&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-8" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-8" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">F.J. Hasert <i>et al.</i> (1974). "Observation of neutrino-like interactions without muon or electron in the Gargamelle neutrino experiment". <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Nuclear_Physics_B" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nuclear Physics B">Nuclear Physics B</a></i> <b>73</b>: 1. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/1974NuPhB..73....1H" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1974NuPhB..73....1H</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1016%2F0550-3213%2874%2990038-8" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1016/0550-3213(74)90038-8</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Observation+of+neutrino-like+interactions+without+muon+or+electron+in+the+Gargamelle+neutrino+experiment&rft.jtitle=%5B%5BNuclear+Physics+B%5D%5D&rft.aulast=F.J.+Hasert+%27%27et+al.%27%27&rft.au=F.J.+Hasert+%27%27et+al.%27%27&rft.date=1974&rft.volume=73&rft.pages=1&rft_id=info:bibcode/1974NuPhB..73....1H&rft_id=info:doi/10.1016%2F0550-3213%2874%2990038-8&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-9" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-9" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation web" style="word-wrap: break-word;">D. Haidt (4 October 2004). <a class="external text" href="http://cerncourier.com/cws/article/cern/29168" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">"The discovery of the weak neutral currents"</a>. <i><a href="http://en.wikipedia.org/wiki/CERN_Courier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="CERN Courier">CERN Courier</a></i><span class="reference-accessdate">. Retrieved 8 May 2008</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=The+discovery+of+the+weak+neutral+currents&rft.atitle=%5B%5BCERN+Courier%5D%5D&rft.aulast=D.+Haidt&rft.au=D.+Haidt&rft.date=4+October+2004&rft_id=http%3A%2F%2Fcerncourier.com%2Fcws%2Farticle%2Fcern%2F29168&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-10" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-10" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">"Details can be worked out if the situation is simple enough for us to make an approximation, which is almost never, but often we can understand more or less what is happening." from <i><a href="http://en.wikipedia.org/wiki/The_Feynman_Lectures_on_Physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="The Feynman Lectures on Physics">The Feynman Lectures on Physics</a></i>, Vol 1. pp. 2–7</span></li>
<li id="cite_note-12" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-12" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">F. Englert, R. Brout (1964). "Broken Symmetry and the Mass of Gauge Vector Mesons". <i><a href="http://en.wikipedia.org/wiki/Physical_Review_Letters" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review Letters">Physical Review Letters</a></i> <b>13</b> (9): 321–323. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1964PhRvL..13..321E" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1964PhRvL..13..321E</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRevLett.13.321" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRevLett.13.321</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Broken+Symmetry+and+the+Mass+of+Gauge+Vector+Mesons&rft.jtitle=%5B%5BPhysical+Review+Letters%5D%5D&rft.aulast=F.+Englert%2C+R.+Brout&rft.au=F.+Englert%2C+R.+Brout&rft.date=1964&rft.volume=13&rft.issue=9&rft.pages=321%E2%80%93323&rft_id=info:bibcode/1964PhRvL..13..321E&rft_id=info:doi/10.1103%2FPhysRevLett.13.321&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-Peter_W._Higgs_1964_508-509-13" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-Peter_W._Higgs_1964_508-509_13-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">P.W. Higgs (1964). "Broken Symmetries and the Masses of Gauge Bosons".<i><a href="http://en.wikipedia.org/wiki/Physical_Review_Letters" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review Letters">Physical Review Letters</a></i> <b>13</b> (16): 508–509. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1964PhRvL..13..508H" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1964PhRvL..13..508H</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRevLett.13.508" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRevLett.13.508</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Broken+Symmetries+and+the+Masses+of+Gauge+Bosons&rft.jtitle=%5B%5BPhysical+Review+Letters%5D%5D&rft.aulast=P.W.+Higgs&rft.au=P.W.+Higgs&rft.date=1964&rft.volume=13&rft.issue=16&rft.pages=508%E2%80%93509&rft_id=info:bibcode/1964PhRvL..13..508H&rft_id=info:doi/10.1103%2FPhysRevLett.13.508&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-14" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-14" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">G.S. Guralnik, C.R. Hagen, T.W.B. Kibble (1964). "Global Conservation Laws and Massless Particles". <i><a href="http://en.wikipedia.org/wiki/Physical_Review_Letters" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review Letters">Physical Review Letters</a></i> <b>13</b> (20): 585–587. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/1964PhRvL..13..585G" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1964PhRvL..13..585G</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRevLett.13.585" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRevLett.13.585</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Global+Conservation+Laws+and+Massless+Particles&rft.jtitle=%5B%5BPhysical+Review+Letters%5D%5D&rft.aulast=G.S.+Guralnik%2C+C.R.+Hagen%2C+T.W.B.+Kibble&rft.au=G.S.+Guralnik%2C+C.R.+Hagen%2C+T.W.B.+Kibble&rft.date=1964&rft.volume=13&rft.issue=20&rft.pages=585%E2%80%93587&rft_id=info:bibcode/1964PhRvL..13..585G&rft_id=info:doi/10.1103%2FPhysRevLett.13.585&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-15" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-15" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">G.S. Guralnik (2009). "The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles". <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/International_Journal_of_Modern_Physics_A" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Journal of Modern Physics A">International Journal of Modern Physics A</a></i> <b>24</b> (14): 2601–2627. <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/0907.3466" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">0907.3466</a>. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/2009IJMPA..24.2601G" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">2009IJMPA..24.2601G</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1142%2FS0217751X09045431" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1142/S0217751X09045431</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+History+of+the+Guralnik%2C+Hagen+and+Kibble+development+of+the+Theory+of+Spontaneous+Symmetry+Breaking+and+Gauge+Particles&rft.jtitle=%5B%5BInternational+Journal+of+Modern+Physics+A%5D%5D&rft.aulast=G.S.+Guralnik&rft.au=G.S.+Guralnik&rft.date=2009&rft.volume=24&rft.issue=14&rft.pages=2601%E2%80%932627&rft_id=info:arxiv/0907.3466&rft_id=info:bibcode/2009IJMPA..24.2601G&rft_id=info:doi/10.1142%2FS0217751X09045431&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-16" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-16" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a class="external free" href="http://cms.web.cern.ch/cms/News/2011/LP11/" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://cms.web.cern.ch/cms/News/2011/LP11/</a></span></li>
<li id="cite_note-17" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-17" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a class="external free" href="http://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2011-135/" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2011-135/</a></span></li>
<li id="cite_note-18" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-18" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a class="external free" href="http://www.zdnet.co.uk/news/emerging-tech/2011/07/25/cern-higgs-boson-answer-to-come-by-end-of-2012-40093510/" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://www.zdnet.co.uk/news/emerging-tech/2011/07/25/cern-higgs-boson-answer-to-come-by-end-of-2012-40093510/</a></span></li>
<li id="cite_note-19" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-19" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a class="external free" href="http://news.cnet.com/8301-30685_3-57342044-264/cern-physicists-find-hint-of-higgs-boson/" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://news.cnet.com/8301-30685_3-57342044-264/cern-physicists-find-hint-of-higgs-boson/</a></span></li>
<li id="cite_note-21" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-21" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a class="external free" href="http://operaweb.lngs.infn.it/spip.php?rubrique14" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://operaweb.lngs.infn.it/spip.php?rubrique14</a> 31May2010 Press Release.</span></li>
<li id="cite_note-22" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-22" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation news" style="word-wrap: break-word;"><a class="external text" href="http://www.bbc.co.uk/news/10364160" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">"Neutrino 'ghost particle' sized up by astronomers"</a>. <i>BBC News</i>. 22 June 2010.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Neutrino+%27ghost+particle%27+sized+up+by+astronomers&rft.jtitle=BBC+News&rft.date=22+June+2010&rft_id=http%3A%2F%2Fwww.bbc.co.uk%2Fnews%2F10364160&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-23" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-23" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">W.-M. Yao <i>et al</i>. (<a href="http://en.wikipedia.org/wiki/Particle_Data_Group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Particle Data Group">Particle Data Group</a>) (2006). <a class="external text" href="http://pdg.lbl.gov/2006/tables/qxxx.pdf" rel="nofollow" style="background-image: url(http://upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif); background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 18px; text-decoration: none;">"Review of Particle Physics: Quarks"</a>. <i><a href="http://en.wikipedia.org/wiki/Journal_of_Physics_G" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Journal of Physics G">Journal of Physics G</a></i> <b>33</b>: 1. <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/astro-ph/0601168" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">astro-ph/0601168</a>. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/2006JPhG...33....1Y" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">2006JPhG...33....1Y</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1088%2F0954-3899%2F33%2F1%2F001" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1088/0954-3899/33/1/001</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Review++of+Particle+Physics%3A+Quarks&rft.jtitle=%5B%5BJournal+of+Physics+G%5D%5D&rft.aulast=W.-M.+Yao+%27%27et+al%27%27.+%28%5B%5BParticle+Data+Group%5D%5D%29&rft.au=W.-M.+Yao+%27%27et+al%27%27.+%28%5B%5BParticle+Data+Group%5D%5D%29&rft.date=2006&rft.volume=33&rft.pages=1&rft_id=info:arxiv/astro-ph%2F0601168&rft_id=info:bibcode/2006JPhG...33....1Y&rft_id=info:doi/10.1088%2F0954-3899%2F33%2F1%2F001&rft_id=http%3A%2F%2Fpdg.lbl.gov%2F2006%2Ftables%2Fqxxx.pdf&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></span></li>
<li id="cite_note-29" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-29" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a class="external text" href="http://www-public.slac.stanford.edu/babar/BaBar-BtoDtaunu.aspx" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">BABAR Data in Tension with the Standard Model (SLAC press-release)</a>.</span></li>
<li id="cite_note-30" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-30" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">BaBar Collaboration, <i>Evidence for an excess of B -> D(*) Tau Nu decays</i>,<a class="external text" href="http://arxiv.org/abs/1205.5442" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">arXiv:1205.5442</a>.</span></li>
<li id="cite_note-31" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-31" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a class="external text" href="http://esciencenews.com/articles/2012/06/18/babar.data.hint.cracks.standard.model" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">BaBar data hint at cracks in the Standard Model (EScienceNews.com)</a>.</span></li>
<li id="cite_note-32" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Standard_model#cite_ref-32" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a class="external text" href="http://press.web.cern.ch/press/PressReleases/Releases2010/PR08.10E.html" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">CERN Press Release</a></span></li>
</ol>
</div>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=27" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Further reading">edit</a>]</span><span class="mw-headline" id="Further_reading">Further reading</span></h2>
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">R. Oerter (2006). <i>The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics</i>. <a href="http://en.wikipedia.org/wiki/Plume_(publisher)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Plume (publisher)">Plume</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Almost+Everything%3A+The+Standard+Model%2C+the+Unsung+Triumph+of+Modern+Physics&rft.aulast=R.+Oerter&rft.au=R.+Oerter&rft.date=2006&rft.pub=%5B%5BPlume+%28publisher%29%7CPlume%5D%5D&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">B.A. Schumm (2004). <i>Deep Down Things: The Breathtaking Beauty of Particle Physics</i>. <a href="http://en.wikipedia.org/wiki/Johns_Hopkins_University_Press" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Johns Hopkins University Press">Johns Hopkins University Press</a>. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-8018-7971-X" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-8018-7971-X">0-8018-7971-X</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Deep+Down+Things%3A++The+Breathtaking+Beauty+of+Particle+Physics&rft.aulast=B.A.+Schumm&rft.au=B.A.+Schumm&rft.date=2004&rft.pub=%5B%5BJohns+Hopkins+University+Press%5D%5D&rft.isbn=0-8018-7971-X&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></li>
</ul>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;">
<dt style="font-weight: bold; margin-bottom: 0.1em;">Introductory textbooks</dt>
</dl>
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">I. Aitchison, A. Hey (2003). <i>Gauge Theories in Particle Physics: A Practical Introduction.</i>. <a href="http://en.wikipedia.org/wiki/Institute_of_Physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Institute of Physics">Institute of Physics</a>. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-585-44550-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-585-44550-2">978-0-585-44550-2</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gauge+Theories+in+Particle+Physics%3A+A+Practical+Introduction.&rft.aulast=I.+Aitchison%2C+A.+Hey&rft.au=I.+Aitchison%2C+A.+Hey&rft.date=2003&rft.pub=%5B%5BInstitute+of+Physics%5D%5D&rft.isbn=978-0-585-44550-2&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">W. Greiner, B. Müller (2000). <i>Gauge Theory of Weak Interactions</i>. <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Springer_(publisher)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Springer (publisher)">Springer</a>. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/3-540-67672-4" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/3-540-67672-4">3-540-67672-4</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gauge+Theory+of+Weak+Interactions&rft.aulast=W.+Greiner%2C+B.+M%C3%BCller&rft.au=W.+Greiner%2C+B.+M%C3%BCller&rft.date=2000&rft.pub=%5B%5BSpringer+%28publisher%29%7CSpringer%5D%5D&rft.isbn=3-540-67672-4&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">G.D. Coughlan, J.E. Dodd, B.M. Gripaios (2006). <i>The Ideas of Particle Physics: An Introduction for Scientists</i>. <a href="http://en.wikipedia.org/wiki/Cambridge_University_Press" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cambridge University Press">Cambridge University Press</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Ideas+of++Particle+Physics%3A+An+Introduction+for+Scientists&rft.aulast=G.D.+Coughlan%2C+J.E.+Dodd%2C+B.M.+Gripaios&rft.au=G.D.+Coughlan%2C+J.E.+Dodd%2C+B.M.+Gripaios&rft.date=2006&rft.pub=%5B%5BCambridge+University+Press%5D%5D&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">D.J. Griffiths (1987). <i>Introduction to Elementary Particles</i>. <a href="http://en.wikipedia.org/wiki/John_Wiley_%26_Sons" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-471-60386-4" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-471-60386-4">0-471-60386-4</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Elementary+Particles&rft.aulast=D.J.+Griffiths&rft.au=D.J.+Griffiths&rft.date=1987&rft.pub=%5B%5BJohn+Wiley+%26+Sons%5D%5D&rft.isbn=0-471-60386-4&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">G.L. Kane (1987). <i>Modern Elementary Particle Physics</i>. <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Perseus_Books" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Perseus Books">Perseus Books</a>. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-201-11749-5" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-201-11749-5">0-201-11749-5</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Elementary+Particle+Physics&rft.aulast=G.L.+Kane&rft.au=G.L.+Kane&rft.date=1987&rft.pub=%5B%5BPerseus+Books%5D%5D&rft.isbn=0-201-11749-5&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></li>
</ul>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;">
<dt style="font-weight: bold; margin-bottom: 0.1em;">Advanced textbooks</dt>
</dl>
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">T.P. Cheng, L.F. Li (2006). <i>Gauge theory of elementary particle physics</i>. <a href="http://en.wikipedia.org/wiki/Oxford_University_Press" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Oxford University Press">Oxford University Press</a>. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-19-851961-3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-19-851961-3">0-19-851961-3</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gauge+theory+of+elementary+particle+physics&rft.aulast=T.P.+Cheng%2C+L.F.+Li&rft.au=T.P.+Cheng%2C+L.F.+Li&rft.date=2006&rft.pub=%5B%5BOxford+University+Press%5D%5D&rft.isbn=0-19-851961-3&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span> Highlights the <a href="http://en.wikipedia.org/wiki/Gauge_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge theory">gauge theory</a> aspects of the Standard Model.</li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">J.F. Donoghue, E. Golowich, B.R. Holstein (1994). <i>Dynamics of the Standard Model</i>. <a href="http://en.wikipedia.org/wiki/Cambridge_University_Press" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cambridge University Press">Cambridge University Press</a>. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-521-47652-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-521-47652-2">978-0-521-47652-2</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamics+of+the+Standard+Model&rft.aulast=J.F.+Donoghue%2C+E.+Golowich%2C+B.R.+Holstein&rft.au=J.F.+Donoghue%2C+E.+Golowich%2C+B.R.+Holstein&rft.date=1994&rft.pub=%5B%5BCambridge+University+Press%5D%5D&rft.isbn=978-0-521-47652-2&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span> Highlights dynamical and<a href="http://en.wikipedia.org/wiki/Phenomenology_(particle_physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Phenomenology (particle physics)">phenomenological</a> aspects of the Standard Model.</li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">L. O'Raifeartaigh (1988). <i>Group structure of gauge theories</i>. <a href="http://en.wikipedia.org/wiki/Cambridge_University_Press" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cambridge University Press">Cambridge University Press</a>. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-521-34785-8" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-521-34785-8">0-521-34785-8</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Group+structure+of+gauge+theories&rft.aulast=L.+O%27Raifeartaigh&rft.au=L.+O%27Raifeartaigh&rft.date=1988&rft.pub=%5B%5BCambridge+University+Press%5D%5D&rft.isbn=0-521-34785-8&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span> Highlights <a href="http://en.wikipedia.org/wiki/Finite_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Finite group">group-theoretical</a> aspects of the Standard Model.</li>
</ul>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;">
<dt style="font-weight: bold; margin-bottom: 0.1em;">Journal articles</dt>
</dl>
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;"><span class="citation Journal" style="word-wrap: break-word;">E.S. Abers, B.W. Lee (1973). "Gauge theories". <i><a href="http://en.wikipedia.org/wiki/Physics_Reports" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physics Reports">Physics Reports</a></i> <b>9</b>: 1–141. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1973PhR.....9....1A" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1973PhR.....9....1A</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1016%2F0370-1573%2873%2990027-6" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1016/0370-1573(73)90027-6</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Gauge+theories&rft.jtitle=%5B%5BPhysics+Reports%5D%5D&rft.aulast=E.S.+Abers%2C+B.W.+Lee&rft.au=E.S.+Abers%2C+B.W.+Lee&rft.date=1973&rft.volume=9&rft.pages=1%E2%80%93141&rft_id=info:bibcode/1973PhR.....9....1A&rft_id=info:doi/10.1016%2F0370-1573%2873%2990027-6&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation Journal" style="word-wrap: break-word;">Y. Hayato <i>et al.</i> (1999). "Search for Proton Decay through <i>p</i> → <i>νK</i><sup style="line-height: 1em;">+</sup> in a Large Water Cherenkov Detector". <i><a href="http://en.wikipedia.org/wiki/Physical_Review_Letters" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review Letters">Physical Review Letters</a></i> <b>83</b> (8): 1529. <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/hep-ex/9904020" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">hep-ex/9904020</a>.<a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1999PhRvL..83.1529H" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1999PhRvL..83.1529H</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRevLett.83.1529" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRevLett.83.1529</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Search+for+Proton+Decay+through+%27%27p%27%27+%E2%86%92+%27%27%CE%BDK%27%27%3Csup%3E%2B%3C%2Fsup%3E+in+a+Large+Water+Cherenkov+Detector&rft.jtitle=%5B%5BPhysical+Review+Letters%5D%5D&rft.aulast=Y.+Hayato+%27%27et+al.%27%27&rft.au=Y.+Hayato+%27%27et+al.%27%27&rft.date=1999&rft.volume=83&rft.issue=8&rft.pages=1529&rft_id=info:arxiv/hep-ex%2F9904020&rft_id=info:bibcode/1999PhRvL..83.1529H&rft_id=info:doi/10.1103%2FPhysRevLett.83.1529&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation Journal" style="word-wrap: break-word;">S.F. Novaes (2000). "Standard Model: An Introduction". <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/hep-ph/0001283" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">hep-ph/0001283</a> [<a class="external text" href="http://arxiv.org/archive/hep-ph" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">hep-ph</a>].</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Standard+Model%3A+An+Introduction&rft.jtitle=%27%27%5B%5BarXiv%5D%5D%3A%5Bhttp%3A%2F%2Farxiv.org%2Fabs%2Fhep-ph%2F0001283+hep-ph%2F0001283%5D%26nbsp%3B%5B%5Bhttp%3A%2F%2Farxiv.org%2Farchive%2Fhep-ph+hep-ph%5D%5D%27%27&rft.aulast=S.F.+Novaes&rft.au=S.F.+Novaes&rft.date=2000&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation Journal" style="word-wrap: break-word;">D.P. Roy (1999). "Basic Constituents of Matter and their Interactions — A Progress Report.". <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/hep-ph/9912523" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">hep-ph/9912523</a> [<a class="external text" href="http://arxiv.org/archive/hep-ph" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">hep-ph</a>].</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Basic+Constituents+of+Matter+and+their+Interactions+%E2%80%94+A+Progress+Report.&rft.jtitle=%27%27%5B%5BarXiv%5D%5D%3A%5Bhttp%3A%2F%2Farxiv.org%2Fabs%2Fhep-ph%2F9912523+hep-ph%2F9912523%5D%26nbsp%3B%5B%5Bhttp%3A%2F%2Farxiv.org%2Farchive%2Fhep-ph+hep-ph%5D%5D%27%27&rft.aulast=D.P.+Roy&rft.au=D.P.+Roy&rft.date=1999&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation Journal" style="word-wrap: break-word;">F. Wilczek (2004). "The Universe Is A Strange Place". <i>Nuclear Physics B - Proceedings Supplements</i> <b>134</b>: 3. <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/astro-ph/0401347" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">astro-ph/0401347</a>. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/2004NuPhS.134....3W" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">2004NuPhS.134....3W</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1016%2Fj.nuclphysbps.2004.08.001" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1016/j.nuclphysbps.2004.08.001</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Universe+Is+A+Strange+Place&rft.jtitle=Nuclear+Physics+B+-+Proceedings+Supplements&rft.aulast=F.+Wilczek&rft.au=F.+Wilczek&rft.date=2004&rft.volume=134&rft.pages=3&rft_id=info:arxiv/astro-ph%2F0401347&rft_id=info:bibcode/2004NuPhS.134....3W&rft_id=info:doi/10.1016%2Fj.nuclphysbps.2004.08.001&rfr_id=info:sid/en.wikipedia.org:Standard_Model"></span></li>
</ul>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Standard_Model&action=edit&section=28" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: External links">edit</a>]</span><span class="mw-headline" id="External_links">External links</span></h2>
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;">"<a class="external text" href="http://www.newscientist.com/article/dn21279-lhc-sees-hint-of-lightweight-higgs-boson.html" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">LHC sees hint of lightweight Higgs boson</a>" "<a href="http://en.wikipedia.org/wiki/New_Scientist" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="New Scientist">New Scientist</a>".</li>
<li style="margin-bottom: 0.1em;">"<a class="external text" href="http://www.newscientist.com/news/news.jsp?id=ns9999404" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Standard Model may be found incomplete,</a>" <i><a href="http://en.wikipedia.org/wiki/New_Scientist" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="New Scientist">New Scientist</a></i>.</li>
<li style="margin-bottom: 0.1em;">"<a class="external text" href="http://www-cdf.fnal.gov/top_status/top.html" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Observation of the Top Quark</a>" at <a href="http://en.wikipedia.org/wiki/Fermilab" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermilab">Fermilab</a>.</li>
<li style="margin-bottom: 0.1em;">"<a class="external text" href="http://cosmicvariance.com/2006/11/23/thanksgiving" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">The Standard Model Lagrangian.</a>" After electroweak <a href="http://en.wikipedia.org/wiki/Symmetry_breaking" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Symmetry breaking">symmetry breaking</a>, with no explicit <a href="http://en.wikipedia.org/wiki/Higgs_boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Higgs boson">Higgs boson</a>.</li>
<li style="margin-bottom: 0.1em;">"<a class="external text" href="http://nuclear.ucdavis.edu/~tgutierr/files/stmL1.html" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Standard Model Lagrangian</a>" with explicit Higgs terms. PDF, PostScript, and LaTeX versions.</li>
<li style="margin-bottom: 0.1em;">"<a class="external text" href="http://particleadventure.org/" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">The particle adventure.</a>" Web tutorial.</li>
<li style="margin-bottom: 0.1em;">Nobes, Matthew (2002) "Introduction to the Standard Model of Particle Physics" on <a href="http://en.wikipedia.org/wiki/Kuro5hin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Kuro5hin">Kuro5hin</a>: <a class="external text" href="http://www.kuro5hin.org/story/2002/5/1/3712/31700" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Part 1,</a> <a class="external text" href="http://www.kuro5hin.org/story/2002/5/14/19363/8142" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Part 2,</a> <a class="external text" href="http://www.kuro5hin.org/story/2002/7/15/173318/784" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Part 3a,</a> <a class="external text" href="http://www.kuro5hin.org/story/2002/8/21/195035/576" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Part 3b.</a></li>
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</div>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com0tag:blogger.com,1999:blog-5303246073824127471.post-22566210993106492212012-06-22T19:38:00.004-04:002012-06-22T19:39:59.176-04:00Quantum Chromodynamics (QED)<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyPHEQO6il5Q7qLdIE3M_QnK4kyjZHGFtEZa4S5wkflcSipLcdJ2O6WDBZ_xPtqI9vM-O_R5Ih6RzzhwxQdW55gqoGVreuXE7OGHa9yQWftnGwcONfi20_2ua_xAW086KWRJgoiUQzvg/s1600/wilczek-300.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="282" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyPHEQO6il5Q7qLdIE3M_QnK4kyjZHGFtEZa4S5wkflcSipLcdJ2O6WDBZ_xPtqI9vM-O_R5Ih6RzzhwxQdW55gqoGVreuXE7OGHa9yQWftnGwcONfi20_2ua_xAW086KWRJgoiUQzvg/s400/wilczek-300.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Frank Wilczek, and yes the Universe is a Strange Place</td></tr>
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<span style="font-family: sans-serif; font-size: 13px; line-height: 20px;">Once QED is in play and Quantum Field Theory continues to develop until three men further Physics considerably by discovering how the strong force interacts with quarks thus forming protons and neutrons and 100's of other particles. They were Frank Wilczek and his mentor David Gross at Princeton Physics, and Hugh David Politzer (half Slovak like me!) at Harvard Physics, with a big assist by Politzer's mentor Sidney Coleman. From Wikipedia:<br /><br />"The discovery of </span><a href="http://en.wikipedia.org/wiki/Asymptotic_freedom" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 20px; text-decoration: none;" title="Asymptotic freedom">asymptotic freedom</a><span style="font-family: sans-serif; font-size: 13px; line-height: 20px;"> in the strong interactions by </span><a href="http://en.wikipedia.org/wiki/David_Gross" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 20px; text-decoration: none;" title="David Gross">David Gross</a><span style="font-family: sans-serif; font-size: 13px; line-height: 20px;">, </span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/David_Politzer" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 20px; text-decoration: none;" title="David Politzer">David Politzer</a><span style="font-family: sans-serif; font-size: 13px; line-height: 20px;"> and </span><a href="http://en.wikipedia.org/wiki/Frank_Wilczek" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 20px; text-decoration: none;" title="Frank Wilczek">Frank Wilczek</a><span style="font-family: sans-serif; font-size: 13px; line-height: 20px;"> allowed physicists to make precise predictions of the results of many high energy experiments using the quantum field theory technique of </span><a href="http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 20px; text-decoration: none;" title="Perturbation theory (quantum mechanics)">perturbation theory</a><span style="font-family: sans-serif; font-size: 13px; line-height: 20px;">. Evidence of gluons was discovered in </span><a href="http://en.wikipedia.org/wiki/Three_jet_event" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 20px; text-decoration: none;" title="Three jet event">three jet events</a><span style="font-family: sans-serif; font-size: 13px; line-height: 20px;"> at </span><a href="http://en.wikipedia.org/wiki/PETRA" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 20px; text-decoration: none;" title="PETRA">PETRA</a><span style="font-family: sans-serif; font-size: 13px; line-height: 20px;"> in 1979. These experiments became more and more precise, culminating in the verification of </span><a href="http://en.wikipedia.org/wiki/Perturbative_QCD" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 20px; text-decoration: none;" title="Perturbative QCD">perturbative QCD</a><span style="font-family: sans-serif; font-size: 13px; line-height: 20px;"> at the level of a few percent at the </span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/LEP" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 20px; text-decoration: none;" title="LEP">LEP</a><span style="font-family: sans-serif; font-size: 13px; line-height: 20px;"> in </span><a href="http://en.wikipedia.org/wiki/CERN" style="background-image: none; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 20px; text-decoration: none;" title="CERN">CERN</a><span style="font-family: sans-serif; font-size: 13px; line-height: 20px;">."<br /><br />And the stage is now set for that greatest of Human achievements to date: The Standard Model of Particle Physics, which Wilzcek likes to call: The Core.<br /><br /><br /><br />From Wikipedia:</span><br />
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In <a href="http://en.wikipedia.org/wiki/Theoretical_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theoretical physics">theoretical physics</a>, <b>quantum chromodynamics</b> (<b>QCD</b>) is a theory of the <a href="http://en.wikipedia.org/wiki/Strong_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Strong interaction">strong interaction</a> (<a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge">color</a> force), a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Fundamental_force" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fundamental force">fundamental force</a>describing the interactions between <a href="http://en.wikipedia.org/wiki/Quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark">quarks</a> and <a href="http://en.wikipedia.org/wiki/Gluon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gluon">gluons</a> which make up <a href="http://en.wikipedia.org/wiki/Hadron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hadron">hadrons</a> (such as the <a href="http://en.wikipedia.org/wiki/Proton" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Proton">proton</a>, <a href="http://en.wikipedia.org/wiki/Neutron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Neutron">neutron</a> or <a href="http://en.wikipedia.org/wiki/Pion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pion">pion</a>). It is the study of the <a href="http://en.wikipedia.org/wiki/Special_unitary_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special unitary group"><b>SU</b>(3)</a> <a href="http://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Yang–Mills theory">Yang–Mills theory</a> of color-charged <a href="http://en.wikipedia.org/wiki/Fermion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermion">fermions</a> (the quarks). QCD is a <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">quantum field theory</a> of a special kind called a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Non-abelian_gauge_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Non-abelian gauge theory">non-abelian gauge theory</a>, consisting of a 'color field' mediated by a set of exchange particles (the gluons). The theory is an important part of the <a href="http://en.wikipedia.org/wiki/Standard_Model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Standard Model">Standard Model</a> of <a href="http://en.wikipedia.org/wiki/Particle_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Particle physics">particle physics</a>. A huge body of <a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Experimental_tests" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum chromodynamics">experimental evidence for QCD</a> has been gathered over the years.</div>
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QCD enjoys two peculiar properties:</div>
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<li style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Color_confinement" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color confinement">Confinement</a></b>, which means that the force between quarks does not diminish as they are separated. Because of this, it would take an infinite amount of energy to separate two quarks; they are forever bound into hadrons such as the <a href="http://en.wikipedia.org/wiki/Proton" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Proton">proton</a> and the<a href="http://en.wikipedia.org/wiki/Neutron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Neutron">neutron</a>. Although analytically unproven, confinement is widely believed to be true because it explains the consistent failure of<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Free_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Free quark">free quark</a> searches, and it is easy to demonstrate in <a href="http://en.wikipedia.org/wiki/Lattice_QCD" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lattice QCD">lattice QCD</a>.</li>
<li style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Asymptotic_freedom" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Asymptotic freedom">Asymptotic freedom</a></b>, which means that in very high-energy reactions, quarks and gluons interact very weakly. This prediction of QCD was first discovered in the early 1970s by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/David_Politzer" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="David Politzer">David Politzer</a> and by <a href="http://en.wikipedia.org/wiki/Frank_Wilczek" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Frank Wilczek">Frank Wilczek</a> and <a href="http://en.wikipedia.org/wiki/David_Gross" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="David Gross">David Gross</a>. For this work they were awarded the 2004 <a href="http://en.wikipedia.org/wiki/Nobel_Prize_in_Physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nobel Prize in Physics">Nobel Prize in Physics</a>.</li>
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There is no known phase-transition line separating these two properties; confinement is dominant in low-energy scales but, as energy increases, asymptotic freedom becomes dominant.</div>
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Contents</h2>
<span class="toctoggle" style="-webkit-user-select: none;"> [<a class="internal" href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#" id="togglelink" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">hide</a>] </span></div>
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<li class="toclevel-1 tocsection-1" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Terminology" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1</span> <span class="toctext">Terminology</span></a></li>
<li class="toclevel-1 tocsection-2" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#History" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2</span> <span class="toctext">History</span></a></li>
<li class="toclevel-1 tocsection-3" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3</span> <span class="toctext">Theory</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin: 0px 0px 0px 2em; padding: 0px;">
<li class="toclevel-2 tocsection-4" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Some_definitions" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.1</span> <span class="toctext">Some definitions</span></a></li>
<li class="toclevel-2 tocsection-5" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Additional_remarks:_duality" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.2</span> <span class="toctext">Additional remarks: duality</span></a></li>
<li class="toclevel-2 tocsection-6" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Symmetry_groups" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.3</span> <span class="toctext">Symmetry groups</span></a></li>
<li class="toclevel-2 tocsection-7" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Lagrangian" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.4</span> <span class="toctext">Lagrangian</span></a></li>
<li class="toclevel-2 tocsection-8" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Fields" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.5</span> <span class="toctext">Fields</span></a></li>
<li class="toclevel-2 tocsection-9" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Dynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.6</span> <span class="toctext">Dynamics</span></a></li>
<li class="toclevel-2 tocsection-10" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Area_law_and_confinement" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.7</span> <span class="toctext">Area law and confinement</span></a></li>
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<li class="toclevel-1 tocsection-11" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Methods" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4</span> <span class="toctext">Methods</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin: 0px 0px 0px 2em; padding: 0px;">
<li class="toclevel-2 tocsection-12" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Perturbative_QCD" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.1</span> <span class="toctext">Perturbative QCD</span></a></li>
<li class="toclevel-2 tocsection-13" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Lattice_QCD" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.2</span> <span class="toctext">Lattice QCD</span></a></li>
<li class="toclevel-2 tocsection-14" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#1.2FN_expansion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.3</span> <span class="toctext">1/N expansion</span></a></li>
<li class="toclevel-2 tocsection-15" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Effective_theories" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.4</span> <span class="toctext">Effective theories</span></a></li>
<li class="toclevel-2 tocsection-16" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#QCD_sum_rules" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.5</span> <span class="toctext">QCD sum rules</span></a></li>
<li class="toclevel-2 tocsection-17" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Nambu-Jona-Lasinio_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.6</span> <span class="toctext">Nambu-Jona-Lasinio model</span></a></li>
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</li>
<li class="toclevel-1 tocsection-18" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Experimental_tests" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5</span> <span class="toctext">Experimental tests</span></a></li>
<li class="toclevel-1 tocsection-19" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Cross-relations_to_solid_state_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6</span> <span class="toctext">Cross-relations to solid state physics</span></a></li>
<li class="toclevel-1 tocsection-20" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#See_also" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">7</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-21" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#References" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">8</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-22" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#Further_reading" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">9</span> <span class="toctext">Further reading</span></a></li>
<li class="toclevel-1 tocsection-23" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#External_links" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">10</span> <span class="toctext">External links</span></a></li>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Terminology">edit</a>]</span><span class="mw-headline" id="Terminology">Terminology</span></h2>
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The word <i>quark</i> was coined by American physicist <a href="http://en.wikipedia.org/wiki/Murray_Gell-Mann" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Murray Gell-Mann">Murray Gell-Mann</a> (b. 1929) in its present sense. It originally comes from the phrase "Three quarks for Muster Mark" in<i><a href="http://en.wikipedia.org/wiki/Finnegans_Wake" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Finnegans Wake">Finnegans Wake</a></i> by <a href="http://en.wikipedia.org/wiki/James_Joyce" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="James Joyce">James Joyce</a>. On June 27, 1978, Gell-Mann wrote a private letter to the editor of the Oxford English Dictionary, in which he related that he had been influenced by Joyce's words: "The allusion to three quarks seemed perfect." (Originally, only three quarks had been discovered.) Gell-Mann, however, wanted to pronounce the word with (ô) not (ä), as Joyce seemed to indicate by rhyming words in the vicinity such as <i>Mark</i>. Gell-Mann got around that "by supposing that one ingredient of the line 'Three quarks for Muster Mark' was a cry of 'Three quarts for Mister . . . ' heard in H.C. Earwicker's pub," a plausible suggestion given the complex punning in Joyce's novel.<sup class="reference" id="cite_ref-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[1]</a></sup></div>
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The three kinds of <a href="http://en.wikipedia.org/wiki/Charge_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Charge (physics)">charge</a> in QCD (as opposed to one in <a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum electrodynamics">quantum electrodynamics</a> or QED) are usually referred to as "<a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge">color charge</a>" by loose analogy to the three kinds of<a href="http://en.wikipedia.org/wiki/Color" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color">color</a> (red, green and blue) <a href="http://en.wikipedia.org/wiki/Color_vision" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color vision">perceived by humans</a>. Other than this nomenclature, the quantum parameter "color" is completely unrelated to the everyday, familiar phenomenon of color.</div>
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Since the theory of electric charge is dubbed "<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Electrodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electrodynamics">electrodynamics</a>", the Greek word "chroma" Χρώμα (meaning color) is applied to the theory of color charge, "chromodynamics".</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: History">edit</a>]</span><span class="mw-headline" id="History">History</span></h2>
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With the invention of <a href="http://en.wikipedia.org/wiki/Bubble_chamber" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bubble chamber">bubble chambers</a> and <a href="http://en.wikipedia.org/wiki/Spark_chamber" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spark chamber">spark chambers</a> in the 1950s, experimental <a href="http://en.wikipedia.org/wiki/Particle_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Particle physics">particle physics</a> discovered a large and ever-growing number of particles called<a href="http://en.wikipedia.org/wiki/Hadron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hadron">hadrons</a>. It seemed that such a large number of particles could not all be <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Fundamental_particles" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fundamental particles">fundamental</a>. First, the particles were classified by <a href="http://en.wikipedia.org/wiki/Charge_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Charge (physics)">charge</a> and <a href="http://en.wikipedia.org/wiki/Isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Isospin">isospin</a> by <a href="http://en.wikipedia.org/wiki/Eugene_Wigner" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eugene Wigner">Eugene Wigner</a> and <a href="http://en.wikipedia.org/wiki/Werner_Heisenberg" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Werner Heisenberg">Werner Heisenberg</a>; then, in 1953, according to <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Strangeness_(particle_physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Strangeness (particle physics)">strangeness</a> by <a href="http://en.wikipedia.org/wiki/Murray_Gell-Mann" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Murray Gell-Mann">Murray Gell-Mann</a> and <a href="http://en.wikipedia.org/wiki/Kazuhiko_Nishijima" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Kazuhiko Nishijima">Kazuhiko Nishijima</a>. To gain greater insight, the hadrons were sorted into groups having similar properties and masses using the <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Eightfold_way_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eightfold way (physics)">eightfold way</a></i>, invented in 1961 by Gell-Mann and <a href="http://en.wikipedia.org/wiki/Yuval_Ne%27eman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Yuval Ne'eman">Yuval Ne'eman</a>. Gell-Mann and <a href="http://en.wikipedia.org/wiki/George_Zweig" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="George Zweig">George Zweig</a>, correcting an earlier approach of <a href="http://en.wikipedia.org/wiki/Shoichi_Sakata" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Shoichi Sakata">Shoichi Sakata</a>, went on to propose in 1963 that the structure of the groups could be explained by the existence of three <a href="http://en.wikipedia.org/wiki/Flavour_(particle_physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Flavour (particle physics)">flavors</a> of smaller particles inside the hadrons: the <a href="http://en.wikipedia.org/wiki/Quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark">quarks</a>.</div>
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Perhaps the first remark that quarks should possess an additional quantum number was made<sup class="reference" id="cite_ref-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup> as a short footnote in the preprint of <a class="new" href="http://en.wikipedia.org/w/index.php?title=Boris_Struminsky&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Boris Struminsky (page does not exist)">Boris Struminsky</a><sup class="reference" id="cite_ref-struminsky_2-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-struminsky-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup> in connection with Ω<sup style="line-height: 1em;">-</sup><a href="http://en.wikipedia.org/wiki/Omega_baryon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Omega baryon">hyperon</a> composed of three <a href="http://en.wikipedia.org/wiki/Strange_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Strange quark">strange quarks</a> with parallel spins (this situation was peculiar, because since quarks are <a href="http://en.wikipedia.org/wiki/Fermion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermion">fermions</a>, such combination is forbidden by the <a href="http://en.wikipedia.org/wiki/Pauli_exclusion_principle" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pauli exclusion principle">Pauli exclusion principle</a>):</div>
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Three identical quarks cannot form an antisymmetric S-state. In order to realize an antisymmetric orbital S-state, it is necessary for the quark to have an additional quantum number.</div>
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— <cite style="font-style: inherit;">B. V. Struminsky, <i>Magnetic moments of barions in the quark model</i>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/JINR" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="JINR">JINR</a>-Preprint P-1939, Dubna, Submitted on January 7, 1965</cite></div>
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Boris Struminsky was a PhD student of <a href="http://en.wikipedia.org/wiki/Nikolay_Bogolyubov" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nikolay Bogolyubov">Nikolay Bogolyubov</a>. The problem considered in this preprint was suggested by Nikolay Bogolyubov, who advised Boris Struminsky in this research.<sup class="reference" id="cite_ref-struminsky_2-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-struminsky-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup> In the beginning of 1965, <a href="http://en.wikipedia.org/wiki/Nikolay_Bogolyubov" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nikolay Bogolyubov">Nikolay Bogolyubov</a>, <a class="new" href="http://en.wikipedia.org/w/index.php?title=Boris_Struminsky&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Boris Struminsky (page does not exist)">Boris Struminsky</a> and <a class="new" href="http://en.wikipedia.org/w/index.php?title=Albert_Tavchelidze&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Albert Tavchelidze (page does not exist)">Albert Tavchelidze</a> wrote a preprint with a more detailed discussion of the additional quark quantum degree of freedom.<sup class="reference" id="cite_ref-3" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[4]</a></sup> This work was also presented by Albert Tavchelidze without obtaining consent of his collaborators for doing so at an international conference in<a href="http://en.wikipedia.org/wiki/Trieste" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Trieste">Trieste</a> (<a href="http://en.wikipedia.org/wiki/Italy" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Italy">Italy</a>), in May 1965.<sup class="reference" id="cite_ref-4" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-4" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[5]</a></sup><sup class="reference" id="cite_ref-5" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-5" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[6]</a></sup></div>
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A similar mysterious situation was with the <a href="http://en.wikipedia.org/wiki/Delta_baryon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Delta baryon">Δ<sup style="line-height: 1em;">++</sup> baryon</a>; in the quark model, it is composed of three <a href="http://en.wikipedia.org/wiki/Up_quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Up quark">up quarks</a> with parallel spins. In 1965, <a href="http://en.wikipedia.org/wiki/Moo-Young_Han" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Moo-Young Han">Moo-Young Han</a> with <a href="http://en.wikipedia.org/wiki/Yoichiro_Nambu" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Yoichiro Nambu">Yoichiro Nambu</a>and <a href="http://en.wikipedia.org/wiki/Oscar_W._Greenberg" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Oscar W. Greenberg">Oscar W. Greenberg</a> independently resolved the problem by proposing that quarks possess an additional <a href="http://en.wikipedia.org/wiki/Special_unitary_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special unitary group"><b>SU</b>(3)</a> <a href="http://en.wikipedia.org/wiki/Gauge_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge theory">gauge</a> <a href="http://en.wikipedia.org/wiki/Degrees_of_freedom_(physics_and_chemistry)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Degrees of freedom (physics and chemistry)">degree of freedom</a>, later called color charge. Han and Nambu noted that quarks might interact via an octet of vector <a href="http://en.wikipedia.org/wiki/Gauge_boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge boson">gauge bosons</a>: the <a href="http://en.wikipedia.org/wiki/Gluon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gluon">gluons</a>.</div>
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Since free quark searches consistently failed to turn up any evidence for the new particles, and because an elementary particle back then was <i>defined</i> as a particle which could be separated and isolated, Gell-Mann often said that quarks were merely convenient mathematical constructs, not real particles. The meaning of this statement was usually clear in context: He meant quarks are confined, but he also was implying that the strong interactions could probably not be fully described by quantum field theory.</div>
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<a href="http://en.wikipedia.org/wiki/Richard_Feynman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard Feynman">Richard Feynman</a> argued that high energy experiments showed quarks are real particles: he called them <i>partons</i> (since they were parts of hadrons). By particles, Feynman meant objects which travel along paths, elementary particles in a field theory.</div>
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The difference between Feynman's and Gell-Mann's approaches reflected a deep split in the theoretical physics community. Feynman thought the quarks have a distribution of position or momentum, like any other particle, and he (correctly) believed that the diffusion of parton momentum explained <a href="http://en.wikipedia.org/wiki/Pomeron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pomeron">diffractive scattering</a>. Although Gell-Mann believed that certain quark charges could be localized, he was open to the possibility that the quarks themselves could not be localized because space and time break down. This was the more radical approach of <a href="http://en.wikipedia.org/wiki/S-matrix_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="S-matrix theory">S-matrix theory</a>.</div>
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<a class="mw-redirect" href="http://en.wikipedia.org/wiki/James_Daniel_Bjorken" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="James Daniel Bjorken">James Bjorken</a> proposed that pointlike partons would imply certain relations should hold in <a href="http://en.wikipedia.org/wiki/Deep_inelastic_scattering" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Deep inelastic scattering">deep inelastic scattering</a> of <a href="http://en.wikipedia.org/wiki/Electron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron">electrons</a> and protons, which were spectacularly verified in experiments at <a class="mw-redirect" href="http://en.wikipedia.org/wiki/SLAC" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="SLAC">SLAC</a> in 1969. This led physicists to abandon the S-matrix approach for the strong interactions.</div>
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The discovery of <a href="http://en.wikipedia.org/wiki/Asymptotic_freedom" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Asymptotic freedom">asymptotic freedom</a> in the strong interactions by <a href="http://en.wikipedia.org/wiki/David_Gross" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="David Gross">David Gross</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/David_Politzer" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="David Politzer">David Politzer</a> and <a href="http://en.wikipedia.org/wiki/Frank_Wilczek" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Frank Wilczek">Frank Wilczek</a> allowed physicists to make precise predictions of the results of many high energy experiments using the quantum field theory technique of <a href="http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Perturbation theory (quantum mechanics)">perturbation theory</a>. Evidence of gluons was discovered in <a href="http://en.wikipedia.org/wiki/Three_jet_event" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Three jet event">three jet events</a> at <a href="http://en.wikipedia.org/wiki/PETRA" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="PETRA">PETRA</a> in 1979. These experiments became more and more precise, culminating in the verification of <a href="http://en.wikipedia.org/wiki/Perturbative_QCD" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Perturbative QCD">perturbative QCD</a> at the level of a few percent at the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/LEP" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="LEP">LEP</a> in <a href="http://en.wikipedia.org/wiki/CERN" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="CERN">CERN</a>.</div>
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The other side of asymptotic freedom is <a href="http://en.wikipedia.org/wiki/Color_confinement" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color confinement">confinement</a>. Since the force between color charges does not decrease with distance, it is believed that quarks and gluons can never be liberated from hadrons. This aspect of the theory is verified within <a href="http://en.wikipedia.org/wiki/Lattice_QCD" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lattice QCD">lattice QCD</a> computations, but is not mathematically proven. One of the <a href="http://en.wikipedia.org/wiki/Millennium_Prize_Problems" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Millennium Prize Problems">Millennium Prize Problems</a>announced by the <a href="http://en.wikipedia.org/wiki/Clay_Mathematics_Institute" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Clay Mathematics Institute">Clay Mathematics Institute</a> requires a claimant to produce such a proof. Other aspects of <a href="http://en.wikipedia.org/wiki/Non-perturbative" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Non-perturbative">non-perturbative</a> QCD are the exploration of phases of <a href="http://en.wikipedia.org/wiki/QCD_matter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="QCD matter">quark matter</a>, including the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Quark-gluon_plasma" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark-gluon plasma">quark-gluon plasma</a>.</div>
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The relation between the short-distance particle limit and the confining long-distance limit is one of the topics recently explored using <a href="http://en.wikipedia.org/wiki/String_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="String theory">string theory</a>, the modern form of S-matrix theory.<sup class="reference" id="cite_ref-6" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-6" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[7]</a></sup><sup class="reference" id="cite_ref-7" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-7" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[8]</a></sup></div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Theory">edit</a>]</span><span class="mw-headline" id="Theory">Theory</span></h2>
<table class="wikitable" style="background-color: #f9f9f9; border-collapse: collapse; border: 1px solid rgb(170, 170, 170); color: black; float: right; font-size: 13px; margin: 0em 1em 1em; width: 300px;"><caption style="font-weight: bold;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Unsolved_problems_in_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Unsolved problems in physics">Unsolved problems in physics</a></caption><tbody>
<tr><td style="border: 1px solid rgb(170, 170, 170); padding: 0.2em;"><i>QCD in the non-<a href="http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Perturbation theory (quantum mechanics)">perturbative</a> regime:</i><br />
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px; text-align: left;">
<li style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Color_confinement" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color confinement">Confinement</a></b>: the equations of QCD remain unsolved at <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Energy_scale" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Energy scale">energy scales</a> relevant for describing <a href="http://en.wikipedia.org/wiki/Atomic_nucleus" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Atomic nucleus">atomic nuclei</a>. How does QCD give rise to the physics of nuclei and nuclear constituents?</li>
<li style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/QCD_matter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="QCD matter">Quark matter</a></b>: the equations of QCD predict that a <a href="http://en.wikipedia.org/wiki/Quark%E2%80%93gluon_plasma" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark–gluon plasma">sea of quarks and gluons</a> should be formed at high temperature and density. What are the properties of this <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Phase_of_matter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Phase of matter">phase of matter</a>?</li>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=4" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Some definitions">edit</a>]</span><span class="mw-headline" id="Some_definitions">Some definitions</span></h3>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Every field theory of <a href="http://en.wikipedia.org/wiki/Particle_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Particle physics">particle physics</a> is based on certain symmetries of nature whose existence is deduced from observations. These can be</div>
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Local_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Local symmetry">local symmetries</a>, that is the symmetry acts independently at each point in <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Space-time" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Space-time">space-time</a>. Each such symmetry is the basis of a <a href="http://en.wikipedia.org/wiki/Gauge_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge theory">gauge theory</a> and requires the introduction of its own <a href="http://en.wikipedia.org/wiki/Gauge_boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge boson">gauge bosons</a>.</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Global_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Global symmetry">global symmetries</a>, which are symmetries whose operations must be simultaneously applied to all points of space-time.</li>
</ul>
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QCD is a gauge theory of the <a href="http://en.wikipedia.org/wiki/Special_unitary_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special unitary group"><b>SU</b>(3)</a> gauge group obtained by taking the <a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge">color charge</a> to define a local symmetry.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Since the strong interaction does not discriminate between different flavors of quark, QCD has approximate <b>flavor symmetry</b>, which is broken by the differing masses of the quarks.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
There are additional global symmetries whose definitions require the notion of <a href="http://en.wikipedia.org/wiki/Chirality_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Chirality (physics)">chirality</a>, discrimination between left and right-handed. If the <a href="http://en.wikipedia.org/wiki/Spin_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin (physics)">spin</a> of a particle has a positive <a href="http://en.wikipedia.org/wiki/Projection_(linear_algebra)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Projection (linear algebra)">projection</a> on its direction of motion then it is called left-handed; otherwise, it is right-handed. Chirality and handedness are not the same, but become approximately equivalent at high energies.</div>
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<li style="margin-bottom: 0.1em;"><b>Chiral</b> symmetries involve independent transformations of these two types of particle.</li>
<li style="margin-bottom: 0.1em;"><b>Vector</b> symmetries (also called diagonal symmetries) mean the same transformation is applied on the two chiralities.</li>
<li style="margin-bottom: 0.1em;"><b>Axial</b> symmetries are those in which one transformation is applied on left-handed particles and the inverse on the right-handed particles.</li>
</ul>
<h3 style="background-image: none; border-bottom-style: none; font-size: 18px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=5" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Additional remarks: duality">edit</a>]</span><span class="mw-headline" id="Additional_remarks:_duality">Additional remarks: duality</span></h3>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
As mentioned, <i>asymptotic freedom</i> means that at large energy - this corresponds also to <i>short distances</i> - there is practically no interaction between the particles. This is in contrast - more precisely one would say <i><a href="http://en.wikipedia.org/wiki/Dualism" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dualism">dual</a></i> - to what one is used to, since usually one connects the absence of interactions with <i>large</i> distances. However, as already mentioned in the original paper of Franz Wegner,<sup class="reference" id="cite_ref-8" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-8" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[9]</a></sup> a solid state theorist who introduced 1971 simple gauge invariant lattice models, the high-temperature behaviour of the<i>original model</i>, e.g. the strong decay of correlations at large distances, corresponds to the low-temperature behaviour of the (usually ordered!) <i>dual model</i>, namely the asymptotic decay of non-trivial correlations, e.g. short-range deviations from almost perfect arrangements, for short distances. Here, in contrast to Wegner, we have only the dual model, which is that one described in this article.<sup class="reference" id="cite_ref-9" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-9" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[10]</a></sup></div>
<h3 style="background-image: none; border-bottom-style: none; font-size: 18px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=6" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Symmetry groups">edit</a>]</span><span class="mw-headline" id="Symmetry_groups">Symmetry groups</span></h3>
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The color group <b>SU</b>(3) corresponds to the local symmetry whose gauging gives rise to QCD. The electric charge labels a representation of the local symmetry group <b>U</b>(1) which is gauged to give <a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum electrodynamics">QED</a>: this is an <a href="http://en.wikipedia.org/wiki/Abelian_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Abelian group">abelian group</a>. If one considers a version of QCD with <i>N<sub style="line-height: 1em;">f</sub></i> flavors of massless quarks, then there is a global (<a href="http://en.wikipedia.org/wiki/Chirality_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Chirality (physics)">chiral</a>) flavor symmetry group <b>SU</b><sub style="line-height: 1em;">L</sub>(<i>N<sub style="line-height: 1em;">f</sub></i>) × <b>SU</b><sub style="line-height: 1em;">R</sub>(<i>N<sub style="line-height: 1em;">f</sub></i>) × <b>U</b><sub style="line-height: 1em;">B</sub>(1) × <b>U</b><sub style="line-height: 1em;">A</sub>(1). The chiral symmetry is <a href="http://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spontaneous symmetry breaking">spontaneously broken</a> by the <a href="http://en.wikipedia.org/wiki/QCD_vacuum" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="QCD vacuum">QCD vacuum</a> to the vector (L+R) <b>SU</b><sub style="line-height: 1em;">V</sub>(<i>N<sub style="line-height: 1em;">f</sub></i>) with the formation of a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Chiral_condensate" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Chiral condensate">chiral condensate</a>. The vector symmetry, <b>U</b><sub style="line-height: 1em;">B</sub>(1) corresponds to the baryon number of quarks and is an exact symmetry. The axial symmetry <b>U</b><sub style="line-height: 1em;">A</sub>(1) is exact in the classical theory, but broken in the quantum theory, an occurrence called an <a href="http://en.wikipedia.org/wiki/Anomaly_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Anomaly (physics)">anomaly</a>. Gluon field configurations called <a href="http://en.wikipedia.org/wiki/Instanton" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Instanton">instantons</a> are closely related to this anomaly.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
There are two different types of <b>SU</b>(3) symmetry: there is the symmetry that acts on the different colors of quarks, and this is an exact gauge symmetry mediated by the gluons, and there is also a flavor symmetry which rotates different flavors of quarks to each other, or <i>flavor <b>SU</b>(3)</i>. Flavor <b>SU</b>(3) is an approximate symmetry of the vacuum of QCD, and is not a fundamental symmetry at all. It is an accidental consequence of the small mass of the three lightest quarks.</div>
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In the <a href="http://en.wikipedia.org/wiki/QCD_vacuum" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="QCD vacuum">QCD vacuum</a> there are vacuum condensates of all the quarks whose mass is less than the QCD scale. This includes the up and down quarks, and to a lesser extent the strange quark, but not any of the others. The vacuum is symmetric under <b>SU</b>(2) <a href="http://en.wikipedia.org/wiki/Isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Isospin">isospin</a> rotations of up and down, and to a lesser extent under rotations of up, down and strange, or full flavor group <b>SU</b>(3), and the observed particles make isospin and <b>SU</b>(3) multiplets.</div>
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The approximate flavor symmetries do have associated gauge bosons, observed particles like the rho and the omega, but these particles are nothing like the gluons and they are not massless. They are emergent gauge bosons in an approximate <a href="http://en.wikipedia.org/wiki/AdS/QCD" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="AdS/QCD">string description of QCD</a>.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=7" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Lagrangian">edit</a>]</span><span class="mw-headline" id="Lagrangian">Lagrangian</span></h3>
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The dynamics of the quarks and gluons are controlled by the quantum chromodynamics Lagrangian. The <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gauge_invariant" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge invariant">gauge invariant</a> QCD <a href="http://en.wikipedia.org/wiki/Lagrangian" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lagrangian">Lagrangian</a> is</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="
\mathcal{L}_\mathrm{QCD}
= \bar{\psi}_i\left(i \gamma^\mu (D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a
" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/b/c/7bc38efc1e630ac15da9dc171cbf2a08.png" style="border: none; vertical-align: middle;" /></dd></dl>
</dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
where <img alt="\psi_i(x) \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/0/9/30909e89716575f9f7a28b7a0c5a68d1.png" style="border: none; margin: 0px; vertical-align: middle;" /> is the quark field, a dynamical function of space-time, in the fundamental representation of the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/SU(3)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="SU(3)"><b>SU</b>(3)</a> gauge <a href="http://en.wikipedia.org/wiki/Group_(mathematics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Group (mathematics)">group</a>, indexed by <img alt="i,\,j,\,\ldots" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/3/a/d3a93ddd77a1112973d9ee2f677f7dea.png" style="border: none; margin: 0px; vertical-align: middle;" />; <img alt="G^a_\mu(x) \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/d/f/9df7881ae7cfdcf328bf0c27061fc842.png" style="border: none; margin: 0px; vertical-align: middle;" /> are the gluon fields, also a dynamical function of space-time, in the adjoint representation of the <b>SU</b>(3) gauge group, indexed by <i>a</i>, <i>b</i>, .... The γ<sup style="line-height: 1em;">μ</sup> are <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Dirac_matrices" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac matrices">Dirac matrices</a> connecting the spinor representation to the vector representation of the <a href="http://en.wikipedia.org/wiki/Lorentz_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lorentz group">Lorentz group</a>.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The symbol <img alt="G^a_{\mu \nu} \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/e/3/1e3ef83e97030ba3f551317b30222b67.png" style="border: none; margin: 0px; vertical-align: middle;" /> represents the gauge invariant gluonic field strength tensor, analogous to the electromagnetic field strength tensor, <i>F</i><sup style="line-height: 1em;">μ, ν</sup>, in <a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum electrodynamics">Electrodynamics</a>. It is given by</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="G^a_{\mu \nu} = \partial_\mu G^a_{\nu} - \partial_\nu G^a_\mu - g f^{abc} G^b_\mu G^c_\nu \,," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/8/c/48c8e790cdafef425c9ebd19f913b2ce.png" style="border: none; vertical-align: middle;" /></dd></dl>
</dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
where <i>f<sub style="line-height: 1em;">abc</sub></i> are the structure constants of <b>SU</b>(3). Note that the rules to move-up or pull-down the <i>a</i>, <i>b</i>, or <i>c</i> indexes are <i>trivial</i>, (+......+), so that <i>f<sup style="line-height: 1em;">abc</sup></i> = <i>f<sub style="line-height: 1em;">abc</sub></i> = <i>f</i><span style="display: inline-block; font-size: 11px; line-height: 1.2em; margin-bottom: -0.3em; text-align: left; vertical-align: -0.4em;"><i>a</i><br /><i>bc</i></span> whereas for the<i>μ</i> or <i>ν</i> indexes one has the non-trivial <i>relativistic</i> rules, corresponding e.g. to the signature (+---). Furthermore, for mathematicians, according to this formula the gluon color field can be represented by a <b>SU</b>(3)-Lie algebra-valued "curvature"-2-form <img alt="\mathbf G=\mathrm d\mathbf {\tilde G}-g\,\mathbf {\tilde G}\wedge \mathbf {\tilde G}\,," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/0/1/70183613ab28265c85d97c9ac609dfe6.png" style="border: none; margin: 0px; vertical-align: middle;" /> where <img alt="\mathbf {\tilde G}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/f/5/5f5c588d5f6f10a114ef77ae58c27156.png" style="border: none; margin: 0px; vertical-align: middle;" /> is a "vector potential"-1-form corresponding to <b>G</b> and <img alt="\wedge" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/b/a/1ba4f06f68614e5da79a8ebd378d532a.png" style="border: none; margin: 0px; vertical-align: middle;" /> is the (antisymmetric) "<a href="http://en.wikipedia.org/wiki/Exterior_algebra" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Exterior algebra">wedge product</a>" of this algebra, producing the "structure constants" <i>f<sup style="line-height: 1em;">abc</sup></i>. The <a href="http://en.wikipedia.org/wiki/%C3%89lie_Cartan" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Élie Cartan">Cartan</a>-derivative of the field form (i.e. essentially the divergence of the field) would be zero in the absence of the "gluon terms", i.e. those ~ g, which represent the non-abelian character of the <b>SU</b>(3).</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The constants <i>m</i> and <i>g</i> control the quark mass and coupling constants of the theory, subject to renormalization in the full quantum theory.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
An important theoretical notion concerning the final term of the above Lagrangian is the <i><a href="http://en.wikipedia.org/wiki/Wilson_loop" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wilson loop">Wilson loop</a></i> variable. This loop variable plays a most-important role in discretized forms of the QCD (see <a href="http://en.wikipedia.org/wiki/Lattice_QCD" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lattice QCD">lattice QCD</a>), and more generally, it distinguishes <a href="http://en.wikipedia.org/wiki/Confinement" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Confinement">confined</a> and deconfined states of a gauge theory. It was introduced by the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Nobel_prize" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nobel prize">Nobel prize</a> winner<a href="http://en.wikipedia.org/wiki/Kenneth_G._Wilson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Kenneth G. Wilson">Kenneth G. Wilson</a> and is treated in a separate article.</div>
<h3 style="background-image: none; border-bottom-style: none; font-size: 18px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=8" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Fields">edit</a>]</span><span class="mw-headline" id="Fields">Fields</span></h3>
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Quarks are massive spin-1/2 <a href="http://en.wikipedia.org/wiki/Fermion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermion">fermions</a> which carry a <a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge">color charge</a> whose gauging is the content of QCD. Quarks are represented by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Dirac_field" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac field">Dirac fields</a> in the <a href="http://en.wikipedia.org/wiki/Fundamental_representation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fundamental representation">fundamental representation</a> <b>3</b> of the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gauge_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge group">gauge group</a> <a class="mw-redirect" href="http://en.wikipedia.org/wiki/SU(3)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="SU(3)"><b>SU</b>(3)</a>. They also carry electric charge (either -1/3 or 2/3) and participate in <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Weak_interactions" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak interactions">weak interactions</a> as part of <a href="http://en.wikipedia.org/wiki/Weak_isospin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak isospin">weak isospin</a> doublets. They carry global quantum numbers including the <a href="http://en.wikipedia.org/wiki/Baryon_number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Baryon number">baryon number</a>, which is 1/3 for each quark, <a href="http://en.wikipedia.org/wiki/Hypercharge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hypercharge">hypercharge</a> and one of the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Flavor_(particle_physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Flavor (particle physics)">flavor quantum numbers</a>.</div>
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Gluons are spin-1 <a href="http://en.wikipedia.org/wiki/Boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Boson">bosons</a> which also carry <a href="http://en.wikipedia.org/wiki/Color_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Color charge">color charges</a>, since they lie in the <a href="http://en.wikipedia.org/wiki/Adjoint_representation_of_a_Lie_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Adjoint representation of a Lie group">adjoint representation</a> <b>8</b> of <b>SU</b>(3). They have no electric charge, do not participate in the weak interactions, and have no flavor. They lie in the <a href="http://en.wikipedia.org/wiki/Singlet_state" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Singlet state">singlet representation</a> <b>1</b> of all these symmetry groups.</div>
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Every quark has its own antiquark. The charge of each antiquark is exactly the opposite of the corresponding quark.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=9" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Dynamics">edit</a>]</span><span class="mw-headline" id="Dynamics">Dynamics</span></h3>
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According to the rules of <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">quantum field theory</a>, and the associated <a href="http://en.wikipedia.org/wiki/Feynman_diagram" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Feynman diagram">Feynman diagrams</a>, the above theory gives rise to three basic interactions: a quark may emit (or absorb) a gluon, a gluon may emit (or absorb) a gluon, and two gluons may directly interact. This contrasts with <a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum electrodynamics">QED</a>, in which only the first kind of interaction occurs, since <a href="http://en.wikipedia.org/wiki/Photon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Photon">photons</a> have no charge. Diagrams involving <a href="http://en.wikipedia.org/wiki/Faddeev%E2%80%93Popov_ghost" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Faddeev–Popov ghost">Faddeev–Popov ghosts</a> must be considered too (except in the <a href="http://en.wikipedia.org/wiki/Unitarity_gauge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Unitarity gauge">unitarity gauge</a>).</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=10" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Area law and confinement">edit</a>]</span><span class="mw-headline" id="Area_law_and_confinement">Area law and confinement</span></h3>
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Detailed computations with the above-mentioned Lagrangian<sup class="reference" id="cite_ref-10" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-10" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[11]</a></sup> show that the effective potential between a quark and its anti-quark in a <a href="http://en.wikipedia.org/wiki/Meson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Meson">meson</a> contains a term <img alt="\propto r" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/6/1/b61ec2d76acbcba46737b1b7e5206295.png" style="border: none; margin: 0px; vertical-align: middle;" />, which represents some kind of "stiffness" of the interaction between the particle and its anti-particle at large distances, similar to the <a href="http://en.wikipedia.org/wiki/Entropic_force" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Entropic force">entropic elasticity</a> of a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Rubber" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Rubber">rubber</a> band (see below). This leads to <i>confinement</i> <sup class="reference" id="cite_ref-11" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-11" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[12]</a></sup> of the quarks to the interior of hadrons, i.e. <a href="http://en.wikipedia.org/wiki/Meson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Meson">mesons</a> and <a href="http://en.wikipedia.org/wiki/Nucleon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nucleon">nucleons</a>, with typical radii R<sub style="line-height: 1em;">c</sub>, corresponding to former "<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Bag_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bag model">Bag models</a>" of the hadrons<sup class="reference" id="cite_ref-12" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-12" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[13]</a></sup> . The order of magnitude of the "bag radius" is 1 fm (=10<sup style="line-height: 1em;">−15</sup> m). Moreover, the above-mentioned stiffness is quantitatively related to the so-called "area law" behaviour of the expectation value of the <a href="http://en.wikipedia.org/wiki/Wilson_loop" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wilson loop">Wilson loop</a> product <i>P<sub style="line-height: 1em;">W</sub></i> of the ordered coupling constants around a closed loop <i>W</i>; i.e. <img alt="\,\langle P_W\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/d/a/2da28650ba09b422acdd9a6997d95f65.png" style="border: none; margin: 0px; vertical-align: middle;" /> is proportional to the <i>area</i> enclosed by the loop. For this behaviour the non-abelian behaviour of the gauge group is essential.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=11" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Methods">edit</a>]</span><span class="mw-headline" id="Methods">Methods</span></h2>
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Further analysis of the content of the theory is complicated. Various techniques have been developed to work with QCD. Some of them are discussed briefly below.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=12" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Perturbative QCD">edit</a>]</span><span class="mw-headline" id="Perturbative_QCD">Perturbative QCD</span></h3>
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This approach is based on asymptotic freedom, which allows <a href="http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Perturbation theory (quantum mechanics)">perturbation theory</a> to be used accurately in experiments performed at very high energies. Although limited in scope, this approach has resulted in the most precise tests of QCD to date.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=13" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Lattice QCD">edit</a>]</span><span class="mw-headline" id="Lattice_QCD">Lattice QCD</span></h3>
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Main article: <a href="http://en.wikipedia.org/wiki/Lattice_QCD" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lattice QCD">Lattice QCD</a></div>
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<a class="image" href="http://en.wikipedia.org/wiki/File:Fluxtube_meson.png" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><img alt="" class="thumbimage" height="259" src="http://upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fluxtube_meson.png/150px-Fluxtube_meson.png" style="background-color: white; border: 1px solid rgb(204, 204, 204); vertical-align: middle;" width="150" /></a><br />
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A quark and an antiquark (red color) are glued together (green color) to form a meson (result of a lattice QCD simulation by M. Cardoso et al. <sup class="reference" id="cite_ref-13" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-13" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[14]</a></sup>)</div>
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Among non-perturbative approaches to QCD, the most well established one is <a href="http://en.wikipedia.org/wiki/Lattice_QCD" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lattice QCD">lattice QCD</a>. This approach uses a discrete set of space-time points (called the lattice) to reduce the analytically intractable path integrals of the continuum theory to a very difficult numerical computation which is then carried out on <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Supercomputers" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Supercomputers">supercomputers</a> like the <a href="http://en.wikipedia.org/wiki/QCDOC" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="QCDOC">QCDOC</a> which was constructed for precisely this purpose. While it is a slow and resource-intensive approach, it has wide applicability, giving insight into parts of the theory inaccessible by other means, in particular into the explicit forces acting between quarks and antiquarks in a meson. However, the <a href="http://en.wikipedia.org/wiki/Numerical_sign_problem" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Numerical sign problem">numerical sign problem</a> makes it difficult to use lattice methods to study QCD at high density and low temperature (e.g. nuclear matter or the interior of neutron stars).</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=14" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: 1/N expansion">edit</a>]</span><span class="mw-headline" id="1.2FN_expansion">1/N expansion</span></h3>
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Main article: <a href="http://en.wikipedia.org/wiki/1/N_expansion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="1/N expansion">1/N expansion</a></div>
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A well-known approximation scheme, the <a href="http://en.wikipedia.org/wiki/1/N_expansion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="1/N expansion">1/N expansion</a>, starts from the premise that the number of colors is infinite, and makes a series of corrections to account for the fact that it is not. Until now, it has been the source of qualitative insight rather than a method for quantitative predictions. Modern variants include the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/AdS/CFT" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="AdS/CFT">AdS/CFT</a> approach.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=15" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Effective theories">edit</a>]</span><span class="mw-headline" id="Effective_theories">Effective theories</span></h3>
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For specific problems effective theories may be written down which give qualitatively correct results in certain limits. In the best of cases, these may then be obtained as systematic expansions in some parameter of the QCD Lagrangian. One such <a href="http://en.wikipedia.org/wiki/Effective_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Effective field theory">effective field theory</a> is <a href="http://en.wikipedia.org/wiki/Chiral_perturbation_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Chiral perturbation theory">chiral perturbation theory</a> or ChiPT, which is the QCD effective theory at low energies. More precisely, it is a low energy expansion based on the spontaneous chiral symmetry breaking of QCD, which is an exact symmetry when quark masses are equal to zero, but for the u,d and s quark, which have small mass, it is still a good approximate symmetry. Depending on the number of quarks which are treated as light, one uses either<b>SU</b>(2) ChiPT or <b>SU</b>(3) ChiPT . Other effective theories are <a class="new" href="http://en.wikipedia.org/w/index.php?title=Heavy_quark_effective_theory&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Heavy quark effective theory (page does not exist)">heavy quark effective theory</a> (which expands around heavy quark mass near infinity), and <a href="http://en.wikipedia.org/wiki/Soft-collinear_effective_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Soft-collinear effective theory">soft-collinear effective theory</a> (which expands around large ratios of energy scales). In addition to effective theories, models like the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Nambu-Jona-Lasinio_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nambu-Jona-Lasinio model">Nambu-Jona-Lasinio model</a> and the<a href="http://en.wikipedia.org/wiki/Chiral_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Chiral model">chiral model</a> are often used when discussing general features.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=16" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: QCD sum rules">edit</a>]</span><span class="mw-headline" id="QCD_sum_rules">QCD sum rules</span></h3>
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Main article: <a href="http://en.wikipedia.org/wiki/QCD_sum_rules" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="QCD sum rules">QCD sum rules</a></div>
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Based on an <a href="http://en.wikipedia.org/wiki/Operator_product_expansion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Operator product expansion">Operator product expansion</a> one can derive sets of relations that connect different observables with each other.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=17" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Nambu-Jona-Lasinio model">edit</a>]</span><span class="mw-headline" id="Nambu-Jona-Lasinio_model"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Nambu-Jona-Lasinio_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nambu-Jona-Lasinio model">Nambu-Jona-Lasinio model</a></span></h3>
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In one of his recent works, <a class="new" href="http://en.wikipedia.org/w/index.php?title=Kei-Ichi_Kondo&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Kei-Ichi Kondo (page does not exist)">Kei-Ichi Kondo</a> derived as a low-energy limit of QCD, a theory linked to the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Nambu-Jona-Lasinio_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nambu-Jona-Lasinio model">Nambu-Jona-Lasinio model</a> since it is basically a particular non-local version of the <a class="new" href="http://en.wikipedia.org/w/index.php?title=Polyakov-Nambu-Jona-Lasinio_model&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Polyakov-Nambu-Jona-Lasinio model (page does not exist)">Polyakov-Nambu-Jona-Lasinio model</a>.<sup class="reference" id="cite_ref-14" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-14" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[15]</a></sup> The later being in its local version, nothing but the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Nambu-Jona-Lasinio_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nambu-Jona-Lasinio model">Nambu-Jona-Lasinio model</a> in which one has included the Polyakov loop effect, in order to describe a 'certain confinement'.</div>
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The <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Nambu-Jona-Lasinio_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nambu-Jona-Lasinio model">Nambu-Jona-Lasinio model</a> in itself is, among many other things, used because it is a 'relatively simple' model of <a href="http://en.wikipedia.org/wiki/Chiral_symmetry" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Chiral symmetry">chiral symmetry breaking</a>, phenomenon present up to certain conditions (Chiral limit i.e. massless fermions) in QCD itself.</div>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=18" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Experimental tests">edit</a>]</span><span class="mw-headline" id="Experimental_tests">Experimental tests</span></h2>
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The notion of quark <a href="http://en.wikipedia.org/wiki/Flavour_(particle_physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Flavour (particle physics)">flavors</a> was prompted by the necessity of explaining the properties of hadrons during the development of the <a href="http://en.wikipedia.org/wiki/Quark_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark model">quark model</a>. The notion of color was necessitated by the puzzle of the <span class="unicode;" style="white-space: nowrap;">Δ<span style="display: inline-block; font-size: 11px; line-height: 1.2em; margin-bottom: -0.3em; text-align: left; vertical-align: 0.8em;">++</span></span>. This has been dealt with in the section on <a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#History" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum chromodynamics">the history of QCD</a>.</div>
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The first evidence for quarks as real constituent elements of hadrons was obtained in <a href="http://en.wikipedia.org/wiki/Deep_inelastic_scattering" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Deep inelastic scattering">deep inelastic scattering</a> experiments at <a class="mw-redirect" href="http://en.wikipedia.org/wiki/SLAC" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="SLAC">SLAC</a>. The first evidence for gluons came in<a href="http://en.wikipedia.org/wiki/Three_jet_event" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Three jet event">three jet events</a> at <a href="http://en.wikipedia.org/wiki/PETRA" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="PETRA">PETRA</a>.</div>
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Good quantitative tests of perturbative QCD are</div>
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;">the <a href="http://en.wikipedia.org/wiki/Coupling_constant#QCD_and_asymptotic_freedom" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Coupling constant">running of the QCD coupling</a> as deduced from many observations</li>
<li style="margin-bottom: 0.1em;"><a class="new" href="http://en.wikipedia.org/w/index.php?title=Scaling_violation&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Scaling violation (page does not exist)">scaling violation</a> in polarized and unpolarized <a href="http://en.wikipedia.org/wiki/Deep_inelastic_scattering" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Deep inelastic scattering">deep inelastic scattering</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Vector_boson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vector boson">vector boson</a> production at colliders (this includes the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Drell-Yan_process" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Drell-Yan process">Drell-Yan process</a>)</li>
<li style="margin-bottom: 0.1em;"><a class="new" href="http://en.wikipedia.org/w/index.php?title=Jet_cross_sections&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Jet cross sections (page does not exist)">jet cross sections</a> in colliders</li>
<li style="margin-bottom: 0.1em;"><a class="new" href="http://en.wikipedia.org/w/index.php?title=Event_shape_observables&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Event shape observables (page does not exist)">event shape observables</a> at the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/LEP" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="LEP">LEP</a></li>
<li style="margin-bottom: 0.1em;"><a class="new" href="http://en.wikipedia.org/w/index.php?title=Heavy-quark_production&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Heavy-quark production (page does not exist)">heavy-quark production</a> in colliders</li>
</ul>
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Quantitative tests of non-perturbative QCD are fewer, because the predictions are harder to make. The best is probably the running of the QCD coupling as probed through<a href="http://en.wikipedia.org/wiki/Lattice_QCD" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lattice QCD">lattice</a> computations of <a class="new" href="http://en.wikipedia.org/w/index.php?title=Heavy-quarkonium_spectra&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Heavy-quarkonium spectra (page does not exist)">heavy-quarkonium spectra</a>. There is a recent claim about the mass of the heavy meson B<sub style="line-height: 1em;">c</sub> <a class="external autonumber" href="http://www.aip.org/pnu/2005/split/731-1.html" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">[4]</a>. Other non-perturbative tests are currently at the level of 5% at best. Continuing work on masses and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Form_factor_(QFT)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Form factor (QFT)">form factors</a> of hadrons and their <a class="new" href="http://en.wikipedia.org/w/index.php?title=Weak_matrix_elements&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Weak matrix elements (page does not exist)">weak matrix elements</a> are promising candidates for future quantitative tests. The whole subject of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Quark_matter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark matter">quark matter</a> and the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Quark-gluon_plasma" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark-gluon plasma">quark-gluon plasma</a> is a non-perturbative test bed for QCD which still remains to be properly exploited.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=19" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Cross-relations to solid state physics">edit</a>]</span><span class="mw-headline" id="Cross-relations_to_solid_state_physics">Cross-relations to <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Solid_state_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Solid state physics">solid state physics</a></span></h2>
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There are unexpected cross-relations to solid state physics. For example, the notion of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gauge_invariance" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge invariance">gauge invariance</a> forms the basis of the well-known Mattis <a href="http://en.wikipedia.org/wiki/Spin_glass" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin glass">spin glasses</a>,<sup class="reference" id="cite_ref-15" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-15" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[16]</a></sup> which are systems with the usual spin degrees of freedom <img alt="s_i=\pm 1\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/d/0/fd0314200ae7c4b0d88a5bf7719b6a7b.png" style="border: none; margin: 0px; vertical-align: middle;" /> for <i>i</i> =1,...,N, with the special fixed "random" couplings <img alt="J_{i,k}=\epsilon_i \,J_0\,\epsilon_k\,." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/7/2/67253e55c27ff19c0a66e5d1b7061d48.png" style="border: none; margin: 0px; vertical-align: middle;" /> Here the ε<sub style="line-height: 1em;">i</sub> and ε<sub style="line-height: 1em;">k</sub> quantities can independently and "randomly" take the values ±1, which corresponds to a most-simple gauge transformation <img alt="(\,s_i\to s_i\cdot\epsilon_i\quad\,J_{i,k}\to \epsilon_i J_{i,k}\epsilon_k\,\quad s_k\to s_k\cdot\epsilon_k \,)\,." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/e/e/4ee88de00c9d3987b6b7feda541ee503.png" style="border: none; margin: 0px; vertical-align: middle;" />This means that thermodynamic expectation values of measurable quantities, e.g. of the energy <img alt="{\mathcal H}:=-\sum s_i\,J_{i,k}\,s_k\,," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/e/3/2e3c947c757793dcca9078b0d9e02f32.png" style="border: none; margin: 0px; vertical-align: middle;" /> are invariant.</div>
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However, here the <i>coupling degrees of freedom</i> <img alt="J_{i,k}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/5/0/f506b8afdaa913b3489662e90c3d7e72.png" style="border: none; margin: 0px; vertical-align: middle;" />, which in the QCD correspond to the <i>gluons</i>, are "frozen" to fixed values (quenching). In contrast, in the QCD they "fluctuate" (annealing), and through the large number of gauge degrees of freedom the <a href="http://en.wikipedia.org/wiki/Entropy" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Entropy">entropy</a> plays an important role (see below).</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
For positive <i>J</i><sub style="line-height: 1em;">0</sub> the thermodynamics of the Mattis spin glass corresponds in fact simply to a ferromagnet, just because these systems have no "<a href="http://en.wikipedia.org/wiki/Geometrical_frustration" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Geometrical frustration">frustration</a>“ at all. This term is a basic measure in spin glass theory.<sup class="reference" id="cite_ref-16" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-16" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[17]</a></sup> Quantitatively it is identical with the loop-product <img alt="P_W:\,=\,J_{i,k}J_{k,l}...J_{n,m}J_{m,i}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/7/f375b1ac04f653977c088be9d232369a.png" style="border: none; margin: 0px; vertical-align: middle;" /> along a closed loop <i>W</i>. However, for a Mattis spin glass - in contrast to "genuine" spin glasses - the quantity <i>P<sub style="line-height: 1em;">W</sub></i> never becomes negative.</div>
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The basic notion "frustration" of the spin-glass is actually similar to the <a href="http://en.wikipedia.org/wiki/Wilson_loop" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wilson loop">Wilson loop</a> quantity of the QCD. The only difference is again that in the QCD one is dealing with SU(3) matrices, and that one is dealing with a "fluctuating" quantity. Energetically, perfect absence of frustration should be non-favorable and atypical for a spin glass, which means that one should add the loop-product to the Hamiltonian, by some kind of term representing a "punishment". - In the QCD the Wilson loop is essential for the Lagrangian rightaway.</div>
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The relation between the QCD and "disordered magnetic systems" (the spin glasses belong to them) were additionally stressed in a paper by Fradkin, Huberman und Shenker,<sup class="reference" id="cite_ref-17" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-17" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[18]</a></sup> which also stresses the notion of <a href="http://en.wikipedia.org/wiki/Dualism" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dualism">duality</a>.</div>
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A further analogy consists in the already mentioned similarity to <a href="http://en.wikipedia.org/wiki/Polymer_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Polymer physics">polymer physics</a>, where, analogously to <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Wilson_Loop" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wilson Loop">Wilson Loops</a>, so-called "entangled nets" appear, which are important for the formation of the <a href="http://en.wikipedia.org/wiki/Entropic_force" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Entropic force">entropy-elasticity</a> (force proportional to the length) of a rubber band. The non-abelian character of the SU(3) corresponds thereby to the non-trivial "chemical links“, which glue different loop segments together, and "<a href="http://en.wikipedia.org/wiki/Asymptotic_freedom" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Asymptotic freedom">asymptotic freedom</a>" means in the polymer analogy simply the fact that in the short-wave limit, i.e. for <img alt="0\leftarrow\lambda_w\ll R_c" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/7/f/e7ffbc2fd8f52c78d25c6ce23f81197a.png" style="border: none; margin: 0px; vertical-align: middle;" /> (where <i>R<sub style="line-height: 1em;">c</sub></i> is a characteristic correlation-length for the glued loops, corresponding to the above-mentioned "bag radius", while λ<sub style="line-height: 1em;">w</sub> is the wavelength of an excitation) any non-trivial correlation vanishes totally, as if the system had crystallized.<sup class="reference" id="cite_ref-18" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-18" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[19]</a></sup></div>
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There is also a correspondence between confinement in QCD - the fact that the color-field is only different from zero in the interior of hadrons - and the behaviour of the usual magnetic field in the theory of <a href="http://en.wikipedia.org/wiki/Type-II_superconductor" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Type-II superconductor">type-II superconductors</a>: there the magnetism is confined to the interiour of the <a href="http://en.wikipedia.org/wiki/Abrikosov_vortex" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Abrikosov vortex">Abrikosov flux-line lattice</a>,<sup class="reference" id="cite_ref-19" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_note-19" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[20]</a></sup> i.e., the London penetration depth<i>λ</i> of that theory is analogous to the confinement radius <i>R<sub style="line-height: 1em;">c</sub></i> of quantum chromodynamics. Mathematically, this correspondendence is supported by the second term, <img alt="\propto g G^a_\mu \bar{\psi}_i \gamma^\mu T^a_{ij} \psi_j\,," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/1/3/2136013bade40cbf1b110f20e1613043.png" style="border: none; margin: 0px; vertical-align: middle;" /> on the r.h.s. of the Lagrangian.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=20" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: See also">edit</a>]</span><span class="mw-headline" id="See_also">See also</span></h2>
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;">For overviews, see <a href="http://en.wikipedia.org/wiki/Standard_Model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Standard Model">Standard Model</a>, its <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Standard_model_(basic_details)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Standard model (basic details)">field theoretical formulation</a>, <a href="http://en.wikipedia.org/wiki/Strong_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Strong interaction">strong interactions</a>, <a href="http://en.wikipedia.org/wiki/Quark" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark">quarks</a> and <a href="http://en.wikipedia.org/wiki/Gluon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gluon">gluons</a>, <a href="http://en.wikipedia.org/wiki/Hadron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hadron">hadrons</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Colour_confinement" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Colour confinement">confinement</a>, <a href="http://en.wikipedia.org/wiki/QCD_matter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="QCD matter">QCD matter</a>, or <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Quark-gluon_plasma" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark-gluon plasma">quark-gluon plasma</a>.</li>
<li style="margin-bottom: 0.1em;">For details, see <a href="http://en.wikipedia.org/wiki/Gauge_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge theory">gauge theory</a>, <a href="http://en.wikipedia.org/wiki/Quantum_gauge_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum gauge theory">quantization procedure</a> including <a href="http://en.wikipedia.org/wiki/BRST_quantization" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="BRST quantization">BRST quantization</a> and <a href="http://en.wikipedia.org/wiki/Faddeev%E2%80%93Popov_ghost" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Faddeev–Popov ghost">Faddeev–Popov ghosts</a>. A more general category is <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">quantum field theory</a>.</li>
<li style="margin-bottom: 0.1em;">For techniques, see <a href="http://en.wikipedia.org/wiki/Lattice_QCD" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lattice QCD">Lattice QCD</a>, <a href="http://en.wikipedia.org/wiki/1/N_expansion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="1/N expansion">1/N expansion</a>, <a href="http://en.wikipedia.org/wiki/Perturbative_QCD" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Perturbative QCD">perturbative QCD</a>, <a href="http://en.wikipedia.org/wiki/Soft-collinear_effective_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Soft-collinear effective theory">Soft-collinear effective theory</a>, <a class="new" href="http://en.wikipedia.org/w/index.php?title=Heavy_quark_effective_theory&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Heavy quark effective theory (page does not exist)">heavy quark effective theory</a>, <a href="http://en.wikipedia.org/wiki/Chiral_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Chiral model">chiral models</a>, and the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Nambu-Jona-Lasinio_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nambu-Jona-Lasinio model">Nambu and Jona-Lasinio model</a>.</li>
<li style="margin-bottom: 0.1em;">For experiments, see <a class="new" href="http://en.wikipedia.org/w/index.php?title=Quark_search_experiment&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Quark search experiment (page does not exist)">quark search experiments</a>, <a href="http://en.wikipedia.org/wiki/Deep_inelastic_scattering" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Deep inelastic scattering">deep inelastic scattering</a>, <a class="new" href="http://en.wikipedia.org/w/index.php?title=Jet_physics&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Jet physics (page does not exist)">jet physics</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Quark-gluon_plasma" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark-gluon plasma">quark-gluon plasma</a>.</li>
</ul>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=21" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: References">edit</a>]</span><span class="mw-headline" id="References">References</span></h2>
<div class="reflist references-column-count references-column-count-2" style="-webkit-column-count: 2; font-size: 12px; list-style-type: decimal; margin-bottom: 0.5em;">
<ol class="references" style="line-height: 1.5em; list-style-image: none; list-style-type: inherit; margin: 0.3em 0px 0.5em 3.2em; padding: 0px;">
<li id="cite_note-0" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Murray_Gell-Mann" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Murray Gell-Mann">Gell-Mann, Murray</a> (1995). <i>The Quark and the Jaguar</i>. <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Owl_Books" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Owl Books">Owl Books</a>. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-8050-7253-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-8050-7253-2">978-0-8050-7253-2</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Quark+and+the+Jaguar&rft.aulast=Gell-Mann&rft.aufirst=Murray&rft.au=Gell-Mann%2C%26%2332%3BMurray&rft.date=1995&rft.pub=%5B%5BOwl+Books%5D%5D&rft.isbn=978-0-8050-7253-2&rfr_id=info:sid/en.wikipedia.org:Quantum_chromodynamics"></span></span></li>
<li id="cite_note-1" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">Fyodor Tkachov (2009). "A contribution to the history of quarks: Boris Struminsky's 1965 JINR publication". <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/0904.0343" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">0904.0343</a> [<a class="external text" href="http://arxiv.org/archive/physics.hist-ph" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">physics.hist-ph</a>].</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+contribution+to+the+history+of+quarks%3A+Boris+Struminsky%27s+1965+JINR+publication&rft.jtitle=%27%27%5B%5BarXiv%5D%5D%3A%5Bhttp%3A%2F%2Farxiv.org%2Fabs%2F0904.0343+0904.0343%5D%26nbsp%3B%5B%5Bhttp%3A%2F%2Farxiv.org%2Farchive%2Fphysics.hist-ph+physics.hist-ph%5D%5D%27%27&rft.aulast=Fyodor+Tkachov&rft.au=Fyodor+Tkachov&rft.date=2009&rfr_id=info:sid/en.wikipedia.org:Quantum_chromodynamics"></span></span></li>
<li id="cite_note-struminsky-2" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink">^ <a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-struminsky_2-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-struminsky_2-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a></span> <span class="reference-text">B. V. Struminsky, Magnetic moments of barions in the quark model. <a class="mw-redirect" href="http://en.wikipedia.org/wiki/JINR" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="JINR">JINR</a>-Preprint P-1939, Dubna, Russia. Submitted on January 7, 1965.</span></li>
<li id="cite_note-3" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a href="http://en.wikipedia.org/wiki/Nikolay_Bogolyubov" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nikolay Bogolyubov">N. Bogolubov</a>, B. Struminsky, A. Tavkhelidze. On composite models in the theory of elementary particles. <a class="mw-redirect" href="http://en.wikipedia.org/wiki/JINR" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="JINR">JINR</a> Preprint D-1968, <a href="http://en.wikipedia.org/wiki/Dubna" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dubna">Dubna</a> 1965.</span></li>
<li id="cite_note-4" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-4" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">A. Tavkhelidze. Proc. Seminar on High Energy Physics and Elementary Particles, Trieste, 1965, Vienna IAEA, 1965, p. 763.</span></li>
<li id="cite_note-5" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-5" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">V. A. Matveev and A. N. Tavkhelidze (INR, RAS, Moscow) <a class="external text" href="http://www.inr.ru/quantum.html" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">The quantum number color, colored quarks and QCD</a> (Dedicated to the 40th Anniversary of the Discovery of the Quantum Number Color). Report presented at the 99th Session of the JINR Scientific Council, Dubna, 19–20 January 2006.</span></li>
<li id="cite_note-6" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-6" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">J. Polchinski, M. Strassler (2002). "Hard Scattering and Gauge/String duality".<i><a href="http://en.wikipedia.org/wiki/Physical_Review_Letters" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review Letters">Physical Review Letters</a></i> <b>88</b> (3): 31601. <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/hep-th/0109174" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">hep-th/0109174</a>. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/2002PhRvL..88c1601P" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">2002PhRvL..88c1601P</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRevLett.88.031601" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRevLett.88.031601</a>. <a class="mw-redirect" href="http://en.wikipedia.org/wiki/PubMed_Identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="PubMed Identifier">PMID</a> <a class="external text" href="http://www.ncbi.nlm.nih.gov/pubmed/11801052" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">11801052</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Hard+Scattering+and+Gauge%2FString+duality&rft.jtitle=%5B%5BPhysical+Review+Letters%5D%5D&rft.aulast=J.+Polchinski%2C+M.+Strassler&rft.au=J.+Polchinski%2C+M.+Strassler&rft.date=2002&rft.volume=88&rft.issue=3&rft.pages=31601&rft_id=info:arxiv/hep-th%2F0109174&rft_id=info:bibcode/2002PhRvL..88c1601P&rft_id=info:doi/10.1103%2FPhysRevLett.88.031601&rft_id=info:pmid/11801052&rfr_id=info:sid/en.wikipedia.org:Quantum_chromodynamics"></span></span></li>
<li id="cite_note-7" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-7" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">Brower, Richard C.; Mathur, Samir D.; Chung-I Tan (2000). "Glueball Spectrum for QCD from AdS Supergravity Duality". <i>Nuclear Physics B</i> <b>587</b>: 249–276. <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/hep-th/0003115" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">hep-th/0003115</a>. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/2000NuPhB.587..249B" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">2000NuPhB.587..249B</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1016%2FS0550-3213%2800%2900435-1" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1016/S0550-3213(00)00435-1</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Glueball+Spectrum+for+QCD+from+AdS+Supergravity+Duality&rft.jtitle=Nuclear+Physics+B&rft.aulast=Brower&rft.aufirst=Richard+C.&rft.au=Brower%2C%26%2332%3BRichard+C.&rft.au=Mathur%2C%26%2332%3BSamir+D.&rft.au=Chung-I+Tan&rft.date=2000&rft.volume=587&rft.pages=249%E2%80%93276&rft_id=info:arxiv/hep-th%2F0003115&rft_id=info:bibcode/2000NuPhB.587..249B&rft_id=info:doi/10.1016%2FS0550-3213%2800%2900435-1&rfr_id=info:sid/en.wikipedia.org:Quantum_chromodynamics"></span></span></li>
<li id="cite_note-8" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-8" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">F. Wegner, <i>Duality in Generalized Ising Models and Phase Transitions without Local Order Parameter</i>, J. Math. Phys. <b>12</b> (1971) 2259-2272.</span>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="reference-text">Reprinted in <a class="new" href="http://en.wikipedia.org/w/index.php?title=Claudio_Rebbi&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Claudio Rebbi (page does not exist)">Claudio Rebbi</a> (ed.), <i>Lattice Gauge Theories and Monte Carlo Simulations</i>, World Scientific, Singapore (1983), p. 60-73. Abstract: <a class="external autonumber" href="http://www.tphys.uni-heidelberg.de/~wegner/Abstracts.html#12" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">[1]</a></span></dd></dl>
</li>
<li id="cite_note-9" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-9" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">Perhaps one can guess that in the "original" model mainly the quarks would fluctuate, whereas in the present one, the "dual" model, mainly the gluons do.</span></li>
<li id="cite_note-10" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-10" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">See all standard textbooks on the QCD, e.g., those noted above</span></li>
<li id="cite_note-11" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-11" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">Only at extremely large pressures and or temperatures, e.g. for <img alt="T\cong 5\cdot 10^{12}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/c/e/cce5d615a7e751a515c1f83b558b2239.png" style="border: none; vertical-align: middle;" /> K or larger, <i>confinement</i> gives way to a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Quark-gluon_plasma" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quark-gluon plasma">quark-gluon plasma</a>.</span></li>
<li id="cite_note-12" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-12" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a class="new" href="http://en.wikipedia.org/w/index.php?title=Kenneth_A._Johnson&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Kenneth A. Johnson (page does not exist)">Kenneth A. Johnson</a>, "The bag model of quark confinement", Scientific American, July 1979</span></li>
<li id="cite_note-13" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-13" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">M. Cardoso et al., "Lattice QCD computation of the colour fields for the static hybrid quark-gluon-antiquark system, and microscopic study of the Casimir scaling", Phys. Rev. D 81, 034504 (2010) ).</span></li>
<li id="cite_note-14" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-14" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">Kei-Ichi Kondo (2010). "Toward a first-principle derivation of confinement and chiral-symmetry-breaking crossover transitions in QCD". <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Physical_Review_D" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review D">Physical Review D</a></i> <b>82</b> (6): 065024.<a href="http://en.wikipedia.org/wiki/ArXiv" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/1005.0314v2" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1005.0314v2</a>. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/2010PhRvD..82f5024K" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">2010PhRvD..82f5024K</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRevD.82.065024" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRevD.82.065024</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Toward+a+first-principle+derivation+of+confinement+and+chiral-symmetry-breaking+crossover+transitions+in+QCD&rft.jtitle=%5B%5BPhysical+Review+D%5D%5D&rft.aulast=Kei-Ichi+Kondo&rft.au=Kei-Ichi+Kondo&rft.date=2010&rft.volume=82&rft.issue=6&rft.pages=065024&rft_id=info:arxiv/1005.0314v2&rft_id=info:bibcode/2010PhRvD..82f5024K&rft_id=info:doi/10.1103%2FPhysRevD.82.065024&rfr_id=info:sid/en.wikipedia.org:Quantum_chromodynamics"></span></span></li>
<li id="cite_note-15" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-15" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">D.C. Mattis, Phys. Lett. 56a (1976) 421</span></li>
<li id="cite_note-16" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-16" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">J. Vanninemus and G. Toulouse, J. Phys. C 10 (1977) 537</span></li>
<li id="cite_note-17" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-17" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">E. Fradkin, B.A. Huberman, S. Shenker, <i>Gauge Symmetries in random magnetic systems</i>, Phys. Rev. B 18 (1978) 4783-4794, <a class="external autonumber" href="http://prb.aps.org/abstract/PRB/v18/i9/p4879-1" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">[2]</a></span></li>
<li id="cite_note-18" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-18" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">A. Bergmann, A. Owen , "Dielectric relaxation spectroscopy of poly[(R)-3-Hydroxybutyrate] (PHD) during crystallization", Polymer International 53 (7) (2004) 863-868, <a class="external autonumber" href="http://www3.interscience.wiley.com/journal/108563755/abstract?CRETRY=1&SRETRY=0" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">[3]</a></span></li>
<li id="cite_note-19" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics#cite_ref-19" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">Mathematically, the flux-line lattices are described by <a href="http://en.wikipedia.org/wiki/Emil_Artin" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Emil Artin">Emil Artin</a>'s braid group, which is nonabelian, since one braid can wind around another one.</span></li>
</ol>
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<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=22" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Further reading">edit</a>]</span><span class="mw-headline" id="Further_reading">Further reading</span></h2>
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">Greiner, Walter;Schäfer, Andreas (1994). <i>Quantum Chromodynamics</i>. Springer. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-387-57103-5" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-387-57103-5">0-387-57103-5</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Chromodynamics&rft.aulast=Greiner%2C+Walter%3BSch%C3%A4fer%2C+Andreas&rft.au=Greiner%2C+Walter%3BSch%C3%A4fer%2C+Andreas&rft.date=1994&rft.pub=Springer&rft.isbn=0-387-57103-5&rfr_id=info:sid/en.wikipedia.org:Quantum_chromodynamics"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">Halzen, Francis; Martin, Alan (1984). <i>Quarks & Leptons: An Introductory Course in Modern Particle Physics</i>. John Wiley & Sons. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-471-88741-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-471-88741-2">0-471-88741-2</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quarks+%26+Leptons%3A+An+Introductory+Course+in+Modern+Particle+Physics&rft.aulast=Halzen%2C+Francis%3B+Martin%2C+Alan&rft.au=Halzen%2C+Francis%3B+Martin%2C+Alan&rft.date=1984&rft.pub=John+Wiley+%26+Sons&rft.isbn=0-471-88741-2&rfr_id=info:sid/en.wikipedia.org:Quantum_chromodynamics"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">Creutz, Michael (1985). <i>Quarks, Gluons and Lattices</i>. Cambridge University Press. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-521-31535-7" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-521-31535-7">978-0-521-31535-7</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quarks%2C+Gluons+and+Lattices&rft.aulast=Creutz%2C+Michael&rft.au=Creutz%2C+Michael&rft.date=1985&rft.pub=Cambridge+University+Press&rft.isbn=978-0-521-31535-7&rfr_id=info:sid/en.wikipedia.org:Quantum_chromodynamics"></span></li>
</ul>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_chromodynamics&action=edit&section=23" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: External links">edit</a>]</span><span class="mw-headline" id="External_links">External links</span></h2>
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<li style="margin-bottom: 0.1em;"><a class="external text" href="http://pdg.lbl.gov/" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Particle data group</a></li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.claymath.org/millennium/" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">The millennium prize</a> for <a class="external text" href="http://www.claymath.org/millennium/" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">proving confinement</a></li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.sciencemag.org/cgi/content/abstract/322/5905/1224" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Ab Initio Determination of Light Hadron Masses</a></li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.sciencemag.org/cgi/content/summary/322/5905/1198" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Andreas S Kronfeld</a> <i>The Weight of the World Is Quantum Chromodynamics</i></li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.iop.org/EJ/article/1742-6596/125/1/012067/jpconf8_125_012067.pdf?request-id=f9ccdf0d-ee26-4856-99fb-ce5bfef07c4c" rel="nofollow" style="background-image: url(http://upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif); background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 18px; text-decoration: none;">Andreas S Kronfeld</a> <i>Quantum chromodynamics with advanced computing</i></li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.sciencenews.org/view/generic/id/38788/title/Standard_model_gets_right_answer_for_proton,_neutron_masses" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Standard model gets right answer</a></li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://arxiv.org/abs/hepph/9505231" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Quantum Chromodynamics</a></li>
</ul>
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Quantum field theories</div>
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<tr><th class="navbox-group" scope="row" style="background-color: #ddddff; background-position: initial initial; background-repeat: initial initial; padding-left: 1em; padding-right: 1em; text-align: right; white-space: nowrap;">Standard</th><td class="navbox-list navbox-odd" style="background-color: transparent; background-position: initial initial; background-repeat: initial initial; border-color: rgb(253, 253, 253); border-left-style: solid; border-left-width: 2px; padding: 0px; text-align: left; width: 839px;"><div style="padding: 0em 0.25em;">
<a href="http://en.wikipedia.org/wiki/Chern%E2%80%93Simons_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Chern–Simons theory">Chern–Simons model</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Chiral_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Chiral model">Chiral model</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Kondo_effect" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Kondo effect">Kondo model</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Conformal_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Conformal field theory">Conformal field theory</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Noncommutative_quantum_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Noncommutative quantum field theory">Noncommutative quantum field theory</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Non-linear_sigma_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Non-linear sigma model">Non-linear sigma model</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Quartic_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Quartic interaction">Quartic interaction</a> <span style="font-weight: bold;">·</span><strong class="selflink" style="white-space: nowrap;">Quantum chromodynamics</strong> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Quantum electrodynamics">Quantum electrodynamics</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Quantum_gauge_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Quantum gauge theory">Quantum Yang–Mills theory</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Schwinger_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Schwinger model">Schwinger model</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Sine%E2%80%93Gordon_equation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Sine–Gordon equation">Sine–Gordon</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Standard_Model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Standard Model">Standard Model</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/String_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="String theory">String theory</a> <span style="font-weight: bold;">·</span><a href="http://en.wikipedia.org/wiki/Toda_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Toda field theory">Toda field theory</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Topological_quantum_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Topological quantum field theory">Topological quantum field theory</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Wess%E2%80%93Zumino_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Wess–Zumino model">Wess–Zumino model</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Wess%E2%80%93Zumino%E2%80%93Witten_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Wess–Zumino–Witten model">Wess–Zumino–Witten model</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Yang–Mills theory">Yang–Mills theory</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Yang%E2%80%93Mills%E2%80%93Higgs_equations" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Yang–Mills–Higgs equations">Yang–Mills–Higgs model</a> <span style="font-weight: bold;">·</span><a href="http://en.wikipedia.org/wiki/Yukawa_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Yukawa interaction">Yukawa model</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Thirring%E2%80%93Wess_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Thirring–Wess model">Thirring–Wess model</a></div>
</td></tr>
<tr style="height: 2px;"><td></td></tr>
<tr><th class="navbox-group" scope="row" style="background-color: #ddddff; background-position: initial initial; background-repeat: initial initial; padding-left: 1em; padding-right: 1em; text-align: right; white-space: nowrap;"><b><a href="http://en.wikipedia.org/wiki/Four-fermion_interactions" style="background-image: none; color: #0b0080; text-decoration: none;" title="Four-fermion interactions">Four-fermion interactions</a></b></th><td class="navbox-list navbox-odd" style="background-color: transparent; background-position: initial initial; background-repeat: initial initial; border-color: rgb(253, 253, 253); border-left-style: solid; border-left-width: 2px; padding: 0px; text-align: left; width: 839px;"><div style="padding: 0em 0.25em;">
<a href="http://en.wikipedia.org/wiki/Thirring_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Thirring model">Thirring model</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Gross%E2%80%93Neveu_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Gross–Neveu model">Gross–Neveu model</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Nambu%E2%80%93Jona-Lasinio_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;" title="Nambu–Jona-Lasinio model">Nambu–Jona-Lasinio model</a></div>
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</div>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com0tag:blogger.com,1999:blog-5303246073824127471.post-41779056106344651902012-06-22T19:07:00.000-04:002012-06-22T19:07:56.179-04:00Quantum Electrodynamics (QED)Once The Dirac Equation (1928) is in play the stage is set for greater revolutions and 19 years later three men use Dirac's and others' contributions to beautifully explain how the fundamental particles the electron and the photon interact.<br />
<br />
In this following one-and-a-quarter lecture one of those three men, Richard Feynman, perhaps the greatest professor of Physics (aka Reality) of all time, explains in his typical entertaining fashion, in a lecture in New Zealand in 1979. Enjoy, and the Wikipedia stuff follows.<br />
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<b>Quantum electrodynamics</b> (<b>QED</b>) is the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Relativity_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Relativity theory">relativistic</a> <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">quantum field theory</a> of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Electrodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electrodynamics">electrodynamics</a>. In essence, it describes how <a href="http://en.wikipedia.org/wiki/Light" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Light">light</a>and <a href="http://en.wikipedia.org/wiki/Matter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Matter">matter</a> interact and is the first theory where full agreement between <a href="http://en.wikipedia.org/wiki/Quantum_mechanics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum mechanics">quantum mechanics</a> and <a href="http://en.wikipedia.org/wiki/Special_relativity" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special relativity">special relativity</a> is achieved. QED mathematically describes all <a href="http://en.wikipedia.org/wiki/Phenomenon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Phenomenon">phenomena</a> involving <a href="http://en.wikipedia.org/wiki/Electric_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electric charge">electrically charged</a> particles interacting by means of exchange of<a href="http://en.wikipedia.org/wiki/Photon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Photon">photons</a> and represents the <a href="http://en.wikipedia.org/wiki/Quantum_mechanics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum mechanics">quantum</a> counterpart of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Classical_electrodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Classical electrodynamics">classical electrodynamics</a> giving a complete account of matter and light interaction. One of the founding fathers of QED, <a href="http://en.wikipedia.org/wiki/Richard_Feynman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard Feynman">Richard Feynman</a>, has called it "the jewel of physics" for its <a href="http://en.wikipedia.org/wiki/Precision_tests_of_QED" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Precision tests of QED">extremely accurate predictions</a> of quantities like the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Anomalous_magnetic_moment" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Anomalous magnetic moment">anomalous magnetic moment</a> of the electron, and the <a href="http://en.wikipedia.org/wiki/Lamb_shift" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lamb shift">Lamb shift</a> of the <a href="http://en.wikipedia.org/wiki/Energy_level" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Energy level">energy levels</a> of<a href="http://en.wikipedia.org/wiki/Hydrogen" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hydrogen">hydrogen</a>.<sup class="reference" id="cite_ref-feynbook1_0-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-feynbook1-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[1]</a></sup></div>
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In technical terms, QED can be described as a <a href="http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Perturbation theory (quantum mechanics)">perturbation theory</a> of the electromagnetic <a href="http://en.wikipedia.org/wiki/Vacuum_state" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vacuum state">quantum vacuum</a>.</div>
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Contents</h2>
<span class="toctoggle" style="-webkit-user-select: none;"> [<a class="internal" href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#" id="togglelink" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">hide</a>] </span></div>
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<li class="toclevel-1 tocsection-1" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#History" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1</span> <span class="toctext">History</span></a></li>
<li class="toclevel-1 tocsection-2" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Feynman.27s_view_of_quantum_electrodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2</span> <span class="toctext">Feynman's view of quantum electrodynamics</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin: 0px 0px 0px 2em; padding: 0px;">
<li class="toclevel-2 tocsection-3" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Introduction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.1</span> <span class="toctext">Introduction</span></a></li>
<li class="toclevel-2 tocsection-4" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Basic_constructions" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.2</span> <span class="toctext">Basic constructions</span></a></li>
<li class="toclevel-2 tocsection-5" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Probability_amplitudes" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.3</span> <span class="toctext">Probability amplitudes</span></a></li>
<li class="toclevel-2 tocsection-6" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Propagators" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.4</span> <span class="toctext">Propagators</span></a></li>
<li class="toclevel-2 tocsection-7" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Mass_renormalization" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.5</span> <span class="toctext">Mass renormalization</span></a></li>
<li class="toclevel-2 tocsection-8" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Conclusions" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.6</span> <span class="toctext">Conclusions</span></a></li>
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<li class="toclevel-1 tocsection-9" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Mathematics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3</span> <span class="toctext">Mathematics</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin: 0px 0px 0px 2em; padding: 0px;">
<li class="toclevel-2 tocsection-10" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Equations_of_motion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.1</span> <span class="toctext">Equations of motion</span></a></li>
<li class="toclevel-2 tocsection-11" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Interaction_picture" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.2</span> <span class="toctext">Interaction picture</span></a></li>
<li class="toclevel-2 tocsection-12" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Feynman_diagrams" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.3</span> <span class="toctext">Feynman diagrams</span></a></li>
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<li class="toclevel-1 tocsection-13" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Renormalizability" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4</span> <span class="toctext">Renormalizability</span></a></li>
<li class="toclevel-1 tocsection-14" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Nonconvergence_of_series" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5</span> <span class="toctext">Nonconvergence of series</span></a></li>
<li class="toclevel-1 tocsection-15" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#See_also" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-16" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#References" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">7</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-17" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Further_reading" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">8</span> <span class="toctext">Further reading</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin: 0px 0px 0px 2em; padding: 0px;">
<li class="toclevel-2 tocsection-18" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Books" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">8.1</span> <span class="toctext">Books</span></a></li>
<li class="toclevel-2 tocsection-19" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#Journals" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">8.2</span> <span class="toctext">Journals</span></a></li>
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<li class="toclevel-1 tocsection-20" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#External_links" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">9</span> <span class="toctext">External links</span></a></li>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: History">edit</a>]</span><span class="mw-headline" id="History">History</span></h2>
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Main article: <a href="http://en.wikipedia.org/wiki/History_of_quantum_mechanics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="History of quantum mechanics">History of quantum mechanics</a></div>
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<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Paul_Adrien_Maurice_Dirac" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Paul Adrien Maurice Dirac">Paul Dirac</a></div>
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The first formulation of a <a href="http://en.wikipedia.org/wiki/Quantum_mechanics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum mechanics">quantum theory</a> describing radiation and matter interaction is due to British scientist <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Paul_Adrien_Maurice_Dirac" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Paul Adrien Maurice Dirac">Paul Dirac</a>, who, during the 1920s, was first able to compute the coefficient of spontaneous emission of an <a href="http://en.wikipedia.org/wiki/Atom" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Atom">atom</a>.<sup class="reference" id="cite_ref-dirac_1-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-dirac-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup></div>
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Dirac described the quantization of the <a href="http://en.wikipedia.org/wiki/Electromagnetic_field" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electromagnetic field">electromagnetic field</a> as an ensemble of <a href="http://en.wikipedia.org/wiki/Harmonic_oscillator" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Harmonic oscillator">harmonic oscillators</a> with the introduction of the concept of <a href="http://en.wikipedia.org/wiki/Creation_and_annihilation_operators" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Creation and annihilation operators">creation and annihilation operators</a> of particles. In the following years, with contributions from <a href="http://en.wikipedia.org/wiki/Wolfgang_Pauli" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wolfgang Pauli">Wolfgang Pauli</a>, <a href="http://en.wikipedia.org/wiki/Eugene_Wigner" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eugene Wigner">Eugene Wigner</a>, <a href="http://en.wikipedia.org/wiki/Pascual_Jordan" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pascual Jordan">Pascual Jordan</a>, <a href="http://en.wikipedia.org/wiki/Werner_Heisenberg" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Werner Heisenberg">Werner Heisenberg</a>and an elegant formulation of quantum electrodynamics due to <a href="http://en.wikipedia.org/wiki/Enrico_Fermi" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Enrico Fermi">Enrico Fermi</a>,<sup class="reference" id="cite_ref-fermi_2-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-fermi-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup> physicists came to believe that, in principle, it would be possible to perform any computation for any physical process involving photons and charged particles. However, further studies by <a href="http://en.wikipedia.org/wiki/Felix_Bloch" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Felix Bloch">Felix Bloch</a> with <a class="new" href="http://en.wikipedia.org/w/index.php?title=Arnold_Nordsieck&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Arnold Nordsieck (page does not exist)">Arnold Nordsieck</a>,<sup class="reference" id="cite_ref-bloch_3-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-bloch-3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[4]</a></sup> and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Victor_Weisskopf" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Victor Weisskopf">Victor Weisskopf</a>,<sup class="reference" id="cite_ref-weisskopf_4-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-weisskopf-4" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[5]</a></sup> in 1937 and 1939, revealed that such computations were reliable only at a first order of <a href="http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Perturbation theory (quantum mechanics)">perturbation theory</a>, a problem already pointed out by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Robert_Oppenheimer" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Robert Oppenheimer">Robert Oppenheimer</a>.<sup class="reference" id="cite_ref-oppenheimer_5-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-oppenheimer-5" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[6]</a></sup> At higher orders in the series infinities emerged, making such computations meaningless and casting serious doubts on the internal consistency of the theory itself. With no solution for this problem known at the time, it appeared that a fundamental incompatibility existed between <a href="http://en.wikipedia.org/wiki/Special_relativity" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special relativity">special relativity</a> and <a href="http://en.wikipedia.org/wiki/Quantum_mechanics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum mechanics">quantum mechanics</a>.</div>
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<a href="http://en.wikipedia.org/wiki/Hans_Bethe" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hans Bethe">Hans Bethe</a></div>
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Difficulties with the theory increased through the end of 1940. Improvements in <a href="http://en.wikipedia.org/wiki/Microwave" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Microwave">microwave</a> technology made it possible to take more precise measurements of the shift of the levels of a <a href="http://en.wikipedia.org/wiki/Hydrogen_atom" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hydrogen atom">hydrogen atom</a>,<sup class="reference" id="cite_ref-lamb_6-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-lamb-6" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[7]</a></sup> now known as the <a href="http://en.wikipedia.org/wiki/Lamb_shift" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lamb shift">Lamb shift</a> and <a href="http://en.wikipedia.org/wiki/Magnetic_moment" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Magnetic moment">magnetic moment</a> of the electron.<sup class="reference" id="cite_ref-foley_7-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-foley-7" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[8]</a></sup> These experiments unequivocally exposed discrepancies which the theory was unable to explain.</div>
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A first indication of a possible way out was given by <a href="http://en.wikipedia.org/wiki/Hans_Bethe" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hans Bethe">Hans Bethe</a>. In 1947, while he was traveling by train to reach <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Schenectady" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schenectady">Schenectady</a> from <a href="http://en.wikipedia.org/wiki/New_York" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="New York">New York</a>,<sup class="reference" id="cite_ref-schweber_8-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-schweber-8" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[9]</a></sup> after giving a talk at the <a href="http://en.wikipedia.org/wiki/Shelter_Island_Conference" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Shelter Island Conference">conference at Shelter Island</a> on the subject, Bethe completed the first non-relativistic computation of the shift of the lines of the hydrogen atom as measured by Lamb and Retherford.<sup class="reference" id="cite_ref-bethe_9-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-bethe-9" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[10]</a></sup> Despite the limitations of the computation, agreement was excellent. The idea was simply to attach infinities to corrections at <a href="http://en.wikipedia.org/wiki/Mass" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mass">mass</a> and <a href="http://en.wikipedia.org/wiki/Charge_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Charge (physics)">charge</a> that were actually fixed to a finite value by experiments. In this way, the infinities get absorbed in those constants and yield a finite result in good agreement with experiments. This procedure was named <a href="http://en.wikipedia.org/wiki/Renormalization" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Renormalization">renormalization</a>.</div>
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<a href="http://en.wikipedia.org/wiki/Richard_Feynman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard Feynman">Feynman</a> (center) and<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Robert_Oppenheimer" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Robert Oppenheimer">Oppenheimer</a> (right) at <a href="http://en.wikipedia.org/wiki/Los_Alamos_National_Laboratory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Los Alamos National Laboratory">Los Alamos</a>.</div>
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Based on Bethe's intuition and fundamental papers on the subject by <a href="http://en.wikipedia.org/wiki/Sin-Itiro_Tomonaga" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sin-Itiro Tomonaga">Sin-Itiro Tomonaga</a>,<sup class="reference" id="cite_ref-tomonaga_10-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-tomonaga-10" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[11]</a></sup> <a href="http://en.wikipedia.org/wiki/Julian_Schwinger" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Julian Schwinger">Julian Schwinger</a>,<sup class="reference" id="cite_ref-schwinger1_11-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-schwinger1-11" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[12]</a></sup><sup class="reference" id="cite_ref-schwinger2_12-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-schwinger2-12" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[13]</a></sup> <a href="http://en.wikipedia.org/wiki/Richard_Feynman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard Feynman">Richard Feynman</a><sup class="reference" id="cite_ref-feynman1_13-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-feynman1-13" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[14]</a></sup><sup class="reference" id="cite_ref-feynman2_14-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-feynman2-14" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[15]</a></sup><sup class="reference" id="cite_ref-feynman3_15-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-feynman3-15" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[16]</a></sup> and<a href="http://en.wikipedia.org/wiki/Freeman_Dyson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Freeman Dyson">Freeman Dyson</a>,<sup class="reference" id="cite_ref-dyson1_16-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-dyson1-16" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[17]</a></sup><sup class="reference" id="cite_ref-dyson2_17-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-dyson2-17" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[18]</a></sup> it was finally possible to get fully <a href="http://en.wikipedia.org/wiki/Lorentz_covariance" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lorentz covariance">covariant</a> formulations that were finite at any order in a perturbation series of quantum electrodynamics. <a href="http://en.wikipedia.org/wiki/Sin-Itiro_Tomonaga" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sin-Itiro Tomonaga">Sin-Itiro Tomonaga</a>, <a href="http://en.wikipedia.org/wiki/Julian_Schwinger" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Julian Schwinger">Julian Schwinger</a> and <a href="http://en.wikipedia.org/wiki/Richard_Feynman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard Feynman">Richard Feynman</a> were jointly awarded with a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Nobel_prize_in_physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nobel prize in physics">Nobel prize in physics</a> in 1965 for their work in this area.<sup class="reference" id="cite_ref-nobel65_18-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-nobel65-18" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[19]</a></sup> Their contributions, and those of <a href="http://en.wikipedia.org/wiki/Freeman_Dyson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Freeman Dyson">Freeman Dyson</a>, were about <a href="http://en.wikipedia.org/wiki/Lorentz_covariance" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lorentz covariance">covariant</a> and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gauge_invariant" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge invariant">gauge invariant</a> formulations of quantum electrodynamics that allow computations of observables at any order of <a href="http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Perturbation theory (quantum mechanics)">perturbation theory</a>. Feynman's mathematical technique, based on his<a href="http://en.wikipedia.org/wiki/Feynman_diagram" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Feynman diagram">diagrams</a>, initially seemed very different from the field-theoretic, <a href="http://en.wikipedia.org/wiki/Operator_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Operator (physics)">operator</a>-based approach of Schwinger and Tomonaga, but <a href="http://en.wikipedia.org/wiki/Freeman_Dyson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Freeman Dyson">Freeman Dyson</a>later showed that the two approaches were equivalent.<sup class="reference" id="cite_ref-dyson1_16-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-dyson1-16" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[17]</a></sup> <a href="http://en.wikipedia.org/wiki/Renormalization" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Renormalization">Renormalization</a>, the need to attach a physical meaning at certain divergences appearing in the theory through <a href="http://en.wikipedia.org/wiki/Integral" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Integral">integrals</a>, has subsequently become one of the fundamental aspects of <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">quantum field theory</a> and has come to be seen as a criterion for a theory's general acceptability. Even though renormalization works very well in practice, Feynman was never entirely comfortable with its mathematical validity, even referring to renormalization as a "shell game" and "hocus pocus".<sup class="reference" id="cite_ref-feynbook2_19-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-feynbook2-19" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[20]</a></sup></div>
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QED has served as the model and template for all subsequent quantum field theories. One such subsequent theory is <a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum chromodynamics">quantum chromodynamics</a>, which began in the early 1960s and attained its present form in the 1975 work by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/H._David_Politzer" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="H. David Politzer">H. David Politzer</a>, <a href="http://en.wikipedia.org/wiki/Sidney_Coleman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sidney Coleman">Sidney Coleman</a>, <a href="http://en.wikipedia.org/wiki/David_Gross" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="David Gross">David Gross</a> and <a href="http://en.wikipedia.org/wiki/Frank_Wilczek" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Frank Wilczek">Frank Wilczek</a>. Building on the pioneering work of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Schwinger" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schwinger">Schwinger</a>, <a href="http://en.wikipedia.org/wiki/Gerald_Guralnik" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gerald Guralnik">Gerald Guralnik</a>, <a href="http://en.wikipedia.org/wiki/C._R._Hagen" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="C. R. Hagen">Dick Hagen</a>, and <a href="http://en.wikipedia.org/wiki/Tom_W._B._Kibble" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tom W. B. Kibble">Tom Kibble</a>,<sup class="reference" id="cite_ref-20" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-20" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[21]</a></sup><sup class="reference" id="cite_ref-21" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-21" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[22]</a></sup> <a href="http://en.wikipedia.org/wiki/Peter_Higgs" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Peter Higgs">Peter Higgs</a>, <a href="http://en.wikipedia.org/wiki/Jeffrey_Goldstone" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Jeffrey Goldstone">Jeffrey Goldstone</a>, and others, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Sheldon_Glashow" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sheldon Glashow">Sheldon Glashow</a>, <a href="http://en.wikipedia.org/wiki/Steven_Weinberg" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Steven Weinberg">Steven Weinberg</a> and <a href="http://en.wikipedia.org/wiki/Abdus_Salam" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Abdus Salam">Abdus Salam</a> independently showed how the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Weak_nuclear_force" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weak nuclear force">weak nuclear force</a> and quantum electrodynamics could be merged into a single <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Electroweak_force" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electroweak force">electroweak force</a>.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Feynman's view of quantum electrodynamics">edit</a>]</span><span class="mw-headline" id="Feynman.27s_view_of_quantum_electrodynamics">Feynman's view of quantum electrodynamics</span></h2>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Introduction">edit</a>]</span><span class="mw-headline" id="Introduction">Introduction</span></h3>
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Near the end of his life, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Richard_P._Feynman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard P. Feynman">Richard P. Feynman</a> gave a series of lectures on QED intended for the lay public. These lectures were transcribed and published as Feynman (1985),<a class="mw-redirect" href="http://en.wikipedia.org/wiki/QED_(book)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="QED (book)"><i>QED: The strange theory of light and matter</i></a>,<sup class="reference" id="cite_ref-feynbook1_0-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-feynbook1-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[1]</a></sup><sup class="reference" id="cite_ref-feynbook2_19-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-feynbook2-19" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[20]</a></sup> a classic non-mathematical exposition of QED from the point of view articulated below.</div>
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The key components of Feynman's presentation of QED are three basic actions.</div>
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<li style="margin-bottom: 0.1em;">A <a href="http://en.wikipedia.org/wiki/Photon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Photon">photon</a> goes from one place and time to another place and time.</li>
<li style="margin-bottom: 0.1em;">An <a href="http://en.wikipedia.org/wiki/Electron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron">electron</a> goes from one place and time to another place and time.</li>
<li style="margin-bottom: 0.1em;">An electron emits or absorbs a photon at a certain place and time.</li>
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<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Feynman_diagrams" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Feynman diagrams">Feynman diagram</a> elements</div>
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These actions are represented in a form of visual shorthand by the three basic elements of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Feynman_diagrams" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Feynman diagrams">Feynman diagrams</a>: a wavy line for the photon, a straight line for the electron and a junction of two straight lines and a wavy one for a vertex representing emission or absorption of a photon by an electron. These can all be seen in the adjacent diagram.</div>
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It is important not to over-interpret these diagrams. Nothing is implied about <i>how</i> a particle gets from one point to another. The diagrams do <i>not</i> imply that the particles are moving in straight or curved lines. They do <i>not</i> imply that the particles are moving with fixed speeds. The fact that the photon is often represented, by convention, by a wavy line and not a straight one does <i>not</i> imply that it is thought that it is more wavelike than is an electron. The images are just symbols to represent the actions above: photons and electrons do, somehow, move from point to point and electrons, somehow, emit and absorb photons. We do not know how these things happen, but the theory tells us about the probabilities of these things happening.</div>
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As well as the visual shorthand for the actions Feynman introduces another kind of shorthand for the numerical quantities which tell us about the probabilities. If a photon moves from one place and time – in shorthand, A – to another place and time – shorthand, B – the associated quantity is written in Feynman's shorthand as P(A to B). The similar quantity for an electron moving from C to D is written E(C to D). The quantity which tells us about the probability for the emission or absorption of a photon he calls 'j'. This is related to, but not the same as, the measured <a href="http://en.wikipedia.org/wiki/Elementary_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Elementary charge">electron charge</a> 'e'.</div>
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QED is based on the assumption that complex interactions of many electrons and photons can be represented by fitting together a suitable collection of the above three building blocks, and then using the probability-quantities to calculate the probability of any such complex interaction. It turns out that the basic idea of QED can be communicated while making the assumption that the quantities mentioned above are just our everyday <a href="http://en.wikipedia.org/wiki/Probability" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability">probabilities</a>. (A simplification of Feynman's book.) Later on this will be corrected to include specifically quantum mathematics, following Feynman.</div>
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The basic rules of probabilities that will be used are that a) if an event can happen in a variety of different ways then its probability is the <b>sum</b> of the probabilities of the possible ways and b) if a process involves a number of independent subprocesses then its probability is the <b>product</b> of the component probabilities.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=4" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Basic constructions">edit</a>]</span><span class="mw-headline" id="Basic_constructions">Basic constructions</span></h3>
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Suppose we start with one electron at a certain place and time (this place and time being given the arbitrary label A) and a photon at another place and time (given the label B). A typical question from a physical standpoint is: 'What is the probability of finding an electron at C (another place and a later time) and a photon at D (yet another place and time)?'. The simplest process to achieve this end is for the electron to move from A to C (an elementary action) and that the photon moves from B to D (another elementary action). From a knowledge of the probabilities of each of these subprocesses – E(A to C) and P(B to D) – then we would expect to calculate the probability of both happening by multiplying them, using rule b) above. This gives a simple estimated answer to our question.</div>
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<a href="http://en.wikipedia.org/wiki/Compton_scattering" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Compton scattering">Compton scattering</a></div>
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But there are other ways in which the end result could come about. The electron might move to a place and time E where it absorbs the photon; then move on before emitting another photon at F; then move on to C where it is detected, while the new photon moves on to D. The probability of this complex process can again be calculated by knowing the probabilities of each of the individual actions: three electron actions, two photon actions and two vertexes – one emission and one absorption. We would expect to find the total probability by multiplying the probabilities of each of the actions, for any chosen positions of E and F. We then, using rule a) above, have to add up all these probabilities for all the alternatives for E and F. (This is not elementary in practice, and involves <a href="http://en.wikipedia.org/wiki/Integral" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Integral">integration</a>.) But there is another possibility: that is that the electron first moves to G where it emits a photon which goes on to D, while the electron moves on to H, where it absorbs the first photon, before moving on to C. Again we can calculate the probability of these possibilities (for all points G and H). We then have a better estimation for the total probability by adding the probabilities of these two possibilities to our original simple estimate. Incidentally the name given to this process of a photon interacting with an electron in this way is <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Compton_Scattering" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Compton Scattering">Compton Scattering</a>.</div>
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There are an <i>infinite number</i> of other intermediate processes in which more and more photons are absorbed and/or emitted. For each of these possibilities there is a Feynman diagram describing it. This implies a complex computation for the resulting probabilities, but provided it is the case that the more complicated the diagram the less it contributes to the result, it is only a matter of time and effort to find as accurate an answer as one wants to the original question. This is the basic approach of QED. To calculate the probability of <b>any</b> interactive process between electrons and photons it is a matter of first noting, with Feynman diagrams, all the possible ways in which the process can be constructed from the three basic elements. Each diagram involves some calculation involving definite rules to find the associated probability.</div>
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That basic scaffolding remains when one moves to a quantum description but some conceptual changes are needed. One is that whereas we might expect in our everyday life that there would be some constraints on the points to which a particle can move, that is <b>not</b> true in full quantum electrodynamics. There is a possibility of an electron at A, or a photon at B, moving as a basic action to <i>any other place and time in the universe</i>. That includes places that could only be reached at speeds greater than that of light and also<i>earlier times</i>. (An electron moving backwards in time can be viewed as a <a href="http://en.wikipedia.org/wiki/Positron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Positron">positron</a> moving forward in time.)</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=5" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Probability amplitudes">edit</a>]</span><span class="mw-headline" id="Probability_amplitudes">Probability amplitudes</span></h3>
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Addition of probability amplitudes as complex numbers</div>
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<a href="http://en.wikipedia.org/wiki/Quantum_mechanics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum mechanics">Quantum mechanics</a> introduces an important change on the way probabilities are computed. It has been found that the quantities which we have to use to represent the probabilities are not the usual real numbers we use for probabilities in our everyday world, but <a href="http://en.wikipedia.org/wiki/Complex_number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Complex number">complex numbers</a> which are called <a href="http://en.wikipedia.org/wiki/Probability_amplitude" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability amplitude">probability amplitudes</a>. Feynman avoids exposing the reader to the mathematics of complex numbers by using a simple but accurate representation of them as arrows on a piece of paper or screen. (These must not be confused with the arrows of Feynman diagrams which are actually simplified representations in two dimensions of a relationship between points in three dimensions of space and one of time.) The amplitude-arrows are fundamental to the description of the world given by quantum theory. No satisfactory reason has been given for <i>why</i> they are needed. But pragmatically we have to accept that they are an essential part of our description of all quantum phenomena. They are related to our everyday ideas of probability by the simple rule that the probability of an event is the <b>square</b> of the length of the corresponding amplitude-arrow. So, for a given process, if two probability amplitudes, <b>v</b>and <b>w</b>, are involved, the probability of the process will be given either by</div>
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or</div>
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The rules as regards adding or multiplying, however, are the same as above. But where you would expect to add or multiply probabilities, instead you add or multiply probability amplitudes that now are complex numbers.</div>
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Multiplication of probability amplitudes as complex numbers</div>
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Addition and multiplication are familiar operations in the theory of complex numbers and are given in the figures. The sum is found as follows. Let the start of the second arrow be at the end of the first. The sum is then a third arrow that goes directly from the start of the first to the end of the second. The product of two arrows is an arrow whose length is the product of the two lengths. The direction of the product is found by adding the angles that each of the two have been turned through relative to a reference direction: that gives the angle that the product is turned relative to the reference direction.</div>
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That change, from probabilities to probability amplitudes, complicates the mathematics without changing the basic approach. But that change is still not quite enough because it fails to take into account the fact that both photons and electrons can be polarized, which is to say that their orientation in space and time have to be taken into account. Therefore P(A to B) actually consists of 16 complex numbers, or probability amplitude arrows. There are also some minor changes to do with the quantity "j", which may have to be rotated by a multiple of 90° for some polarizations, which is only of interest for the detailed bookkeeping.</div>
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Associated with the fact that the electron can be polarized is another small necessary detail which is connected with the fact that an electron is a <a href="http://en.wikipedia.org/wiki/Fermion" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermion">Fermion</a> and obeys <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Fermi-Dirac_statistics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermi-Dirac statistics">Fermi-Dirac statistics</a>. The basic rule is that if we have the probability amplitude for a given complex process involving more than one electron, then when we include (as we always must) the complementary Feynman diagram in which we just exchange two electron events, the resulting amplitude is the reverse – the negative – of the first. The simplest case would be two electrons starting at A and B ending at C and D. The amplitude would be calculated as the "difference", E(A to B)xE(C to D) – E(A to C)xE(B to D), where we would expect, from our everyday idea of probabilities, that it would be a sum.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=6" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Propagators">edit</a>]</span><span class="mw-headline" id="Propagators">Propagators</span></h3>
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Finally, one has to compute P(A to B) and E (C to D) corresponding to the probability amplitudes for the photon and the electron respectively. These are essentially the solutions of the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Dirac_Equation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac Equation">Dirac Equation</a> which describes the behavior of the electron's probability amplitude and the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Klein-Gordon_equation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Klein-Gordon equation">Klein-Gordon equation</a> which describes the behavior of the photon's probability amplitude. These are called <a href="http://en.wikipedia.org/wiki/Propagator" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Propagator">Feynman propagators</a>. The translation to a notation commonly used in the standard literature is as follows:</div>
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where a shorthand symbol such as <img alt="x_A" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/c/4/3c430fc59ecfcc331a864e17196f7c3b.png" style="border: none; margin: 0px; vertical-align: middle;" /> stands for the four real numbers which give the time and position in three dimensions of the point labeled A.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=7" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Mass renormalization">edit</a>]</span><span class="mw-headline" id="Mass_renormalization">Mass renormalization</span></h3>
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A problem arose historically which held up progress for twenty years: although we start with the assumption of three basic "simple" actions, the rules of the game say that if we want to calculate the probability amplitude for an electron to get from A to B we must take into account <b>all</b> the possible ways: all possible Feynman diagrams with those end points. Thus there will be a way in which the electron travels to C, emits a photon there and then absorbs it again at D before moving on to B. Or it could do this kind of thing twice, or more. In short we have a <a href="http://en.wikipedia.org/wiki/Fractal" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fractal">fractal-like situation</a> in which if we look closely at a line it breaks up into a collection of "simple" lines, each of which, if looked at closely, are in turn composed of "simple" lines, and so on <i>ad infinitum</i>. This is a very difficult situation to handle. If adding that detail only altered things slightly then it would not have been too bad, but disaster struck when it was found that the simple correction mentioned above led to <i>infinite</i> probability amplitudes. In time this problem was "fixed" by the technique of <a href="http://en.wikipedia.org/wiki/Renormalization" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Renormalization">renormalization</a> (see below and the article on <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Self-Energy" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Self-Energy">mass renormalization</a>). However, Feynman himself remained unhappy about it, calling it a "dippy process".<sup class="reference" id="cite_ref-feynbook2_19-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-feynbook2-19" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[20]</a></sup></div>
<h3 style="background-image: none; border-bottom-style: none; font-size: 18px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=8" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Conclusions">edit</a>]</span><span class="mw-headline" id="Conclusions">Conclusions</span></h3>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Within the above framework physicists were then able to calculate to a high degree of accuracy some of the properties of electrons, such as the <a href="http://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Anomalous magnetic dipole moment">anomalous magnetic dipole moment</a>. However, as Feynman points out, it fails totally to explain why particles such as the electron have the masses they do. "There is no theory that adequately explains these numbers. We use the numbers in all our theories, but we don't understand them – what they are, or where they come from. I believe that from a fundamental point of view, this is a very interesting and serious problem."<sup class="reference" id="cite_ref-feynbook3_22-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-feynbook3-22" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[23]</a></sup></div>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=9" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Mathematics">edit</a>]</span><span class="mw-headline" id="Mathematics">Mathematics</span></h2>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Mathematically, QED is an <a href="http://en.wikipedia.org/wiki/Abelian_group" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Abelian group">abelian</a> <a href="http://en.wikipedia.org/wiki/Gauge_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge theory">gauge theory</a> with the symmetry group <a class="mw-redirect" href="http://en.wikipedia.org/wiki/U(1)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="U(1)">U(1)</a>. The <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gauge_field" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge field">gauge field</a>, which mediates the interaction between the charged <a href="http://en.wikipedia.org/wiki/Spin_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin (physics)">spin-1/2</a> <a href="http://en.wikipedia.org/wiki/Field_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Field (physics)">fields</a>, is the<a href="http://en.wikipedia.org/wiki/Electromagnetic_field" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electromagnetic field">electromagnetic field</a>. The QED <a href="http://en.wikipedia.org/wiki/Lagrangian" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lagrangian">Lagrangian</a> for a spin-1/2 field interacting with the electromagnetic field is given by the real part of</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><table cellpadding="5" style="background-color: #ecfcf4; background-position: initial initial; background-repeat: initial initial; border: 2px solid rgb(80, 200, 120); font-size: 13px; text-align: center;"><tbody>
<tr><td><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
<img alt="\mathcal{L}=\bar\psi(i\gamma^\mu D_\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/e/f/0ef7214b5093dbe29546f6ae93f97e51.png" style="border: none; margin: 0px; vertical-align: middle;" /></div>
</td></tr>
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</dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
where</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \gamma^\mu " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/6/0/d60917d1e76e82865bc6b29078c627c5.png" style="border: none; vertical-align: middle;" /> are <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Dirac_matrices" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac matrices">Dirac matrices</a>;</dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\psi" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/9/d/19df1c2726ed43128440c1157f72a937.png" style="border: none; vertical-align: middle;" /> a <a href="http://en.wikipedia.org/wiki/Bispinor" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bispinor">bispinor</a> <a href="http://en.wikipedia.org/wiki/Field_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Field (physics)">field</a> of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Spin-1/2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin-1/2">spin-1/2</a> particles (e.g. <a href="http://en.wikipedia.org/wiki/Electron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron">electron</a>-<a href="http://en.wikipedia.org/wiki/Positron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Positron">positron</a> field);</dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\bar\psi\equiv\psi^\dagger\gamma_0" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/a/f3a879eb983a31c4092a83ffd52911b5.png" style="border: none; vertical-align: middle;" />, called "psi-bar", is sometimes referred to as <a href="http://en.wikipedia.org/wiki/Dirac_adjoint" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac adjoint">Dirac adjoint</a>;</dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="D_\mu \equiv \partial_\mu+ieA_\mu+ieB_\mu \,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/4/7/e4757a2e48b4b1763d5b0fa5835bc6de.png" style="border: none; vertical-align: middle;" /> is the <a href="http://en.wikipedia.org/wiki/Gauge_covariant_derivative" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge covariant derivative">gauge covariant derivative</a>;</dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><i>e</i> is the <a href="http://en.wikipedia.org/wiki/Fine-structure_constant" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fine-structure constant">coupling constant</a>, equal to the <a href="http://en.wikipedia.org/wiki/Electric_charge" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electric charge">electric charge</a> of the bispinor field;</dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><i>A</i><sub style="line-height: 1em;">μ</sub> is the <a href="http://en.wikipedia.org/wiki/Covariance" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Covariance">covariant</a> <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Four-potential" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Four-potential">four-potential</a> of the electromagnetic field generated by the electron itself;</dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><i>B</i><sub style="line-height: 1em;">μ</sub> is the external field imposed by external source;</dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/3/b/63baedbcac688d1ff42ae2d8c076e0db.png" style="border: none; vertical-align: middle;" /> is the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Electromagnetic_field_tensor" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electromagnetic field tensor">electromagnetic field tensor</a>.</dd></dl>
<h3 style="background-image: none; border-bottom-style: none; font-size: 18px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=10" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Equations of motion">edit</a>]</span><span class="mw-headline" id="Equations_of_motion">Equations of motion</span></h3>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
To begin, substituting the definition of <i>D</i> into the Lagrangian gives us</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\mathcal{L} = i \bar\psi \gamma^\mu \partial_\mu \psi - e\bar{\psi}\gamma_\mu (A^\mu+B^\mu) \psi -m \bar{\psi} \psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}. \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/7/1/a71073c9798671c3b9a1027f39893c1a.png" style="border: none; vertical-align: middle;" /></dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Next, we can substitute this Lagrangian into the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Euler-Lagrange_equation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Euler-Lagrange equation">Euler-Lagrange equation</a> of motion for a field:</div>
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<img alt=" \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi )} \right) - \frac{\partial \mathcal{L}}{\partial \psi} = 0 \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/9/5/9951590a2d1ae79b258286c4589e6fe7.png" style="border: none; margin: 0px; vertical-align: middle;" /></div>
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</div>
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<br /></div>
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<br /></div>
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<br /></div>
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</td><td nowrap="nowrap" style="border: none; padding: 0.08em; vertical-align: middle;"><div style="line-height: 1.5em; margin: 0pt;">
<b>(<cite id="math_2" style="font-style: inherit;"></cite><span class="reference plainlinksneverexpand"><cite id="math_2" style="font-style: inherit;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#equation_2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">2</a></cite><b><cite id="math_2" style="font-style: inherit;"></cite>)</b></span></b></div>
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<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
to find the field equations for QED.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The two terms from this Lagrangian are then</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \psi )} \right) = \partial_\mu \left( i \bar{\psi} \gamma^\mu \right), \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/2/2/a226c5f7bca017b2e1291473f6512dd5.png" style="border: none; vertical-align: middle;" /></dd></dl>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\frac{\partial \mathcal{L}}{\partial \psi} = -e\bar{\psi}\gamma_\mu (A^\mu+B^\mu) - m \bar{\psi}. \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/1/d/81deca0fc3b427dc3c9a91a5444d2331.png" style="border: none; vertical-align: middle;" /></dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Substituting these two back into the Euler-Lagrange equation (<cite id="equation_2" style="font-style: normal;"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#math_2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">2</a></b></cite>) results in</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="i \partial_\mu \bar{\psi} \gamma^\mu + e\bar{\psi}\gamma_\mu (A^\mu+B^\mu) + m \bar{\psi} = 0 \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/3/b/b3bada1ef08b628d5a11843642d23b1f.png" style="border: none; vertical-align: middle;" /></dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
with complex conjugate</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="i \gamma^\mu \partial_\mu \psi - e \gamma_\mu (A^\mu+B^\mu) \psi - m \psi = 0. \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/7/a/97a479b024fa2b0c2490098fca627520.png" style="border: none; vertical-align: middle;" /></dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Bringing the middle term to the right-hand side transforms this second equation into</div>
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<img alt="i \gamma^\mu \partial_\mu \psi - m \psi = e \gamma_\mu (A^\mu+B^\mu) \psi \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/c/6/4c6c603619c7cf8a0c17240b2a44ca4d.png" style="border: none; margin: 0px; vertical-align: middle;" /></div>
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</dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The left-hand side is like the original <a href="http://en.wikipedia.org/wiki/Dirac_equation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac equation">Dirac equation</a> and the right-hand side is the interaction with the electromagnetic field.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
One further important equation can be found by substituting the Lagrangian into another Euler-Lagrange equation, this time for the field, <i>A</i><sup style="line-height: 1em;">μ</sup>:</div>
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<tr><td nowrap="nowrap" style="border: none; padding: 0.08em; vertical-align: middle;"><div style="line-height: 1.5em;">
<img alt=" \partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) - \frac{\partial \mathcal{L}}{\partial A_\mu} = 0\,. " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/c/a/0cae87b1a6e1e6b4601e8a4c509b7f69.png" style="border: none; margin: 0px; vertical-align: middle;" /></div>
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</div>
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<br /></div>
</td><td style="border: none; padding: 0.08em; width: 730px;"><div style="font-size: 1pt; line-height: 1.5em;">
<br /></div>
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<br /></div>
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<br /></div>
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</td><td nowrap="nowrap" style="border: none; padding: 0.08em; vertical-align: middle;"><div style="line-height: 1.5em; margin: 0pt;">
<b>(<cite id="math_3" style="font-style: inherit;"></cite><span class="reference plainlinksneverexpand"><cite id="math_3" style="font-style: inherit;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#equation_3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">3</a></cite><b><cite id="math_3" style="font-style: inherit;"></cite>)</b></span></b></div>
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<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
The two terms this time are</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) = \partial_\nu \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right), \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/c/f/acfa5691fca506b163a3d9b426e97650.png" style="border: none; vertical-align: middle;" /></dd></dl>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\frac{\partial \mathcal{L}}{\partial A_\mu} = -e\bar{\psi} \gamma^\mu \psi \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/d/b/ddbf5a5a8748d1e668c169ce31680014.png" style="border: none; vertical-align: middle;" /></dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
and these two terms, when substituted back into (<cite id="equation_3" style="font-style: normal;"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#math_3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">3</a></b></cite>) give us</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><table cellpadding="5" style="background-color: mintcream; background-position: initial initial; background-repeat: initial initial; border: 2px solid rgb(0, 115, 207); font-size: 13px; text-align: center;"><tbody>
<tr><td><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
<img alt="\partial_\nu F^{\nu \mu} = e \bar{\psi} \gamma^\mu \psi \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/6/b/c6b89f25c507bc03b080699445747eb8.png" style="border: none; margin: 0px; vertical-align: middle;" /></div>
</td></tr>
</tbody></table>
</dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Now, if we impose the Lorenz-Gauge condition, i.e., that the divergence of the four potential vanishes then we get</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\Box A^{\mu}=e\bar{\psi} \gamma^{\mu} \psi. " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/f/6/6f68645b521fffb743babab41dede1db.png" style="border: none; vertical-align: middle;" /></dd></dl>
<h3 style="background-image: none; border-bottom-style: none; font-size: 18px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=11" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Interaction picture">edit</a>]</span><span class="mw-headline" id="Interaction_picture">Interaction picture</span></h3>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
This theory can be straightforwardly quantized by treating bosonic and fermionic sectors as free. This permits us to build a set of asymptotic states which can be used to start a computation of the probability amplitudes for different processes. In order to do so, we have to compute an <a href="http://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hamiltonian (quantum mechanics)">evolution operator</a> that, for a given initial state <img alt="|i\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/7/a/f7abcf2a267cc27325cdbc4d9a21303f.png" style="border: none; margin: 0px; vertical-align: middle;" />, will give a final state <img alt="\langle f|" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/5/4/554eb614d30f236d62b247f6ce389af3.png" style="border: none; margin: 0px; vertical-align: middle;" /> in such a way to have</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="M_{fi}=\langle f|U|i\rangle." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/8/c/d8c0ca1da53e529f77f3779bfc404466.png" style="border: none; vertical-align: middle;" /></dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
This technique is also known as the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/S-Matrix" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="S-Matrix">S-Matrix</a>. The evolution operator is obtained in the <a href="http://en.wikipedia.org/wiki/Interaction_picture" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Interaction picture">interaction picture</a> where time evolution is given by the interaction Hamiltonian, which is the integral over space of the second term in the Lagrangian density given above:</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="V=e\int d^3x\bar\psi\gamma^\mu\psi A_\mu" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/3/b/73b1ca254f3113adafbb40ce3c6e97ce.png" style="border: none; vertical-align: middle;" /></dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
and so, one has</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="U=T\exp\left[-\frac{i}{\hbar}\int_{t_0}^tdt'V(t')\right]" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/f/c/ffce1a89c334d137bda73d0ccba5aa1a.png" style="border: none; vertical-align: middle;" /></dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
where <i>T</i> is the <a href="http://en.wikipedia.org/wiki/Path-ordering" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Path-ordering">time ordering</a> operator. This evolution operator only has meaning as a series, and what we get here is a <a href="http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Perturbation theory (quantum mechanics)">perturbation series</a> with the <a href="http://en.wikipedia.org/wiki/Fine-structure_constant" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fine-structure constant">fine structure constant</a> as the development parameter. This series is called the <a href="http://en.wikipedia.org/wiki/Dyson_series" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dyson series">Dyson series</a>.</div>
<h3 style="background-image: none; border-bottom-style: none; font-size: 18px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=12" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Feynman diagrams">edit</a>]</span><span class="mw-headline" id="Feynman_diagrams">Feynman diagrams</span></h3>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Despite the conceptual clarity of this Feynman approach to QED, almost no textbooks follow him in their presentation. When performing calculations it is much easier to work with the <a href="http://en.wikipedia.org/wiki/Fourier_transform" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fourier transform">Fourier transforms</a> of the <a href="http://en.wikipedia.org/wiki/Propagator" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Propagator">propagators</a>. Quantum physics considers particle's <a href="http://en.wikipedia.org/wiki/Momentum" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Momentum">momenta</a> rather than their positions, and it is convenient to think of particles as being created or annihilated when they interact. Feynman diagrams then <i>look</i> the same, but the lines have different interpretations. The electron line represents an electron with a given energy and momentum, with a similar interpretation of the photon line. A vertex diagram represents the annihilation of one electron and the creation of another together with the absorption or creation of a photon, each having specified energies and momenta.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Using <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Wick_theorem" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wick theorem">Wick theorem</a> on the terms of the Dyson series, all the terms of the <a href="http://en.wikipedia.org/wiki/S-matrix" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="S-matrix">S-matrix</a> for quantum electrodynamics can be computed through the technique of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Feynman_diagrams" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Feynman diagrams">Feynman diagrams</a>. In this case rules for drawing are the following</div>
<br />
<center><a class="image" href="http://en.wikipedia.org/wiki/File:Qed_rules.jpg" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><img alt="Qed rules.jpg" height="286" src="http://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Qed_rules.jpg/488px-Qed_rules.jpg" style="border: none; vertical-align: middle;" width="488" /></a></center><center><a class="image" href="http://en.wikipedia.org/wiki/File:Qed2e.jpg" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><img alt="Qed2e.jpg" height="446" src="http://upload.wikimedia.org/wikipedia/commons/thumb/9/91/Qed2e.jpg/488px-Qed2e.jpg" style="border: none; vertical-align: middle;" width="488" /></a></center><br />
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
To these rules we must add a further one for closed loops that implies an integration on momenta <img alt="\int d^4p/(2\pi)^4" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/b/e/abee01bb18096d7d0b4caa9ccb610d54.png" style="border: none; margin: 0px; vertical-align: middle;" />, since these internal ("virtual") particles are not constrained to any specific energy-momentum - even that usually required by special relativity (see <a href="http://en.wikipedia.org/wiki/Propagator#Propagators_in_Feynman_diagrams" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Propagator">this article</a> for details). From them, computations of <a href="http://en.wikipedia.org/wiki/Probability_amplitude" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability amplitude">probability amplitudes</a>are straightforwardly given. An example is <a href="http://en.wikipedia.org/wiki/Compton_scattering" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Compton scattering">Compton scattering</a>, with an <a href="http://en.wikipedia.org/wiki/Electron" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron">electron</a> and a <a href="http://en.wikipedia.org/wiki/Photon" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Photon">photon</a> undergoing <a href="http://en.wikipedia.org/wiki/Elastic_scattering" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Elastic scattering">elastic scattering</a>. Feynman diagrams are in this case</div>
<br />
<center><a class="image" href="http://en.wikipedia.org/wiki/File:Compton_qed.jpg" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><img alt="Compton qed.jpg" height="411" src="http://upload.wikimedia.org/wikipedia/commons/thumb/5/51/Compton_qed.jpg/300px-Compton_qed.jpg" style="border: none; vertical-align: middle;" width="300" /></a></center><br />
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
and so we are able to get the corresponding amplitude at the first order of a <a href="http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Perturbation theory (quantum mechanics)">perturbation series</a> for the <a href="http://en.wikipedia.org/wiki/S-matrix" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="S-matrix">S-matrix</a>:</div>
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="M_{fi}=(ie)^{2}\overline{u}(\vec{p}\,',s')\epsilon\!\!\!/\,'(\vec{k}\,',\lambda')^{*}{p\!\!\!/+k\!\!\!/+m_{e}
\over (p+k)^{2}-m^{2}_{e}}\epsilon\!\!\!/(\vec{k},\lambda)u(\vec{p},s)+(ie)^{2}\overline{u}(\vec{p}\,',s')\epsilon\!\!\!/(\vec{k},\lambda){p\!\!\!/-k\!\!\!/'+m_{e}\over (p-k')^{2}-m^{2}_{e}}\epsilon\!\!\!/\,'(\vec{k}\,',\lambda')^{*}u(\vec{p},s)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/f/3/9f36a6ce0ea69c67b1ad3fa0a4f17d15.png" style="border: none; vertical-align: middle;" /></dd></dl>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
from which we are able to compute the <a href="http://en.wikipedia.org/wiki/Cross_section_(physics)" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cross section (physics)">cross section</a> for this scattering.</div>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=13" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Renormalizability">edit</a>]</span><span class="mw-headline" id="Renormalizability">Renormalizability</span></h2>
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Higher order terms can be straightforwardly computed for the evolution operator but these terms display diagrams containing the following simpler ones</div>
<br />
<center><ul class="gallery" style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 2px; padding: 2px; zoom: 1;">
<li class="gallerybox" style="display: inline-block; margin-bottom: 0.1em; vertical-align: top; width: 155px; zoom: 1;"><div style="width: 155px;">
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<a class="image" href="http://en.wikipedia.org/wiki/File:Vacuum_polarization.svg" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><img alt="" height="63" src="http://upload.wikimedia.org/wikipedia/commons/thumb/9/99/Vacuum_polarization.svg/120px-Vacuum_polarization.svg.png" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border: none; display: block; margin: 0px auto; vertical-align: middle;" width="120" /></a></div>
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<div class="gallerytext" style="font-size: 12px; overflow: hidden; padding: 2px 4px; word-wrap: break-word;">
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
One-loop contribution to the<a href="http://en.wikipedia.org/wiki/Vacuum_polarization" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vacuum polarization">vacuum polarization</a>function <img alt="\Pi\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/0/6/206932cf05272e53cc4b051c110e58f0.png" style="border: none; margin: 0px; vertical-align: middle;" /></div>
</div>
</div>
</li>
<li class="gallerybox" style="display: inline-block; margin-bottom: 0.1em; vertical-align: top; width: 155px; zoom: 1;"><div style="width: 155px;">
<div class="thumb" style="background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); margin: 2px; width: 150px;">
<div style="margin: 55px auto;">
<a class="image" href="http://en.wikipedia.org/wiki/File:Electron_self_energy.svg" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><img alt="" height="40" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Electron_self_energy.svg/120px-Electron_self_energy.svg.png" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border: none; display: block; margin: 0px auto; vertical-align: middle;" width="120" /></a></div>
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<div class="gallerytext" style="font-size: 12px; overflow: hidden; padding: 2px 4px; word-wrap: break-word;">
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
One-loop contribution to the electron <a href="http://en.wikipedia.org/wiki/Self-energy" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Self-energy">self-energy</a> function<img alt="\Sigma \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/b/4/cb4efae84f23aaf41fa73a2bf19e9068.png" style="border: none; margin: 0px; vertical-align: middle;" /></div>
</div>
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</li>
<li class="gallerybox" style="display: inline-block; margin-bottom: 0.1em; vertical-align: top; width: 155px; zoom: 1;"><div style="width: 155px;">
<div class="thumb" style="background-color: #f9f9f9; border: 1px solid rgb(204, 204, 204); margin: 2px; width: 150px;">
<div style="margin: 15px auto;">
<a class="image" href="http://en.wikipedia.org/wiki/File:Vertex_correction.svg" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><img alt="" height="120" src="http://upload.wikimedia.org/wikipedia/commons/thumb/8/87/Vertex_correction.svg/114px-Vertex_correction.svg.png" style="background-color: white; background-position: initial initial; background-repeat: initial initial; border: none; display: block; margin: 0px auto; vertical-align: middle;" width="114" /></a></div>
</div>
<div class="gallerytext" style="font-size: 12px; overflow: hidden; padding: 2px 4px; word-wrap: break-word;">
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
One-loop contribution to the<a href="http://en.wikipedia.org/wiki/Vertex_function" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vertex function">vertex function</a> <img alt="\Gamma\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/c/0/9c057f7e7e7e8a781beff7d4a3f30980.png" style="border: none; margin: 0px; vertical-align: middle;" /></div>
</div>
</div>
</li>
</ul>
</center><br />
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
that, being closed loops, imply the presence of diverging <a href="http://en.wikipedia.org/wiki/Integral" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Integral">integrals</a> having no mathematical meaning. To overcome this difficulty, a technique like <a href="http://en.wikipedia.org/wiki/Renormalization" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Renormalization">renormalization</a> has been devised, producing finite results in very close agreement with experiments. It is important to note that a criterion for theory being meaningful after renormalization is that the number of diverging diagrams is finite. In this case the theory is said <b>renormalizable</b>. The reason for this is that to get observables renormalized one needs a finite number of constants to maintain the predictive value of the theory untouched. This is exactly the case of quantum electrodynamics displaying just three diverging diagrams. This procedure gives observables in very close agreement with experiment as seen e.g. for electron <a href="http://en.wikipedia.org/wiki/Gyromagnetic_ratio" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gyromagnetic ratio">gyromagnetic ratio</a>.</div>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
Renormalizability has become an essential criterion for a <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">quantum field theory</a> to be considered as a viable one. All the theories describing <a href="http://en.wikipedia.org/wiki/Fundamental_interaction" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fundamental interaction">fundamental interactions</a>, except<a href="http://en.wikipedia.org/wiki/Gravitation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gravitation">gravitation</a> whose quantum counterpart is presently under very active research, are renormalizable theories.</div>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=14" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Nonconvergence of series">edit</a>]</span><span class="mw-headline" id="Nonconvergence_of_series">Nonconvergence of series</span></h2>
<div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
An argument by <a href="http://en.wikipedia.org/wiki/Freeman_Dyson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Freeman Dyson">Freeman Dyson</a> shows that the <a href="http://en.wikipedia.org/wiki/Radius_of_convergence" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Radius of convergence">radius of convergence</a> of the perturbation series in QED is zero.<sup class="reference" id="cite_ref-23" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_note-23" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[24]</a></sup> The basic argument goes as follows: if the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Fine_structure_constant" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fine structure constant">coupling constant</a> were negative, this would be equivalent to the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Coulomb_force_constant" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Coulomb force constant">Coulomb force constant</a> being negative. This would "reverse" the electromagnetic interaction so that <i>like</i> charges would<i>attract</i> and <i>unlike</i> charges would <i>repel</i>. This would render the vacuum unstable against decay into a cluster of electrons on one side of the universe and a cluster of positrons on the other side of the universe. Because the theory is 'sick' for any negative value of the coupling constant, the series do not converge, but are an <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Asymptotic_series" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Asymptotic series">asymptotic series</a>. This can be taken as a need for a new theory, a problem with <a href="http://en.wikipedia.org/wiki/Perturbation_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Perturbation theory">perturbation theory</a>, or ignored by taking a "shut-up-and-calculate" approach.</div>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=15" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: See also">edit</a>]</span><span class="mw-headline" id="See_also">See also</span></h2>
<table cellpadding="0" cellspacing="0" class="multicol" style="background-color: transparent; background-position: initial initial; background-repeat: initial initial; font-size: 13px; width: 1000px;"><tbody>
<tr><td align="left" valign="top" width="50%"><ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Abraham-Lorentz_force" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Abraham-Lorentz force">Abraham-Lorentz force</a></li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Anomalous_magnetic_moment" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Anomalous magnetic moment">Anomalous magnetic moment</a></li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Basics_of_quantum_mechanics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Basics of quantum mechanics">Basics of quantum mechanics</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Bhabha_scattering" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bhabha scattering">Bhabha scattering</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Cavity_quantum_electrodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cavity quantum electrodynamics">Cavity quantum electrodynamics</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Compton_scattering" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Compton scattering">Compton scattering</a></li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Euler-Heisenberg_Lagrangian" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Euler-Heisenberg Lagrangian">Euler-Heisenberg Lagrangian</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Path_integral_formulation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Path integral formulation">Feynman path integrals</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Gauge_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge theory">Gauge theory</a></li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gupta-Bleuler_formalism" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gupta-Bleuler formalism">Gupta-Bleuler formalism</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Lamb_shift" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lamb shift">Lamb shift</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Landau_pole" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Landau pole">Landau pole</a></li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Moeller_scattering" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Moeller scattering">Moeller scattering</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Photon_dynamics_in_the_double-slit_experiment" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Photon dynamics in the double-slit experiment">Photon dynamics in the double-slit experiment</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Photon_polarization" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Photon polarization">Photon polarization</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Positronium" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Positronium">Positronium</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Propagator" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Propagator">Propagators</a></li>
</ul>
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</div>
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/QED_vacuum" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="QED vacuum">QED vacuum</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/QED:_The_Strange_Theory_of_Light_and_Matter" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="QED: The Strange Theory of Light and Matter">QED: The Strange Theory of Light and Matter</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_chromodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum chromodynamics">Quantum chromodynamics</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">Quantum field theory</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_gauge_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum gauge theory">Quantum gauge theory</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Renormalization" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Renormalization">Renormalization</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Scalar_electrodynamics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Scalar electrodynamics">Scalar electrodynamics</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schrödinger equation">Schrödinger equation</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Schwinger_model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schwinger model">Schwinger model</a></li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Schwinger-Dyson_equation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schwinger-Dyson equation">Schwinger-Dyson equation</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Self-energy" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Self-energy">Self-energy</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Standard_Model" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Standard Model">Standard Model</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Theoretical_and_experimental_justification_for_the_Schr%C3%B6dinger_equation" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theoretical and experimental justification for the Schrödinger equation">Theoretical and experimental justification for the Schrödinger equation</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Vacuum_polarization" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vacuum polarization">Vacuum polarization</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Vertex_function" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vertex function">Vertex function</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Ward%E2%80%93Takahashi_identity" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Ward–Takahashi identity">Ward–Takahashi identity</a></li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Wheeler-Feynman_absorber_theory" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wheeler-Feynman absorber theory">Wheeler-Feynman absorber theory</a></li>
</ul>
</td></tr>
</tbody></table>
<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=16" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: References">edit</a>]</span><span class="mw-headline" id="References">References</span></h2>
<div class="reflist references-column-count references-column-count-2" style="-webkit-column-count: 2; font-size: 12px; list-style-type: decimal; margin-bottom: 0.5em;">
<ol class="references" style="line-height: 1.5em; list-style-image: none; list-style-type: inherit; margin: 0.3em 0px 0.5em 3.2em; padding: 0px;">
<li id="cite_note-feynbook1-0" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink">^ <a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-feynbook1_0-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-feynbook1_0-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a></span> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Richard_Feynman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard Feynman">Feynman, Richard</a> (1985). "Chapter 1". <i>QED: The Strange Theory of Light and Matter</i>. Princeton University Press. p. 6. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-691-12575-6" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-691-12575-6">978-0-691-12575-6</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Chapter+1&rft.atitle=QED%3A+The+Strange+Theory+of+Light+and+Matter&rft.aulast=Feynman&rft.aufirst=Richard&rft.au=Feynman%2C%26%2332%3BRichard&rft.date=1985&rft.pages=p.%26nbsp%3B6&rft.pub=Princeton+University+Press&rft.isbn=978-0-691-12575-6&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-dirac-1" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-dirac_1-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Paul_Adrien_Maurice_Dirac" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Paul Adrien Maurice Dirac">P.A.M. Dirac</a> (1927). "The Quantum Theory of the Emission and Absorption of Radiation". <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Proceedings_of_the_Royal_Society_of_London_A" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Proceedings of the Royal Society of London A">Proceedings of the Royal Society of London A</a></i> <b>114</b> (767): 243–265. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/1927RSPSA.114..243D" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1927RSPSA.114..243D</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1098%2Frspa.1927.0039" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1098/rspa.1927.0039</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Quantum+Theory+of+the+Emission+and+Absorption+of+Radiation&rft.jtitle=%5B%5BProceedings+of+the+Royal+Society+of+London+A%5D%5D&rft.aulast=P.A.M.+Dirac&rft.au=P.A.M.+Dirac&rft.date=1927&rft.volume=114&rft.issue=767&rft.pages=243%E2%80%93265&rft_id=info:bibcode/1927RSPSA.114..243D&rft_id=info:doi/10.1098%2Frspa.1927.0039&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-fermi-2" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-fermi_2-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Enrico_Fermi" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Enrico Fermi">E. Fermi</a> (1932). "Quantum Theory of Radiation". <i><a href="http://en.wikipedia.org/wiki/Reviews_of_Modern_Physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Reviews of Modern Physics">Reviews of Modern Physics</a></i> <b>4</b>: 87–132. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1932RvMP....4...87F" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1932RvMP....4...87F</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FRevModPhys.4.87" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/RevModPhys.4.87</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Quantum+Theory+of+Radiation&rft.jtitle=%5B%5BReviews+of+Modern+Physics%5D%5D&rft.aulast=E.+Fermi&rft.au=E.+Fermi&rft.date=1932&rft.volume=4&rft.pages=87%E2%80%93132&rft_id=info:bibcode/1932RvMP....4...87F&rft_id=info:doi/10.1103%2FRevModPhys.4.87&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-bloch-3" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-bloch_3-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Felix_Bloch" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Felix Bloch">F. Bloch</a>; <a class="new" href="http://en.wikipedia.org/w/index.php?title=Arnold_Nordsieck&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Arnold Nordsieck (page does not exist)">A. Nordsieck</a> (1937). "Note on the Radiation Field of the Electron".<i><a href="http://en.wikipedia.org/wiki/Physical_Review" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review">Physical Review</a></i> <b>52</b> (2): 54–59. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1937PhRv...52...54B" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1937PhRv...52...54B</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRev.52.54" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRev.52.54</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Note+on+the+Radiation+Field+of+the+Electron&rft.jtitle=%5B%5BPhysical+Review%5D%5D&rft.aulast=F.+Bloch&rft.au=F.+Bloch&rft.au=A.+Nordsieck&rft.date=1937&rft.volume=52&rft.issue=2&rft.pages=54%E2%80%9359&rft_id=info:bibcode/1937PhRv...52...54B&rft_id=info:doi/10.1103%2FPhysRev.52.54&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-weisskopf-4" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-weisskopf_4-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Victor_Weisskopf" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Victor Weisskopf">V. F. Weisskopf</a> (1939). "On the Self-Energy and the Electromagnetic Field of the Electron". <i><a href="http://en.wikipedia.org/wiki/Physical_Review" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review">Physical Review</a></i> <b>56</b>: 72–85. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1939PhRv...56...72W" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1939PhRv...56...72W</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRev.56.72" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRev.56.72</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+the+Self-Energy+and+the+Electromagnetic+Field+of+the+Electron&rft.jtitle=%5B%5BPhysical+Review%5D%5D&rft.aulast=V.+F.+Weisskopf&rft.au=V.+F.+Weisskopf&rft.date=1939&rft.volume=56&rft.pages=72%E2%80%9385&rft_id=info:bibcode/1939PhRv...56...72W&rft_id=info:doi/10.1103%2FPhysRev.56.72&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-oppenheimer-5" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-oppenheimer_5-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Robert_Oppenheimer" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Robert Oppenheimer">R. Oppenheimer</a> (1930). "Note on the Theory of the Interaction of Field and Matter".<i><a href="http://en.wikipedia.org/wiki/Physical_Review" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review">Physical Review</a></i> <b>35</b> (5): 461–477. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1930PhRv...35..461O" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1930PhRv...35..461O</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRev.35.461" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRev.35.461</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Note+on+the+Theory+of+the+Interaction+of+Field+and+Matter&rft.jtitle=%5B%5BPhysical+Review%5D%5D&rft.aulast=R.+Oppenheimer&rft.au=R.+Oppenheimer&rft.date=1930&rft.volume=35&rft.issue=5&rft.pages=461%E2%80%93477&rft_id=info:bibcode/1930PhRv...35..461O&rft_id=info:doi/10.1103%2FPhysRev.35.461&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-lamb-6" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-lamb_6-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Willis_Lamb" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Willis Lamb">W. E. Lamb</a>; <a href="http://en.wikipedia.org/wiki/Robert_Retherford" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Robert Retherford">R. C. Retherford</a> (1947). "Fine Structure of the Hydrogen Atom by a Microwave Method,". <i><a href="http://en.wikipedia.org/wiki/Physical_Review" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review">Physical Review</a></i> <b>72</b> (3): 241–243. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/1947PhRv...72..241L" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1947PhRv...72..241L</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRev.72.241" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRev.72.241</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Fine+Structure+of+the+Hydrogen+Atom+by+a+Microwave+Method%2C&rft.jtitle=%5B%5BPhysical+Review%5D%5D&rft.aulast=W.+E.+Lamb&rft.au=W.+E.+Lamb&rft.au=R.+C.+Retherford&rft.date=1947&rft.volume=72&rft.issue=3&rft.pages=241%E2%80%93243&rft_id=info:bibcode/1947PhRv...72..241L&rft_id=info:doi/10.1103%2FPhysRev.72.241&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-foley-7" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-foley_7-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Polykarp_Kusch" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Polykarp Kusch">P. Kusch</a>; <a class="new" href="http://en.wikipedia.org/w/index.php?title=H._M._Foley&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="H. M. Foley (page does not exist)">H. M. Foley</a> (1948). "On the Intrinsic Moment of the Electron". <i><a href="http://en.wikipedia.org/wiki/Physical_Review" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review">Physical Review</a></i> <b>73</b> (3): 412. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1948PhRv...73..412F" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1948PhRv...73..412F</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRev.73.412" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRev.73.412</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+the+Intrinsic+Moment+of+the+Electron&rft.jtitle=%5B%5BPhysical+Review%5D%5D&rft.aulast=P.+Kusch&rft.au=P.+Kusch&rft.au=H.+M.+Foley&rft.date=1948&rft.volume=73&rft.issue=3&rft.pages=412&rft_id=info:bibcode/1948PhRv...73..412F&rft_id=info:doi/10.1103%2FPhysRev.73.412&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-schweber-8" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-schweber_8-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;"><a class="new" href="http://en.wikipedia.org/w/index.php?title=Silvan_Schweber&action=edit&redlink=1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Silvan Schweber (page does not exist)">Schweber, Silvan</a> (1994). "Chapter 5". <i>QED and the Men Who Did it: Dyson, Feynman, Schwinger, and Tomonaga</i>. Princeton University Press. p. 230. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-691-03327-3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-691-03327-3">978-0-691-03327-3</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Chapter+5&rft.atitle=QED+and+the+Men+Who+Did+it%3A+Dyson%2C+Feynman%2C+Schwinger%2C+and+Tomonaga&rft.aulast=Schweber&rft.aufirst=Silvan&rft.au=Schweber%2C%26%2332%3BSilvan&rft.date=1994&rft.pages=p.%26nbsp%3B230&rft.pub=Princeton+University+Press&rft.isbn=978-0-691-03327-3&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-bethe-9" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-bethe_9-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Hans_Bethe" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hans Bethe">H. Bethe</a> (1947). "The Electromagnetic Shift of Energy Levels". <i><a href="http://en.wikipedia.org/wiki/Physical_Review" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review">Physical Review</a></i> <b>72</b>(4): 339–341. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1947PhRv...72..339B" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1947PhRv...72..339B</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRev.72.339" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRev.72.339</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Electromagnetic+Shift+of+Energy+Levels&rft.jtitle=%5B%5BPhysical+Review%5D%5D&rft.aulast=H.+Bethe&rft.au=H.+Bethe&rft.date=1947&rft.volume=72&rft.issue=4&rft.pages=339%E2%80%93341&rft_id=info:bibcode/1947PhRv...72..339B&rft_id=info:doi/10.1103%2FPhysRev.72.339&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-tomonaga-10" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-tomonaga_10-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Sin-Itiro_Tomonaga" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sin-Itiro Tomonaga">S. Tomonaga</a> (1946). "On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields". <i><a href="http://en.wikipedia.org/wiki/Progress_of_Theoretical_Physics" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Progress of Theoretical Physics">Progress of Theoretical Physics</a></i> <b>1</b> (2): 27–42.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1143%2FPTP.1.27" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1143/PTP.1.27</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+a+Relativistically+Invariant+Formulation+of+the+Quantum+Theory+of+Wave+Fields&rft.jtitle=%5B%5BProgress+of+Theoretical+Physics%5D%5D&rft.aulast=S.+Tomonaga&rft.au=S.+Tomonaga&rft.date=1946&rft.volume=1&rft.issue=2&rft.pages=27%E2%80%9342&rft_id=info:doi/10.1143%2FPTP.1.27&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-schwinger1-11" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-schwinger1_11-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Julian_Schwinger" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Julian Schwinger">J. Schwinger</a> (1948). "On Quantum-Electrodynamics and the Magnetic Moment of the Electron". <i><a href="http://en.wikipedia.org/wiki/Physical_Review" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review">Physical Review</a></i> <b>73</b> (4): 416–417. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1948PhRv...73..416S" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1948PhRv...73..416S</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRev.73.416" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRev.73.416</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+Quantum-Electrodynamics+and+the+Magnetic+Moment+of+the+Electron&rft.jtitle=%5B%5BPhysical+Review%5D%5D&rft.aulast=J.+Schwinger&rft.au=J.+Schwinger&rft.date=1948&rft.volume=73&rft.issue=4&rft.pages=416%E2%80%93417&rft_id=info:bibcode/1948PhRv...73..416S&rft_id=info:doi/10.1103%2FPhysRev.73.416&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-schwinger2-12" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-schwinger2_12-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Julian_Schwinger" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Julian Schwinger">J. Schwinger</a> (1948). "Quantum Electrodynamics. I. A Covariant Formulation".<i><a href="http://en.wikipedia.org/wiki/Physical_Review" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review">Physical Review</a></i> <b>74</b> (10): 1439–1461. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1948PhRv...74.1439S" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1948PhRv...74.1439S</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRev.74.1439" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRev.74.1439</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Quantum+Electrodynamics.+I.+A+Covariant+Formulation&rft.jtitle=%5B%5BPhysical+Review%5D%5D&rft.aulast=J.+Schwinger&rft.au=J.+Schwinger&rft.date=1948&rft.volume=74&rft.issue=10&rft.pages=1439%E2%80%931461&rft_id=info:bibcode/1948PhRv...74.1439S&rft_id=info:doi/10.1103%2FPhysRev.74.1439&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-feynman1-13" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-feynman1_13-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Richard_Feynman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard Feynman">R. P. Feynman</a> (1949). "Space-Time Approach to Quantum Electrodynamics".<i><a href="http://en.wikipedia.org/wiki/Physical_Review" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review">Physical Review</a></i> <b>76</b> (6): 769–789. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1949PhRv...76..769F" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1949PhRv...76..769F</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRev.76.769" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRev.76.769</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Space-Time+Approach+to+Quantum+Electrodynamics&rft.jtitle=%5B%5BPhysical+Review%5D%5D&rft.aulast=R.+P.+Feynman&rft.au=R.+P.+Feynman&rft.date=1949&rft.volume=76&rft.issue=6&rft.pages=769%E2%80%93789&rft_id=info:bibcode/1949PhRv...76..769F&rft_id=info:doi/10.1103%2FPhysRev.76.769&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-feynman2-14" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-feynman2_14-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Richard_Feynman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard Feynman">R. P. Feynman</a> (1949). "The Theory of Positrons". <i><a href="http://en.wikipedia.org/wiki/Physical_Review" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review">Physical Review</a></i> <b>76</b> (6): 749–759.<a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1949PhRv...76..749F" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1949PhRv...76..749F</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRev.76.749" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRev.76.749</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Theory+of+Positrons&rft.jtitle=%5B%5BPhysical+Review%5D%5D&rft.aulast=R.+P.+Feynman&rft.au=R.+P.+Feynman&rft.date=1949&rft.volume=76&rft.issue=6&rft.pages=749%E2%80%93759&rft_id=info:bibcode/1949PhRv...76..749F&rft_id=info:doi/10.1103%2FPhysRev.76.749&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-feynman3-15" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-feynman3_15-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Richard_Feynman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard Feynman">R. P. Feynman</a> (1950). "Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction". <i><a href="http://en.wikipedia.org/wiki/Physical_Review" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review">Physical Review</a></i> <b>80</b> (3): 440–457. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/1950PhRv...80..440F" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1950PhRv...80..440F</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRev.80.440" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRev.80.440</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Mathematical+Formulation+of+the+Quantum+Theory+of+Electromagnetic+Interaction&rft.jtitle=%5B%5BPhysical+Review%5D%5D&rft.aulast=R.+P.+Feynman&rft.au=R.+P.+Feynman&rft.date=1950&rft.volume=80&rft.issue=3&rft.pages=440%E2%80%93457&rft_id=info:bibcode/1950PhRv...80..440F&rft_id=info:doi/10.1103%2FPhysRev.80.440&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-dyson1-16" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink">^ <a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-dyson1_16-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-dyson1_16-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Freeman_Dyson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Freeman Dyson">F. Dyson</a> (1949). "The Radiation Theories of Tomonaga, Schwinger, and Feynman". <i><a href="http://en.wikipedia.org/wiki/Physical_Review" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review">Physical Review</a></i> <b>75</b> (3): 486–502. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1949PhRv...75..486D" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1949PhRv...75..486D</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRev.75.486" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRev.75.486</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Radiation+Theories+of+Tomonaga%2C+Schwinger%2C+and+Feynman&rft.jtitle=%5B%5BPhysical+Review%5D%5D&rft.aulast=F.+Dyson&rft.au=F.+Dyson&rft.date=1949&rft.volume=75&rft.issue=3&rft.pages=486%E2%80%93502&rft_id=info:bibcode/1949PhRv...75..486D&rft_id=info:doi/10.1103%2FPhysRev.75.486&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-dyson2-17" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-dyson2_17-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Freeman_Dyson" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Freeman Dyson">F. Dyson</a> (1949). "The S Matrix in Quantum Electrodynamics". <i><a href="http://en.wikipedia.org/wiki/Physical_Review" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review">Physical Review</a></i> <b>75</b>(11): 1736–1755. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1949PhRv...75.1736D" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1949PhRv...75.1736D</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRev.75.1736" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRev.75.1736</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+S+Matrix+in+Quantum+Electrodynamics&rft.jtitle=%5B%5BPhysical+Review%5D%5D&rft.aulast=F.+Dyson&rft.au=F.+Dyson&rft.date=1949&rft.volume=75&rft.issue=11&rft.pages=1736%E2%80%931755&rft_id=info:bibcode/1949PhRv...75.1736D&rft_id=info:doi/10.1103%2FPhysRev.75.1736&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-nobel65-18" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-nobel65_18-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation web" style="word-wrap: break-word;"><a class="external text" href="http://nobelprize.org/nobel_prizes/physics/laureates/1965/index.html" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">"The Nobel Prize in Physics 1965"</a>. Nobel Foundation<span class="reference-accessdate">. Retrieved 2008-10-09</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=The+Nobel+Prize+in+Physics+1965&rft.atitle=&rft.pub=Nobel+Foundation&rft_id=http%3A%2F%2Fnobelprize.org%2Fnobel_prizes%2Fphysics%2Flaureates%2F1965%2Findex.html&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-feynbook2-19" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink">^ <a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-feynbook2_19-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-feynbook2_19-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-feynbook2_19-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>c</b></i></sup></a></span> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Richard_Feynman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard Feynman">Feynman, Richard</a> (1985). <i>QED: The Strange Theory of Light and Matter</i>. Princeton University Press. p. 128. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-691-12575-6" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-691-12575-6">978-0-691-12575-6</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=QED%3A+The+Strange+Theory+of+Light+and+Matter&rft.aulast=Feynman&rft.aufirst=Richard&rft.au=Feynman%2C%26%2332%3BRichard&rft.date=1985&rft.pages=p.%26nbsp%3B128&rft.pub=Princeton+University+Press&rft.isbn=978-0-691-12575-6&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-20" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-20" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">G.S. Guralnik, C.R. Hagen, T.W.B. Kibble (1964). "Global Conservation Laws and Massless Particles". <i><a href="http://en.wikipedia.org/wiki/Physical_Review_Letters" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical Review Letters">Physical Review Letters</a></i> <b>13</b> (20): 585–587. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/1964PhRvL..13..585G" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1964PhRvL..13..585G</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRevLett.13.585" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRevLett.13.585</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Global+Conservation+Laws+and+Massless+Particles&rft.jtitle=%5B%5BPhysical+Review+Letters%5D%5D&rft.aulast=G.S.+Guralnik%2C+C.R.+Hagen%2C+T.W.B.+Kibble&rft.au=G.S.+Guralnik%2C+C.R.+Hagen%2C+T.W.B.+Kibble&rft.date=1964&rft.volume=13&rft.issue=20&rft.pages=585%E2%80%93587&rft_id=info:bibcode/1964PhRvL..13..585G&rft_id=info:doi/10.1103%2FPhysRevLett.13.585&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-21" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-21" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">G.S. Guralnik (2009). "The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles". <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/International_Journal_of_Modern_Physics_A" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Journal of Modern Physics A">International Journal of Modern Physics A</a></i> <b>24</b> (14): 2601–2627. <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/0907.3466" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">0907.3466</a>. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/2009IJMPA..24.2601G" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">2009IJMPA..24.2601G</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1142%2FS0217751X09045431" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1142/S0217751X09045431</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+History+of+the+Guralnik%2C+Hagen+and+Kibble+development+of+the+Theory+of+Spontaneous+Symmetry+Breaking+and+Gauge+Particles&rft.jtitle=%5B%5BInternational+Journal+of+Modern+Physics+A%5D%5D&rft.aulast=G.S.+Guralnik&rft.au=G.S.+Guralnik&rft.date=2009&rft.volume=24&rft.issue=14&rft.pages=2601%E2%80%932627&rft_id=info:arxiv/0907.3466&rft_id=info:bibcode/2009IJMPA..24.2601G&rft_id=info:doi/10.1142%2FS0217751X09045431&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-feynbook3-22" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-feynbook3_22-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Richard_Feynman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard Feynman">Feynman, Richard</a> (1985). <i>QED: The Strange Theory of Light and Matter</i>. Princeton University Press. p. 152. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-691-12575-6" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-691-12575-6">978-0-691-12575-6</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=QED%3A+The+Strange+Theory+of+Light+and+Matter&rft.aulast=Feynman&rft.aufirst=Richard&rft.au=Feynman%2C%26%2332%3BRichard&rft.date=1985&rft.pages=p.%26nbsp%3B152&rft.pub=Princeton+University+Press&rft.isbn=978-0-691-12575-6&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
<li id="cite_note-23" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics#cite_ref-23" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation web" style="word-wrap: break-word;">Kinoshita, Toichiro. <a class="external text" href="http://www.lassp.cornell.edu/sethna/Cracks/QED.html" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">"Quantum Electrodynamics has Zero Radius of Convergence Summarized from</a> <a href="http://en.wikipedia.org/wiki/Toichiro_Kinoshita" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Toichiro Kinoshita">Toichiro Kinoshita</a>"<span class="reference-accessdate">. Retrieved 06-10-2010</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Quantum+Electrodynamics+has+Zero+Radius+of+Convergence+Summarized+from+%5B%5BToichiro+Kinoshita%5D%5D&rft.atitle=&rft.aulast=Kinoshita&rft.aufirst=Toichiro&rft.au=Kinoshita%2C%26%2332%3BToichiro&rft_id=http%3A%2F%2Fwww.lassp.cornell.edu%2Fsethna%2FCracks%2FQED.html&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></span></li>
</ol>
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<h2 style="background-image: none; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 20px; font-weight: normal; margin: 0px 0px 0.6em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=17" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Further reading">edit</a>]</span><span class="mw-headline" id="Further_reading">Further reading</span></h2>
<h3 style="background-image: none; border-bottom-style: none; font-size: 18px; margin: 0px 0px 0.3em; overflow: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=18" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Books">edit</a>]</span><span class="mw-headline" id="Books">Books</span></h3>
<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Louis_de_Broglie" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Louis de Broglie">De Broglie, Louis</a> (1925). <i>Recherches sur la theorie des quanta [Research on quantum theory]</i>. France: Wiley-Interscience.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Recherches+sur++la+theorie+des+quanta+%5BResearch+on+quantum+theory%5D&rft.aulast=De+Broglie&rft.aufirst=Louis&rft.au=De+Broglie%2C%26%2332%3BLouis&rft.date=1925&rft.place=France&rft.pub=Wiley-Interscience&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Richard_Feynman" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard Feynman">Feynman, Richard Phillips</a> (1998). <i>Quantum Electrodynamics</i>. Westview Press; New Ed edition. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-201-36075-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-201-36075-2">978-0-201-36075-2</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Electrodynamics&rft.aulast=Feynman&rft.aufirst=Richard+Phillips&rft.au=Feynman%2C%26%2332%3BRichard+Phillips&rft.date=1998&rft.pub=Westview+Press%3B+New+Ed+edition&rft.isbn=978-0-201-36075-2&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">Jauch, J.M.; Rohrlich, F. (1980). <i>The Theory of Photons and Electrons</i>. Springer-Verlag. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-387-07295-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-387-07295-1">978-0-387-07295-1</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Photons+and+Electrons&rft.aulast=Jauch&rft.aufirst=J.M.&rft.au=Jauch%2C%26%2332%3BJ.M.&rft.date=1980&rft.pub=Springer-Verlag&rft.isbn=978-0-387-07295-1&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">Greiner, Walter; Bromley, D.A.,Müller, Berndt. (2000). <i>Gauge Theory of Weak Interactions</i>. Springer. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-3-540-67672-0" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-3-540-67672-0">978-3-540-67672-0</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gauge+Theory+of+Weak+Interactions&rft.aulast=Greiner&rft.aufirst=Walter&rft.au=Greiner%2C%26%2332%3BWalter&rft.date=2000&rft.pub=Springer&rft.isbn=978-3-540-67672-0&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">Kane, Gordon, L. (1993). <i>Modern Elementary Particle Physics</i>. Westview Press. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-201-62460-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-201-62460-1">978-0-201-62460-1</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Elementary+Particle+Physics&rft.aulast=Kane&rft.aufirst=Gordon%2C+L.&rft.au=Kane%2C%26%2332%3BGordon%2C+L.&rft.date=1993&rft.pub=Westview+Press&rft.isbn=978-0-201-62460-1&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">Miller, Arthur I. (1995). <i>Early Quantum Electrodynamics : A Sourcebook</i>. Cambridge University Press. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-521-56891-3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-521-56891-3">978-0-521-56891-3</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Early+Quantum+Electrodynamics+%3A+A+Sourcebook&rft.aulast=Miller&rft.aufirst=Arthur+I.&rft.au=Miller%2C%26%2332%3BArthur+I.&rft.date=1995&rft.pub=Cambridge+University+Press&rft.isbn=978-0-521-56891-3&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></li>
<li style="margin-bottom: 0.1em;">Milonni, Peter W., (1994) <i>The quantum vacuum - an introduction to quantum electrodynamics</i>. Academic Press. <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/0124980805" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 0-12-498080-5</a></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">Schweber, Silvian, S. (1994). <i>QED and the Men Who Made It</i>. Princeton University Press. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-691-03327-3" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-691-03327-3">978-0-691-03327-3</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=QED+and+the+Men+Who+Made+It&rft.aulast=Schweber&rft.aufirst=Silvian%2C+S.&rft.au=Schweber%2C%26%2332%3BSilvian%2C+S.&rft.date=1994&rft.pub=Princeton+University+Press&rft.isbn=978-0-691-03327-3&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Julian_Schwinger" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Julian Schwinger">Schwinger, Julian</a> (1958). <i>Selected Papers on Quantum Electrodynamics</i>. Dover Publications. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-486-60444-2" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-486-60444-2">978-0-486-60444-2</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Selected+Papers+on+Quantum+Electrodynamics&rft.aulast=Schwinger&rft.aufirst=Julian&rft.au=Schwinger%2C%26%2332%3BJulian&rft.date=1958&rft.pub=Dover+Publications&rft.isbn=978-0-486-60444-2&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Claude_Cohen-Tannoudji" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Claude Cohen-Tannoudji">Tannoudji-Cohen, Claude</a>; Dupont-Roc, Jacques, and Grynberg, Gilbert (1997). <i>Photons and Atoms: Introduction to Quantum Electrodynamics</i>. Wiley-Interscience.<a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-471-18433-1" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-471-18433-1">978-0-471-18433-1</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Photons+and+Atoms%3A+Introduction+to+Quantum+Electrodynamics&rft.aulast=Tannoudji-Cohen&rft.aufirst=Claude&rft.au=Tannoudji-Cohen%2C%26%2332%3BClaude&rft.date=1997&rft.pub=Wiley-Interscience&rft.isbn=978-0-471-18433-1&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></li>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=19" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Journals">edit</a>]</span><span class="mw-headline" id="Journals">Journals</span></h3>
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<li style="margin-bottom: 0.1em;"><span class="citation Journal" style="word-wrap: break-word;">Dudley, J.M.; Kwan, A.M. (1996). "Richard Feynman's popular lectures on quantum electrodynamics: The 1979 Robb Lectures at Auckland University". <i>American Journal of Physics</i> <b>64</b> (6): 694–698. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1996AmJPh..64..694D" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1996AmJPh..64..694D</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">DOI</a>:<a class="external text" href="http://dx.doi.org/10.1119%2F1.18234" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1119/1.18234</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Richard+Feynman%27s+popular+lectures+on+quantum+electrodynamics%3A+The+1979+Robb+Lectures+at+Auckland+University&rft.jtitle=American+Journal+of+Physics&rft.aulast=Dudley&rft.aufirst=J.M.&rft.au=Dudley%2C%26%2332%3BJ.M.&rft.au=Kwan%2C%26%2332%3BA.M.&rft.date=1996&rft.volume=64&rft.issue=6&rft.pages=694%E2%80%93698&rft_id=info:bibcode/1996AmJPh..64..694D&rft_id=info:doi/10.1119%2F1.18234&rfr_id=info:sid/en.wikipedia.org:Quantum_electrodynamics"></span></li>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Quantum_electrodynamics&action=edit&section=20" style="background-image: none; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: External links">edit</a>]</span><span class="mw-headline" id="External_links">External links</span></h2>
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<li style="margin-bottom: 0.1em;"><a class="external text" href="http://nobelprize.org/physics/laureates/1965/feynman-lecture.html" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Feynman's Nobel Prize lecture describing the evolution of QED and his role in it</a></li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.vega.org.uk/video/subseries/8" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Feynman's New Zealand lectures on QED for non-physicists</a></li>
<li style="margin-bottom: 0.1em;"><a class="external free" href="http://qed.wikina.org/" rel="nofollow" style="background-image: url(data:image/png; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://qed.wikina.org/</a> - Animations demonstrating QED</li>
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<strong class="selflink" style="white-space: nowrap;">Quantum electrodynamics</strong></div>
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Quantum field theories</div>
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</div>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com0tag:blogger.com,1999:blog-5303246073824127471.post-47950950512619962152012-05-08T08:27:00.000-04:002012-05-08T08:38:19.373-04:00The Dirac Equation (1928)<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZh7X3iUS4L-3pWceMDXX3FXSkIgOXalz_E8gv9ceclbW3nleqEvkuIvzY_ilhOI2igDrbXc00OWTKCp6GZgdmDaof1-cX1ArhlSnZXgfBhuRvnN8iq9Q5A-pXAVpSFXAiiTb9EsWSZQ/s1600/Dirac+34.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZh7X3iUS4L-3pWceMDXX3FXSkIgOXalz_E8gv9ceclbW3nleqEvkuIvzY_ilhOI2igDrbXc00OWTKCp6GZgdmDaof1-cX1ArhlSnZXgfBhuRvnN8iq9Q5A-pXAVpSFXAiiTb9EsWSZQ/s320/Dirac+34.jpg" width="240" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The bust of Paul Dirac at Princeton University's Fine Hall Library I snapped Apr. 23, 2012 </td></tr>
</tbody></table>
Paul Dirac's Equation stands as the most beautiful equation in Science and not just in my opinion. Once developed, Quantum Field Theory (QED and QCD) took off like a shot. We don't live in the world we live in today with all its technological marvels,without it.
It's not as simple as E = m c squared, and understanding it requires 4-5 semesters of college level Calculus as well as the mathematics and study of both Special Relativity and Quantum Mechanics, but it's wonderful nevertheless.
<br />
<br />
<br />
"Although Dirac did not at first fully appreciate what his own equation was telling him, his resolute faith in the logic of mathematics as a means to physical reasoning, his explanation of spin as a consequence of the union of quantum mechanics and special relativity, and the eventual discovery of the positron, represents one of the great triumphs of theoretical physics."<br />
<br />
From Wiki:<br />
<br />
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
In <a href="http://en.wikipedia.org/wiki/Physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physics">physics</a>, more specifically <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Relativistic_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Relativistic quantum mechanics">relativistic quantum mechanics</a>, the <b>Dirac equation</b><sup class="reference" id="cite_ref-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_note-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[1]</a></sup> is a <a href="http://en.wikipedia.org/wiki/Relativistic_wave_equations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Relativistic wave equations">wave equation</a>, formulated by <a href="http://en.wikipedia.org/wiki/British_people" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="British people">British</a><a href="http://en.wikipedia.org/wiki/Physicist" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physicist">physicist</a> <a href="http://en.wikipedia.org/wiki/Paul_Dirac" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Paul Dirac">Paul Dirac</a> in 1928. It provided a description of <a href="http://en.wikipedia.org/wiki/Elementary_particle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Elementary particle">elementary</a> <a href="http://en.wikipedia.org/wiki/Spin-%C2%BD" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin-½">spin-½</a> particles, such as <a href="http://en.wikipedia.org/wiki/Electron" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron">electrons</a>, consistent with both the principles of <a href="http://en.wikipedia.org/wiki/Quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum mechanics">quantum mechanics</a> and the theory of <a href="http://en.wikipedia.org/wiki/Special_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special relativity">special relativity</a>, and made relativistic corrections to quantum mechanics. It accounted for the <a href="http://en.wikipedia.org/wiki/Fine_structure" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fine structure">fine structure</a> of the hydrogen spectrum in a rigorous way. The equation also implied the existence of a new form of matter, <i><a href="http://en.wikipedia.org/wiki/Antimatter" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Antimatter">antimatter</a></i>, hitherto unsuspected and unobserved, later discovered experimentally. It also provided a <i>theoretical</i>justification for the introduction of several-component wave functions in <a href="http://en.wikipedia.org/wiki/Pauli_exclusion_principle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pauli exclusion principle">Pauli's phenomenological theory</a> of <a href="http://en.wikipedia.org/wiki/Spin_(physics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin (physics)">spin</a>. Although Dirac did not at first fully appreciate what his own equation was telling him, his resolute faith in the <a href="http://en.wikipedia.org/wiki/Mathematical_logic" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mathematical logic">logic</a> of <a href="http://en.wikipedia.org/wiki/Mathematics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mathematics">mathematics</a> as a means to physical reasoning, his explanation of spin as a consequence of the union of quantum mechanics and special relativity, and the eventual discovery of the <a href="http://en.wikipedia.org/wiki/Positron" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Positron">positron</a>, represents one of the great triumphs of <a href="http://en.wikipedia.org/wiki/Theoretical_physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theoretical physics">theoretical physics</a>.</div>
<table class="toc" id="toc" style="background-color: #f9f9f9; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; border-image: initial; border-left-color: rgb(170, 170, 170); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(170, 170, 170); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(170, 170, 170); border-top-style: solid; border-top-width: 1px; color: black; font-family: sans-serif; font-size: 12px; line-height: 20px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;"><tbody>
<tr><td><div id="toctitle" style="direction: ltr; text-align: center;">
<h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; display: inline; font-size: 12px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
Contents</h2>
<span class="toctoggle" style="-webkit-user-select: none;"> [<a class="internal" href="http://en.wikipedia.org/wiki/Dirac_equation#" id="togglelink" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">hide</a>] </span></div>
<ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;">
<li class="toclevel-1 tocsection-1" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#The_Dirac_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1</span> <span class="toctext">The Dirac equation</span></a></li>
<li class="toclevel-1 tocsection-2" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Background_and_development" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2</span> <span class="toctext">Background and development</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-3" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Making_the_Schr.C3.B6dinger_equation_relativistic" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.1</span> <span class="toctext">Making the Schrödinger equation relativistic</span></a></li>
<li class="toclevel-2 tocsection-4" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Square_root_of_the_Klein-Gordon_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.2</span> <span class="toctext">Square root of the Klein-Gordon equation</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-5" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Mathematical_formulation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3</span> <span class="toctext">Mathematical formulation</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-6" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#The_Dirac_.CE.B1_and_.CE.B2_matrices" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.1</span> <span class="toctext">The Dirac α and β matrices</span></a></li>
<li class="toclevel-2 tocsection-7" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#The_Dirac_.CE.B3_matrices" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.2</span> <span class="toctext">The Dirac γ matrices</span></a></li>
<li class="toclevel-2 tocsection-8" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#The_Pauli_spin_.CF.83_matrices" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.3</span> <span class="toctext">The Pauli spin σ matrices</span></a></li>
<li class="toclevel-2 tocsection-9" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Dirac_equation_in_curved_spacetime" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.4</span> <span class="toctext">Dirac equation in curved spacetime</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-10" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Physical_interpretation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4</span> <span class="toctext">Physical interpretation</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-11" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Identification_of_observables" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.1</span> <span class="toctext">Identification of observables</span></a></li>
<li class="toclevel-2 tocsection-12" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Hole_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.2</span> <span class="toctext">Hole theory</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-13" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Properties" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5</span> <span class="toctext">Properties</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-14" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Covariant_form_and_relativistic_invariance" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5.1</span> <span class="toctext">Covariant form and relativistic invariance</span></a></li>
<li class="toclevel-2 tocsection-15" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Relativistic_eigenvalue_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5.2</span> <span class="toctext">Relativistic eigenvalue equation</span></a></li>
<li class="toclevel-2 tocsection-16" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Spinor_transformations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5.3</span> <span class="toctext">Spinor transformations</span></a></li>
<li class="toclevel-2 tocsection-17" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Adjoint_equation_and_probability_conservation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5.4</span> <span class="toctext">Adjoint equation and probability conservation</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-18" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#See_also" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-19" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#References" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">7</span> <span class="toctext">References</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-20" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Selected_papers" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">7.1</span> <span class="toctext">Selected papers</span></a></li>
<li class="toclevel-2 tocsection-21" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#Textbooks" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">7.2</span> <span class="toctext">Textbooks</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-22" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#External_links" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">8</span> <span class="toctext">External links</span></a></li>
</ul>
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<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: The Dirac equation">edit</a>]</span><span class="mw-headline" id="The_Dirac_equation">The Dirac equation</span></h2>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
The equation in the form originally proposed by Dirac is:<sup class="reference" id="cite_ref-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_note-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup><sup class="reference" id="cite_ref-McMahon_2-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_note-McMahon-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup></div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\left( c \boldsymbol{\alpha}\cdot \mathbf{\hat{p}}+\beta mc^2 \right ) \psi = i\hbar\frac{\partial \psi}{\partial t}\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/0/d/a0de9c540c7e75a9771cfb9723ed0a4f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <i>ψ</i> = <i>ψ</i>(<b>r</b>, <i>t</i>) is a complex four-component <a href="http://en.wikipedia.org/wiki/Fermionic_field" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermionic field">field</a> <i>ψ</i> that Dirac thought of as the <a href="http://en.wikipedia.org/wiki/Wave_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wave function">wave function</a> for the <a href="http://en.wikipedia.org/wiki/Electron" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron">electron</a>, <b>r</b> and <i>t</i> are the <a href="http://en.wikipedia.org/wiki/Space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Space">space</a> and <a href="http://en.wikipedia.org/wiki/Time" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Time">time</a> coordinates, <i>m</i> is the <a href="http://en.wikipedia.org/wiki/Mass" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mass">rest mass</a>of the electron, <img alt="\hat{p}\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/4/8/3485426f66806a6a663889a9ef2ce777.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is the <a href="http://en.wikipedia.org/wiki/Momentum_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Momentum operator">momentum operator</a>, <i>c</i> is the <a href="http://en.wikipedia.org/wiki/Speed_of_light" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Speed of light">speed of light</a>, and <i>ħ</i> is the reduced <a href="http://en.wikipedia.org/wiki/Planck_constant" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Planck constant">Planck constant</a> (<i>h</i>/2<i>π</i>). Furthermore, <b>α</b> is a vector operator whose components are 4 × 4 matricies: <b>α</b> = (<i>α</i><sub style="line-height: 1em;">1</sub>, <i>α</i><sub style="line-height: 1em;">2</sub>, <i>α</i><sub style="line-height: 1em;">3</sub>), and <i>β</i> is another 4 × 4 matrix.</div>
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This single symbolic equation unravels into four coupled linear first-order partial differential equations for the four quantities that make up the field. These matrices, and the form of the field, have a deep mathematical significance. The algebraic structure represented by the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Dirac_matrices" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac matrices">Dirac matrices</a> had been created some 50 years earlier by the English mathematician <a href="http://en.wikipedia.org/wiki/William_Kingdon_Clifford" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="William Kingdon Clifford">W. K. Clifford</a>. In turn, Clifford's ideas had emerged from the mid-19th century work of the German mathematician <a href="http://en.wikipedia.org/wiki/Hermann_Grassmann" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermann Grassmann">Hermann Grassmann</a> in his "Lineale Ausdehnungslehre" (Theory of Linear Extensions). The latter had been regarded as well-nigh incomprehensible by most of his contemporaries. The appearance of something so seemingly abstract, at such a late date, and in such a direct physical manner, is one of the most remarkable chapters in the history of physics.</div>
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Dirac's purpose in casting this equation was to explain the behavior of the relativistically moving electron, and so to allow the <a href="http://en.wikipedia.org/wiki/Atom" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Atom">atom</a> to be treated in a manner consistent with relativity. His rather modest hope was that the corrections introduced this way might have bearing on the problem of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Atomic_spectra" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Atomic spectra">atomic spectra</a>. Up until that time, attempts to make the <a href="http://en.wikipedia.org/wiki/Old_quantum_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Old quantum theory">old quantum theory</a> of the atom compatible with the theory of relativity by discretizing the <a href="http://en.wikipedia.org/wiki/Angular_momentum#Angular_momentum_in_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Angular momentum">angular momentum</a> of the electron's <a href="http://en.wikipedia.org/wiki/Orbit" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Orbit">orbit</a> had failed - and the new quantum mechanics of<a href="http://en.wikipedia.org/wiki/Werner_Heisenberg" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Werner Heisenberg">Heisenberg</a>, <a href="http://en.wikipedia.org/wiki/Wolfgang_Pauli" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wolfgang Pauli">Pauli</a>, Jordan, <a href="http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Erwin Schrödinger">Schrödinger</a>, and Dirac himself had not developed sufficiently to treat this problem. Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of <a href="http://en.wikipedia.org/wiki/Matter" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Matter">matter</a>, and introduced new mathematical classes of objects that are now essential elements of fundamental physics.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Background and development">edit</a>]</span><span class="mw-headline" id="Background_and_development">Background and development</span></h2>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Making the Schrödinger equation relativistic">edit</a>]</span><span class="mw-headline" id="Making_the_Schr.C3.B6dinger_equation_relativistic">Making the Schrödinger equation relativistic</span></h3>
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The Dirac equation was motivated by the <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schrödinger equation">Schrödinger equation</a> for a massive <a href="http://en.wikipedia.org/wiki/Free_particle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Free particle">free particle</a>:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="-\frac{\hbar^2}{2m}\nabla^2\psi = i\hbar\frac{\partial}{\partial t}\psi." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/a/c/8ac14f5f0208978bcb29803a4954098f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The left side, the non-relativistic kinetic energy, is the square of the <a href="http://en.wikipedia.org/wiki/Momentum_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Momentum operator">momentum operator</a> divided by twice the mass <i>m</i>. Relativity treats <a href="http://en.wikipedia.org/wiki/Space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Space">space</a> and <a href="http://en.wikipedia.org/wiki/Time" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Time">time</a> as a unified spacetime, so a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically, as they do in the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Maxwell_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Maxwell equation">Maxwell equations</a> that govern the behavior of light — the equations must be differentially of the <i>same order</i> in space and time. In relativity, the momentum and the energy are the space and time parts of a space-time vector, the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/4-momentum" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="4-momentum">4-momentum</a>, and they are related by the relativistically invariant relation</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\frac{E^2}{c^2} - p^2 = m^2c^2" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/2/2/7227448cc1693dd50549796a4020bf96.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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which says that the length of this vector is proportional to the <a href="http://en.wikipedia.org/wiki/Invariant_mass" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Invariant mass">invariant mass</a> <i>m</i>. Substituting the operator equivalents of the energy and momentum from the Schrödinger theory, we get an equation describing the propagation of waves, constructed from relativistically invariant objects, the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Klein-Gordon_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Klein-Gordon equation">Klein-Gordon equation</a>:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\left ( -\frac{1}{c^2}\frac{\partial^2}{\partial t^2} + \nabla^2 \right ) \psi = \frac{m^2c^2}{\hbar^2}\psi" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/d/d/dddb492e43d0f24b35b9d0cc01510205.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where the wave function <i>ψ</i> is a relativistic scalar: a complex number which has the same numerical value in all frames of reference. The space and time derivatives both enter to second order. This has an important consequence for the interpretation of the equation: the expression for the density is no longer positive definite - the initial values of both <i>ψ</i> and <img alt="\scriptstyle \partial \psi/\partial t" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/d/2/2d23d71ccd69b085dd2eb385f184df31.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> may be freely chosen, and the density may thus become negative, something that is impossible if the density is to be a legitimate probability density, as it is for the Schrödinger equation. Thus we cannot get a relativistic generalization of the Schrödinger equation under the naive assumption that the wave function is a scalar.</div>
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Although the Klein-Gordon equation is not a successful relativistic generalization of the Schrödinger equation, this equation is a valid field equation in the context of quantum field theory, describing a spinless particle field (e.g. pi meson). Historically, Schrödinger himself arrived at this equation before the one that bears his name, but soon discarded it. In the context of quantum field theory, the indefinite density is understood to correspond to the <i>charge</i> density, which can be positive or negative, and not the probability density. Finding a relativistic field equation with first order derivatives required a more elaborate construction.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Square root of the Klein-Gordon equation">edit</a>]</span><span class="mw-headline" id="Square_root_of_the_Klein-Gordon_equation">Square root of the Klein-Gordon equation</span></h3>
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Dirac thought to try an equation that was <i>first order</i> in both space and time. One could, for example, formally take the relativistic expression for the energy</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="E = c\sqrt{p^2 + m^2c^2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/1/9/d194400306a8433eb2f86188c8695826.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />,</dd></dl>
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expand the square root in an <a href="http://en.wikipedia.org/wiki/Series_(mathematics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Series (mathematics)">infinite series</a>, replace <i>p</i> and <i>E</i> by their operator equivalents, set up an <a href="http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eigenvalues and eigenvectors">eigenvalue problem</a>, then solve the equation formally by iterations. Most physicists had little faith in such a formidable process, even if it were technically possible.</div>
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As the story goes, Dirac was staring into the fireplace at Cambridge, pondering this problem, when he hit upon the idea of taking the square root of the wave operator thus:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} = \left ( A \frac{\partial }{\partial x} + B \frac{\partial }{\partial y} + C \frac{\partial }{\partial z} + \frac{i}{c}D \frac{\partial }{\partial t} \right ) \left ( A \frac{\partial }{\partial x} + B \frac{\partial }{\partial y} + C \frac{\partial }{\partial z} + \frac{i}{c}D \frac{\partial }{\partial t} \right )." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/2/3/3239b61265ce223be7286a39eadc609e.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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On multiplying out the right side, it can be noticed that the cross-terms, such as <img alt="\scriptstyle \partial^2 /\partial x \partial y " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/c/4/2c40d7ef2d0a8377c90743879a5efae5.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, will vanish if we assume that for every different pair of coefficents their<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Anticommutator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Anticommutator">anticommutator</a> vanishes:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" [A,B]_+ = 0, [A,C]_+ = 0, \cdots \, , " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/9/f/89fe6cb86cf53b9f5c753803ac21bef4.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where the brackets [, ]<sub style="line-height: 1em;">+</sub> denote the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Anticommutator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Anticommutator">anticommutator</a>:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="[A,B]_+ = AB + BA \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/7/a/f7ad33578fe04c0ddc8ba54b6198d92d.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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and that they each square to the 4 × 4 identity:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="A^2 = B^2 = C^2 = D^2 = 1 \, .\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/7/0/070976190ea74f5b31b16bcaa5959c2a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Dirac, who had just then been intensely involved with working out the foundations of Heisenberg's <a href="http://en.wikipedia.org/wiki/Matrix_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Matrix mechanics">matrix mechanics</a>, immediately understood that these conditions could be met if <i>A</i>, <i>B</i>, <i>C</i> and <i>D</i> are <i>matrices</i>, with the implication that the wave function has <i>multiple components</i>. This immediately explained the appearance of two-component wave functions in Pauli's phenomenological theory of <a href="http://en.wikipedia.org/wiki/Spin_(physics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin (physics)">spin</a>, something that up until then had been regarded as mysterious, even to Pauli himself. However, one needs at least 4 × 4 matrices to set up a system with the properties required — so the wave function had <i>four</i> components, not two, as in the Pauli theory, or one, as in the bare Schrödinger theory. The four-component wave function represents a new class of mathematical object in physical theories, <a href="http://en.wikipedia.org/wiki/Spinor" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spinor">spinors</a>, that makes its first appearance here.</div>
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Given the factorization in terms of these matrices, the Dirac equation can be obtained from one of the factors, an equation first order in space and time (as given above).</div>
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<tr><th><span class="collapseButton" style="float: right; font-weight: normal; margin-left: 0.5em; text-align: right; width: auto;">[<a href="http://en.wikipedia.org/wiki/Dirac_equation#" id="collapseButton0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">show</a>]</span>Derivation</th></tr>
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</dd></dl>
<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Mathematical formulation">edit</a>]</span><span class="mw-headline" id="Mathematical_formulation">Mathematical formulation</span></h2>
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The Dirac equation can take several different forms, relating to the nature of the matrices.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: The Dirac α and β matrices">edit</a>]</span><span class="mw-headline" id="The_Dirac_.CE.B1_and_.CE.B2_matrices">The Dirac <span style="font-family: 'Times new roman';">α</span> and <span style="font-family: 'Times new roman';">β</span> matrices</span></h3>
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Starting from the original form of Dirac's equation:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\left( c \boldsymbol{\alpha}\cdot \mathbf{\hat{p}}+\beta mc^2 \right ) \psi = i\hbar\frac{\partial \psi}{\partial t}\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/0/d/a0de9c540c7e75a9771cfb9723ed0a4f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The matrices <i>α</i><sub style="line-height: 1em;">1</sub>, <i>α</i><sub style="line-height: 1em;">2</sub>, <i>α</i><sub style="line-height: 1em;">3</sub>, and <i>β</i>, are 4 × 4 matrices. Some properties are as follows:</div>
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They are all <a href="http://en.wikipedia.org/wiki/Hermitian_matrix" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermitian matrix">Hermitian</a> so that the Dirac Hamiltonian is Hermitian.</div>
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They have squares equal to the 4 × 4 <a href="http://en.wikipedia.org/wiki/Identity_matrix" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Identity matrix">identity matrix</a> <i>I</i><sub style="line-height: 1em;">4</sub>:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="(\alpha_i)^2=\beta^2=I_4" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/6/3/963b55915b03c38189ab91259c6da2bd.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
and they all mutually <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Anticommute" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Anticommute">anticommute</a>:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="[\alpha_i,\alpha_j]_+ = 0 \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/4/5/b456d01cdf2de95c5a5834f9356b5f6a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="[\alpha_i,\beta]_+ = 0 \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/d/8/ed8ee081141d1f39a868f1abf4dd93e3.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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for all <i>i</i> and <i>j</i> not equal to each other.</div>
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Dirac defined these matrices (in the chiral representation) as the following:<sup class="reference" id="cite_ref-3" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_note-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[4]</a></sup></div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="
\beta = \begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
\end{pmatrix} ,\quad
\alpha_1 = \begin{pmatrix}
0 & -1 & 0 & 0 \\
-1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
\end{pmatrix},\quad
\alpha_2 = \begin{pmatrix}
0 & i & 0 & 0 \\
-i & 0 & 0 & 0 \\
0 & 0 & 0 & -i \\
0 & 0 & i & 0 \\
\end{pmatrix},\quad
\alpha_3 = \begin{pmatrix}
-1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1 \\
\end{pmatrix} " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/6/d/76d5c0215516cc54f5ef7b2c2b6617d3.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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NB: In the literature and this context, all matrices are usually written in italic like <a href="http://en.wikipedia.org/wiki/Scalar_(mathematics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Scalar (mathematics)">scalars</a>, bold is used for a vector whose <i>components</i> are matrices. Superscript and subscript indices are used to label components of the vectors of matrices. See <a href="http://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a> - except for identity matrices. Also it is conventional not to write identity matrices, or write them as 1, as they can be revealed from their positions in the equation. If a matrix is shown as 2 × 2 when it is known to be 4 × 4, then the missing identities are the 2 × 2 identity matrix, <i>I</i><sub style="line-height: 1em;">2</sub>. If no matrix is shown at all in the full Dirac equation, then it is understood that the missing identity is 4 × 4 identity matrix, <i>I</i><sub style="line-height: 1em;">4</sub>.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: The Dirac γ matrices">edit</a>]</span><span class="mw-headline" id="The_Dirac_.CE.B3_matrices">The Dirac <span style="font-family: 'Times new roman';">γ</span> matrices</span></h3>
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It is useful to define new matrices:<sup class="reference" id="cite_ref-McMahon_2-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_note-McMahon-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup></div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\gamma^0 = \beta \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/0/9/909e94fc232c1d810b8ff64668a1a1a9.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\gamma^j = \beta \alpha_j. \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/8/5/d8523304db393c3346d086500244d379.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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These matrices are known as the <a href="http://en.wikipedia.org/wiki/Gamma_matrices" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gamma matrices">gamma matrices</a>, and there are many different representations of them. In the <i>Pauli-Dirac representation (and basis)</i>:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="
\gamma^0 = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \\
\end{pmatrix} ,\quad
\gamma^1 = \begin{pmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & -1 & 0 & 0 \\
-1 & 0 & 0 & 0 \\
\end{pmatrix},\quad
\gamma^2 = \begin{pmatrix}
0 & 0 & 0 & -i \\
0 & 0 & i & 0 \\
0 & i & 0 & 0 \\
-i & 0 & 0 & 0 \\
\end{pmatrix},\quad
\gamma^3 = \begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1 \\
-1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
\end{pmatrix} " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/c/c/ccc84f99c6b4bab172bd5b623c1e73e7.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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While in the <i>chiral representation (and basis)</i>, also known as the <i>Weyl representation</i>:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="
\gamma^0 = \begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
\end{pmatrix} " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/c/2/dc246b4b158c8b617ac6c904299574e4.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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and the spatial gamma matrices are the same as in the Pauli-Dirac representation. The gamma matrices are representative basis elements of a <a href="http://en.wikipedia.org/wiki/Clifford_algebra" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Clifford algebra">Clifford algebra</a>, satisfying the defining relationship</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\frac{1}{2} ( \gamma_{a} \gamma_{b} + \gamma_{b} \gamma_{a} ) = \eta_{a b} " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/9/f/39fa06cf8b56b297d7f43b65cd716752.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
in which <img alt="\eta_{a b}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/6/9/169a272655b69039bd0273e8f86e587e.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Minkowski_metric" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Minkowski metric">Minkowski metric</a> of signature (+---). Using gamma matrices, the Dirac equation becomes:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><table cellpadding="5" style="background-attachment: initial; background-clip: initial; background-color: #ecfcf4; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(80, 200, 120); border-bottom-style: solid; border-bottom-width: 2px; border-image: initial; border-left-color: rgb(80, 200, 120); border-left-style: solid; border-left-width: 2px; border-right-color: rgb(80, 200, 120); border-right-style: solid; border-right-width: 2px; border-top-color: rgb(80, 200, 120); border-top-style: solid; border-top-width: 2px; font-size: 13px; text-align: center;"><tbody>
<tr><td><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
<img alt=" \left( c\boldsymbol{\gamma}\cdot\bold{\hat{p}}+ mc^2\right)\psi = i\hbar\gamma^0\frac{\partial}{\partial t}\psi " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/4/f/64fbfaa8450ade82b3bb87e029d2b166.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /></div>
</td></tr>
</tbody></table>
</dd></dl>
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This is a particularly useful way to write the equation, since it can be immediately translated into the language of 4-vectors and relativistic covariance can be demonstrated (see below), while it resembles a similar form to the original.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: The Pauli spin σ matrices">edit</a>]</span><span class="mw-headline" id="The_Pauli_spin_.CF.83_matrices">The Pauli spin <span style="font-family: 'Times new roman';">σ</span> matrices</span></h3>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
The Dirac matrices are <a href="http://en.wikipedia.org/wiki/Block_matrix" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Block matrix">block matrices</a>; where the partitions are the 2 × 2 <a href="http://en.wikipedia.org/wiki/Zero_matrix" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Zero matrix">zero matrix</a>, the 2 × 2 Identity matrix <i>I</i><sub style="line-height: 1em;">2</sub>, and the <a href="http://en.wikipedia.org/wiki/Pauli_matrices" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pauli matrices">Pauli matrices</a> <i>σ<sub style="line-height: 1em;">x</sub>, σ<sub style="line-height: 1em;">y</sub>, σ<sub style="line-height: 1em;">z</sub></i> (equivalently written <i>σ</i><sub style="line-height: 1em;">1</sub>, <i>σ</i><sub style="line-height: 1em;">2</sub>,<i>σ</i><sub style="line-height: 1em;">3</sub>). In practice these rather large matrices can be written in the following standard representations: the <i>α</i> and <i>β</i> matrices are</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="
\beta = \begin{pmatrix}
0 & I_2 \\
I_2 & 0 \\
\end{pmatrix} ,\quad
\alpha_1 = \begin{pmatrix}
-\sigma_x & 0 \\
0 & \sigma_x \\
\end{pmatrix},\quad
\alpha_2 = \begin{pmatrix}
-\sigma_y & 0 \\
0 & \sigma_y \\
\end{pmatrix},\quad
\alpha_3 = \begin{pmatrix}
-\sigma_z & 0 \\
0 & \sigma_z \\
\end{pmatrix} " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/5/d/65d3d47070164f53f45e8bd378fc7bcf.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
the Pauli-Dirac basis is</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="
\gamma^0 = \begin{pmatrix}
I_2 & 0 \\
0 & -I_2 \\
\end{pmatrix},\quad
\gamma^1 = \begin{pmatrix}
0 & \sigma_x \\
-\sigma_x & 0
\end{pmatrix},\quad
\gamma^2 = \begin{pmatrix}
0 & \sigma_y \\
-\sigma_y & 0
\end{pmatrix},\quad
\gamma^3 = \begin{pmatrix}
0 & \sigma_z \\
-\sigma_z & 0
\end{pmatrix}
\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/4/6/4467de98cd28cdf8f582ddfef9a8b183.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
and the chiral basis is:</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
<img alt="
\gamma^0 = \begin{pmatrix}
0 & I_2 \\
I_2 & 0 \\
\end{pmatrix},\quad
\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/f/a/cfafd74875d210039b4d6004169f1372.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /></div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
where <img alt=" \scriptstyle \gamma^1, \gamma^2, \gamma^3 \,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/c/9/3c9101ea69fce590a4afb1c70f9b600a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> are as before.</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
These can be written in terms of the <a href="http://en.wikipedia.org/wiki/Kronecker_product" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Kronecker product">Kronecker product</a> (aka direct product, denoted by <img alt="\scriptstyle\otimes\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/f/3/3f3ce296c368287c121cb8e0d7861059.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> or sometimes <img alt="\scriptstyle\times\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/0/c/90c9768f5ab80c3f1f3c9fa5edd17297.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />)<sup class="reference" id="cite_ref-4" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_note-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[5]</a></sup><sup class="reference" id="cite_ref-5" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_note-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[6]</a></sup> of the matrices</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="
\beta = \gamma^0 = \begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix}\otimes I_2 , \quad
\boldsymbol{\alpha} = \begin{pmatrix}
1 & 0 \\
0 & -1 \\
\end{pmatrix}\otimes\boldsymbol{\sigma}
\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/7/a/67a755fb0877574f5ee32d15fb1da3e6.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
and</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="
\gamma^0 = \begin{pmatrix}
1 & 0 \\
0 & -1 \\
\end{pmatrix}\otimes I_2 , \quad
\boldsymbol{\gamma} = \begin{pmatrix}
0 & 1 \\
-1 & 0 \\
\end{pmatrix}\otimes\boldsymbol{\sigma}
\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/7/9/679bad598164d9eb79fb64becefe932f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
where</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\boldsymbol{\sigma} = \left(\sigma_x,\sigma_y,\sigma_z\right)= \left(\sigma_1,\sigma_2,\sigma_3\right)\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/3/0/e307ea5a1d8eeb57ba3155465766f7d7.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
is a vector whose components are the Pauli matrices.</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
The Dirac equation can then be written directly in terms of the Pauli σ matrices, illustrating how the Dirac theory accounts for Pauli's theory of spin. Substituting the <i>α</i> and <i>β</i>matrices leads to</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><table cellpadding="10" style="background-attachment: initial; background-clip: initial; background-color: #ecfcf4; background-image: initial; background-origin: initial; background-position: initial initial; background-repeat: initial initial; border-bottom-color: rgb(80, 200, 120); border-bottom-style: solid; border-bottom-width: 2px; border-image: initial; border-left-color: rgb(80, 200, 120); border-left-style: solid; border-left-width: 2px; border-right-color: rgb(80, 200, 120); border-right-style: solid; border-right-width: 2px; border-top-color: rgb(80, 200, 120); border-top-style: solid; border-top-width: 2px; font-size: 13px; text-align: center;"><tbody>
<tr><td><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">
<img alt="
\begin{pmatrix}
\hat{E} - c\boldsymbol{\sigma}\cdot\bold{\hat{p}} & 0 \\
0 & \hat{E} + c\boldsymbol{\sigma}\cdot\bold{\hat{p}} \\
\end{pmatrix}\psi
= mc^2
\begin{pmatrix}
0 & I_2 \\
I_2 & 0 \\
\end{pmatrix}
\psi
\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/9/0/f902ff5a79a79e652f783947374010ee.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /></div>
</td></tr>
</tbody></table>
</dd></dl>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><table class="toccolours collapsible collapsed" id="collapsibleTable1" style="background-color: mintcream; border-bottom-color: rgb(8, 64, 128); border-bottom-style: solid; border-bottom-width: 2px; border-image: initial; border-left-color: rgb(8, 64, 128); border-left-style: solid; border-left-width: 2px; border-right-color: rgb(8, 64, 128); border-right-style: solid; border-right-width: 2px; border-top-color: rgb(8, 64, 128); border-top-style: solid; border-top-width: 2px; font-size: 12px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;"><tbody>
<tr><th><span class="collapseButton" style="float: right; font-weight: normal; margin-left: 0.5em; text-align: right; width: auto;">[<a href="http://en.wikipedia.org/wiki/Dirac_equation#" id="collapseButton1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">show</a>]</span>Proof of equivalence</th></tr>
</tbody></table>
</dd></dl>
<h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 18px; line-height: 20px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Dirac equation in curved spacetime">edit</a>]</span><span class="mw-headline" id="Dirac_equation_in_curved_spacetime">Dirac equation in curved spacetime</span></h3>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
The Dirac equation in <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Curved_spacetime" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Curved spacetime">curved spacetime</a> can be written by using <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Vierbein" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vierbein">vierbein</a> fields and the gravitational <a href="http://en.wikipedia.org/wiki/Spin_connection" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin connection">spin connection</a>. The vierbein defines a local rest <a href="http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Frame fields in general relativity">frame</a>, allowing the constant <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Dirac_matrices" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac matrices">Dirac matrices</a> to act at each spacetime point. In this way, Dirac's equation takes the following form in curved spacetime<sup class="reference" id="cite_ref-6" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_note-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[7]</a></sup>:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="-i\gamma^a e_a^\mu D_\mu \Psi + m \Psi = 0." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/b/1/fb13217d82696849188f59c5d4414950.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
Here <img alt="\scriptstyle e_a^\mu" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/2/1/42198a0fd5a3a58ee91d7b84640a737a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Vierbein" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vierbein">vierbein</a> and <img alt="\scriptstyle D_\mu" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/2/4/024a27a58d52db16d48418ccc62116e4.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is the <a href="http://en.wikipedia.org/wiki/Covariant_derivative" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Covariant derivative">covariant derivative</a> for fermion fields, defined as follows</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="D_\mu = \partial_\mu - \frac{i}{4} \omega_{\mu}^{ab} \sigma_{ab}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/1/3/c1343b6bea733bd4962c95c353cd1eca.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
where <img alt="\sigma_{ab}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/4/1/64149222988ea3d12a70e6391458bfe0.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is the commutator of Dirac matrices:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\sigma_{ab}=\frac{i}{2} \left[\gamma_{a},\gamma_{b}\right] " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/1/4/91483818aedf428ca45d4af9251ae3bd.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
and <img alt="\omega_{\mu}^{ab}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/5/2/852e7bc7f64acc61a5c607f9617ba250.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> are the <a href="http://en.wikipedia.org/wiki/Spin_connection" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin connection">spin connection</a> components.</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
Note that here Latin indices denote the "Lorentzian" vierbein labels while Greek indices denote <a href="http://en.wikipedia.org/wiki/Manifold" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Manifold">manifold</a> coordinate indices.</div>
<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Physical interpretation">edit</a>]</span><span class="mw-headline" id="Physical_interpretation">Physical interpretation</span></h2>
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The Dirac theory, while providing a wealth of information that is accurately confirmed by experiments, nevertheless introduces a new physical paradigm that appears at first difficult to interpret and even paradoxical. Some of these issues of interpretation must be regarded as open questions. The Dirac theory brilliantly answered some of the outstanding issues in physics at the time it was put forward, while posing others that are still the subject of debate. Many of these issues were resolved in modern quantum field theory by considering the Dirac equation not as a relativistic description of quantum mechanics but merely as another relativistic field equation, on the same footing as the Klein-Gordon equation or Maxwell's equations, in which <i>ψ</i> is not interpreted as a wave function but rather as a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Fermion_field" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermion field">fermion field</a>, similar to the Klein-Gordon scalar field or electromagnetic field. Nevertheless, considering Dirac's equation as a relativistic version of Schrödinger's equation is extremely computationally useful, and raises important issues.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Identification of observables">edit</a>]</span><span class="mw-headline" id="Identification_of_observables">Identification of observables</span></h3>
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The critical physical question in a quantum theory is - what are the physically observable quantities defined by the theory? According to general principles, such quantities are defined by Hermitian operators that act on the Hilbert space of possible states of a system. The eigenvalues of these operators are then the possible results of measuring the corresponding physical quantity. In the Schrödinger theory, the simplest such object is the overall Hamiltonian, which represents the total energy of the system. If we wish to maintain this interpretation on passing to the Dirac theory, we must take the Hamiltonian to be</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="H = \gamma^0 \left [ mc^2 + c \sum_{k = 1}^3 \gamma^k \left ( p_k-\frac{q}{c}A_k \right ) \right ] + qA^0." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/0/2/e0246468a405e774241b975194f88085.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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This looks promising, because we see by inspection the rest energy of the particle and, in case <i>A</i> = 0, the energy of a charge placed in an electric potential <i>qA</i><sup style="line-height: 1em;">0</sup>. What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="H = c\sqrt{\left ( p - \frac{q}{c}A \right )^2 + m^2c^2} + qA^0." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/9/0/0901555fc874645fa5da6f05d940b6a9.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Thus the Dirac Hamiltonian is <i>fundamentally distinguished</i> from its classical counterpart, and we must take great care to correctly identify what is an observable in this theory. Much of the apparent paradoxical behavior implied by the Dirac equation amounts to a misidentification of these observables. The following issues arise with the Dirac equation, which are not immediately easy to interpret:</div>
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<a href="http://en.wikipedia.org/wiki/Klein_paradox" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Klein paradox">Klein paradox</a>: when a Dirac electron interacts with an electric potential, the total probability is not conserved. Also, the electron can tunnel into high potential barriers, unlike the case in quantum mechanics as described by the Schrödinger equation.</div>
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<a href="http://en.wikipedia.org/wiki/Zitterbewegung" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Zitterbewegung">Zitterbewegung</a>: there is an apparent fluctuation (at the speed of light) of the position of an electron around the median.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=12" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Hole theory">edit</a>]</span><span class="mw-headline" id="Hole_theory">Hole theory</span></h3>
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The negative <i>E</i> solutions of Dirac's equation were problematic, for it was assumed that the particle has a positive energy. Mathematically, however, there seemed to be no reason to reject the negative-energy solutions. Since they exist, we cannot simply ignore them, for once we include the interaction between the electron and the electromagnetic field, any electron placed in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy by emitting excess energy in the form of <a href="http://en.wikipedia.org/wiki/Photon" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Photon">photons</a>. Real electrons obviously do not behave in this way.</div>
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To cope with this problem, Dirac introduced the hypothesis, known as <b>hole theory</b>: that the <a href="http://en.wikipedia.org/wiki/Vacuum" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vacuum">vacuum</a> is the many-body quantum state in which all the negative-energy electron eigenstates are occupied. This description of the vacuum as a "sea" of electrons is called the <a href="http://en.wikipedia.org/wiki/Dirac_sea" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac sea">Dirac sea</a>. Since the <a href="http://en.wikipedia.org/wiki/Pauli_exclusion_principle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pauli exclusion principle">Pauli exclusion principle</a> forbids electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy eigenstate, and positive-energy electrons would be forbidden from decaying into negative-energy eigenstates.</div>
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Dirac further reasoned that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate – called a <b>hole</b> – would behave like a positively charged particle. The hole possesses a <i>positive</i> energy, since energy is required to create a particle–hole pair from the vacuum. As noted above, Dirac initially thought that the hole might be the proton, but <a href="http://en.wikipedia.org/wiki/Hermann_Weyl" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermann Weyl">Hermann Weyl</a> pointed out that the hole should behave as if it had the same mass as an electron, whereas the proton is over 1800 times heavier. The hole was eventually identified as the <a href="http://en.wikipedia.org/wiki/Positron" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Positron">positron</a>, experimentally discovered by <a href="http://en.wikipedia.org/wiki/Carl_David_Anderson" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Carl David Anderson">Carl Anderson</a> in 1932.</div>
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It is not entirely satisfactory to describe the "vacuum" using an infinite sea of negative-energy electrons. The infinitely negative contributions from the sea of negative-energy electrons has to be canceled by an infinite positive "bare" energy and the contribution to the charge density and current coming from the sea of negative-energy electrons is exactly canceled by an infinite positive "<a href="http://en.wikipedia.org/wiki/Jellium" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Jellium">jellium</a>" background so that the net electric charge density of the vacuum is zero. In <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">quantum field theory</a>, a <a href="http://en.wikipedia.org/wiki/Bogoliubov_transformation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bogoliubov transformation">Bogoliubov transformation</a>on the creation and annihilation operators (turning an occupied negative-energy electron state into an unoccupied positive energy positron state and an unoccupied negative-energy electron state into an occupied positive energy positron state) allows us to bypass the Dirac sea formalism even though, formally, it is equivalent to it.</div>
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In certain applications of <a href="http://en.wikipedia.org/wiki/Condensed_matter_physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Condensed matter physics">condensed matter physics</a>, however, the underlying concepts of "hole theory" are valid. The sea of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Conduction_electron" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Conduction electron">conduction electrons</a> in an <a href="http://en.wikipedia.org/wiki/Electrical_conductor" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electrical conductor">electrical conductor</a>, called a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Fermi_sea" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermi sea">Fermi sea</a>, contains electrons with energies up to the <a href="http://en.wikipedia.org/wiki/Chemical_potential" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Chemical potential">chemical potential</a> of the system. An unfilled state in the Fermi sea behaves like a positively-charged electron, though it is referred to as a "hole" rather than a "positron". The negative charge of the Fermi sea is balanced by the positively-charged ionic lattice of the material.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=13" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Properties">edit</a>]</span><span class="mw-headline" id="Properties">Properties</span></h2>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=14" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Covariant form and relativistic invariance">edit</a>]</span><span class="mw-headline" id="Covariant_form_and_relativistic_invariance">Covariant form and relativistic invariance</span></h3>
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Main articles: <a href="http://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a> and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Relativistic_invariance" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Relativistic invariance">relativistic invariance</a></div>
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To demonstrate the relativistic invariance of the equation, it is advantageous to cast it into a form in which the space and time derivatives appear on an equal footing. Using the gamma-matrix form above, the covariant form can be obtained by inserting the gradient operator and collecting all space and time derivatives together (dividing by <i>c</i> for convenience):</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \begin{align}
& c\boldsymbol{\gamma}\cdot\bold{\hat{p}}\psi - i\hbar\gamma^0\frac{\partial}{\partial t}\psi + mc^2 \psi = 0 \\
& -i\hbar \boldsymbol{\gamma}\cdot\nabla\psi - \gamma^0\frac{i\hbar}{c}\frac{\partial}{\partial t}\psi + mc \psi = 0 \\
& -i\hbar \left(\boldsymbol{\gamma}\cdot\nabla + \gamma^0\frac{1}{c}\frac{\partial}{\partial t}\right)\psi + mc \psi = 0 \\
\end{align}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/5/c/a5c1fb274078e58f7318404525410c89.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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then using the <a href="http://en.wikipedia.org/wiki/Four-vector" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Four-vector">4-position</a> (as above) and (+−−−) <a href="http://en.wikipedia.org/wiki/Metric_signature" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Metric signature">metric signature</a> to gain the <a href="http://en.wikipedia.org/wiki/Tensor_contraction#Contraction_in_index_notation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tensor contraction">contraction</a> between the gamma matrices and the 4-position derivatives;</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \boldsymbol{\gamma}\cdot\nabla + \gamma^0\frac{1}{c}\frac{\partial}{\partial t} = \gamma^1\frac{\partial}{\partial x^1} + \gamma^2\frac{\partial}{\partial x^2} + \gamma^3\frac{\partial}{\partial x^3} + \gamma^0\frac{1}{c}\frac{\partial}{\partial x^0} = \gamma^\mu\partial_\mu " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/1/f/f1f5f54e2412c84c3b2341dd49e5ca76.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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so we have<sup class="reference" id="cite_ref-7" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_note-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[8]</a></sup></div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" -i\hbar \gamma^\mu\partial_\mu\psi + mc\psi = 0 " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/b/f/9bfd4077a284f4f2bab4cfb7d10c34b4.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Using the <a href="http://en.wikipedia.org/wiki/Feynman_slash_notation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Feynman slash notation">Feynman slash notation</a> the equation becomes</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" -i\hbar \partial\!\!\!/\psi + mc\psi = 0 " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/6/7/8678c183721b536f7a29c7f0d51984f8.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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This covariant form has further relativistic implications:</div>
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<li style="margin-bottom: 0.1em;">the Dirac equation <i>is</i> the square root of the Klein-Gordon equation. The Klein-Gordon equation is based on <img alt="E^2 = (pc)^2 + (mc^2)^2" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/d/5/dd55d440393b8af0f98d04326f457b95.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />, meaning the Dirac equation is based on its square root <img alt="E= \sqrt{(pc)^2 + (mc^2)^2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/3/4/2340c3b6a1e9795c5a17d66efbf92344.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />.</li>
<li style="margin-bottom: 0.1em;">Any solution to the Dirac equation is automatically a solution to the Klein-Gordon equation, but not vice versa, i.e. there are solutions to the Klein-Gordon equation that do not solve the Dirac equation.<sup class="reference" id="cite_ref-McMahon_2-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_note-McMahon-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup></li>
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This can be found by factoring the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Klein-Gordon_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Klein-Gordon equation">Klein-Gordon equation</a> (in the slash notation):</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="0 = [\hbar^2\partial^\mu \partial_\mu + (mc)^2]\psi = [(\hbar\partial\!\!\!/)^2 + (mc)^2]\psi = (i\hbar\partial\!\!\!/ + mc)(-i\hbar\partial\!\!\!/ + mc)\psi \,." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/5/f/25f43c1a22b15c86cf12fcafafd60bf3.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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and noticing the last factor, <img alt="(-i\hbar\partial\!\!\!/ + mc)\psi \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/d/f/5dfbc34e9a20015570e8e394afab3a7f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, is simply the Dirac equation. In this sense, the Dirac equation takes an extra step forward into relativistic quantum mechanics compared with Klein–Gordon equation.</div>
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The complete system is summarized using the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Minkowski_metric" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Minkowski metric">Minkowski metric</a> on spacetime in the form</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="[\gamma^\mu,\gamma^\nu ]_+ = 2 g^{\mu\nu} \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/b/2/4b28fd3b8889337069a4fddc546b1f39.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where again [, ]<sub style="line-height: 1em;">+</sub> denotes the anticommutator. These are the defining relations of a <a href="http://en.wikipedia.org/wiki/Clifford_algebra" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Clifford algebra">Clifford algebra</a> over a pseudo-orthogonal 4-d space with metric signature (+−−−). The specific Clifford algebra employed in the Dirac equation is known today as the <a href="http://en.wikipedia.org/wiki/Dirac_algebra" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac algebra">Dirac algebra</a>. Although not recognized as such by Dirac at the time the equation was formulated, in hindsight the introduction of this <i>geometric algebra</i> also represents a step forward in the development of relativistic quantum theory.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=15" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Relativistic eigenvalue equation">edit</a>]</span><span class="mw-headline" id="Relativistic_eigenvalue_equation">Relativistic eigenvalue equation</span></h3>
<div class="rellink boilerplate seealso" style="background-color: white; font-family: sans-serif; font-size: 13px; font-style: italic; line-height: 20px; margin-bottom: 0.5em; padding-left: 1.6em;">
See also: <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Eigenvalue" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eigenvalue">eigenvalue</a></div>
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Further, the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/4-momentum" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="4-momentum">4-momentum</a> vector is</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="P^\mu = \left(\frac{E}{c},-\bold{p}\right) \,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/6/a/46acb010eba476c2c4ce53a6c03a7d14.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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so inserting the quantum operators obtains the 4-momentum operator;</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\hat{P}^\mu = \left(\frac{1}{c}\hat{E},-\bold{\hat{p}}\right) = i\hbar\left(\frac{1}{c}\frac{\partial}{\partial t},\nabla\right) = i\hbar\partial_\mu \,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/a/7/1a78a38e75a308766c71bb4770aeaf1a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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(the −<i>iħ</i> becomes +<i>iħ</i> preceding the 3-momentum operator). Contraction of this operator with the gamma matrices (using <a href="http://en.wikipedia.org/wiki/Feynman_slash_notation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Feynman slash notation">Feynman slash notation</a>) gives</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\gamma^\mu\hat{P}^\mu = i\hbar\gamma^\mu\partial_\mu = /\!\!\!\!\hat{P} \,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/f/5/ff55edac4cab2f715b7a6a5f297140bc.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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which dramatically shortens the Dirac equation to the familiar structure of momentum;</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="/\!\!\!\!\hat{P} \psi = mc\psi \,. \,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/d/3/0d3cdc177e8f24f9343200a75661a800.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The Dirac equation may now be interpreted as an <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Eigenvalue" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eigenvalue">eigenvalue</a> equation, where the rest mass is proportional to an eigenvalue of the 4-momentum operator, the proportionality constant being the speed of light <i>c</i>.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=16" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Spinor transformations">edit</a>]</span><span class="mw-headline" id="Spinor_transformations">Spinor transformations</span></h3>
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Main article: <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Spinors" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spinors">spinors</a></div>
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In practice, physicists often use units of measure such that <i>ħ</i> = <i>c</i> = 1, known as "<a href="http://en.wikipedia.org/wiki/Natural_units" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Natural units">natural units</a>". The equation then takes the simple form</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="(-i\gamma^\mu\partial_\mu + m) \psi = 0\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/8/e/28e00fe9cb0823b06f7121db466fe419.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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A fundamental theorem states that if two distinct sets of matrices are given that both satisfy the Clifford relations, then they are connected to each other by a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Similarity_transformation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Similarity transformation">similarity transformation</a>:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\gamma^{\mu\prime} = S^{-1} \gamma^\mu S." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/e/1/8e1c1891ae617749e59ba65d927a69e3.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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If in addition the matrices are all <a href="http://en.wikipedia.org/wiki/Unitary_transformation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Unitary transformation">unitary</a>, as are the Dirac set, then <i>S</i> itself is unitary;</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\gamma^{\mu\prime} = U^\dagger \gamma^\mu U." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/8/a/a8aadda530af33a3504a5566f7c63b9e.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The transformation <i>U</i> is unique up to a multiplicative factor of absolute value 1. Let us now imagine a Lorentz transformation to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector. For the operator <img alt="\scriptstyle \gamma^\mu\,\partial_\mu" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/5/e/25e218276d8bc5368c61b2053f25974b.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> to remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index. These new gammas will themselves satisfy the Clifford relations, because of the orthogonality of the Lorentz transformation. By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation. In the new frame, remembering that the rest mass is a relativistic scalar, the Dirac equation will then take the form</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="( -iU^\dagger \gamma^\mu U\partial_\mu^\prime + m)\psi(x^\prime,t^\prime) = 0" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/b/7/cb71c236fe81e7ee18ae05c21f1c5f07.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="U^\dagger(-i\gamma^\mu\partial_\mu^\prime + m)U \psi(x^\prime,t^\prime) = 0." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/1/d/71d81b1da538fadeed08ca57ed4acb37.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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If we now define the transformed spinor</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\psi^\prime = U\psi" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/5/0/e50bc67787ae6018b6f7bf0f40cb3b89.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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then we have the transformed Dirac equation in a way that demonstrates manifest relativistic invariance:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="(-i\gamma^\mu\partial_\mu^\prime + m)\psi^\prime(x^\prime,t^\prime) = 0." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/2/c/32cacfed3f5a76c61be67762ede945c7.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Thus, once we settle on any unitary representation of the gammas, it is final provided we transform the spinor according to the unitary transformation that corresponds to the given Lorentz transformation. The various representations of the Dirac matrices employed will bring into focus particular aspects of the physical content in the Dirac field (see below). The representation shown here is known as the <i>standard</i> representation - in it, the upper two components go over into Pauli's 2-spinor wave function in the limit of low energies and small velocities in comparison to light.</div>
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The considerations above reveal the origin of the gammas in <i>geometry</i>, harking back to Grassmann's original motivation - they represent a fixed basis of unit vectors in spacetime. Similarly, products of the gammas such as <img alt="\gamma_\mu\,\gamma_\nu" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/b/c/5bc3f9c61a81b7970d833a78a285f7a1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> represent <i>oriented surface elements</i>, and so on. With this in mind, we can find the form of the unit volume element in spacetime in terms of the gammas as follows. By definition, it is</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="V = \frac{1}{4!}\epsilon_{\mu\nu\alpha\beta}\gamma^\mu\gamma^\nu\gamma^\alpha\gamma^\beta." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/3/e/b3ed02fe006b387027ed529bdb0bde25.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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For this to be an invariant, the epsilon symbol must be a tensor, and so must contain a factor of <img alt="\sqrt{g}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/a/3/1a30f0eb69d02245f99bf209372d5fa1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, where g is the determinant of the metric tensor. Since this is negative, that factor is <i>imaginary</i>. Thus</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="V = i \gamma^0\gamma^1\gamma^2\gamma^3.\ " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/5/3/b531ab25f1d446b1a459834b239b80cc.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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This matrix is given the special symbol <img alt="\scriptstyle \gamma^5" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/b/6/fb6ffcc27a7b669582947bed73b8135f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, owing to its importance when one is considering improper transformations of spacetime, that is, those that change the orientation of the basis vectors. In the standard representation it is</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\gamma^5 = \begin{pmatrix} 0 & I_{2} \\ I_{2} & 0 \end{pmatrix}." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/1/e/91e853974c1824f80cc283fc46c0057e.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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This matrix will also be found to anticommute with the other four Dirac matrices. It takes on a leading role when questions of <i>parity</i> arise, because the volume element as a directed magnitude changes sign under a space-time reflection. Taking the positive square root above thus amounts to choosing a handedness convention on space-time.</div>
<h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 18px; line-height: 20px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=17" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Adjoint equation and probability conservation">edit</a>]</span><span class="mw-headline" id="Adjoint_equation_and_probability_conservation">Adjoint equation and probability conservation</span></h3>
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Main articles: <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Conservation_of_probability" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Conservation of probability">Conservation of probability</a> and <a href="http://en.wikipedia.org/wiki/Probability_current" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability current">probability current</a></div>
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In the Schrödinger theory, the <a href="http://en.wikipedia.org/wiki/Probability_density_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability density function">probability density</a> is given by the positive definite expression</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\rho=\psi^*\psi\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/a/5/5a5f2561b7285a9c56d0727ba44c9c4a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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and this density is convected according to the <a href="http://en.wikipedia.org/wiki/Probability_current" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability current">probability current</a> vector</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\bold{J} = -\frac{i\hbar}{2m}\left(\psi^*\nabla\psi - \psi\nabla\psi^*\right)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/1/9/c199546f8288648a52f4076b38266ad6.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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according to a <a href="http://en.wikipedia.org/wiki/Continuity_equation#Quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Continuity equation">continuity equation</a> for probability. For a relativistic theory, these may be incorporated into a <a href="http://en.wikipedia.org/wiki/Probability_current#Definition_.28relativistic_4-current.29" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability current">probability 4-current</a>, which has the relativistically covariant expression</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="J^\mu = \frac{i\hbar}{2m}(\psi^*\partial^\mu\psi - \psi\partial^\mu\psi^*) \, . " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/d/6/2d6d98563f7402dedef4e33f46759ad6.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where (translating usual cartesian-subscript notation into vector indices):<sup class="reference" id="cite_ref-McMahon_2-3" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_note-McMahon-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup></div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\begin{align} & J^\mu = (J^0, J^1, J^2, J^3) = (\rho, J_x, J_y, J_z) \\
& (x^0, x^1, x^2, x^3) = (t, x, y, z) \\
\end{align} \, " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/0/a/90ae4711ab06856052f75d77d54ee05c.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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By defining the <a href="http://en.wikipedia.org/wiki/Dirac_adjoint" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac adjoint">adjoint spinor</a></div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\bar{\psi} = \psi^\dagger\gamma^0" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/b/b/2bb6f9607d53af0b9d37828b86afb475.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <img alt="\scriptstyle \psi^\dagger " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/3/b/63bf3ff61aab97a46eee29f2255253c9.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is the <a href="http://en.wikipedia.org/wiki/Conjugate_transpose" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Conjugate transpose">conjugate transpose</a> of <img alt="\scriptstyle \psi " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/2/e/d2e79802c0615b1460d3934878f3fd5f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, and noticing that</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="(\gamma^\mu)^\dagger\gamma^0 = \gamma^0\gamma^\mu \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/f/b/6fbc50df0afef68b6af31c555657acec.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />,</dd></dl>
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we obtain, by taking the Hermitian conjugate of the Dirac equation and multiplying from the right by <img alt="\scriptstyle \gamma^0" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/a/5/3a564e6a531260a8763b3eeb16219056.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, the adjoint equation:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\bar{\psi}(i\gamma^\mu\partial_\mu + m) = 0 \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/e/b/5eb4aba374a7c58f7022ae1a450f705d.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <img alt="\scriptstyle \partial_\mu" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/f/5/3f5d1ed21090b13bafefff06bc4dd53a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is understood to act to the left. Multiplying the Dirac equation by <img alt="\scriptstyle \bar{\psi}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/2/a/32a19441574f9ee2adfc81f533e6143f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> from the left, and the adjoint equation by <img alt="\psi" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/9/d/19df1c2726ed43128440c1157f72a937.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> from the right, and subtracting, produces the law of conservation of the Dirac current:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\partial_\mu \left ( \bar{\psi}\gamma^\mu\psi \right ) = 0." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/b/e/7beb0537393ca3af39827cdf0dc8e3e4.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Now we see the great advantage of the first-order equation over the one Schrödinger had tried - this is the conserved current density required by relativistic invariance, only now its 4th component is <i>positive definite</i> and thus suitable for the role of a probability density:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="J^0 = \bar{\psi}\gamma^0\psi = \psi^\dagger\psi." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/f/a/bfaffff93e1a95bc61e07a5b4e60c941.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Because the probability density now appears as the fourth component of a relativistic vector, and not a simple scalar as in the Schrödinger equation, it will be subject to the usual effects of the Lorentz transformations such as time dilation. Thus for example atomic processes that are observed as rates, will necessarily be adjusted in a way consistent with relativity, while those involving the measurement of energy and momentum, which themselves form a relativistic vector, will undergo parallel adjustment which preserves the relativistic covariance of the observed values.</div>
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In a general case (if a certain linear function of electromagnetic field does not vanish identically), three out of four components of the spinor function in the Dirac equation can be algebraically eliminated, yielding an equivalent fourth-order partial differential equation for just one component. Furthermore, this remaining component can be made real by a gauge transform.<sup class="reference" id="cite_ref-8" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_note-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[9]</a></sup></div>
<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=18" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: See also">edit</a>]</span><span class="mw-headline" id="See_also">See also</span></h2>
<ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Old_quantum_theory#Relativistic_orbit" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Old quantum theory">Bohr–Sommerfeld theory</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Breit_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Breit equation">Breit equation</a></li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Dirac_field" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac field">Dirac field</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Einstein-Maxwell-Dirac_equations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Einstein-Maxwell-Dirac equations">Einstein-Maxwell-Dirac equations</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Feynman_checkerboard" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Feynman checkerboard">Feynman checkerboard</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Foldy%E2%80%93Wouthuysen_transformation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Foldy–Wouthuysen transformation">Foldy–Wouthuysen transformation</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Klein–Gordon equation">Klein–Gordon equation</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_electrodynamics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum electrodynamics">Quantum electrodynamics</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Rarita%E2%80%93Schwinger_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Rarita–Schwinger equation">Rarita–Schwinger equation</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Theoretical_and_experimental_justification_for_the_Schr%C3%B6dinger_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theoretical and experimental justification for the Schrödinger equation">Theoretical and experimental justification for the Schrödinger equation</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation_in_the_algebra_of_physical_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac equation in the algebra of physical space">Dirac equation in the algebra of physical space</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/EPR_paradox" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="EPR paradox">EPR paradox</a></li>
<li style="margin-bottom: 0.1em;">The Dirac Equation appears on the floor of <a href="http://en.wikipedia.org/wiki/Westminster_Abbey" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Westminster Abbey">Westminster Abbey</a>. It appears on the plaque commemorating Paul Dirac's life which was inaugurated on November 13, 1995.<sup class="reference" id="cite_ref-9" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_note-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[10]</a></sup></li>
</ul>
<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=19" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: References">edit</a>]</span><span class="mw-headline" id="References">References</span></h2>
<div class="reflist" style="background-color: white; font-family: sans-serif; font-size: 12px; line-height: 20px; list-style-type: decimal; margin-bottom: 0.5em;">
<ol class="references" style="line-height: 1.5em; list-style-image: none; list-style-type: inherit; margin-bottom: 0.5em; margin-left: 3.2em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li id="cite_note-0" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_ref-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">B Hatfield, <i>Quantum Field Theory of Point Particles and Strings</i>, Addison-Wesley, Reading, MA, 1989.</span></li>
<li id="cite_note-1" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_ref-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">Particle Physics (3rd Edition), B. R. Martin, G.Shaw, Manchester Physics Series, John Wiley & Sons, <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/9780470032947" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 978-0-470-03294-7</a></span></li>
<li id="cite_note-McMahon-2" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink">^ <a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_ref-McMahon_2-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_ref-McMahon_2-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_ref-McMahon_2-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>c</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_ref-McMahon_2-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>d</b></i></sup></a></span> <span class="reference-text">Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/9780071543828" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 978-0-07-154382-8</a></span></li>
<li id="cite_note-3" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_ref-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/9780131461000" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 978-0-13-146100-0</a></span></li>
<li id="cite_note-4" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_ref-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a class="external free" href="http://mathworld.wolfram.com/KroneckerProduct.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://mathworld.wolfram.com/KroneckerProduct.html</a></span></li>
<li id="cite_note-5" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_ref-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) 0-89573-752-3</span></li>
<li id="cite_note-6" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_ref-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;">Lawrie, Ian D.. <i>A Unified Grand Tour of Theoretical Physics</i>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Unified+Grand+Tour+of+Theoretical+Physics&rft.aulast=Lawrie&rft.aufirst=Ian+D.&rft.au=Lawrie%2C%26%2332%3BIan+D.&rfr_id=info:sid/en.wikipedia.org:Dirac_equation"></span></span></li>
<li id="cite_note-7" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_ref-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/9780521575072" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 978-0-521-57507-2</a></span></li>
<li id="cite_note-8" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_ref-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">Journal of Mathematical Physics, 52, 082303 (2011) (<a class="external free" href="http://jmp.aip.org/resource/1/jmapaq/v52/i8/p082303_s1" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://jmp.aip.org/resource/1/jmapaq/v52/i8/p082303_s1</a> or <a class="external free" href="http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif); background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 18px; text-decoration: none;">http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf</a> )</span></li>
<li id="cite_note-9" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Dirac_equation#cite_ref-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a class="external free" href="http://www.dirac.ch/PaulDirac.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://www.dirac.ch/PaulDirac.html</a></span></li>
</ol>
</div>
<h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 18px; line-height: 20px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=20" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Selected papers">edit</a>]</span><span class="mw-headline" id="Selected_papers">Selected papers</span></h3>
<ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;"><span class="citation Journal" style="word-wrap: break-word;">Dirac, P. A. M. (1928). "The Quantum Theory of the Electron". <i>Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences</i> <b>117</b> (778): 610.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1098%2Frspa.1928.0023" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1098/rspa.1928.0023</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Quantum+Theory+of+the+Electron&rft.jtitle=Proceedings+of+the+Royal+Society+A%3A+Mathematical%2C+Physical+and+Engineering+Sciences&rft.aulast=Dirac&rft.aufirst=P.+A.+M.&rft.au=Dirac%2C%26%2332%3BP.+A.+M.&rft.date=1928&rft.volume=117&rft.issue=778&rft.pages=610&rft_id=info:doi/10.1098%2Frspa.1928.0023&rfr_id=info:sid/en.wikipedia.org:Dirac_equation"></span></li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k562109/f646" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">P.A.M. Dirac "The Quantum Theory of the Electron", Proc. R. Soc. <b>A117</b>)</a> link to the volume of the Proceedings of the Royal Society of London containing the article at page 610</li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k56219d/f388.table" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">P.A.M. Dirac "A Theory of Electrons and Protons", Proc. R. Soc. <b>A126</b>)</a> link to the volume of the Proceedings of the Royal Society of London containing the article at page 360</li>
<li style="margin-bottom: 0.1em;">C.D. Anderson, Phys. Rev. <b>43</b>, 491 (1933)</li>
<li style="margin-bottom: 0.1em;">R. Frisch and O. Stern, Z. Phys. <b>85</b>, 4 (1933)</li>
</ul>
<h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 18px; line-height: 20px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=21" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Textbooks">edit</a>]</span><span class="mw-headline" id="Textbooks">Textbooks</span></h3>
<ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">Halzen, Francis; Martin, Alan (1984). <i>Quarks & Leptons: An Introductory Course in Modern Particle Physics</i>. John Wiley & Sons. ISBN.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quarks+%26+Leptons%3A+An+Introductory+Course+in+Modern+Particle+Physics&rft.aulast=Halzen%2C+Francis%3B+Martin%2C+Alan&rft.au=Halzen%2C+Francis%3B+Martin%2C+Alan&rft.date=1984&rft.pub=John+Wiley+%26+Sons&rfr_id=info:sid/en.wikipedia.org:Dirac_equation"></span></li>
<li style="margin-bottom: 0.1em;">Dirac, P.A.M., <i>Principles of Quantum Mechanics</i>, 4th edition (Clarendon, 1982)</li>
<li style="margin-bottom: 0.1em;">Shankar, R., <i>Principles of Quantum Mechanics</i>, 2nd edition (Plenum, 1994)</li>
<li style="margin-bottom: 0.1em;">Bjorken, J D & Drell, S, <i>Relativistic Quantum mechanics</i></li>
<li style="margin-bottom: 0.1em;">Thaller, B., <i>The Dirac Equation</i>, Texts and Monographs in Physics (Springer, 1992)</li>
<li style="margin-bottom: 0.1em;">Schiff, L.I., <i>Quantum Mechanics</i>, 3rd edition (McGraw-Hill, 1968)</li>
<li style="margin-bottom: 0.1em;">Griffiths, D.J., <i>Introduction to Elementary Particles</i>, 2nd edition (Wiley-VCH, 2008) <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/9783527406012" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 978-3-527-40601-2</a>.</li>
</ul>
<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Dirac_equation&action=edit&section=22" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: External links">edit</a>]</span><span class="mw-headline" id="External_links">External links</span></h2>
<ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.mathpages.com/home/kmath654/kmath654.htm" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">The Dirac Equation</a> at MathPages</li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.mc.maricopa.edu/~kevinlg/i256/Nature_Dirac.pdf" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif); background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 18px; text-decoration: none;">The Nature of the Dirac Equation, its solutions and Spin</a></li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://electron6.phys.utk.edu/qm2/modules/m9/dirac.htm" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Dirac equation for a spin ½ particle</a></li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.quantumfieldtheory.info/" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Pedagogic Aids to Quantum Field Theory</a> click on Chap. 4 for a step-by-small-step introduction to the Dirac equation, spinors, and relativistic spin/helicity operators.</li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.sjsu.edu/faculty/watkins/spinor.htm" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Nature of spinors</a> Introductory explanation of spinors.</li>
</ul>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com0tag:blogger.com,1999:blog-5303246073824127471.post-27829145058608299562012-05-06T12:00:00.000-04:002012-05-06T12:00:42.714-04:00SpacetimeWe are building to the most important equation in Science, The Dirac Equation, that which united Special Relativity and Quantum Mechanics. Read this and back through the posts of this month May 2012, and be prepared.
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYRgXfE7yqmik8Pd7-QBEP7jTNzUyXV4Z4h2TcdLVSF0rSAj5DBml_elc5bxM1wozsegu5k3Hfcctoy7WfQGGhxI-4Fhc4-Iu6x34E64c-07XBASvR3xBaZ9GWPAC6ubSIR3wAJORWFg/s1600/Spacetime.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYRgXfE7yqmik8Pd7-QBEP7jTNzUyXV4Z4h2TcdLVSF0rSAj5DBml_elc5bxM1wozsegu5k3Hfcctoy7WfQGGhxI-4Fhc4-Iu6x34E64c-07XBASvR3xBaZ9GWPAC6ubSIR3wAJORWFg/s1600/Spacetime.png" /></a></div>
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<br /><br />In <a href="http://en.wikipedia.org/wiki/Physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physics">physics</a>, <b>spacetime</b> (or <b>space-time</b>, <b>space time</b>, <b>space-time continuum</b>) is any <a href="http://en.wikipedia.org/wiki/Mathematical_model" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mathematical model">mathematical model</a> that combines<a href="http://en.wikipedia.org/wiki/Space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Space">space</a> and <a href="http://en.wikipedia.org/wiki/Time_in_physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Time in physics">time</a> into a single <a href="http://en.wikipedia.org/wiki/Continuum_(theory)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Continuum (theory)">continuum</a>. Spacetime is usually interpreted with space as being <a href="http://en.wikipedia.org/wiki/Three-dimensional_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Three-dimensional space">three-dimensional</a> and time playing the role of a <a href="http://en.wikipedia.org/wiki/Minkowski_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Minkowski space">fourth dimension</a> that is of a different sort from the spatial dimensions. From a <a href="http://en.wikipedia.org/wiki/Euclidean_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Euclidean space">Euclidean space</a>perspective, the <a href="http://en.wikipedia.org/wiki/Universe" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Universe">universe</a> has three <a href="http://en.wikipedia.org/wiki/Dimension" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dimension">dimensions</a> of space and one dimension of time. By combining space and time into a single <a href="http://en.wikipedia.org/wiki/Manifold" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Manifold">manifold</a>, physicists have significantly simplified a large number of <a href="http://en.wikipedia.org/wiki/Theoretical_physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theoretical physics">physical theories</a>, as well as described in a more uniform way the workings of the universe at both the <a href="http://en.wikipedia.org/wiki/Physical_cosmology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical cosmology">supergalactic</a> and <a href="http://en.wikipedia.org/wiki/Quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum mechanics">subatomic</a> levels.</div>
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In non-relativistic <a href="http://en.wikipedia.org/wiki/Classical_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Classical mechanics">classical mechanics</a>, the use of Euclidean space instead of spacetime is appropriate, as time is treated as universal and constant, being independent of the state of motion of an observer. In <a href="http://en.wikipedia.org/wiki/Theory_of_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theory of relativity">relativistic</a> contexts, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object's <a href="http://en.wikipedia.org/wiki/Velocity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Velocity">velocity</a> relative to the observer and also on the strength of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gravitational_fields" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gravitational fields">gravitational fields</a>, which can slow the passage of time.</div>
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In <a href="http://en.wikipedia.org/wiki/Cosmology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cosmology">cosmology</a>, the concept of spacetime combines space and time to a single abstract <a href="http://en.wikipedia.org/wiki/Universe" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Universe">universe</a>. Mathematically it is a<a href="http://en.wikipedia.org/wiki/Manifold" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Manifold">manifold</a> consisting of "events" which are described by some type of <a href="http://en.wikipedia.org/wiki/Coordinate_system" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Coordinate system">coordinate system</a>. Typically three spatial dimensions (length, width, height), and one temporal dimension (<a href="http://en.wikipedia.org/wiki/Time" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Time">time</a>) are required. Dimensions are independent components of a coordinate grid needed to locate a point in a certain defined "space". For example, on the globe the <a href="http://en.wikipedia.org/wiki/Latitude" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Latitude">latitude</a> and <a href="http://en.wikipedia.org/wiki/Longitude" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Longitude">longitude</a> are two independent coordinates which together uniquely determine a location. In spacetime, a coordinate grid that spans the 3+1 dimensions locates <a href="http://en.wikipedia.org/wiki/Event_(relativity)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Event (relativity)">events</a> (rather than just points in space), i.e. time is added as another dimension to the coordinate grid. This way the coordinates specify <i>where</i>and <i>when</i> events occur. However, the unified nature of spacetime and the freedom of coordinate choice it allows imply that to express the temporal coordinate in one coordinate system requires both temporal and spatial coordinates in another coordinate system. Unlike in normal spatial coordinates, there are still restrictions for how measurements can be made spatially and temporally (see <a href="http://en.wikipedia.org/wiki/Spacetime#Spacetime_intervals" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">Spacetime intervals</a>). These restrictions correspond roughly to a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Lorentzian_manifold#Lorentzian_manifold" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lorentzian manifold">particular mathematical model</a> which differs from Euclidean space in its manifest<a href="http://en.wikipedia.org/wiki/Symmetry" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Symmetry">symmetry</a>.</div>
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Until the beginning of the 20th century, time was believed to be independent of motion, progressing at a fixed rate in all <a href="http://en.wikipedia.org/wiki/Frame_of_reference" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Frame of reference">reference frames</a>; however, later experiments revealed that time slowed down at higher speeds of the reference frame relative to another reference frame (with such slowing called "<a href="http://en.wikipedia.org/wiki/Time_dilation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Time dilation">time dilation</a>" explained in the theory of "<a href="http://en.wikipedia.org/wiki/Special_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special relativity">special relativity</a>"). Many experiments have confirmed time dilation, such as <a href="http://en.wikipedia.org/wiki/Atomic_clock" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Atomic clock">atomic clocks</a> onboard a <a href="http://en.wikipedia.org/wiki/Space_Shuttle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Space Shuttle">Space Shuttle</a> running slower than synchronized Earth-bound inertial clocks and the relativistic<a href="http://en.wikipedia.org/wiki/Particle_decay" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Particle decay">decay</a> of <a href="http://en.wikipedia.org/wiki/Muon" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Muon">muons</a> from <a href="http://en.wikipedia.org/wiki/Cosmic_ray" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cosmic ray">cosmic ray</a> showers. The duration of time can therefore vary for various events and various <a href="http://en.wikipedia.org/wiki/Frame_of_reference" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Frame of reference">reference frames</a>.</div>
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When dimensions are understood as mere components of the grid system, rather than physical attributes of space, it is easier to understand the alternate dimensional views as being simply the result of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Coordinate_transformation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Coordinate transformation">coordinate transformations</a>.</div>
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The term <i>spacetime</i> has taken on a generalized meaning beyond treating spacetime events with the normal 3+1 dimensions. It is really the combination of space and time. Other proposed spacetime theories include additional dimensions—normally spatial but there exist some speculative theories that include additional temporal dimensions and even some that include dimensions that are neither temporal nor spatial. How many dimensions are needed to describe the universe is still an open question. Speculative theories such as<a href="http://en.wikipedia.org/wiki/String_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="String theory">string theory</a> predict 10 or 26 dimensions (with <a href="http://en.wikipedia.org/wiki/M-theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="M-theory">M-theory</a> predicting 11 dimensions: 10 spatial and 1 temporal), but the existence of more than four dimensions would only appear to make a difference at the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Subatomic" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Subatomic">subatomic</a> level.</div>
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Contents</h2>
<span class="toctoggle" style="-webkit-user-select: none;"> [<a class="internal" href="http://en.wikipedia.org/wiki/Spacetime#" id="togglelink" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">hide</a>] </span></div>
<ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;">
<li class="toclevel-1 tocsection-1" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Spacetime_in_literature" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1</span> <span class="toctext">Spacetime in literature</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-2" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Mathematical_concept" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1.1</span> <span class="toctext">Mathematical concept</span></a></li>
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<li class="toclevel-1 tocsection-3" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Basic_concepts" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2</span> <span class="toctext">Basic concepts</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-4" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Spacetime_intervals" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.1</span> <span class="toctext">Spacetime intervals</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-3 tocsection-5" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Time-like_interval" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.1.1</span> <span class="toctext">Time-like interval</span></a></li>
<li class="toclevel-3 tocsection-6" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Light-like_interval" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.1.2</span> <span class="toctext">Light-like interval</span></a></li>
<li class="toclevel-3 tocsection-7" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Space-like_interval" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.1.3</span> <span class="toctext">Space-like interval</span></a></li>
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<li class="toclevel-1 tocsection-8" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Mathematics_of_spacetimes" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3</span> <span class="toctext">Mathematics of spacetimes</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-9" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Topology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.1</span> <span class="toctext">Topology</span></a></li>
<li class="toclevel-2 tocsection-10" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Spacetime_symmetries" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.2</span> <span class="toctext">Spacetime symmetries</span></a></li>
<li class="toclevel-2 tocsection-11" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Causal_structure" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.3</span> <span class="toctext">Causal structure</span></a></li>
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</li>
<li class="toclevel-1 tocsection-12" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Spacetime_in_special_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4</span> <span class="toctext">Spacetime in special relativity</span></a></li>
<li class="toclevel-1 tocsection-13" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Spacetime_in_general_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5</span> <span class="toctext">Spacetime in general relativity</span></a></li>
<li class="toclevel-1 tocsection-14" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Quantized_spacetime" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6</span> <span class="toctext">Quantized spacetime</span></a></li>
<li class="toclevel-1 tocsection-15" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Privileged_character_of_3.2B1_spacetime" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">7</span> <span class="toctext">Privileged character of 3+1 spacetime</span></a></li>
<li class="toclevel-1 tocsection-16" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#See_also" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">8</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-17" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#Notes" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">9</span> <span class="toctext">Notes</span></a></li>
<li class="toclevel-1 tocsection-18" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#References" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">10</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-19" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#External_links" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">11</span> <span class="toctext">External links</span></a></li>
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<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Spacetime in literature">edit</a>]</span><span class="mw-headline" id="Spacetime_in_literature">Spacetime in literature</span></h2>
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<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Incas" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Incas">Incas</a> regarded space and time as a single concept, named <i><b>pacha</b></i> (<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Quechua_language" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quechua language">Quechua</a>: <span lang="qu" xml:lang="qu"><i>pacha</i></span>, <a href="http://en.wikipedia.org/wiki/Aymara_language" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Aymara language">Aymara</a>: <span lang="ay" xml:lang="ay"><i>pacha</i></span>).<sup class="reference" id="cite_ref-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[1]</a></sup><sup class="reference" id="cite_ref-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup> The peoples of the <a href="http://en.wikipedia.org/wiki/Andes" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Andes">Andes</a> have kept this understanding until now.<sup class="reference" id="cite_ref-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup></div>
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<a href="http://en.wikipedia.org/wiki/Arthur_Schopenhauer" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Arthur Schopenhauer">Arthur Schopenhauer</a> wrote in §18 of <i><a href="http://en.wikipedia.org/wiki/On_the_Fourfold_Root_of_the_Principle_of_Sufficient_Reason" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="On the Fourfold Root of the Principle of Sufficient Reason">On the Fourfold Root of the Principle of Sufficient Reason</a></i> (1813): "...the representation of coexistence is impossible in Time alone; it depends, for its completion, upon the representation of Space; because, in mere Time, all things follow one another, and in mere Space all things are side by side; it is accordingly only by the<b>combination of Time and Space</b> that the representation of coexistence arises."</div>
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The idea of a unified spacetime is stated by <a href="http://en.wikipedia.org/wiki/Edgar_Allan_Poe" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edgar Allan Poe">Edgar Allan Poe</a> in his essay on cosmology titled <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Eureka_(Edgar_Allan_Poe)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eureka (Edgar Allan Poe)">Eureka</a></i> (1848) that "Space and duration are one." In 1895, in his novel <i><a href="http://en.wikipedia.org/wiki/The_Time_Machine" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="The Time Machine">The Time Machine</a></i>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/H.G._Wells" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="H.G. Wells">H.G. Wells</a> wrote, "There is no difference between time and any of the three dimensions of space except that our consciousness moves along it", and that "any real body must have extension in four directions: it must have Length, Breadth, Thickness, and Duration."</div>
<h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 18px; line-height: 20px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Mathematical concept">edit</a>]</span><span class="mw-headline" id="Mathematical_concept">Mathematical concept</span></h3>
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The first reference to spacetime as a mathematical concept was in 1754 by <a href="http://en.wikipedia.org/wiki/Jean_le_Rond_d%27Alembert" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Jean le Rond d'Alembert">Jean le Rond d'Alembert</a> in the article <i>Dimension</i> in <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Encyclopedie" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Encyclopedie">Encyclopedie</a>. Another early venture was by <a href="http://en.wikipedia.org/wiki/Joseph_Louis_Lagrange" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Joseph Louis Lagrange">Joseph Louis Lagrange</a> in his <i>Theory of Analytic Functions</i> (1797, 1813). He said, "One may view mechanics as a geometry of four dimensions, and mechanical analysis as an extension of geometric analysis".<sup class="reference" id="cite_ref-3" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[4]</a></sup></div>
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After discovering <a href="http://en.wikipedia.org/wiki/Quaternion" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quaternion">quaternions</a>,<sup class="reference" id="cite_ref-4" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[5]</a></sup> <a href="http://en.wikipedia.org/wiki/William_Rowan_Hamilton" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="William Rowan Hamilton">William Rowan Hamilton</a> commented, "Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be 'time plus space', or 'space plus time': and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, <a href="http://en.wikipedia.org/wiki/Classical_Hamiltonian_quaternions" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Classical Hamiltonian quaternions">Might in the Chain</a> of Symbols girdled be." Hamilton's <a href="http://en.wikipedia.org/wiki/Biquaternion" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Biquaternion">biquaternions</a>, which have algebraic properties sufficient to model spacetime and its symmetry, were in play for more than a half-century before formal relativity. For instance, <a href="http://en.wikipedia.org/wiki/William_Kingdon_Clifford" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="William Kingdon Clifford">William Kingdon Clifford</a> noted their relevance.</div>
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Another important antecedent to spacetime was the work of <a href="http://en.wikipedia.org/wiki/James_Clerk_Maxwell" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="James Clerk Maxwell">James Clerk Maxwell</a> as he used <a href="http://en.wikipedia.org/wiki/Partial_differential_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Partial differential equation">partial differential equations</a> to develop electrodynamics with the four parameters.<a href="http://en.wikipedia.org/wiki/Hendrik_Lorentz" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hendrik Lorentz">Lorentz</a> discovered some <a href="http://en.wikipedia.org/wiki/Lorentz_transformation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lorentz transformation">invariances</a> of <a href="http://en.wikipedia.org/wiki/Maxwell%27s_equations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Maxwell's equations">Maxwell's equations</a> late in the 19th century which were to become the basis of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Einstein" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Einstein">Einstein</a>'s theory of special relativity. Fiction authors were also involved, as mentioned above. It has always been the case that time and space are measured using real numbers, and the suggestion that the dimensions of space and time are comparable could have been raised by the first people to have formalized physics, but ultimately, the contradictions between Maxwell's laws and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Galilean_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Galilean relativity">Galilean relativity</a> had to come to a head with the realization of the import of finitude of the <a href="http://en.wikipedia.org/wiki/Speed_of_light" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Speed of light">speed of light</a>.</div>
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While spacetime can be viewed as a consequence of <a href="http://en.wikipedia.org/wiki/Albert_Einstein" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Albert Einstein">Albert Einstein</a>'s 1905 theory of <a href="http://en.wikipedia.org/wiki/Special_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special relativity">special relativity</a>, it was first explicitly proposed mathematically by one of his teachers, the mathematician <a href="http://en.wikipedia.org/wiki/Hermann_Minkowski" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermann Minkowski">Hermann Minkowski</a>, in a 1908 essay<sup class="reference" id="cite_ref-5" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[6]</a></sup> building on and extending Einstein's work. His concept of <a href="http://en.wikipedia.org/wiki/Minkowski_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Minkowski space">Minkowski space</a> is the earliest treatment of space and time as two aspects of a unified whole, the essence of <a href="http://en.wikipedia.org/wiki/Special_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special relativity">special relativity</a>. The idea of Minkowski space also led to special relativity being viewed in a more geometrical way, this geometric viewpoint of spacetime being important in general relativity too. (For an English translation of Minkowski's article, see Lorentz et al. 1952.) The 1926 thirteenth edition of the<i><a href="http://en.wikipedia.org/wiki/Encyclop%C3%A6dia_Britannica" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Encyclopædia Britannica">Encyclopædia Britannica</a></i> included an article by Einstein titled "Space-Time".<sup class="reference" id="cite_ref-6" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[7]</a></sup></div>
<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Basic concepts">edit</a>]</span><span class="mw-headline" id="Basic_concepts">Basic concepts</span></h2>
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Spacetimes are the arenas in which all physical events take place—an event is a point in spacetime specified by its time and place. For example, the motion of <a href="http://en.wikipedia.org/wiki/Planet" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Planet">planets</a> around the<a href="http://en.wikipedia.org/wiki/Sun" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sun">sun</a> may be described in a particular type of spacetime, or the motion of <a href="http://en.wikipedia.org/wiki/Light" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Light">light</a> around a rotating <a href="http://en.wikipedia.org/wiki/Star" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Star">star</a> may be described in another type of spacetime. The basic elements of spacetime are events. In any given spacetime, an event is a unique position at a unique time. Because events are spacetime points, an example of an event in classical relativistic physics is <img alt="(x,y,z,t)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/6/3/e63eff7873285a5fafce3488c0dc3734.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, the location of an elementary (point-like) particle at a particular time. A spacetime itself can be viewed as the union of all events in the same way that a line is the union of all of its points, formally organized into a <a href="http://en.wikipedia.org/wiki/Manifold" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Manifold">manifold</a>, a space which can be described at small scales using coordinates systems.</div>
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A spacetime is independent of any observer.<sup class="reference" id="cite_ref-7" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[8]</a></sup> However, in describing physical phenomena (which occur at certain moments of time in a given region of space), each observer chooses a convenient metrical <a href="http://en.wikipedia.org/wiki/Coordinate_system" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Coordinate system">coordinate system</a>. Events are specified by four <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Real_numbers" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Real numbers">real numbers</a> in any such coordinate system. The trajectories of elementary (point-like) particles through space and time are thus a continuum of events called the <a href="http://en.wikipedia.org/wiki/World_line" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="World line">world line</a> of the particle. Extended or composite objects (consisting of many elementary particles) are thus a union of many world lines twisted together by virtue of their interactions through spacetime into a "world-braid".</div>
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However, in physics, it is common to treat an extended object as a "particle" or "field" with its own unique (e.g. center of mass) position at any given time, so that the world line of a particle or light beam is the path that this particle or beam takes in the spacetime and represents the history of the particle or beam. The world line of the orbit of the Earth (in such a description) is depicted in two spatial dimensions <i>x</i> and <i>y</i> (the plane of the Earth's orbit) and a time dimension orthogonal to <i>x</i> and <i>y</i>. The orbit of the Earth is an <a href="http://en.wikipedia.org/wiki/Ellipse" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Ellipse">ellipse</a> in space alone, but its world line is a <a href="http://en.wikipedia.org/wiki/Helix" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Helix">helix</a> in spacetime.<sup class="reference" id="cite_ref-8" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[9]</a></sup></div>
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The unification of space and time is exemplified by the common practice of selecting a metric (the measure that specifies the <a href="http://en.wikipedia.org/wiki/Interval_(mathematics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Interval (mathematics)">interval</a> between two events in spacetime) such that all four dimensions are measured in terms of <a href="http://en.wikipedia.org/wiki/Units_of_measurement" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Units of measurement">units</a> of distance: representing an event as <img alt="(x_0,x_1,x_2,x_3) = (ct,x,y,z)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/e/2/ae290ddfe131bccead418d700ce59a23.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> (in the Lorentz metric) or <img alt="(x_1,x_2,x_3,x_4) = (x,y,z,ict)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/3/2/c32389568e301026adcc074bb450bac9.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> (in the original Minkowski metric)<sup class="reference" id="cite_ref-9" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[10]</a></sup> where <img alt="c" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/a/8/4a8a08f09d37b73795649038408b5f33.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is the <a href="http://en.wikipedia.org/wiki/Speed_of_light" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Speed of light">speed of light</a>. The metrical descriptions of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Minkowski_Space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Minkowski Space">Minkowski Space</a> and spacelike, lightlike, and timelike intervals given below follow this convention, as do the conventional formulations of the <a href="http://en.wikipedia.org/wiki/Lorentz_transformation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lorentz transformation">Lorentz transformation</a>.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Spacetime intervals">edit</a>]</span><span class="mw-headline" id="Spacetime_intervals">Spacetime intervals</span></h3>
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In a <a href="http://en.wikipedia.org/wiki/Euclidean_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Euclidean space">Euclidean space</a>, the separation between two points is measured by the distance between the two points. A distance is purely spatial, and is always positive. In spacetime, the separation between two events is measured by the <i><a href="http://en.wikipedia.org/wiki/Invariant_interval" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Invariant interval">invariant interval</a></i> between the two events, which takes into account not only the spatial separation between the events, but also their temporal separation. The interval between two events is defined as:</div>
<center style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px;"><img alt="s^2 = \Delta r^2 - c^2\Delta t^2 \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/e/6/3e6502fa4b2e9d9c0852e7a42b50e80f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /> (spacetime interval),</center><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
where <i>c</i> is the speed of light, and Δ<i>r</i> and Δ<i>t</i> denote differences of the space and time coordinates, respectively, between the events.</div>
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(Note that the choice of signs for <img alt="s^2" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/5/e/75e1b7b3dcc7c16d135aa0bb6cf5b32a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> above follows the <a href="http://en.wikipedia.org/wiki/Sign_convention#Relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sign convention">space-like convention (-+++)</a>. Other treatments reverse the sign of <img alt="s^2" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/5/e/75e1b7b3dcc7c16d135aa0bb6cf5b32a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />.)</div>
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Spacetime intervals may be classified into three distinct types based on whether the temporal separation (<img alt="c^2 \Delta t^2" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/1/8/4187c132b3a52ef7c52c6fedaf754c21.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />) or the spatial separation (<img alt="\Delta r^2" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/0/e/30ef95ca1b71fc5b5b1ffdc7ff88283a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />) of the two events is greater.</div>
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Certain types of <a href="http://en.wikipedia.org/wiki/World_line" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="World line">world lines</a> (called <a href="http://en.wikipedia.org/wiki/Geodesic" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Geodesic">geodesics</a> of the spacetime) are the shortest paths between any two events, with <i>distance</i> being defined in terms of spacetime intervals. The concept of geodesics becomes critical in <a href="http://en.wikipedia.org/wiki/General_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="General relativity">general relativity</a>, since geodesic motion may be thought of as "pure motion" (<a href="http://en.wikipedia.org/wiki/Fictitious_force" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fictitious force">inertial motion</a>) in spacetime, that is, free from any external influences.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Time-like interval">edit</a>]</span><span class="mw-headline" id="Time-like_interval">Time-like interval</span></h4>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><b><img alt="\begin{align} \\
c^2\Delta t^2 &> \Delta r^2 \\
s^2 &< 0 \\
\end{align}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/d/4/7d40606e17197f22c8343aa6fd354115.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></b></dd></dl>
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For two events separated by a time-like interval, enough time passes between them for there to be a cause-effect relationship between the two events. For a particle traveling through space at less than the speed of light, any two events which occur to or by the particle must be separated by a time-like interval. Event pairs with time-like separation define a negative squared spacetime interval (<img alt="s^2 < 0" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/1/c/61c019b207c743577d85d4a65f5a583c.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />) and may be said to occur in each other's future or past. There exists a <a href="http://en.wikipedia.org/wiki/Frame_of_reference" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Frame of reference">reference frame</a> such that the two events are observed to occur in the same spatial location, but there is no reference frame in which the two events can occur at the same time.</div>
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The measure of a time-like spacetime interval is described by the <a href="http://en.wikipedia.org/wiki/Proper_time" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Proper time">proper time</a>:</div>
<center style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px;"><img alt="\Delta\tau = \sqrt{\Delta t^2 - \frac{\Delta r^2}{c^2}}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/5/b/a5bc803586a523e9585e65af3a898969.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /> (proper time).</center><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
The proper time interval would be measured by an observer with a clock traveling between the two events in an <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Inertial" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Inertial">inertial</a> reference frame, when the observer's path intersects each event as that event occurs. (The proper time defines a <a href="http://en.wikipedia.org/wiki/Real_number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Real number">real number</a>, since the interior of the square root is positive.)</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Light-like interval">edit</a>]</span><span class="mw-headline" id="Light-like_interval">Light-like interval</span></h4>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><b><img alt="\begin{align}
c^2\Delta t^2 &= \Delta r^2 \\
s^2 &= 0 \\
\end{align}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/1/c/01c44bf0ebe255781c049339e27028f5.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></b></dd></dl>
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In a light-like interval, the spatial distance between two events is exactly balanced by the time between the two events. The events define a squared spacetime interval of zero (<img alt="s^2 = 0" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/6/0/6602be59bc8c0b4a73716ac3e62afa31.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />). Light-like intervals are also known as "null" intervals.</div>
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Events which occur to or are initiated by a <a href="http://en.wikipedia.org/wiki/Photon" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Photon">photon</a> along its path (i.e., while traveling at <img alt="c" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/a/8/4a8a08f09d37b73795649038408b5f33.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, the speed of light) all have light-like separation. Given one event, all those events which follow at light-like intervals define the propagation of a <a href="http://en.wikipedia.org/wiki/Light_cone" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Light cone">light cone</a>, and all the events which preceded from a light-like interval define a second (graphically inverted, which is to say "<i>pastward</i>") light cone.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Space-like interval">edit</a>]</span><span class="mw-headline" id="Space-like_interval">Space-like interval</span></h4>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><b><img alt="\begin{align} \\
c^2\Delta t^2 &< \Delta r^2 \\
s^2 &> 0 \\
\end{align}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/6/c/36c24ff1c4a2d6564ac46e804382e12f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></b></dd></dl>
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When a space-like interval separates two events, not enough time passes between their occurrences for there to exist a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Causal" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Causal">causal</a> relationship crossing the spatial distance between the two events at the speed of light or slower. Generally, the events are considered not to occur in each other's future or past. There exists a <a href="http://en.wikipedia.org/wiki/Frame_of_reference" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Frame of reference">reference frame</a> such that the two events are observed to occur at the same time, but there is no reference frame in which the two events can occur in the same spatial location.</div>
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For these space-like event pairs with a positive squared spacetime interval (<img alt="s^2 > 0" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/3/0/630969e26cd6cbfdbda5ffe131096b10.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />), the measurement of space-like separation is the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Proper_distance" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Proper distance">proper distance</a>:</div>
<center style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px;"><img alt="\Delta\sigma = \sqrt{\Delta r^2 - c^2\Delta t^2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/f/0/1f0abc5a99b580f44b3c5d35e7970d9f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /> (proper distance).</center><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
Like the proper time of time-like intervals, the proper distance (<img alt="\Delta\sigma" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/7/d/e7d12a405a591e43b2bd20cc74de2411.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />) of space-like spacetime intervals is a real number value.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Mathematics of spacetimes">edit</a>]</span><span class="mw-headline" id="Mathematics_of_spacetimes">Mathematics of spacetimes</span></h2>
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For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Lorentzian_manifold" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lorentzian manifold">Lorentzian manifold</a> <img alt="(M,g)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/9/e/e9eae9f79a7f233d826505675f155b0a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. This means the smooth <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Lorentz_metric" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lorentz metric">Lorentz metric</a> <img alt="g" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/2/f/b2f5ff47436671b6e533d8dc3614845d.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> has signature <img alt="(3,1)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/4/9/14934aec3cac03b43841a2c7799a544b.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. The metric determines the geometry of spacetime, as well as determining the <a href="http://en.wikipedia.org/wiki/Geodesic" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Geodesic">geodesics</a> of particles and light beams. About each point (event) on this manifold, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Coordinate_charts" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Coordinate charts">coordinate charts</a> are used to represent observers in reference frames. Usually, Cartesian coordinates <img alt="(x, y, z, t)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/6/3/e63eff7873285a5fafce3488c0dc3734.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> are used. Moreover, for simplicity's sake, the speed of light <img alt="c" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/a/8/4a8a08f09d37b73795649038408b5f33.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is usually assumed to be unity.</div>
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A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event <img alt="p" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/3/8/83878c91171338902e0fe0fb97a8c47a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. Another reference frame may be identified by a second coordinate chart about <img alt="p" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/3/8/83878c91171338902e0fe0fb97a8c47a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. Two observers (one in each reference frame) may describe the same event <img alt="p" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/3/8/83878c91171338902e0fe0fb97a8c47a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> but obtain different descriptions.</div>
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Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing <img alt="p" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/3/8/83878c91171338902e0fe0fb97a8c47a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> (representing an observer) and another containing <img alt="q" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/6/9/7694f4a66316e53c8cdd9d9954bd611d.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />(representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Non-singular" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Non-singular">non-singular</a> coordinate transformation on this intersection. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally.</div>
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For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event <img alt="p" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/3/8/83878c91171338902e0fe0fb97a8c47a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples <img alt="(x, y, z, t)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/6/3/e63eff7873285a5fafce3488c0dc3734.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> (as they are using different coordinate systems). Although their kinematic descriptions will differ, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these (and all other physical) laws must take the same form in all coordinate systems. This introduces <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Tensors" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tensors">tensors</a> into relativity, by which all physical quantities are represented.</div>
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Geodesics are said to be time-like, null, or space-like if the tangent vector to one point of the geodesic is of this nature. Paths of particles and light beams in spacetime are represented by time-like and null (light-like) geodesics, respectively.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Topology">edit</a>]</span><span class="mw-headline" id="Topology">Topology</span></h3>
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Main article: <a href="http://en.wikipedia.org/wiki/Spacetime_topology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spacetime topology">Spacetime topology</a></div>
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The assumptions contained in the definition of a spacetime are usually justified by the following considerations.</div>
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The <a href="http://en.wikipedia.org/wiki/Connectedness" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Connectedness">connectedness</a> assumption serves two main purposes. First, different observers making measurements (represented by coordinate charts) should be able to compare their observations on the non-empty intersection of the charts. If the connectedness assumption were dropped, this would not be possible. Second, for a manifold, the properties of connectedness and path-connectedness are equivalent, and one requires the existence of paths (in particular, <a href="http://en.wikipedia.org/wiki/Geodesic" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Geodesic">geodesics</a>) in the spacetime to represent the motion of particles and radiation.</div>
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Every spacetime is <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Paracompact" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Paracompact">paracompact</a>. This property, allied with the smoothness of the spacetime, gives rise to a smooth <a href="http://en.wikipedia.org/wiki/Connection_(principal_bundle)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Connection (principal bundle)">linear connection</a>, an important structure in general relativity. Some important theorems on constructing spacetimes from compact and non-compact manifolds include the following:<sup class="Template-Fact" style="line-height: 1em; white-space: nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources from November 2007">citation needed</span></a></i>]</sup></div>
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<li style="margin-bottom: 0.1em;">A <a href="http://en.wikipedia.org/wiki/Compact_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Compact space">compact</a> manifold can be turned into a spacetime if, and only if, its <a href="http://en.wikipedia.org/wiki/Euler_characteristic" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Euler characteristic">Euler characteristic</a> is 0. (Proof idea: the existence of a Lorentzian metric is shown to be equivalent to the existence of a nonvanishing vector field.)</li>
<li style="margin-bottom: 0.1em;">Any non-compact 4-manifold can be turned into a spacetime.</li>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Spacetime symmetries">edit</a>]</span><span class="mw-headline" id="Spacetime_symmetries">Spacetime symmetries</span></h3>
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Main article: <a href="http://en.wikipedia.org/wiki/Spacetime_symmetries" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spacetime symmetries">Spacetime symmetries</a></div>
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Often in relativity, spacetimes that have some form of symmetry are studied. As well as helping to classify spacetimes, these symmetries usually serve as a simplifying assumption in specialized work. Some of the most popular ones include:</div>
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<li style="margin-bottom: 0.1em;">Axisymmetric spacetimes</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Spherically_symmetric_spacetime" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spherically symmetric spacetime">Spherically symmetric spacetimes</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Static_spacetime" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Static spacetime">Static spacetimes</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Stationary_spacetime" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Stationary spacetime">Stationary spacetimes</a>.</li>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Causal structure">edit</a>]</span><span class="mw-headline" id="Causal_structure">Causal structure</span></h3>
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Main article: <a href="http://en.wikipedia.org/wiki/Causal_structure" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Causal structure">Causal structure</a></div>
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The causal structure of a spacetime describes causal relationships between pairs of points in the spacetime based on the existence of certain types of curves joining the points.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=12" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Spacetime in special relativity">edit</a>]</span><span class="mw-headline" id="Spacetime_in_special_relativity">Spacetime in special relativity</span></h2>
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Main article: <a href="http://en.wikipedia.org/wiki/Minkowski_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Minkowski space">Minkowski space</a></div>
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The geometry of spacetime in special relativity is described by the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Minkowski_metric" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Minkowski metric">Minkowski metric</a> on R<sup style="line-height: 1em;">4</sup>. This spacetime is called Minkowski space. The Minkowski metric is usually denoted by <img alt="\eta" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/1/7/7174cbd6aeaaa56e37102b72386bb2b9.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />and can be written as a four-by-four matrix:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\eta_{ab} \, = \operatorname{diag}(1, -1, -1, -1)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/a/e/cae3707016a34a4fc2f4f4b388ba64e0.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where the <a href="http://en.wikipedia.org/wiki/Sign_convention#Relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sign convention">Landau–Lifshitz space-like convention</a> is being used. A basic assumption of relativity is that coordinate transformations must leave spacetime intervals invariant. Intervals are <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Lorentz_invariance" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lorentz invariance">invariant</a> under <a href="http://en.wikipedia.org/wiki/Lorentz_transformation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lorentz transformation">Lorentz transformations</a>. This invariance property leads to the use of <a href="http://en.wikipedia.org/wiki/Four-vector" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Four-vector">four-vectors</a> (and other tensors) in describing physics.</div>
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Strictly speaking, one can also consider events in Newtonian physics as a single spacetime. This is <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Galilean-Newtonian_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Galilean-Newtonian relativity">Galilean-Newtonian relativity</a>, and the coordinate systems are related by <a href="http://en.wikipedia.org/wiki/Galilean_transformation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Galilean transformation">Galilean transformations</a>. However, since these preserve spatial and temporal distances independently, such a spacetime can be decomposed into spatial coordinates plus temporal coordinates, which is not possible in the general case.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=13" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Spacetime in general relativity">edit</a>]</span><span class="mw-headline" id="Spacetime_in_general_relativity">Spacetime in general relativity</span></h2>
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In <a href="http://en.wikipedia.org/wiki/General_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="General relativity">general relativity</a>, it is assumed that spacetime is curved by the presence of matter (energy), this curvature being represented by the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Riemann_tensor" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Riemann tensor">Riemann tensor</a>. In <a href="http://en.wikipedia.org/wiki/Special_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special relativity">special relativity</a>, the Riemann tensor is identically zero, and so this concept of "non-curvedness" is sometimes expressed by the statement <i>Minkowski spacetime is flat.</i></div>
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The earlier discussed notions of time-like, light-like and space-like intervals in special relativity can similarly be used to classify one-dimensional <a href="http://en.wikipedia.org/wiki/Causal_structure#Curves" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Causal structure">curves</a> through curved spacetime. A time-like curve can be understood as one where the interval between any two <a href="http://en.wikipedia.org/wiki/Infinitesimal" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Infinitesimal">infinitesimally</a> close events on the curve is time-like, and likewise for light-like and space-like curves. Technically the three types of curves are usually defined in terms of whether the <a href="http://en.wikipedia.org/wiki/Causal_structure#Tangent_vectors" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Causal structure">tangent vector</a> at each point on the curve is time-like, light-like or space-like. The <a href="http://en.wikipedia.org/wiki/World_line" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="World line">world line</a> of a slower-than-light object will always be a time-like curve, the world line of a massless particle such as a photon will be a light-like curve, and a space-like curve could be the world line of a hypothetical <a href="http://en.wikipedia.org/wiki/Tachyon" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tachyon">tachyon</a>. In the local neighborhood of any event, time-like curves that pass through the event will remain inside that event's past and future <a href="http://en.wikipedia.org/wiki/Light_cone" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Light cone">light cones</a>, light-like curves that pass through the event will be on the surface of the light cones, and space-like curves that pass through the event will be outside the light cones. One can also define the notion of a 3-dimensional "spacelike hypersurface", a continuous 3-dimensional "slice" through the 4-dimensional property with the property that every curve that is contained entirely within this hypersurface is a space-like curve.<sup class="reference" id="cite_ref-10" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[11]</a></sup></div>
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Many spacetime continua have physical interpretations which most physicists would consider bizarre or unsettling. For example, a <a href="http://en.wikipedia.org/wiki/Compact_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Compact space">compact</a> spacetime has <a href="http://en.wikipedia.org/wiki/Closed_timelike_curve" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Closed timelike curve">closed timelike curves</a>, which violate our usual ideas of causality (that is, future events could affect past ones). For this reason, mathematical physicists usually consider only restricted subsets of all the possible spacetimes. One way to do this is to study "realistic" solutions of the equations of general relativity. Another way is to add some additional "physically reasonable" but still fairly general geometric restrictions and try to prove interesting things about the resulting spacetimes. The latter approach has led to some important results, most notably the<a href="http://en.wikipedia.org/wiki/Penrose%E2%80%93Hawking_singularity_theorems" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Penrose–Hawking singularity theorems">Penrose–Hawking singularity theorems</a>.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=14" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Quantized spacetime">edit</a>]</span><span class="mw-headline" id="Quantized_spacetime">Quantized spacetime</span></h2>
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Main article: <a href="http://en.wikipedia.org/wiki/Quantum_spacetime" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum spacetime">Quantum spacetime</a></div>
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In general relativity, spacetime is assumed to be smooth and continuous—and not just in the mathematical sense. In the theory of quantum mechanics, there is an inherent discreteness present in physics. In attempting to reconcile these two theories, it is sometimes postulated that spacetime should be quantized at the very smallest scales. Current theory is focused on the nature of spacetime at the <a href="http://en.wikipedia.org/wiki/Planck_scale" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Planck scale">Planck scale</a>. <a href="http://en.wikipedia.org/wiki/Causal_sets" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Causal sets">Causal sets</a>, <a href="http://en.wikipedia.org/wiki/Loop_quantum_gravity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Loop quantum gravity">loop quantum gravity</a>, <a href="http://en.wikipedia.org/wiki/String_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="String theory">string theory</a>, and <a href="http://en.wikipedia.org/wiki/Black_hole_thermodynamics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Black hole thermodynamics">black hole thermodynamics</a> all predict a <a href="http://en.wikipedia.org/wiki/Quantization_(physics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantization (physics)">quantized</a> spacetime with agreement on the order of magnitude. Loop quantum gravity makes precise predictions about the geometry of spacetime at the Planck scale.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=15" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Privileged character of 3+1 spacetime">edit</a>]</span><span class="mw-headline" id="Privileged_character_of_3.2B1_spacetime">Privileged character of 3+1 spacetime</span></h2>
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There are two kinds of dimensions, spatial (bidirectional) and temporal (unidirectional). Let the number of spatial dimensions be <i>N</i> and the number of temporal dimensions be <i>T</i>. That <i>N</i> = 3 and <i>T</i> = 1, setting aside the compactified dimensions invoked by <a href="http://en.wikipedia.org/wiki/String_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="String theory">string theory</a> and undetectable to date, can be explained by appealing to the physical consequences of letting <i>N</i> differ from 3 and <i>T</i> differ from 1. The argument is often of an <a href="http://en.wikipedia.org/wiki/Anthropic_principle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Anthropic principle">anthropic</a> character.</div>
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<a href="http://en.wikipedia.org/wiki/Immanuel_Kant" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Immanuel Kant">Immanuel Kant</a> argued that 3-dimensional space was a consequence of the inverse square <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Law_of_universal_gravitation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Law of universal gravitation">law of universal gravitation</a>. While Kant's argument is historically important, <a href="http://en.wikipedia.org/wiki/John_D._Barrow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John D. Barrow">John D. Barrow</a> says that it "...gets the punch-line back to front: it is the three-dimensionality of space that explains why we see inverse-square force laws in Nature, not vice-versa." (Barrow 2002: 204). This is because the law of gravitation (or any other <a href="http://en.wikipedia.org/wiki/Inverse-square_law" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Inverse-square law">inverse-square law</a>) follows from the concept of <a href="http://en.wikipedia.org/wiki/Flux" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Flux">flux</a> and the proportional relationship of flux density and the strength of field. If <i>N</i> = 3, then 3-dimensional solid objects have surface areas proportional to the square of their size in any selected spatial dimension. In particular, a sphere of <a href="http://en.wikipedia.org/wiki/Radius" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Radius">radius</a> <i>r</i> has area of 4π<i>r</i> ². More generally, in a space of <i>N</i> dimensions, the strength of the gravitational attraction between two bodies separated by a distance of <i>r</i> would be inversely proportional to <i>r</i><sup style="line-height: 1em;"><i>N</i>−1</sup>.</div>
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In 1920, <a href="http://en.wikipedia.org/wiki/Paul_Ehrenfest" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Paul Ehrenfest">Paul Ehrenfest</a> showed that if we fix <i>T</i> = 1 and let <i>N</i> > 3, the <a href="http://en.wikipedia.org/wiki/Orbit" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Orbit">orbit</a> of a <a href="http://en.wikipedia.org/wiki/Planet" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Planet">planet</a> about its sun cannot remain stable. The same is true of a star's orbit around the center of its<a href="http://en.wikipedia.org/wiki/Galaxy" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Galaxy">galaxy</a>.<sup class="reference" id="cite_ref-11" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[12]</a></sup> Ehrenfest also showed that if <i>N</i> is even, then the different parts of a <a href="http://en.wikipedia.org/wiki/Wave" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wave">wave</a> impulse will travel at different speeds. If <i>N</i> > 3 and odd, then wave impulses become distorted. Only when <i>N</i> = 3 or 1 are both problems avoided. In 1922, <a href="http://en.wikipedia.org/wiki/Hermann_Weyl" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermann Weyl">Hermann Weyl</a> showed that <a href="http://en.wikipedia.org/wiki/James_Clerk_Maxwell" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="James Clerk Maxwell">Maxwell</a>'s theory of <a href="http://en.wikipedia.org/wiki/Electromagnetism" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electromagnetism">electromagnetism</a> works only when <i>N</i> = 3 and <i>T</i> = 1, writing that this fact "...not only leads to a deeper understanding of Maxwell's theory, but also of the fact that the world is four dimensional, which has hitherto always been accepted as merely 'accidental,' become intelligible through it."<sup class="reference" id="cite_ref-12" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-12" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[13]</a></sup> Finally, Tangherlini<sup class="reference" id="cite_ref-13" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-13" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[14]</a></sup> showed in 1963 that when <i>N</i> > 3, electron <a href="http://en.wikipedia.org/wiki/Atomic_orbital" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Atomic orbital">orbitals</a> around nuclei cannot be stable; electrons would either fall into the <a href="http://en.wikipedia.org/wiki/Atomic_nucleus" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Atomic nucleus">nucleus</a> or disperse.</div>
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Properties of <i>n</i>+<i>m</i>-dimensional spacetimes</div>
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<a href="http://en.wikipedia.org/wiki/Max_Tegmark" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Max Tegmark">Max Tegmark</a><sup class="reference" id="cite_ref-tegmark-dim_14-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-tegmark-dim-14" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[15]</a></sup> expands on the preceding argument in the following <a href="http://en.wikipedia.org/wiki/Anthropic_principle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Anthropic principle">anthropic</a> manner. If <i>T</i> differs from 1, the behavior of physical systems could not be predicted reliably from knowledge of the relevant <a href="http://en.wikipedia.org/wiki/Partial_differential_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Partial differential equation">partial differential equations</a>. In such a universe, intelligent life capable of manipulating technology could not emerge. Moreover, if <i>T</i> > 1, Tegmark maintains that<a href="http://en.wikipedia.org/wiki/Proton" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Proton">protons</a> and <a href="http://en.wikipedia.org/wiki/Electron" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron">electrons</a> would be unstable and could decay into particles having greater mass than themselves. (This is not a problem if the particles have a sufficiently low temperature.) If <i>N</i> > 3, Ehrenfest's argument above holds; atoms as we know them (and probably more complex structures as well) could not exist. If <i>N</i> < 3, gravitation of any kind becomes problematic, and the universe is probably too simple to contain observers. For example, when <i>N</i> < 3, nerves cannot cross without intersecting.</div>
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In general, it is not clear how physical law could function if <i>T</i> differed from 1. If <i>T</i> > 1, subatomic particles which decay after a fixed period would not behave predictably, because time-like <a href="http://en.wikipedia.org/wiki/Geodesic" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Geodesic">geodesics</a> would not be necessarily maximal.<sup class="reference" id="cite_ref-15" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-15" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[16]</a></sup> <i>N</i> = 1 and <i>T</i>= 3 has the peculiar property that the <a href="http://en.wikipedia.org/wiki/Speed_of_light" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Speed of light">speed of light</a> in a vacuum is a <i>lower bound</i> on the velocity of matter; all matter consists of <a href="http://en.wikipedia.org/wiki/Tachyon" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tachyon">tachyons</a>.<sup class="reference" id="cite_ref-tegmark-dim_14-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-tegmark-dim-14" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[15]</a></sup></div>
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Hence anthropic and other arguments rule out all cases except <i>N</i> = 3 and <i>T</i> = 1—which happens to describe the world about us. Curiously, the cases <i>N</i> = 3 or 4 have the richest and most difficult <a href="http://en.wikipedia.org/wiki/Geometry" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Geometry">geometry</a> and <a href="http://en.wikipedia.org/wiki/Topology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Topology">topology</a>. There are, for example, geometric statements whose truth or falsity is known for all <i>N</i> except one or both of 3 and 4.<sup class="Template-Fact" style="line-height: 1em; white-space: nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources from July 2009">citation needed</span></a></i>]</sup> <i>N</i> = 3 was the last case of the <a href="http://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Poincaré conjecture">Poincaré conjecture</a> to be proved.</div>
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For an elementary treatment of the privileged status of <i>N</i> = 3 and <i>T</i> = 1, see chpt. 10 (esp. Fig. 10.12) of Barrow;<sup class="reference" id="cite_ref-16" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-16" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[17]</a></sup> for deeper treatments, see §4.8 of Barrow and Tipler (1986) and Tegmark.<sup class="reference" id="cite_ref-tegmark-dim_14-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-tegmark-dim-14" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[15]</a></sup> Barrow has repeatedly cited the work of<a href="http://en.wikipedia.org/wiki/Gerald_James_Whitrow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gerald James Whitrow">Whitrow</a>.<sup class="reference" id="cite_ref-17" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-17" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[18]</a></sup></div>
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<a href="http://en.wikipedia.org/wiki/String_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="String theory">String theory</a> hypothesizes that matter and energy are composed of tiny vibrating strings of various types, most of which are embedded in dimensions that exist only on a scale no larger than the <a href="http://en.wikipedia.org/wiki/Planck_length" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Planck length">Planck length</a>. Hence <i>N</i> = 3 and <i>T</i> = 1 do not characterize string theory, which embeds vibrating strings in coordinate grids having 10, or even 26, dimensions.</div>
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The <a href="http://en.wikipedia.org/wiki/Causal_dynamical_triangulation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Causal dynamical triangulation">Causal dynamical triangulation</a> (CDT) theory is a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Background_independent" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Background independent">background independent</a> theory which derives the observed 3+1 spacetime from a minimal set of assumptions, and needs no adjusting factors. It does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves. It shows spacetime to be 2-d near the <a href="http://en.wikipedia.org/wiki/Planck_scale" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Planck scale">Planck scale</a>, and reveals a <a href="http://en.wikipedia.org/wiki/Fractal" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fractal">fractal</a> structure on slices of constant time, but spacetime becomes 3+1-d in scales significantly larger than Planck. So, CDT may become the first theory which doesn't postulate but really explains observed number of spacetime dimensions.<sup class="reference" id="cite_ref-18" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Spacetime#cite_note-18" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[19]</a></sup></div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=16" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: See also">edit</a>]</span><span class="mw-headline" id="See_also">See also</span></h2>
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<tr><td style="text-align: center;"><a class="image" href="http://en.wikipedia.org/wiki/File:MontreGousset001.jpg" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><img alt="Portal icon" height="28" src="http://upload.wikimedia.org/wikipedia/commons/thumb/4/45/MontreGousset001.jpg/32px-MontreGousset001.jpg" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" width="32" /></a></td><td style="font-style: italic; font-weight: bold; padding-bottom: 0px; padding-left: 0.2em; padding-right: 0.2em; padding-top: 0px; vertical-align: middle;"><a href="http://en.wikipedia.org/wiki/Portal:Time" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Portal:Time">Time portal</a></td></tr>
<tr valign="middle"><td style="text-align: center;"><a class="image" href="http://en.wikipedia.org/wiki/File:Earth-moon.jpg" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><img alt="Portal icon" height="26" src="http://upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Earth-moon.jpg/32px-Earth-moon.jpg" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" width="32" /></a></td><td style="font-style: italic; font-weight: bold; padding-bottom: 0px; padding-left: 0.2em; padding-right: 0.2em; padding-top: 0px; vertical-align: middle;"><a href="http://en.wikipedia.org/wiki/Portal:Space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Portal:Space">Space portal</a></td></tr>
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<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Basic_introduction_to_the_mathematics_of_curved_spacetime" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Basic introduction to the mathematics of curved spacetime">Basic introduction to the mathematics of curved spacetime</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Four-vector" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Four-vector">Four-vector</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Frame-dragging" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Frame-dragging">Frame-dragging</a></li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Global_spacetime_structure" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Global spacetime structure">Global spacetime structure</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Hole_argument" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hole argument">Hole argument</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/List_of_mathematical_topics_in_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="List of mathematical topics in relativity">List of mathematical topics in relativity</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Local_spacetime_structure" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Local spacetime structure">Local spacetime structure</a></li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Lorentz_invariance" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lorentz invariance">Lorentz invariance</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Manifold" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Manifold">Manifold</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematics_of_general_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mathematics of general relativity">Mathematics of general relativity</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Metric_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Metric space">Metric space</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Philosophy_of_space_and_time" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Philosophy of space and time">Philosophy of space and time</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Relativity_of_simultaneity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Relativity of simultaneity">Relativity of simultaneity</a></li>
</ul>
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<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=17" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Notes">edit</a>]</span><span class="mw-headline" id="Notes">Notes</span></h2>
<div class="reflist references-column-width" style="-webkit-column-width: 35em; background-color: white; font-family: sans-serif; font-size: 12px; line-height: 20px; list-style-type: decimal; margin-bottom: 0.5em;">
<ol class="references" style="line-height: 1.5em; list-style-image: none; list-style-type: inherit; margin-bottom: 0.5em; margin-left: 3.2em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li id="cite_note-0" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">Atuq Eusebio Manga Qespi, Instituto de lingüística y Cultura Amerindia de la Universidad de Valencia. <a class="external text" href="http://revistas.ucm.es/ghi/05566533/articulos/REAA9494110155A.PDF" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif); background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 18px; text-decoration: none;"><i>Pacha: un concepto andino de espacio y tiempo</i></a>. Revísta española de Antropología Americana, 24, p. 155-189. Edit. Complutense, Madrid. 1994</span></li>
<li id="cite_note-1" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">Paul Richard Steele, Catherine J. Allen, <i>Handbook of Inca mythology</i>, p. 86, (<a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/1576073548" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 1-57607-354-8</a>)</span></li>
<li id="cite_note-2" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">Shirley Ardener, University of Oxford, <i>Women and space: ground rules and social maps</i>, p. 36 (<a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/0854967281" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 0-85496-728-1</a>)</span></li>
<li id="cite_note-3" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">R.C. Archibald (1914) <a class="external text" href="http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.bams/1183422749&page=record" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;"><i>Time as a fourth dimension</i></a> <i>Bulletin of the American Mathematical Society 20:409.</i></span></li>
<li id="cite_note-4" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;">Gallier, Jean H. (2001). <a class="external text" href="http://books.google.com/books?id=B4JtblR1lkMC" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;"><i>Geometric methods and applications: for computer science and engineering</i></a>. Springer. p. 249. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-387-95044-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-387-95044-3">0-387-95044-3</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+methods+and+applications%3A+for+computer+science+and+engineering&rft.aulast=Gallier&rft.aufirst=Jean+H.&rft.au=Gallier%2C%26%2332%3BJean+H.&rft.date=2001&rft.pages=p.%26nbsp%3B249&rft.pub=Springer&rft.isbn=0-387-95044-3&rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DB4JtblR1lkMC&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span>, <a class="external text" href="http://books.google.com/books?id=B4JtblR1lkMC&pg=PA249" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Chapter 8, page 249</a></span></li>
<li id="cite_note-5" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a href="http://en.wikipedia.org/wiki/Hermann_Minkowski" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermann Minkowski">Hermann Minkowski</a>, <a class="external text" href="http://de.wikisource.org/wiki/Raum_und_Zeit_(Minkowski)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">"Raum und Zeit"</a>, 80. Versammlung Deutscher Naturforscher (Köln, 1908). Published in Physikalische Zeitschrift <b>10</b> 104–111 (1909) and Jahresbericht der Deutschen Mathematiker-Vereinigung <b>18</b> 75-88 (1909). For an English translation, see Lorentz et al. (1952).</span></li>
<li id="cite_note-6" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a href="http://en.wikipedia.org/wiki/Albert_Einstein" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Albert Einstein">Einstein, Albert</a>, 1926, "<a class="external text" href="http://www.britannica.com/eb/article-9117889" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Space-Time,</a>" <i>Encyclopædia Britannica</i>, 13th ed.</span></li>
<li id="cite_note-7" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;">Matolcsi, Tamás (1994). <i>Spacetime Without Reference Frames</i>. Budapest: Akadémiai Kiadó.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spacetime+Without+Reference+Frames&rft.aulast=Matolcsi&rft.aufirst=Tam%C3%A1s&rft.au=Matolcsi%2C%26%2332%3BTam%C3%A1s&rft.date=1994&rft.place=Budapest&rft.pub=Akad%C3%A9miai+Kiad%C3%B3&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span></span></li>
<li id="cite_note-8" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;">Ellis, G. F. R.; Williams, Ruth M. (2000). <a class="external text" href="http://books.google.com/books?id=LKfvAAAAMAAJ" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;"><i>Flat and curved space-times</i></a> (2nd ed.). Oxford University Press. p. 9. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-198-50657-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-198-50657-0">0-198-50657-0</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Flat+and+curved+space-times&rft.aulast=Ellis&rft.aufirst=G.+F.+R.&rft.au=Ellis%2C%26%2332%3BG.+F.+R.&rft.au=Williams%2C%26%2332%3BRuth+M.&rft.date=2000&rft.pages=p.%26nbsp%3B9&rft.edition=2nd&rft.pub=Oxford+University+Press&rft.isbn=0-198-50657-0&rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLKfvAAAAMAAJ&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span></span></li>
<li id="cite_note-9" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;">Petkov, Vesselin (2010). <a class="external text" href="http://books.google.com/?id=trlsrl4mF3YC" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;"><i>Minkowski Spacetime: A Hundred Years Later</i></a>. Springer. p. 70. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/9-048-13474-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/9-048-13474-9">9-048-13474-9</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Minkowski+Spacetime%3A+A+Hundred+Years+Later&rft.aulast=Petkov&rft.aufirst=Vesselin&rft.au=Petkov%2C%26%2332%3BVesselin&rft.date=2010&rft.pages=p.%26nbsp%3B70&rft.pub=Springer&rft.isbn=9-048-13474-9&rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3Dtrlsrl4mF3YC&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span>, <a class="external text" href="http://books.google.com/books?id=trlsrl4mF3YC&pg=PA70" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Section 3.4, p. 70</a></span></li>
<li id="cite_note-10" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">See "Quantum Spacetime and the Problem of Time in Quantum Gravity" by Leszek M. Sokolowski, where on <a class="external text" href="http://books.google.com/books?id=cQGGk2MlUyYC&lpg=PP1&pg=PA32#v=onepage&q&f=false" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">this page</a> he writes "Each of these hypersurfaces is spacelike, in the sense that every curve, which entirely lies on one of such hypersurfaces, is a spacelike curve." More commonly a space-like hypersurface is defined technically as a surface such that the <a href="http://en.wikipedia.org/wiki/Normal_(geometry)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Normal (geometry)">normal vector</a> at every point is time-like, but the definition above may be somewhat more intuitive.</span></li>
<li id="cite_note-11" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Paul_Ehrenfest" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Paul Ehrenfest">Ehrenfest, Paul</a> (1920). "How do the fundamental laws of physics make manifest that Space has 3 dimensions?". <i>Annalen der Physik</i> <b>61</b> (5): 440. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/1920AnP...366..440E" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1920AnP...366..440E</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1002%2Fandp.19203660503" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1002/andp.19203660503</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=How+do+the+fundamental+laws+of+physics+make+manifest+that+Space+has+3+dimensions%3F&rft.jtitle=Annalen+der+Physik&rft.aulast=Ehrenfest&rft.aufirst=Paul&rft.au=Ehrenfest%2C%26%2332%3BPaul&rft.date=1920&rft.volume=61&rft.issue=5&rft.pages=440&rft_id=info:bibcode/1920AnP...366..440E&rft_id=info:doi/10.1002%2Fandp.19203660503&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span>. Also see Ehrenfest, P. (1917) "In what way does it become manifest in the fundamental laws of physics that space has three dimensions?" <i>Proceedings of the Amsterdam Academy</i> 20: 200.</span></li>
<li id="cite_note-12" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-12" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text">Weyl, H. (1922) <i>Space, time, and matter</i>. Dover reprint: 284.</span></li>
<li id="cite_note-13" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-13" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">Tangherlini, F. R. (1963). "Atoms in Higher Dimensions". <i>Nuovo Cimento</i> <b>14</b> (27): 636.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Atoms+in+Higher+Dimensions&rft.jtitle=Nuovo+Cimento&rft.aulast=Tangherlini&rft.aufirst=F.+R.&rft.au=Tangherlini%2C%26%2332%3BF.+R.&rft.date=1963&rft.volume=14&rft.issue=27&rft.pages=636&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span></span></li>
<li id="cite_note-tegmark-dim-14" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink">^ <a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-tegmark-dim_14-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-tegmark-dim_14-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-tegmark-dim_14-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>c</b></i></sup></a></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Max Tegmark">Tegmark, Max</a> (April 1997). <a class="external text" href="http://space.mit.edu/home/tegmark/dimensions.pdf" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif); background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 18px; text-decoration: none;">"On the dimensionality of spacetime"</a>. <i>Classical and Quantum Gravity</i> <b>14</b> (4): L69–L75. <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/gr-qc/9702052" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">gr-qc/9702052</a>. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/1997CQGra..14L..69T" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1997CQGra..14L..69T</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1088%2F0264-9381%2F14%2F4%2F002" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1088/0264-9381/14/4/002</a><span class="reference-accessdate">. Retrieved 2006-12-16</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+the+dimensionality+of+spacetime&rft.jtitle=Classical+and+Quantum+Gravity&rft.aulast=Tegmark&rft.aufirst=Max&rft.au=Tegmark%2C%26%2332%3BMax&rft.date=April+1997&rft.volume=14&rft.issue=4&rft.pages=L69%E2%80%93L75&rft_id=info:arxiv/gr-qc%2F9702052&rft_id=info:bibcode/1997CQGra..14L..69T&rft_id=info:doi/10.1088%2F0264-9381%2F14%2F4%2F002&rft_id=http%3A%2F%2Fspace.mit.edu%2Fhome%2Ftegmark%2Fdimensions.pdf&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span></span></li>
<li id="cite_note-15" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-15" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">Dorling, J. (1970). <a class="external text" href="http://link.aip.org/link/?AJP/38/539/1" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">"The Dimensionality of Time"</a>. <i>American Journal of Physics</i> <b>38</b> (4): 539–40. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1970AmJPh..38..539D" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1970AmJPh..38..539D</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1119%2F1.1976386" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1119/1.1976386</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Dimensionality+of+Time&rft.jtitle=American+Journal+of+Physics&rft.aulast=Dorling&rft.aufirst=J.&rft.au=Dorling%2C%26%2332%3BJ.&rft.date=1970&rft.volume=38&rft.issue=4&rft.pages=539%E2%80%9340&rft_id=info:bibcode/1970AmJPh..38..539D&rft_id=info:doi/10.1119%2F1.1976386&rft_id=http%3A%2F%2Flink.aip.org%2Flink%2F%3FAJP%2F38%2F539%2F1&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span></span></li>
<li id="cite_note-16" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-16" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/John_D._Barrow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John D. Barrow">Barrow, J. D.</a> (2002). <i>The Constants of Nature</i>. Pantheon Books. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-375-42221-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-375-42221-8">0-375-42221-8</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Constants+of+Nature&rft.aulast=Barrow&rft.aufirst=J.+D.&rft.au=Barrow%2C%26%2332%3BJ.+D.&rft.date=2002&rft.pub=Pantheon+Books&rft.isbn=0-375-42221-8&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span></span></li>
<li id="cite_note-17" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-17" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a href="http://en.wikipedia.org/wiki/Gerald_James_Whitrow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gerald James Whitrow">Whitrow, G. J.</a> (1955) " ," <i>British Journal of the Philosophy of Science</i> 6: 13. Also see his (1959) <i>The Structure and Evolution of the Universe</i>. London: Hutchinson.</span></li>
<li id="cite_note-18" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink"><b><a href="http://en.wikipedia.org/wiki/Spacetime#cite_ref-18" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b></span> <span class="reference-text"><a href="http://en.wikipedia.org/wiki/Jan_Ambj%C3%B8rn" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Jan Ambjørn">Jan Ambjørn</a>, <a class="new" href="http://en.wikipedia.org/w/index.php?title=Jerzy_Jurkiewicz&action=edit&redlink=1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Jerzy Jurkiewicz (page does not exist)">Jerzy Jurkiewicz</a>, and <a href="http://en.wikipedia.org/wiki/Renate_Loll" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Renate Loll">Renate Loll</a> - <a class="external text" href="http://www.sciam.com/article.cfm?id=the-self-organizing-quantum-universe" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">"The Self-Organizing Quantum Universe"</a>, <a href="http://en.wikipedia.org/wiki/Scientific_American" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Scientific American">Scientific American</a>, July 2008</span></li>
</ol>
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<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=18" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: References">edit</a>]</span><span class="mw-headline" id="References">References</span></h2>
<ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;"><div class="paradoctor-refeditlink" style="float: right;">
<sup style="line-height: 1em;">[<span class="plainlinks"><a class="external text" href="http://en.wikipedia.org/w/index.php?title=Template:BarrowTipler1986&action=edit" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 0% 0%; background-repeat: no-repeat no-repeat; color: #663366; padding-bottom: 0px !important; padding-left: 0px !important; padding-right: 13px; padding-top: 0px !important; text-decoration: none;">Edit this reference</a></span>]</sup></div>
<span class="citation book" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/John_D._Barrow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John D. Barrow">Barrow, John D.</a>; <a href="http://en.wikipedia.org/wiki/Frank_J._Tipler" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Frank J. Tipler">Tipler, Frank J.</a> (19 May 1988). <a class="external text" href="http://books.google.com/books?id=uSykSbXklWEC&printsec=frontcover" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;"><i>The Anthropic Cosmological Principle</i></a>. foreword by <a href="http://en.wikipedia.org/wiki/John_Archibald_Wheeler" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John Archibald Wheeler">John A. Wheeler</a>. Oxford: Oxford University Press.<a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/9780192821478" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/9780192821478">9780192821478</a>. <a class="external text" href="http://lccn.loc.gov/87028148" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">LC 87-28148</a><span class="reference-accessdate">. Retrieved 31 December 2009</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Anthropic+Cosmological+Principle&rft.aulast=Barrow&rft.aufirst=John+D.&rft.au=Barrow%2C%26%2332%3BJohn+D.&rft.au=Tipler%2C%26%2332%3BFrank+J.&rft.date=19+May+1988&rft.place=Oxford&rft.pub=Oxford+University+Press&rft.isbn=9780192821478&rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuSykSbXklWEC%26printsec%3Dfrontcover&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Paul_Ehrenfest" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Paul Ehrenfest">Ehrenfest, Paul</a> (1920) "How do the fundamental laws of physics make manifest that Space has 3 dimensions?" <i>Annalen der Physik 61</i>: 440.</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/George_F._Ellis" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="George F. Ellis">George F. Ellis</a> and Ruth M. Williams (1992) <i>Flat and curved space-times</i>. Oxford Univ. Press. <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/0198511647" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 0-19-851164-7</a></li>
<li style="margin-bottom: 0.1em;"><span class="citation Journal" style="word-wrap: break-word;">Isenberg, J. A. (1981). "Wheeler-Einstein-Mach spacetimes". <i>Phys. Rev. D</i> <b>24</b> (2): 251–256. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1981PhRvD..24..251I" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1981PhRvD..24..251I</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRevD.24.251" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRevD.24.251</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Wheeler-Einstein-Mach+spacetimes&rft.jtitle=Phys.+Rev.+D&rft.aulast=Isenberg&rft.aufirst=J.+A.&rft.au=Isenberg%2C%26%2332%3BJ.+A.&rft.date=1981&rft.volume=24&rft.issue=2&rft.pages=251%E2%80%93256&rft_id=info:bibcode/1981PhRvD..24..251I&rft_id=info:doi/10.1103%2FPhysRevD.24.251&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Immanuel_Kant" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Immanuel Kant">Kant, Immanuel</a> (1929) "Thoughts on the true estimation of living forces" in J. Handyside, trans., <i>Kant's Inaugural Dissertation and Early Writings on Space</i>. Univ. of Chicago Press.</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Hendrik_Lorentz" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hendrik Lorentz">Lorentz, H. A.</a>, <a href="http://en.wikipedia.org/wiki/Albert_Einstein" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Albert Einstein">Einstein, Albert</a>, <a href="http://en.wikipedia.org/wiki/Hermann_Minkowski" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermann Minkowski">Minkowski, Hermann</a>, and <a href="http://en.wikipedia.org/wiki/Hermann_Weyl" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermann Weyl">Weyl, Hermann</a> (1952) <i>The Principle of Relativity: A Collection of Original Memoirs</i>. Dover.</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/John_Lucas_(philosopher)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John Lucas (philosopher)">Lucas, John Randolph</a> (1973) <i>A Treatise on Time and Space</i>. London: Methuen.</li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Roger_Penrose" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Roger Penrose">Penrose, Roger</a> (2004). <i><a href="http://en.wikipedia.org/wiki/The_Road_to_Reality" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="The Road to Reality">The Road to Reality</a></i>. Oxford: Oxford University Press. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0679454438" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0679454438">0679454438</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%5B%5BThe+Road+to+Reality%5D%5D&rft.aulast=Penrose&rft.aufirst=Roger&rft.au=Penrose%2C%26%2332%3BRoger&rft.date=2004&rft.place=Oxford&rft.pub=Oxford+University+Press&rft.isbn=0679454438&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span> Chpts. 17–18.</li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Edgar_Allan_Poe" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edgar Allan Poe">Poe, Edgar A.</a> (1848). <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Eureka_(Edgar_Allan_Poe)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eureka (Edgar Allan Poe)">Eureka; An Essay on the Material and Spiritual Universe</a></i>. <a href="http://en.wikipedia.org/wiki/Hesperus_Press" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hesperus Press">Hesperus Press</a> Limited. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/1-84391-009-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/1-84391-009-8">1-84391-009-8</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%5B%5BEureka+%28Edgar+Allan+Poe%29%7CEureka%3B+An+Essay+on+the+Material+and+Spiritual+Universe%5D%5D&rft.aulast=Poe&rft.aufirst=Edgar+A.&rft.au=Poe%2C%26%2332%3BEdgar+A.&rft.date=1848&rft.pub=%5B%5BHesperus+Press%5D%5D+Limited&rft.isbn=1-84391-009-8&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">Robb, A. A. (1936). <i>Geometry of Time and Space</i>. University Press.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+of+Time+and+Space&rft.aulast=Robb&rft.aufirst=A.+A.&rft.au=Robb%2C%26%2332%3BA.+A.&rft.date=1936&rft.pub=University+Press&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Erwin Schrödinger">Erwin Schrödinger</a> (1950) <i>Space-time structure</i>. Cambridge Univ. Press.</li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">Schutz, J. W. (1997). <i>Independent axioms for Minkowski Space-time</i>. Addison-Wesley Longman. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0582317606" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0582317606">0582317606</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Independent+axioms+for+Minkowski+Space-time&rft.aulast=Schutz&rft.aufirst=J.+W.&rft.au=Schutz%2C%26%2332%3BJ.+W.&rft.date=1997&rft.pub=Addison-Wesley+Longman&rft.isbn=0582317606&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation Journal" style="word-wrap: break-word;">Tangherlini, F. R. (1963). "Atoms in Higher Dimensions". <i>Nuovo Cimento</i> <b>14</b> (27): 636.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Atoms+in+Higher+Dimensions&rft.jtitle=Nuovo+Cimento&rft.aulast=Tangherlini&rft.aufirst=F.+R.&rft.au=Tangherlini%2C%26%2332%3BF.+R.&rft.date=1963&rft.volume=14&rft.issue=27&rft.pages=636&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;">Taylor, E. F.; <a class="mw-redirect" href="http://en.wikipedia.org/wiki/John_A._Wheeler" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John A. Wheeler">Wheeler, John A.</a> (1963). <i>Spacetime Physics</i>. W. H. Freeman. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0716723271" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0716723271">0716723271</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spacetime+Physics&rft.aulast=Taylor&rft.aufirst=E.+F.&rft.au=Taylor%2C%26%2332%3BE.+F.&rft.date=1963&rft.pub=W.+H.+Freeman&rft.isbn=0716723271&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/H.G._Wells" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="H.G. Wells">Wells, H.G.</a> (2004). <i><a href="http://en.wikipedia.org/wiki/The_Time_Machine" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="The Time Machine">The Time Machine</a></i>. New York: Pocket Books. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0671575546" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0671575546">0671575546</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%5B%5BThe+Time+Machine%5D%5D&rft.aulast=Wells&rft.aufirst=H.G.&rft.au=Wells%2C%26%2332%3BH.G.&rft.date=2004&rft.place=New+York&rft.pub=Pocket+Books&rft.isbn=0671575546&rfr_id=info:sid/en.wikipedia.org:Spacetime"></span> (pp. 5–6)</li>
</ul>
<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Spacetime&action=edit&section=19" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: External links">edit</a>]</span><span class="mw-headline" id="External_links">External links</span></h2>
<table class="metadata mbox-small plainlinks" style="background-color: #f9f9f9; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; border-image: initial; border-left-color: rgb(170, 170, 170); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(170, 170, 170); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(170, 170, 170); border-top-style: solid; border-top-width: 1px; clear: right; color: black; float: right; font-family: sans-serif; font-size: 11px; line-height: 1.25em; margin-bottom: 4px; margin-left: 1em; margin-right: 0px; margin-top: 4px; width: 238px;"><tbody>
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<i><b><a class="external text" href="http://en.wikibooks.org/wiki/Special_Relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 0% 0%; background-repeat: no-repeat no-repeat; color: #663366; padding-bottom: 0px !important; padding-left: 0px !important; padding-right: 13px; padding-top: 0px !important; text-decoration: none;">Special Relativity</a></b></i></div>
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<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Stanford_Encyclopedia_of_Philosophy" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>: "<a class="external text" href="http://plato.stanford.edu/entries/spacetime-iframes/" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Space and Time: Inertial Frames</a>" by Robert DiSalle.</li>
</ul>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com0tag:blogger.com,1999:blog-5303246073824127471.post-9404421563063536772012-05-05T23:59:00.001-04:002012-05-05T23:59:45.011-04:00Mathematical Formulation Of Quantum Mechanics<br />
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<br /><br />The <b>mathematical formulations of quantum mechanics</b> are those <a href="http://en.wikipedia.org/wiki/Formalism_(mathematics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Formalism (mathematics)">mathematical formalisms</a> that permit a rigorous description of <a href="http://en.wikipedia.org/wiki/Quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum mechanics">quantum mechanics</a>. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional <a href="http://en.wikipedia.org/wiki/Hilbert_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hilbert space">Hilbert spaces</a> and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Linear_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear operator">operators</a> on these spaces. Many of these structures are drawn from<a href="http://en.wikipedia.org/wiki/Functional_analysis" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Functional analysis">functional analysis</a>, a research area within <a href="http://en.wikipedia.org/wiki/Pure_mathematics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pure mathematics">pure mathematics</a> that was influenced in part by the needs of quantum mechanics. In brief, values of physical observables such as <a href="http://en.wikipedia.org/wiki/Energy" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Energy">energy</a> and <a href="http://en.wikipedia.org/wiki/Momentum" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Momentum">momentum</a> were no longer considered as values of <a href="http://en.wikipedia.org/wiki/Function_(mathematics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Function (mathematics)">functions</a> on <a href="http://en.wikipedia.org/wiki/Phase_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Phase space">phase space</a>, but as<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Eigenvalue" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eigenvalue">eigenvalues</a>; more precisely: as <a href="http://en.wikipedia.org/wiki/Spectrum_(functional_analysis)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spectrum (functional analysis)">spectral values</a> (point spectrum plus absolute continuous plus singular continuous spectrum) of linear<a href="http://en.wikipedia.org/wiki/Operator_(physics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Operator (physics)">operators</a> in Hilbert space.<sup class="reference" id="cite_ref-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#cite_note-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[1]</a></sup></div>
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These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas of <i>quantum state</i> and <i>quantum observable</i> which are radically different from those used in previous <a href="http://en.wikipedia.org/wiki/Mathematical_model" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mathematical model">models</a> of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured. This limitation was first elucidated by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Heisenberg_uncertainty_relations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Heisenberg uncertainty relations">Heisenberg</a> through a <a href="http://en.wikipedia.org/wiki/Thought_experiment" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Thought experiment">thought experiment</a>, and is represented mathematically in the new formalism by the<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Non-commutative" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Non-commutative">non-commutativity</a> of quantum observables.</div>
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Prior to the emergence of quantum mechanics as a separate <a href="http://en.wikipedia.org/wiki/Theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theory">theory</a>, the mathematics used in physics consisted mainly of formal <a href="http://en.wikipedia.org/wiki/Mathematical_analysis" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mathematical analysis">mathematical analysis</a>, beginning with <a href="http://en.wikipedia.org/wiki/Calculus" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Calculus">calculus</a>; and, increasing in complexity up to <a href="http://en.wikipedia.org/wiki/Differential_geometry" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Differential geometry">differential geometry</a> and <a href="http://en.wikipedia.org/wiki/Partial_differential_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Partial differential equation">partial differential equations</a>. <a href="http://en.wikipedia.org/wiki/Probability_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability theory">Probability theory</a>was used in <a href="http://en.wikipedia.org/wiki/Statistical_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Statistical mechanics">statistical mechanics</a>. Geometric intuition clearly played a strong role in the first two and, accordingly, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Relativity_physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Relativity physics">theories of relativity</a> were formulated entirely in terms of geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the emergence of quantum theory (around 1925) physicists continued to think of quantum theory within the confines of what is now called <a href="http://en.wikipedia.org/wiki/Classical_physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Classical physics">classical physics</a>, and in particular within the same mathematical structures. The most sophisticated example of this is the<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Sommerfeld%E2%80%93Wilson%E2%80%93Ishiwara_quantization" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sommerfeld–Wilson–Ishiwara quantization">Sommerfeld–Wilson–Ishiwara quantization</a> rule, which was formulated entirely on the classical <a href="http://en.wikipedia.org/wiki/Phase_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Phase space">phase space</a>.</div>
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Contents</h2>
<span class="toctoggle" style="-webkit-user-select: none;"> [<a class="internal" href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#" id="togglelink" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">hide</a>] </span></div>
<ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;">
<li class="toclevel-1 tocsection-1" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#History_of_the_formalism" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1</span> <span class="toctext">History of the formalism</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-2" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#The_.22old_quantum_theory.22_and_the_need_for_new_mathematics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1.1</span> <span class="toctext">The "old quantum theory" and the need for new mathematics</span></a></li>
<li class="toclevel-2 tocsection-3" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#The_.22new_quantum_theory.22" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1.2</span> <span class="toctext">The "new quantum theory"</span></a></li>
<li class="toclevel-2 tocsection-4" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#Later_developments" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1.3</span> <span class="toctext">Later developments</span></a></li>
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<li class="toclevel-1 tocsection-5" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#Mathematical_structure_of_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2</span> <span class="toctext">Mathematical structure of quantum mechanics</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-6" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#Postulates_of_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.1</span> <span class="toctext">Postulates of quantum mechanics</span></a></li>
<li class="toclevel-2 tocsection-7" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#Pictures_of_dynamics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.2</span> <span class="toctext">Pictures of dynamics</span></a></li>
<li class="toclevel-2 tocsection-8" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#Representations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.3</span> <span class="toctext">Representations</span></a></li>
<li class="toclevel-2 tocsection-9" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#Time_as_an_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.4</span> <span class="toctext">Time as an operator</span></a></li>
<li class="toclevel-2 tocsection-10" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#Spin" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.5</span> <span class="toctext">Spin</span></a></li>
<li class="toclevel-2 tocsection-11" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#Pauli.27s_principle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.6</span> <span class="toctext">Pauli's principle</span></a></li>
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<li class="toclevel-1 tocsection-12" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#The_problem_of_measurement" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3</span> <span class="toctext">The problem of measurement</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-13" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#The_relative_state_interpretation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.1</span> <span class="toctext">The <i>relative state</i> interpretation</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-14" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#List_of_mathematical_tools" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4</span> <span class="toctext">List of mathematical tools</span></a></li>
<li class="toclevel-1 tocsection-15" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#References" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-16" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#Notes" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6</span> <span class="toctext">Notes</span></a></li>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: History of the formalism">edit</a>]</span><span class="mw-headline" id="History_of_the_formalism">History of the formalism</span></h2>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: The "old quantum theory" and the need for new mathematics">edit</a>]</span><span class="mw-headline" id="The_.22old_quantum_theory.22_and_the_need_for_new_mathematics">The "old quantum theory" and the need for new mathematics</span></h3>
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Main article: <a href="http://en.wikipedia.org/wiki/Old_quantum_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Old quantum theory">Old quantum theory</a></div>
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In the 1890s, <a href="http://en.wikipedia.org/wiki/Max_Planck" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Max Planck">Planck</a> was able to derive the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Blackbody_spectrum" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Blackbody spectrum">blackbody spectrum</a> which was later used to avoid the classical <a href="http://en.wikipedia.org/wiki/Ultraviolet_catastrophe" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Ultraviolet catastrophe">ultraviolet catastrophe</a> by making the unorthodox assumption that, in the interaction of <a href="http://en.wikipedia.org/wiki/Radiation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Radiation">radiation</a> with <a href="http://en.wikipedia.org/wiki/Matter" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Matter">matter</a>, energy could only be exchanged in discrete units which he called <a href="http://en.wikipedia.org/wiki/Quantum" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum">quanta</a>. Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, <i>h</i>, is now called <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Planck%27s_constant" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Planck's constant">Planck's constant</a> in his honor.</div>
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In 1905, <a href="http://en.wikipedia.org/wiki/Albert_Einstein" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Albert Einstein">Einstein</a> explained certain features of the <a href="http://en.wikipedia.org/wiki/Photoelectric_effect" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Photoelectric effect">photoelectric effect</a> by assuming that Planck's energy quanta were actual particles, which were later dubbed <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Photons" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Photons">photons</a>.</div>
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<a class="image" href="http://en.wikipedia.org/wiki/File:Bohr_atom_model_English.svg" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="light at the right frequency."><img alt="light at the right frequency." height="400" src="http://upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bohr_atom_model_English.svg/420px-Bohr_atom_model_English.svg.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" width="420" /></a></div>
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All of these developments were <a href="http://en.wikipedia.org/wiki/Phenomenology_(science)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Phenomenology (science)">phenomenological</a> and flew in the face of the theoretical physics of the time. <a href="http://en.wikipedia.org/wiki/Old_quantum_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Old quantum theory">Bohr and Sommerfeld</a> went on to modify classical mechanics in an attempt to deduce the <a href="http://en.wikipedia.org/wiki/Bohr_model" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bohr model">Bohr model</a>from first principles. They proposed that, of all closed classical orbits traced by a mechanical system in its<a href="http://en.wikipedia.org/wiki/Phase_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Phase space">phase space</a>, only the ones that enclosed an area which was a multiple of Planck's constant were actually allowed. The most sophisticated version of this formalism was the so-called <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Sommerfeld%E2%80%93Wilson%E2%80%93Ishiwara_quantization" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sommerfeld–Wilson–Ishiwara quantization">Sommerfeld–Wilson–Ishiwara quantization</a>. Although the Bohr model of the hydrogen atom could be explained in this way, the spectrum of the helium atom (classically an unsolvable <a class="mw-redirect" href="http://en.wikipedia.org/wiki/3-body_problem" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="3-body problem">3-body problem</a>) could not be predicted. The mathematical status of quantum theory remained uncertain for some time.</div>
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In 1923 <a href="http://en.wikipedia.org/wiki/Louis_de_Broglie" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Louis de Broglie">de Broglie</a> proposed that <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Wave-particle_duality" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wave-particle duality">wave-particle duality</a> applied not only to photons but to electrons and every other physical system.</div>
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The situation changed rapidly in the years 1925–1930, when working mathematical foundations were found through the groundbreaking work of <a href="http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Erwin Schrödinger">Erwin Schrödinger</a>, <a href="http://en.wikipedia.org/wiki/Werner_Heisenberg" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Werner Heisenberg">Werner Heisenberg</a>, <a href="http://en.wikipedia.org/wiki/Max_Born" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Max Born">Max Born</a>, <a href="http://en.wikipedia.org/wiki/Pascual_Jordan" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pascual Jordan">Pascual Jordan</a>, and the foundational work of <a href="http://en.wikipedia.org/wiki/John_von_Neumann" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John von Neumann">John von Neumann</a>, <a href="http://en.wikipedia.org/wiki/Hermann_Weyl" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermann Weyl">Hermann Weyl</a> and <a href="http://en.wikipedia.org/wiki/Paul_Dirac" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Paul Dirac">Paul Dirac</a>, and it became possible to unify several different approaches in terms of a fresh set of ideas. The physical interpretation of the theory was also clarified in these years after <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Heisenberg" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Heisenberg">Werner Heisenberg</a> discovered the uncertainty relations and <a href="http://en.wikipedia.org/wiki/Niels_Bohr" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Niels Bohr">Niels Bohr</a> introduced the idea of complementarity.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: The "new quantum theory"">edit</a>]</span><span class="mw-headline" id="The_.22new_quantum_theory.22">The "new quantum theory"</span></h3>
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<a href="http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Erwin Schrödinger">Erwin Schrödinger's</a> <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schrödinger equation">wave mechanics</a> originally was the first successful attempt at replicating the observed quantization of atomic spectra with the help of a precise mathematical realization of de Broglie's wave-particle duality. Schrödinger's wave mechanics were created independently, uniquely based on de Broglie's concepts, less formal and easier to understand, visualize and exploit. Within a year, it was shown that the two theories were equivalent. Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the <a href="http://en.wikipedia.org/wiki/Absolute_value#Complex_numbers" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Absolute value">absolute square</a> of the wave function of an <a href="http://en.wikipedia.org/wiki/Electron" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron">electron</a> should be interpreted as the<a href="http://en.wikipedia.org/wiki/Charge_density" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Charge density">charge density</a> of an object smeared out over an extended, possibly infinite, volume of space, but <a href="http://en.wikipedia.org/wiki/Max_Born" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Max Born">Max Born</a> introduced the interpretation of the <a href="http://en.wikipedia.org/wiki/Absolute_value#Complex_numbers" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Absolute value">absolute square</a> of the wave function as the probability distribution of the position of a <i>pointlike</i> object. Born's idea was soon taken over by Niels Bohr in Copenhagen, who then became the "father" of the <a href="http://en.wikipedia.org/wiki/Copenhagen_interpretation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Copenhagen interpretation">Copenhagen interpretation</a> of quantum mechanics. Schrödinger's <a href="http://en.wikipedia.org/wiki/Wave_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wave function">wave function</a> can be seen to be closely related to the classical <a href="http://en.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a>. The correspondence to classical mechanics was even more explicit, although somewhat more formal, in Heisenberg's matrix mechanics. I.e., the equation for the operators in the Heisenberg representation, as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, where one uses <a href="http://en.wikipedia.org/wiki/Poisson_bracket" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Poisson bracket">Poisson brackets</a>.</div>
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To be more precise: already before Schrödinger the young student <a href="http://en.wikipedia.org/wiki/Werner_Heisenberg" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Werner Heisenberg">Werner Heisenberg</a> invented his <a href="http://en.wikipedia.org/wiki/Matrix_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Matrix mechanics">matrix mechanics</a>, which was the first correct quantum mechanics, i.e. the essential breakthrough. Heisenberg's <a href="http://en.wikipedia.org/wiki/Matrix_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Matrix mechanics">matrix mechanics</a> formulation was based on algebras of infinite matrices, being certainly very radical in light of the mathematics of classical physics, although he started from the index-terminology of the experimentalists of that time, not even knowing that his "index-schemes" were matrices. In fact, in these early years<a href="http://en.wikipedia.org/wiki/Linear_algebra" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear algebra">linear algebra</a> was not generally known to physicists in its present form.</div>
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Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches is generally attributed to <a href="http://en.wikipedia.org/wiki/Paul_Dirac" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Paul Dirac">Paul Dirac</a>, who wrote a lucid account in his 1930 classic <i>Principles of Quantum Mechanics</i>, being the third, and perhaps most important, person working independently in that field (he soon was the only one, who found a relativistic generalization of the theory). In his above-mentioned account, he introduced the <a href="http://en.wikipedia.org/wiki/Bra-ket_notation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bra-ket notation">bra-ket notation</a>, together with an abstract formulation in terms of the <a href="http://en.wikipedia.org/wiki/Hilbert_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hilbert space">Hilbert space</a> used in <a href="http://en.wikipedia.org/wiki/Functional_analysis" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Functional analysis">functional analysis</a>; he showed that Schrödinger's and Heisenberg's approaches were two different representations of the same theory and found a third, most general one, which represented the dynamics of the system. His work was particularly fruitful in all kind of generalizations of the field. Concerning quantum mechanics, <a href="http://en.wikipedia.org/wiki/Paul_Dirac" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Paul Dirac">Dirac</a>'s method is now called <a href="http://en.wikipedia.org/wiki/Canonical_quantization" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Canonical quantization">canonical quantization</a>.</div>
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The first complete mathematical formulation of this approach is generally credited to <a href="http://en.wikipedia.org/wiki/John_von_Neumann" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John von Neumann">John von Neumann</a>'s 1932 book <i>Mathematical Foundations of Quantum Mechanics</i>, although<a href="http://en.wikipedia.org/wiki/Hermann_Weyl" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermann Weyl">Hermann Weyl</a> had already referred to Hilbert spaces (which he called <i>unitary spaces</i>) in his 1927 classic book. It was developed in parallel with a new approach to the mathematical<a href="http://en.wikipedia.org/wiki/Spectral_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spectral theory">spectral theory</a> based on <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Linear_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear operator">linear operators</a> rather than the <a href="http://en.wikipedia.org/wiki/Quadratic_form" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quadratic form">quadratic forms</a> that were <a href="http://en.wikipedia.org/wiki/David_Hilbert" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="David Hilbert">David Hilbert</a>'s approach a generation earlier. Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formulation of quantum mechanics which underlies most approaches and can be traced back to the mathematical work of <a href="http://en.wikipedia.org/wiki/John_von_Neumann" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John von Neumann">John von Neumann</a>. In other words, discussions about <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Interpretation_of_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Interpretation of quantum mechanics"><i>interpretation</i> of the theory</a>, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Later developments">edit</a>]</span><span class="mw-headline" id="Later_developments">Later developments</span></h3>
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The application of the new quantum theory to electromagnetism resulted in <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">quantum field theory</a>, which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the one presented here is a simple special case. In fact, the difficulties involved in implementing any of the following formulations cannot be said yet to have been solved in a satisfactory fashion except for ordinary quantum mechanics.</div>
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<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Feynman_path_integral" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Feynman path integral">Feynman path integrals</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wightman_axioms" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wightman axioms">axiomatic</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Local_quantum_physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Local quantum physics">algebraic</a> and <a href="http://en.wikipedia.org/wiki/Constructive_quantum_field_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Constructive quantum field theory">constructive quantum field theory</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Weyl_quantization" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Weyl quantization">Weyl quantization</a> & <a href="http://en.wikipedia.org/wiki/Geometric_quantization" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Geometric quantization">geometric quantization</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_field_theory_in_curved_spacetime" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory in curved spacetime">quantum field theory in curved spacetime</a></li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/C*_algebra" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="C* algebra">C* algebra</a> <a href="http://en.wikipedia.org/wiki/Formalism_(mathematics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Formalism (mathematics)">formalism</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/POVM" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="POVM">Generalized Statistical Model of Quantum Mechanics</a></li>
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On a different front, von Neumann originally dispatched <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Quantum_measurement" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum measurement">quantum measurement</a> with his infamous postulate on the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Collapse_of_the_wavefunction" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Collapse of the wavefunction">collapse of the wavefunction</a>, raising a host of philosophical problems. Over the intervening 70 years, the <i>problem of measurement</i> became an active research area and itself spawned some new formulations of quantum mechanics.</div>
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<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Many-worlds_interpretation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Many-worlds interpretation">Relative state/Many-worlds interpretation</a> of quantum mechanics</li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Decoherence" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Decoherence">Decoherence</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Consistent_histories" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Consistent histories">Consistent histories</a> formulation of quantum mechanics</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_logic" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum logic">Quantum logic</a> formulation of quantum mechanics</li>
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A related topic is the relationship to classical mechanics. Any new physical theory is supposed to reduce to successful old theories in some approximation. For quantum mechanics, this translates into the need to study the so-called <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Classical_limit_of_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Classical limit of quantum mechanics">classical limit of quantum mechanics</a>. Also, as Bohr emphasized, human cognitive abilities and language are inextricably linked to the classical realm, and so classical descriptions are intuitively more accessible than quantum ones. In particular, <a href="http://en.wikipedia.org/wiki/Quantization_(physics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantization (physics)">quantization</a>, namely the construction of a quantum theory whose classical limit is a given and known classical theory, becomes an important area of quantum physics in itself.</div>
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Finally, some of the originators of quantum theory (notably Einstein and Schrödinger) were unhappy with what they thought were the philosophical implications of quantum mechanics. In particular, Einstein took the position that quantum mechanics must be incomplete, which motivated research into so-called <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Hidden-variable" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hidden-variable">hidden-variable</a> theories. The issue of hidden variables has become in part an experimental issue with the help of <a href="http://en.wikipedia.org/wiki/Quantum_optics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum optics">quantum optics</a>.</div>
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<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Louis,_7th_duc_de_Broglie" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Louis, 7th duc de Broglie">de Broglie</a>–<a href="http://en.wikipedia.org/wiki/David_Bohm" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="David Bohm">Bohm</a>–<a href="http://en.wikipedia.org/wiki/John_Stewart_Bell" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John Stewart Bell">Bell</a> <a href="http://en.wikipedia.org/wiki/Pilot_wave" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pilot wave">pilot wave</a> formulation of quantum mechanics</li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Bell%27s_inequalities" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bell's inequalities">Bell's inequalities</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Kochen%E2%80%93Specker_theorem" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Kochen–Specker theorem">Kochen–Specker theorem</a></li>
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<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Mathematical structure of quantum mechanics">edit</a>]</span><span class="mw-headline" id="Mathematical_structure_of_quantum_mechanics">Mathematical structure of quantum mechanics</span></h2>
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A physical system is generally described by three basic ingredients: <a href="http://en.wikipedia.org/wiki/Quantum_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum state">states</a>; <a href="http://en.wikipedia.org/wiki/Observable" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Observable">observables</a>; and <a href="http://en.wikipedia.org/wiki/Dynamics_(mechanics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dynamics (mechanics)">dynamics</a> (or law of <a href="http://en.wikipedia.org/wiki/Time_evolution" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Time evolution">time evolution</a>) or, more generally, a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gauge_invariance" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gauge invariance">group of physical symmetries</a>. A classical description can be given in a fairly direct way by a <a href="http://en.wikipedia.org/wiki/Phase_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Phase space">phase space</a> <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Model_(abstract)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Model (abstract)">model</a> of mechanics: states are points in a <a href="http://en.wikipedia.org/wiki/Symplectic_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Symplectic space">symplectic</a> phase space, observables are real-valued functions on it, time evolution is given by a one-parameter <a href="http://en.wikipedia.org/wiki/Group_(mathematics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Group (mathematics)">group</a> of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description consists of a <a href="http://en.wikipedia.org/wiki/Hilbert_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hilbert space">Hilbert space</a> of states, observables are <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Self_adjoint_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Self adjoint operator">self adjoint operators</a> on the space of states, time evolution is given by a <a href="http://en.wikipedia.org/wiki/Stone%27s_theorem_on_one-parameter_unitary_groups" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Stone's theorem on one-parameter unitary groups">one-parameter group</a> of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Postulates of quantum mechanics">edit</a>]</span><span class="mw-headline" id="Postulates_of_quantum_mechanics">Postulates of quantum mechanics</span></h3>
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The following summary of the mathematical framework of quantum mechanics can be partly traced back to von Neumann's postulates.</div>
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<li style="margin-bottom: 0.1em;">Each physical system is associated with a (topologically) <a href="http://en.wikipedia.org/wiki/Separable_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Separable space">separable</a> <a href="http://en.wikipedia.org/wiki/Complex_number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Complex number">complex</a> <a href="http://en.wikipedia.org/wiki/Hilbert_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hilbert space">Hilbert space</a> <i>H</i> with inner product <img alt="\scriptstyle \langle\phi\mid\psi\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/d/e/4de14180ff2b174da7e241c8ec690fc7.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />. <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Ray_(quantum_theory)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Ray (quantum theory)">Rays</a> (one-dimensional subspaces) in <i>H</i> are associated with<a href="http://en.wikipedia.org/wiki/Quantum_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum state">states</a> of the system. In other words, physical states can be identified with equivalence classes of vectors of length 1 in <i>H</i>, where two vectors represent the same state if they differ only by a <a href="http://en.wikipedia.org/wiki/Phase_factor" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Phase factor">phase factor</a>. <i>Separability</i> is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state.</li>
<li style="margin-bottom: 0.1em;">The Hilbert space of a composite system is the Hilbert space <a href="http://en.wikipedia.org/wiki/Tensor_product" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Tensor product">tensor product</a> of the state spaces associated with the component systems (for instance, J.M. Jauch, <i>Foundations of quantum mechanics</i>, section 11-7). For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.</li>
<li style="margin-bottom: 0.1em;">Physical symmetries act on the Hilbert space of quantum states <a href="http://en.wikipedia.org/wiki/Unitary_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Unitary operator">unitarily</a> or <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Antiunitary" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Antiunitary">antiunitarily</a> due to <a href="http://en.wikipedia.org/wiki/Wigner%27s_theorem" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wigner's theorem">Wigner's theorem</a> (<a href="http://en.wikipedia.org/wiki/Supersymmetry" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Supersymmetry">supersymmetry</a> is another matter entirely).</li>
<li style="margin-bottom: 0.1em;">Physical <a href="http://en.wikipedia.org/wiki/Observable" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Observable">observables</a> are represented by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Densely-defined" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Densely-defined">densely-defined</a> <a href="http://en.wikipedia.org/wiki/Self-adjoint_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Self-adjoint operator">self-adjoint operators</a> on <i>H</i>.</li>
</ul>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">The <a href="http://en.wikipedia.org/wiki/Expected_value" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Expected value">expected value</a> (in the sense of probability theory) of the observable <i>A</i> for the system in state represented by the unit vector <img alt="\scriptstyle \left|\psi\right\rangle\in " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/8/a/08a1bd2d5be9173cde30f8a91213c640.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /> <i>H</i> is
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\langle\psi\mid A\mid\psi\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/0/5/b05b5cb7d1b4b2caff8c7cf4bb39fe72.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
</dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">By <a href="http://en.wikipedia.org/wiki/Spectral_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spectral theory">spectral theory</a>, we can associate a <a href="http://en.wikipedia.org/wiki/Probability_measure" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability measure">probability measure</a> to the values of <i>A</i> in any state ψ. We can also show that the possible values of the observable <i>A</i> in any state must belong to the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Spectrum_of_an_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spectrum of an operator">spectrum</a> of <i>A</i>. In the special case <i>A</i> has only <a href="http://en.wikipedia.org/wiki/Discrete_spectrum" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Discrete spectrum">discrete spectrum</a>, the possible outcomes of measuring <i>A</i> are its <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Eigenvalue" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eigenvalue">eigenvalues</a>.</dd></dl>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">More generally, a state can be represented by a so-called <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Density_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Density operator">density operator</a>, which is a <a href="http://en.wikipedia.org/wiki/Trace_class" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Trace class">trace class</a>, nonnegative self-adjoint operator <img alt="\rho" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/7/f/f7f177957cf064a93e9811df8fe65ed1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /> normalized to be of trace 1. The expected value of <i>A</i> in the state <img alt="\rho" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/7/f/f7f177957cf064a93e9811df8fe65ed1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /> is
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \operatorname{tr}(A\rho)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/f/7/af79b14cd4135d6bc1516c131b8097f6.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
</dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">If <img alt="\rho_\psi" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/0/0/c00fb19a81d7cfecaea3d6b052c08b4b.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /> is the orthogonal projector onto the one-dimensional subspace of <i>H</i> spanned by <img alt="\scriptstyle \left|\psi\right\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/5/1/c51800ac5328dd7ffff1a6aa62a51abf.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />, then
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \operatorname{tr}(A\rho_\psi)=\left\langle\psi\mid A\mid\psi\right\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/e/3/be37e47ee375656a53e491bc778169bc.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
</dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">Density operators are those that are in the closure of the <a href="http://en.wikipedia.org/wiki/Convex_hull" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Convex hull">convex hull</a> of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are<a href="http://en.wikipedia.org/wiki/Extreme_point" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Extreme point">extreme points</a> of the set of density operators. Physicists also call one-dimensional orthogonal projectors <i>pure states</i> and other density operators <i>mixed states</i>.</dd></dl>
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One can in this formalism state Heisenberg's <a href="http://en.wikipedia.org/wiki/Uncertainty_principle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Uncertainty principle">uncertainty principle</a> and prove it as a theorem, although the exact historical sequence of events, concerning who derived what and under which framework, is the subject of historical investigations outside the scope of this article.</div>
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Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of <a href="http://en.wikipedia.org/wiki/Spin_(physics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin (physics)">spin</a> and Pauli's <a href="http://en.wikipedia.org/wiki/Pauli_exclusion_principle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pauli exclusion principle">exclusion principle</a>, see below.</div>
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<b>Superselection sectors</b>. The correspondence between states and rays needs to be refined somewhat to take into account so-called <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Superselection_sector" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Superselection sector">superselection sectors</a>. States in different superselection sectors cannot influence each other, and the relative phases between them are unobservable.</div>
<h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 18px; line-height: 20px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Pictures of dynamics">edit</a>]</span><span class="mw-headline" id="Pictures_of_dynamics">Pictures of dynamics</span></h3>
<ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;">In the so-called <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_picture" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schrödinger picture">Schrödinger picture</a> of quantum mechanics, the dynamics is given as follows:</li>
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The <a href="http://en.wikipedia.org/wiki/Time_evolution" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Time evolution">time evolution</a> of the state is given by a differentiable function from the real numbers <b>R</b>, representing instants of time, to the Hilbert space of system states. This map is characterized by a differential equation as follows: If <img alt="\left|\psi\left(t\right)\right\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/a/7/3a7ea3423a633f28f5a11ee0bac0d94b.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> denotes the state of the system at any one time <i>t</i>, the following <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schrödinger equation">Schrödinger equation</a> holds:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" i\hbar\frac{d}{d t}\left|\psi(t)\right\rangle=H\left|\psi(t)\right\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/0/b/70bebea574563fcd24fe4b9f5066e76c.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where H is a densely-defined self-adjoint operator, called the system <a href="http://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>, <i>i</i> is the <a href="http://en.wikipedia.org/wiki/Complex_number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Complex number">imaginary unit</a> and <img alt="\hbar" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/d/f/9dfd055ef1683b053f1b5bf9ed6dbbb4.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Reduced_Planck_constant" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Reduced Planck constant">reduced Planck constant</a>. As an observable, H corresponds to the total <a href="http://en.wikipedia.org/wiki/Energy" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Energy">energy</a> of the system.</div>
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Alternatively, by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Stone%27s_theorem" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Stone's theorem">Stone's theorem</a> one can state that there is a strongly continuous one-parameter unitary group <i>U</i>(<i>t</i>): <i>H</i> → <i>H</i> such that</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\left|\psi(t+s)\right\rangle=U(t)\left|\psi(s)\right\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/e/c/1ecac8bf7cc2a40b15253d66405474f4.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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for all times <i>s</i>, <i>t</i>. The existence of a self-adjoint Hamiltonian H such that</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="U(t)=e^{-(i/\hbar)t H}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/b/1/8b12bb3f46dab9949f7fa85e83dba771.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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is a consequence of <a href="http://en.wikipedia.org/wiki/Stone%27s_theorem_on_one-parameter_unitary_groups" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Stone's theorem on one-parameter unitary groups">Stone's theorem on one-parameter unitary groups</a>. (It is assumed that <i>H</i> does not depend on time and that the perturbation starts at <img alt="t_0=0" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/1/7/117ed34abf0ea194a6bbd386056beaa8.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />; otherwise one must use the <a href="http://en.wikipedia.org/wiki/Dyson_series" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dyson series">Dyson series</a>, formally written as</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="U(t)={\mathcal{T}}\,\,\{\exp{-(i/\hbar )\int\limits_{t_0}^t \,{\rm d}t'\, H(t')}\}\,\,," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/f/0/2f02c8ebe7aa847369866d2beeefe12e.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <img alt="{\mathcal{T}}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/d/3/9d3b226b86733462a55081e60ab20541.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is Dyson's time-ordering symbol.</div>
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(This symbol permutes a product of noncommuting operators of the form</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" B_1(t_1)\cdot B_2(t_2)\cdot\dots \cdot B_n(t_n)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/6/8/3682dd111c3d2764199fb89724586737.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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into the uniquely determined re-ordered expression</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="B_{i_1}(t_{i_1})\cdot B_{i_2}(t_{i_2})\cdot\dots \cdot B_{i_n}(t_{i_n})" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/e/3/7e3b78338d941eb410a96c92b2d78676.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /> with <img alt="t_{i_1}\ge t_{i_2}\ge\dots\ge t_{i_n}\,." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/8/9/2898a32af4d850a9b0b55bc0b4d6c9a8.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The result is a causal chain, the primary <i>cause</i> in the past on the utmost r.h.s., and finally the present <i>effect</i> on the utmost l.h.s. .)</div>
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<li style="margin-bottom: 0.1em;">The <a href="http://en.wikipedia.org/wiki/Heisenberg_picture" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Heisenberg picture">Heisenberg picture</a> of quantum mechanics focuses on observables and instead of considering states as varying in time, it regards the states as fixed and the observables as changing. To go from the Schrödinger to the Heisenberg picture one needs to define time-independent states and time-dependent operators thus:</li>
</ul>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\left|\psi\right\rangle = \left|\psi(0)\right\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/0/6/906e2e19716b985f768321c159bb5f4c.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="A(t) = U(-t)AU(t). \quad" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/d/e/0de7af2a24f03486a575d5216b15b1ea.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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It is then easily checked that the expected values of all observables are the same in both pictures</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\langle\psi\mid A(t)\mid\psi\rangle=\langle\psi(t)\mid A\mid\psi(t)\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/b/b/6bb459f6d6890e4fff700e97f8b224cf.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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and that the time-dependent Heisenberg operators satisfy</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="i\hbar{d\over dt}A(t) = [A(t),H]." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/e/d/5edb983049674d776f43ef2d785d9cd2.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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This assumes A is not time dependent in the Schrödinger picture. Notice the commutator expression is purely formal when one of the operators is unbounded. One would specify a representation for the expression to make sense of it.</div>
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<li style="margin-bottom: 0.1em;">The so-called <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Dirac_picture" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac picture">Dirac picture</a> or <a href="http://en.wikipedia.org/wiki/Interaction_picture" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Interaction picture">interaction picture</a> has time-dependent <i>states</i> and observables, evolving with respect to different Hamiltonians. This picture is most useful when the evolution of the observables can be solved exactly, confining any complications to the evolution of the states. For this reason, the Hamiltonian for the observables is called "free Hamiltonian" and the Hamiltonian for the states is called "interaction Hamiltonian". In symbols:</li>
</ul>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" i\hbar\frac{d }{dt}\left|\psi(t)\right\rangle ={H}_{\rm int}(t) \left|\psi(t)\right\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/2/a/e2afbd9eb3f0da1a88d57472409ecb85.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="i\hbar{d \over d t}A(t) = [A(t),H_{0}]." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/2/4/72440eae4b1d5a21f64ff31aa58ecfd8.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The interaction picture does not always exist, though. In interacting quantum field theories, <a href="http://en.wikipedia.org/wiki/Haag%27s_theorem" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Haag's theorem">Haag's theorem</a> states that the interaction picture does not exist. This is because the Hamiltonian cannot be split into a free and an interacting part within a superselection sector. Moreover, even if in the Schrödinger picture the Hamiltonian does not depend on time, e.g. <img alt=" H=H_0+V" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/a/b/2ab6b574491b225c0d642fac0c133d78.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, in the interaction picture it does, at least, if <i>V</i> does not commute with <img alt="H_0" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/c/5/7c5081abe6c2100f0e44396b6ac51661.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, since</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="H_{\rm int}(t)\equiv e^{{(i/\hbar})tH_0}\,V\,e^{{(-i/\hbar})tH_0}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/4/0/e40f9ca0aece30901140e759141610a3.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />.</dd></dl>
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So the above-mentioned Dyson-series has to be used anyhow.</div>
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The Heisenberg picture is the closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly translate into the classical<a href="http://en.wikipedia.org/wiki/Poisson_bracket" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Poisson bracket">Poisson brackets</a>); but this is already rather "high-browed", and the Schrödinger picture is considered easiest to visualize and understand by most people, to judge from pedagogical accounts of quantum mechanics. The Dirac picture is the one used in perturbation theory, and is specially associated to <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">quantum field theory</a> and <a href="http://en.wikipedia.org/wiki/Many-body_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Many-body theory">many-body physics</a>.</div>
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Similar equations can be written for any one-parameter unitary group of symmetries of the physical system. Time would be replaced by a suitable coordinate parameterizing the unitary group (for instance, a rotation angle, or a translation distance) and the Hamiltonian would be replaced by the conserved quantity associated to the symmetry (for instance, angular or linear momentum).</div>
<h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 18px; line-height: 20px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Representations">edit</a>]</span><span class="mw-headline" id="Representations">Representations</span></h3>
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The original form of the <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schrödinger equation">Schrödinger equation</a> depends on choosing a particular representation of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Heisenberg" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Heisenberg">Heisenberg</a>'s <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Canonical_commutation_relations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Canonical commutation relations">canonical commutation relations</a>. The <a href="http://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Stone–von Neumann theorem">Stone–von Neumann theorem</a>states all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. This is related to <a href="http://en.wikipedia.org/wiki/Quantization_(physics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantization (physics)">quantization</a> and the correspondence between classical and quantum mechanics, and is therefore not strictly part of the general mathematical framework.</div>
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The <a href="http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum harmonic oscillator">quantum harmonic oscillator</a> is an exactly-solvable system where the possibility of choosing among more than one representation can be seen in all its glory. There, apart from the Schrödinger (position or momentum) representation one encounters the Fock (number) representation and the <a class="new" href="http://en.wikipedia.org/w/index.php?title=Segal%E2%80%93Bargmann_representation&action=edit&redlink=1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Segal–Bargmann representation (page does not exist)">Segal–Bargmann (phase space or coherent state) representation</a> (named after <a href="http://en.wikipedia.org/wiki/Irving_Segal" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Irving Segal">Irving Segal</a> and <a href="http://en.wikipedia.org/wiki/Valentine_Bargmann" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Valentine Bargmann">Valentine Bargmann</a>). All three are unitarily equivalent.</div>
<h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 18px; line-height: 20px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Time as an operator">edit</a>]</span><span class="mw-headline" id="Time_as_an_operator">Time as an operator</span></h3>
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The framework presented so far singles out time as <i>the</i> parameter that everything depends on. It is possible to formulate mechanics in such a way that time becomes itself an observable associated to a self-adjoint operator. At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter <i>s</i>, and in that case the time <i>t</i> becomes an additional generalized coordinate of the physical system. At the quantum level, translations in <i>s</i> would be generated by a "Hamiltonian" <i>H</i> − <i>E</i>, where <i>E</i> is the energy operator and <i>H</i> is the "ordinary" Hamiltonian. However, since <i>s</i> is an unphysical parameter, <i>physical</i> states must be left invariant by "<i>s</i>-evolution", and so the physical state space is the kernel of <i>H</i> − <i>E</i> (this requires the use of a <a href="http://en.wikipedia.org/wiki/Rigged_Hilbert_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Rigged Hilbert space">rigged Hilbert space</a> and a renormalization of the norm).</div>
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This is related to <a class="new" href="http://en.wikipedia.org/w/index.php?title=Quantization_of_constrained_systems&action=edit&redlink=1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Quantization of constrained systems (page does not exist)">quantization of constrained systems</a> and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Quantization_of_gauge_theories" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantization of gauge theories">quantization of gauge theories</a>. It is also possible to formulate a quantum theory of "events" where time becomes an observable (see D. Edwards).</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Spin">edit</a>]</span><span class="mw-headline" id="Spin">Spin</span></h3>
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In addition to their other properties all particles possess a quantity, which has no correspondence at all in conventional physics, namely the <a href="http://en.wikipedia.org/wiki/Spin_(physics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin (physics)">spin</a>, which is some kind of <i>intrinsic angular momentum</i> (therefore the name). In the position representation, instead of a wavefunction without spin, <img alt="\psi = \psi(\mathbf r)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/a/b/aab027478871020336caf5e08c97e684.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, one has with spin: <img alt="\psi =\psi(\mathbf r,\sigma)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/d/c/0dc526606191837c1beddcf1a8c9a523.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, where <img alt="\sigma " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/d/4/9d43cb8bbcb702e9d5943de477f099e2.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />belongs to the following discrete set of values</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\sigma \in\{ -S\cdot\hbar , -(S-1)\cdot\hbar , \dots ,+(S-1)\cdot\hbar ,+S\cdot\hbar\}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/3/c/83c57d54a6ced4ca39efa5bc25556ecd.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />.</dd></dl>
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One distinguishes <a href="http://en.wikipedia.org/wiki/Boson" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Boson">bosons</a> (<i>S</i> = 0 or 1 or 2 or ...) and <a href="http://en.wikipedia.org/wiki/Fermion" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermion">fermions</a> (<i>S</i> = 1/2 or 3/2 or 5/2 or ...)</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Pauli's principle">edit</a>]</span><span class="mw-headline" id="Pauli.27s_principle">Pauli's principle</span></h3>
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The property of spin relates to another basic property concerning systems of N identical particles: Pauli's <a href="http://en.wikipedia.org/wiki/Exclusion_principle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Exclusion principle">exclusion principle</a>, which is a consequence of the following permutation behaviour of an N-particle wave function; again in the position representation one must postulate that for the transposition of any two of the N particles one always should have</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\psi ( \,\dots\, ;\,\mathbf r_i,\sigma_i\,;\, \dots\,;\mathbf r_j,\sigma_j\,;\,\dots) \stackrel{!}{=}(-1)^{2S}\cdot \psi ( \,\dots\, ;\,\mathbf r_j,\sigma_j\,;\, \dots\,;\mathbf r_i,\sigma_i\,;\,\dots)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/0/8/f0866aef3667ebb4242961561bca4b9a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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i.e., on transposition of the arguments of any two particles the wavefunction should <i>reproduce</i>, apart from a prefactor (−1)<sup style="line-height: 1em;">2<i>S</i></sup> which is +1 for <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Bosons" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bosons">bosons</a>, but (−1) for <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Fermions" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermions">fermions</a>. Electrons are fermions with <i>S</i> = 1/2; quanta of light are bosons with <i>S</i> = 1. In nonrelativistic quantum mechanics all particles are either <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Bosons" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bosons">bosons</a> or <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Fermions" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermions">fermions</a>; in relativistic quantum theories also<a href="http://en.wikipedia.org/wiki/Supersymmetry" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Supersymmetry">"supersymmetric"</a> theories exist, where a particle is a linear combination of a bosonic and a fermionic part. Only in dimension <i>d=2</i> can one construct entities where <img alt="(-1)^{2S}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/7/3/f73814ba636f06f6926cfafebce4bc25.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is replaced by an arbitrary complex number with magnitude 1 ( -> <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Anyons" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Anyons">anyons</a>).</div>
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Although <i>spin</i> and the <i>Pauli principle</i> can only be derived from relativistic generalizations of quantum mechanics the properties mentioned in the last two paragraphs belong to the basic postulates already in the non-relativistic limit. Especially, many important properties in natural science, e.g. the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Periodic_system" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Periodic system">periodic system</a> of chemistry, are consequences of the two properties.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=12" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: The problem of measurement">edit</a>]</span><span class="mw-headline" id="The_problem_of_measurement">The problem of measurement</span></h2>
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The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is the effects of <a href="http://en.wikipedia.org/wiki/Measurement" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Measurement">measurement</a>.<sup class="reference" id="cite_ref-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#cite_note-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup> The von Neumann description of quantum measurement of an observable <i>A</i>, when the system is prepared in a pure state <i>ψ</i> is the following (note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time – more specifically the <a href="http://en.wikipedia.org/wiki/Compton_scattering" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Compton scattering">Compton–Simon experiment</a>; it is not applicable to most present-day measurements within the quantum domain):</div>
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<li style="margin-bottom: 0.1em;">Let <i>A</i> have spectral resolution</li>
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<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" A = \int \lambda \, d \operatorname{E}_A(\lambda)," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/f/0/8f05dd7bba569539cdf32505fe3de3e7.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where E<sub style="line-height: 1em;"><i>A</i></sub> is the resolution of the identity (also called <a href="http://en.wikipedia.org/wiki/Projection-valued_measure" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Projection-valued measure">projection-valued measure</a>) associated to <i>A</i>. Then the probability of the measurement outcome lying in an interval <i>B</i> of <b>R</b> is |E<sub style="line-height: 1em;"><i>A</i></sub>(<i>B</i>) <i>ψ</i>|<sup style="line-height: 1em;">2</sup>. In other words, the probability is obtained by integrating the characteristic function of <i>B</i> against the countably additive measure</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \langle \psi \mid \operatorname{E}_A \psi \rangle. " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/4/7/d471770e592ffc68382afc957ee25d6c.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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<li style="margin-bottom: 0.1em;">If the measured value is contained in <i>B</i>, then immediately after the measurement, the system will be in the (generally non-normalized) state E<sub style="line-height: 1em;"><i>A</i></sub>(<i>B</i>) <i>ψ</i>. If the measured value does not lie in <i>B</i>, replace <i>B</i> by its complement for the above state.</li>
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For example, suppose the state space is the <i>n</i>-dimensional complex Hilbert space <b>C</b><sup style="line-height: 1em;"><i>n</i></sup> and <i>A</i> is a Hermitian matrix with eigenvalues <i>λ</i><sub style="line-height: 1em;"><i>i</i></sub>, with corresponding eigenvectors <i>ψ</i><sub style="line-height: 1em;"><i>i</i></sub>. The projection-valued measure associated with <i>A</i>, E<sub style="line-height: 1em;"><i>A</i></sub>, is then</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \operatorname{E}_A (B) = | \psi_i\rangle \langle \psi_i|, " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/e/7/ce7120cbeaa241c797a110502b04a224.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <i>B</i> is a Borel set containing only the single eigenvalue <i>λ</i><sub style="line-height: 1em;"><i>i</i></sub>. If the system is prepared in state</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="| \psi \rangle \, " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/7/3/a73a4d80849ffd9c4d9dd45f1efaa998.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Then the probability of a measurement returning the value <i>λ</i><sub style="line-height: 1em;"><i>i</i></sub> can be calculated by integrating the spectral measure</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \langle \psi \mid \operatorname{E}_A \psi \rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/6/2/b629190fc1c1895c09882f7148eec32a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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over <i>B</i><sub style="line-height: 1em;"><i>i</i></sub>. This gives trivially</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \langle \psi| \psi_i\rangle \langle \psi_i \mid \psi \rangle = | \langle \psi \mid \psi_i\rangle | ^2. " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/2/6/d264e45593f74c0cd49e41d65a47cb19.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the <i>projection postulate</i>.</div>
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A more general formulation replaces the projection-valued measure with a <a href="http://en.wikipedia.org/wiki/POVM" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="POVM">positive-operator valued measure (POVM)</a>. To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" | \psi_i\rangle \langle \psi_i| \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/7/e/b7e759d8628ee430730e13a4ded8088e.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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by a finite set of positive operators</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" F_i F_i^* \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/4/e/b4ebe07e20bb8040180fef46ac8124e6.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes {<i>λ</i><sub style="line-height: 1em;">1</sub> ... <i>λ<sub style="line-height: 1em;">n</sub></i>} is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is <i>λ<sub style="line-height: 1em;">i</sub></i>. Instead of collapsing to the (unnormalized) state</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" | \psi_i\rangle \langle \psi_i |\psi\rangle \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/7/c/17c6e1f4945e8781a20004f8fd7b8685.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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after the measurement, the system now will be in the state</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" F_i |\psi\rangle. \, " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/4/7/c47e3912400a7048df96e760025c4e02.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Since the <i>F<sub style="line-height: 1em;">i</sub> F<sub style="line-height: 1em;">i</sub>*</i> 's need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds.</div>
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The same formulation applies to general <a href="http://en.wikipedia.org/wiki/Mixed_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mixed state">mixed states</a>.</div>
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In von Neumann's approach, the state transformation due to measurement is distinct from that due to <a href="http://en.wikipedia.org/wiki/Time_evolution" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Time evolution">time evolution</a> in several ways. For example, time evolution is deterministic and unitary whereas measurement is non-deterministic and non-unitary. However, since both types of state transformation take one quantum state to another, this difference was viewed by many as unsatisfactory. The POVM formalism views measurement as one among many other <a href="http://en.wikipedia.org/wiki/Quantum_operation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum operation">quantum operations</a>, which are described by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Completely_positive_map" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Completely positive map">completely positive maps</a> which do not increase the trace.</div>
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In any case it seems that the above-mentioned problems can only be resolved if the time evolution included not only the quantum system, but also, and essentially, the classical measurement apparatus (see above).</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=13" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: The relative state interpretation">edit</a>]</span><span class="mw-headline" id="The_relative_state_interpretation">The <i>relative state</i> interpretation</span></h3>
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An alternative interpretation of measurement is Everett's <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Relative_state_interpretation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Relative state interpretation">relative state interpretation</a>, which was later dubbed the "<a href="http://en.wikipedia.org/wiki/Many-worlds_interpretation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Many-worlds interpretation">many-worlds interpretation</a>" of quantum mechanics.</div>
<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=14" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: List of mathematical tools">edit</a>]</span><span class="mw-headline" id="List_of_mathematical_tools">List of mathematical tools</span></h2>
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Part of the <a href="http://en.wikipedia.org/wiki/Folklore" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Folklore">folklore</a> of the subject concerns the <a href="http://en.wikipedia.org/wiki/Mathematical_physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mathematical physics">mathematical physics</a> textbook <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Methods_of_Mathematical_Physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Methods of Mathematical Physics">Methods of Mathematical Physics</a> put together by <a href="http://en.wikipedia.org/wiki/Richard_Courant" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard Courant">Richard Courant</a> from <a href="http://en.wikipedia.org/wiki/David_Hilbert" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="David Hilbert">David Hilbert</a>'s <a class="mw-redirect" href="http://en.wikipedia.org/wiki/G%C3%B6ttingen_University" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Göttingen University">Göttingen University</a> courses. The story is told (by mathematicians) that physicists had dismissed the material as not interesting in the current research areas, until the advent of Schrödinger's equation. At that point it was realised that the mathematics of the new quantum mechanics was already laid out in it. It is also said that Heisenberg had consulted Hilbert about his<a href="http://en.wikipedia.org/wiki/Matrix_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Matrix mechanics">matrix mechanics</a>, and Hilbert observed that his own experience with infinite-dimensional matrices had derived from differential equations, advice which Heisenberg ignored, missing the opportunity to unify the theory as Weyl and Dirac did a few years later. Whatever the basis of the anecdotes, the mathematics of the theory was conventional at the time, whereas the physics was radically new.</div>
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The main tools include:</div>
<ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Linear_algebra" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear algebra">linear algebra</a>: <a href="http://en.wikipedia.org/wiki/Complex_number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Complex number">complex numbers</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Eigenvector" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eigenvector">eigenvectors</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Eigenvalue" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eigenvalue">eigenvalues</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Functional_analysis" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Functional analysis">functional analysis</a>: <a href="http://en.wikipedia.org/wiki/Hilbert_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hilbert space">Hilbert spaces</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Linear_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear operator">linear operators</a>, <a href="http://en.wikipedia.org/wiki/Spectral_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spectral theory">spectral theory</a></li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Differential_equations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Differential equations">differential equations</a>: <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Partial_differential_equations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Partial differential equations">partial differential equations</a>, <a href="http://en.wikipedia.org/wiki/Separation_of_variables" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Separation of variables">separation of variables</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Ordinary_differential_equations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Ordinary differential equations">ordinary differential equations</a>, <a href="http://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sturm–Liouville theory">Sturm–Liouville theory</a>, <a href="http://en.wikipedia.org/wiki/Eigenfunction" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eigenfunction">eigenfunctions</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Harmonic_analysis" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Harmonic analysis">harmonic analysis</a>: <a href="http://en.wikipedia.org/wiki/Fourier_transform" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fourier transform">Fourier transforms</a></li>
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See also: <a href="http://en.wikipedia.org/wiki/List_of_mathematical_topics_in_quantum_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="List of mathematical topics in quantum theory">list of mathematical topics in quantum theory</a>.</div>
<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=15" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: References">edit</a>]</span><span class="mw-headline" id="References">References</span></h2>
<ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Thomas_Samuel_Kuhn" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Thomas Samuel Kuhn">T.S. Kuhn</a>, <i><a href="http://en.wikipedia.org/wiki/Black-Body_Theory_and_the_Quantum_Discontinuity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Black-Body Theory and the Quantum Discontinuity">Black-Body Theory and the Quantum Discontinuity</a>, 1894–1912</i>, Clarendon Press, Oxford and Oxford University Press, New York, 1978.</li>
<li style="margin-bottom: 0.1em;">S. Auyang, <i>How is Quantum Field Theory Possible?</i>, Oxford University Press, 1995.</li>
<li style="margin-bottom: 0.1em;">D. Edwards, <i>The Mathematical Foundations of Quantum Mechanics</i>, Synthese, 42 (1979),pp. 1–70.</li>
<li style="margin-bottom: 0.1em;">G. Emch, <i>Algebraic Methods in Statistical Mechanics and Quantum Field Theory</i>, Wiley-Interscience, 1972.</li>
<li style="margin-bottom: 0.1em;">J.M. Jauch, <i>Foundations of quantum mechanics</i>, Addison-Wesley Publ. Cy., Reading, Mass., 1968.</li>
<li style="margin-bottom: 0.1em;">R. Jost, <i>The General Theory of Quantized Fields</i>, American Mathematical Society, 1965.</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Andrew_Gleason" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Andrew Gleason">A. Gleason</a>, <i>Measures on the Closed Subspaces of a Hilbert Space</i>, Journal of Mathematics and Mechanics, 1957.</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/George_Mackey" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="George Mackey">G. Mackey</a>, <i>Mathematical Foundations of Quantum Mechanics</i>, W. A. Benjamin, 1963 (paperback reprint by Dover 2004).</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/John_von_Neumann" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John von Neumann">J. von Neumann</a>, <i>Mathematical Foundations of Quantum Mechanics</i>, Princeton University Press, 1955. Reprinted in paperback form.</li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/R._F._Streater" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="R. F. Streater">R. F. Streater</a> and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/A._S._Wightman" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="A. S. Wightman">A. S. Wightman</a>, <i>PCT, Spin and Statistics and All That</i>, Benjamin 1964 (Reprinted by Princeton University Press)</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Michael_C._Reed" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Michael C. Reed">M. Reed</a> and <a href="http://en.wikipedia.org/wiki/Barry_Simon" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Barry Simon">B. Simon</a>, <i>Methods of Mathematical Physics</i>, vols I–IV, Academic Press 1972.</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Gerald_Teschl" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gerald Teschl">G. Teschl</a>, <i>Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators</i>, <a class="external free" href="http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/</a>, American Mathematical Society, 2009.</li>
<li style="margin-bottom: 0.1em;">N. Weaver, "Mathematical Quantization", Chapman & Hall/CRC 2001.</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Hermann_Weyl" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermann Weyl">H. Weyl</a>, <i>The Theory of Groups and Quantum Mechanics</i>, Dover Publications, 1950.</li>
</ul>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=16" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Notes">edit</a>]</span><span class="mw-headline" id="Notes">Notes</span></h2>
<ol class="references" style="background-color: white; font-family: sans-serif; font-size: 12px; line-height: 1.5em; list-style-image: none; margin-bottom: 0.5em; margin-left: 3.2em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li id="cite_note-0" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#cite_ref-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Frederick W. Byron, Robert W. Fuller; <a class="external text" href="http://books.google.com/books?id=D2Xs8NUKecAC&pg=PA277&lpg=PA277&dq=mathematical+formulation+of+quantum+mechanics&source=bl&ots=hV5VX7FfDj&sig=JuVbSojzBrKJ9MLKrEGGvqLn9SE&hl=en&ei=BTsvSq6CL6amM8PVhIcK&sa=X&oi=book_result&ct=result&resnum=8" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Mathematics of classical and quantum physics</a>; Courier Dover Publications, 1992.</span></li>
<li id="cite_note-1" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#cite_ref-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external free" href="http://books.google.com/books?id=5t0tm0FB1CsC&pg=PA215&lpg=PA215&dq=wave+function+collapse&source=bl&ots=a7iUGurRDC&sig=o1ddjY7lQrj4EQdvS49xcceWq2M&hl=en&ei=RfgtSsDNL4WgM8u-rf4J&sa=X&oi=book_result&ct=result&resnum=7#PPA215,M1" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://books.google.com/books?id=5t0tm0FB1CsC&pg=PA215&lpg=PA215&dq=wave+function+collapse&source=bl&ots=a7iUGurRDC&sig=o1ddjY7lQrj4EQdvS49xcceWq2M&hl=en&ei=RfgtSsDNL4WgM8u-rf4J&sa=X&oi=book_result&ct=result&resnum=7#PPA215,M1</a></span></li>
</ol>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com0tag:blogger.com,1999:blog-5303246073824127471.post-44332395110244842502012-05-03T09:36:00.001-04:002012-05-03T09:38:48.995-04:00Density Matrices and Density Operators<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiUxR19dVuwWD36d2TijVFT6deEDzDqaBXAzKQ4XPYNUDZRjCa2IU_DOsZ1N0U3jIPoeKansdMkjj6SvabzbddPwHuvnP6z2IBP5I0kuFxESEvm2bTG52APeqQkNrSClY1f6rpNy6FaQ/s1600/JohnvonNeumann-LosAlamos.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiUxR19dVuwWD36d2TijVFT6deEDzDqaBXAzKQ4XPYNUDZRjCa2IU_DOsZ1N0U3jIPoeKansdMkjj6SvabzbddPwHuvnP6z2IBP5I0kuFxESEvm2bTG52APeqQkNrSClY1f6rpNy6FaQ/s320/JohnvonNeumann-LosAlamos.gif" width="245" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">John von Neumann at Los Alamos in the 1940's</td></tr>
</tbody></table>
<span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px;">John von Neumann wrote: "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of </span><a href="http://en.wikipedia.org/wiki/Operator_theory" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 20px; text-decoration: none;" title="Operator theory">operator theory</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px;">, Berlin 1930 and Princeton 1935–1939; on the </span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Ergodic_theorem" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 20px; text-decoration: none;" title="Ergodic theorem">ergodic theorem</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px;">, Princeton, 1931–1932."</span><br />
<span style="background-color: white;"><br /><br /><span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 20px;">In Phyisics graduate school, 1st year 2nd semester, when they students take Quantum Mechanics, they are taught the important subject of Density Operators and Matrices. This may be over the head of most of you my dear readers, it is a bit over mine (simply because I haven't studied it in detail ... yet), but we're basically talking about linear algebra. At a minimum to understand this important subject that describes our real world, 5 semesters of College Math (Calculus I-IV + Transforms), Linear Algebra, and Wavefunctions should have been studied first. Then it's easy. :-)</span></span><br /><br /><span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 20px;">The operative sentence is this:</span></span><br /><br /><span style="font-family: sans-serif; font-size: x-small;"><span style="line-height: 20px;">"Just as the </span></span></span><a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 20px; text-decoration: none;" title="Schrödinger equation">Schrödinger equation</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px;"> describes how pure states evolve in time, the </span><i style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px;">von Neumann equation</i><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px;"> (also known as Liouville-von Neumann equation) describes how a density operator evolves in time. In fact, the two equations are equivalent, in the sense that either can be derived from the other."<br /><br />from Wikipedia:</span><br />
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A <b>density matrix</b> is a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Matrix_(math)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Matrix (math)">matrix</a> that describes a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Quantum_system" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum system">quantum system</a> in a <i>mixed state</i>, a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Statistical_ensemble" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Statistical ensemble">statistical ensemble</a> of several <a href="http://en.wikipedia.org/wiki/Quantum_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum state">quantum states</a>. (In contrast, a <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Pure_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pure state">pure state</a></i> is described by a single<a href="http://en.wikipedia.org/wiki/Quantum_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum state">state vector</a>). The density matrix is the quantum-mechanical analogue to a <a href="http://en.wikipedia.org/wiki/Phase_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Phase space">phase-space</a> <a href="http://en.wikipedia.org/wiki/Probability_measure" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability measure">probability measure</a> (probability distribution of position and momentum) in classical<a href="http://en.wikipedia.org/wiki/Statistical_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Statistical mechanics">statistical mechanics</a>.</div>
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Mixed states arise in situations where there is classical uncertainty, <i>i.e.</i>, when the experimenter does not know which particular states are being manipulated. (This should not be confused with <i>quantum</i> uncertainty, which dictates that even if the experimenter knows which states are being manipulated, the results of some measurements cannot be predicted.) Examples include a system in thermal equilibrium (at finite temperatures) or a system with an uncertain or randomly-varying preparation history (so one does not know which pure state the system is in). Also, if a quantum system has two or more subsystems that are <a href="http://en.wikipedia.org/wiki/Quantum_entanglement" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum entanglement">entangled</a>, then each subsystem must be treated as a mixed state even if the complete system is in a pure state. The density matrix is also a crucial tool in <a href="http://en.wikipedia.org/wiki/Quantum_decoherence" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum decoherence">quantum decoherence</a> theory.</div>
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The density matrix is a representation of a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Linear_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear operator">linear operator</a> called the <i>density operator</i>. (The close relationship between matrices and operators is a basic concept in <a href="http://en.wikipedia.org/wiki/Linear_algebra" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear algebra">linear algebra</a>.) In practice, the terms "density matrix" and "density operator" are often used interchangeably. Both matrix and operator are <a href="http://en.wikipedia.org/wiki/Self-adjoint" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Self-adjoint">self-adjoint</a> (or <a href="http://en.wikipedia.org/wiki/Hermitian_matrix" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermitian matrix">Hermitian</a>), <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Positive-semidefinite_matrix" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Positive-semidefinite matrix">positive semi-definite</a>, of <a href="http://en.wikipedia.org/wiki/Trace_class" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Trace class">trace</a>one, and may be <a href="http://en.wikipedia.org/wiki/Dimension_(vector_space)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dimension (vector space)">infinite-dimensional</a>.<sup class="reference" id="cite_ref-0" style="background-color: #ddeeff; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_note-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[1]</a></sup> The formalism was introduced by <a href="http://en.wikipedia.org/wiki/John_von_Neumann" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John von Neumann">John von Neumann</a><sup class="reference" id="cite_ref-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_note-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup> (and independently but less systematically by <a href="http://en.wikipedia.org/wiki/Lev_Landau" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lev Landau">Lev Landau</a> and <a href="http://en.wikipedia.org/wiki/Felix_Bloch" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Felix Bloch">Felix Bloch</a> in 1927).<sup class="reference" id="cite_ref-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_note-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup><sup class="reference" id="cite_ref-3" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_note-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[4]</a></sup><sup class="noprint Template-Fact" style="line-height: 1em; white-space: nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:No_original_research#Primary.2C_secondary_and_tertiary_sources" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:No original research"><span title="This claim needs references to reliable secondary sources. from April 2012">non-primary</span></a> source <a href="http://en.wikipedia.org/wiki/Wikipedia:Verifiability" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:Verifiability">needed</a></i>]</sup></div>
<table class="toc" id="toc" style="background-color: #f9f9f9; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; border-image: initial; border-left-color: rgb(170, 170, 170); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(170, 170, 170); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(170, 170, 170); border-top-style: solid; border-top-width: 1px; color: black; font-family: sans-serif; font-size: 12px; line-height: 20px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;"><tbody>
<tr><td><div id="toctitle" style="direction: ltr; text-align: center;">
<h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; display: inline; font-size: 12px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: auto;">
Contents</h2>
<span class="toctoggle" style="-webkit-user-select: none;"> [<a class="internal" href="http://en.wikipedia.org/wiki/Density_matrix#" id="togglelink" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">hide</a>] </span></div>
<ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;">
<li class="toclevel-1 tocsection-1" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#Pure_and_mixed_states" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1</span> <span class="toctext">Pure and mixed states</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-2" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#Example:_Light_polarization" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1.1</span> <span class="toctext">Example: Light polarization</span></a></li>
<li class="toclevel-2 tocsection-3" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#Mathematical_description" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1.2</span> <span class="toctext">Mathematical description</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-4" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#Formulation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2</span> <span class="toctext">Formulation</span></a></li>
<li class="toclevel-1 tocsection-5" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#Measurement" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3</span> <span class="toctext">Measurement</span></a></li>
<li class="toclevel-1 tocsection-6" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#Entropy" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4</span> <span class="toctext">Entropy</span></a></li>
<li class="toclevel-1 tocsection-7" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#The_Von_Neumann_equation_for_time_evolution" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5</span> <span class="toctext">The Von Neumann equation for time evolution</span></a></li>
<li class="toclevel-1 tocsection-8" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#.22Quantum_Liouville.22.2C_Moyal.27s_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6</span> <span class="toctext">"Quantum Liouville", Moyal's equation</span></a></li>
<li class="toclevel-1 tocsection-9" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#Composite_Systems" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">7</span> <span class="toctext">Composite Systems</span></a></li>
<li class="toclevel-1 tocsection-10" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#C.2A-algebraic_formulation_of_states" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">8</span> <span class="toctext">C*-algebraic formulation of states</span></a></li>
<li class="toclevel-1 tocsection-11" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#See_also" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">9</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-12" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#Notes_and_references" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">10</span> <span class="toctext">Notes and references</span></a></li>
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<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Density_matrix&action=edit&section=1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Pure and mixed states">edit</a>]</span><span class="mw-headline" id="Pure_and_mixed_states">Pure and mixed states</span></h2>
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In <a href="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#Mathematical_structure_of_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mathematical formulation of quantum mechanics">quantum mechanics</a>, a quantum system is represented by a <a href="http://en.wikipedia.org/wiki/Quantum_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum state">state vector</a> (or <a href="http://en.wikipedia.org/wiki/Bra-ket_notation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bra-ket notation">ket</a>) <img alt="| \psi \rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/0/5/605b44d27aad3e8e841b1dd43053faa1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. A quantum system with a state vector <img alt="| \psi \rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/0/5/605b44d27aad3e8e841b1dd43053faa1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is called a <i>pure state</i>. However, it is also possible for a system to be in a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Statistical_ensemble" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Statistical ensemble">statistical ensemble</a> of different state vectors: For example, there may be a 50% probability that the state vector is <img alt="| \psi_1 \rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/4/8/c48a5522f4dcc87b53b677de09bb0827.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> and a 50% chance that the state vector is <img alt="| \psi_2 \rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/b/3/eb3d893757e666569a11f393543c947b.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. This system would be in a <i>mixed state</i>. The density matrix is especially useful for mixed states, because any state, pure or mixed, can be characterized by a single density matrix.</div>
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A mixed state is different from a <a href="http://en.wikipedia.org/wiki/Quantum_superposition" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum superposition">quantum superposition</a>. In fact, a quantum superposition of pure states is another pure state, for example <img alt="| \psi \rangle = (| \psi_1 \rangle + | \psi_2 \rangle)/\sqrt{2} " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/9/8/69864777f02e5bcb8658f6de3f85316c.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />.</div>
<h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 18px; line-height: 20px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Density_matrix&action=edit&section=2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Example: Light polarization">edit</a>]</span><span class="mw-headline" id="Example:_Light_polarization">Example: Light polarization</span></h3>
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An example of pure and mixed states is <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Light_polarization" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Light polarization">light polarization</a>. Photons can have two <a href="http://en.wikipedia.org/wiki/Circular_polarization" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Circular polarization">helicities</a>, corresponding to two orthogonal quantum states, <img alt="|R\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/8/6/b8647e6d08cac85d169b1c8a91cbc949.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> (right <a href="http://en.wikipedia.org/wiki/Circular_polarization" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Circular polarization">circular polarization</a>) and <img alt="|L\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/c/c/6cc36dc85e3cbb0762f53959a43a0385.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />(left <a href="http://en.wikipedia.org/wiki/Circular_polarization" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Circular polarization">circular polarization</a>). A photon can also be in a superposition state, such as <img alt="(|R\rangle+|L\rangle)/\sqrt{2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/6/7/767aeac274a0e474405d699bd0019197.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> (vertical polarization) or <img alt="(|R\rangle-|L\rangle)/\sqrt{2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/6/4/164497dde9b0f9474a45086cbc826a58.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> (horizontal polarization). More generally, it can be in any state <img alt="\alpha|R\rangle+\beta|L\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/2/e/32ee9f65815dc8e2d2d1f6fcec737ad3.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, corresponding to <a href="http://en.wikipedia.org/wiki/Linear_polarization" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear polarization">linear</a>, <a href="http://en.wikipedia.org/wiki/Circular_polarization" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Circular polarization">circular</a>, or <a href="http://en.wikipedia.org/wiki/Elliptical_polarization" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Elliptical polarization">elliptical polarization</a>. If we pass <img alt="(|R\rangle+|L\rangle)/\sqrt{2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/6/7/767aeac274a0e474405d699bd0019197.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> polarized light through a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Circular_polarizer" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Circular polarizer">circular polarizer</a> which allows either only <img alt="|R\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/8/6/b8647e6d08cac85d169b1c8a91cbc949.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> polarized light, or only <img alt="|L\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/c/c/6cc36dc85e3cbb0762f53959a43a0385.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> polarized light, intensity would be reduced by half in both cases. This may make it <i>seem</i> like half of the photons are in state <img alt="|R\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/8/6/b8647e6d08cac85d169b1c8a91cbc949.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> and the other half in state <img alt="|L\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/c/c/6cc36dc85e3cbb0762f53959a43a0385.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. But this is not correct: Both <img alt="|R\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/8/6/b8647e6d08cac85d169b1c8a91cbc949.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> and <img alt="|L\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/c/c/6cc36dc85e3cbb0762f53959a43a0385.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> photons are partly absorbed by a vertical <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Linear_polarizer" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear polarizer">linear polarizer</a>, but the <img alt="(|R\rangle+|L\rangle)/\sqrt{2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/6/7/767aeac274a0e474405d699bd0019197.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> light will pass through that polarizer with no absorption whatsoever.</div>
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However, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Unpolarized_light" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Unpolarized light">unpolarized light</a> (such as the light from an <a href="http://en.wikipedia.org/wiki/Incandescent_light_bulb" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Incandescent light bulb">incandescent light bulb</a>) is different from any state like <img alt="\alpha|R\rangle+\beta|L\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/2/e/32ee9f65815dc8e2d2d1f6fcec737ad3.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> (linear, circular, or elliptical polarization). Unlike linearly or elliptically polarized light, it passes through a polarizer with 50% intensity loss whatever the orientation of the polarizer; and unlike circularly polarized light, it cannot be made linearly polarized with any <a href="http://en.wikipedia.org/wiki/Wave_plate" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wave plate">wave plate</a>. Indeed, unpolarized light cannot be described as <i>any</i> state of the form <img alt="\alpha|R\rangle+\beta|L\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/2/e/32ee9f65815dc8e2d2d1f6fcec737ad3.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. However, unpolarized light <i>can</i> be described perfectly by assuming that each photon is either <img alt="| R \rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/8/6/b8647e6d08cac85d169b1c8a91cbc949.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> with 50% probability or <img alt="| L \rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/c/c/6cc36dc85e3cbb0762f53959a43a0385.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> with 50% probability. The same behavior would occur if each photon was either vertically polarized with 50% probability or horizontally polarized with 50% probability.</div>
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Therefore, unpolarized light cannot be described by any pure state, but can be described as a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Statistical_ensemble" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Statistical ensemble">statistical ensemble</a> of pure states in at least two ways (the ensemble of half left and half right circularly polarized, or the ensemble of half vertically and half horizontally linearly polarized). These two ensembles are completely indistinguishable experimentally, and therefore they are considered the same mixed state. One of the advantages of the density matrix is that there is just one density matrix for each mixed state, whereas there are many statistical ensembles of pure states for each mixed state. Nevertheless, the density matrix contains all the information necessary to calculate any measurable property of the mixed state.</div>
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Where do mixed states come from? To answer that, consider how to generate unpolarized light. One way is to use a system in <a href="http://en.wikipedia.org/wiki/Thermal_equilibrium" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Thermal equilibrium">thermal equilibrium</a>, a statistical mixture of enormous numbers of <a href="http://en.wikipedia.org/wiki/Microstate_(statistical_mechanics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Microstate (statistical mechanics)">microstates</a>, each with a certain probability (the <a href="http://en.wikipedia.org/wiki/Boltzmann_factor" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Boltzmann factor">Boltzmann factor</a>), switching rapidly from one to the next due to <a href="http://en.wikipedia.org/wiki/Thermal_fluctuations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Thermal fluctuations">thermal fluctuations</a>. Thermal randomness explains why an <a href="http://en.wikipedia.org/wiki/Incandescent_light_bulb" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Incandescent light bulb">incandescent light bulb</a>, for example, emits unpolarized light. A second way to generate unpolarized light is to introduce uncertainty in the preparation of the system, for example, passing it through a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Birefringent_crystal" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Birefringent crystal">birefringent crystal</a> with a rough surface, so that slightly different parts of the beam acquire different polarizations. A third way to generate unpolarized light uses an <a href="http://en.wikipedia.org/wiki/EPR_paradox" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="EPR paradox">EPR</a> setup: A radioactive decay can emit two photons traveling in opposite directions, in the quantum state <img alt="(|R,L\rangle+|L,R\rangle)/\sqrt{2}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/e/9/3e929ce6bb06579b20338dacdb85dc25.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. The two photons <i>together</i> are in a pure state, but if you only look at one of the photons and ignore the other, the photon behaves just like unpolarized light.</div>
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More generally, mixed states commonly arise from a statistical mixture of the starting state (such as in thermal equilibrium), from uncertainty in the preparation procedure (such as slightly different paths that a photon can travel), or from looking at a subsystem entangled with something else.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Density_matrix&action=edit&section=3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Mathematical description">edit</a>]</span><span class="mw-headline" id="Mathematical_description">Mathematical description</span></h3>
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The <a href="http://en.wikipedia.org/wiki/Quantum_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum state">state vector</a> <img alt="| \psi \rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/0/5/605b44d27aad3e8e841b1dd43053faa1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> of a pure state completely determines the statistical behavior of a measurement. As an example, take an observable quantity, and let <i>A</i> be the associated<a href="http://en.wikipedia.org/wiki/Observable" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Observable">observable operator</a> that has a representation on the <a href="http://en.wikipedia.org/wiki/Hilbert_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hilbert space">Hilbert space</a> <img alt=" \mathcal{H} " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/5/f/35fbfe11f730130acc2af6d8d8f69056.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> of the quantum system. For any real-valued function <i>F</i> defined on the real numbers,<sup class="reference" id="cite_ref-4" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_note-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[5]</a></sup> suppose that <i>F</i>(<i>A</i>) is the result of applying <i>F</i> to the outcome of a measurement. The expectation value of <i>F</i>(<i>A</i>) is</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \langle \psi | F(A) | \psi \rangle\, . " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/a/0/5a06eaad561904561e877c910c89f6f5.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Now consider a mixed state prepared by statistically combining two different pure states <img alt=" | \psi \rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/0/5/605b44d27aad3e8e841b1dd43053faa1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> and <img alt=" |\phi\rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/4/7/647e7b8a3252e5f6eac8022f9ca5de17.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, with the associated probabilities <i>p</i> and <span class="nowrap" style="white-space: nowrap;">1 − <i>p</i></span>, respectively. The associated probabilities mean that the preparation process for the quantum system ends in the state <img alt="|\psi\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/0/5/605b44d27aad3e8e841b1dd43053faa1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> with probability <i>p</i> and in the state <img alt=" |\phi\rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/4/7/647e7b8a3252e5f6eac8022f9ca5de17.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> with probability <span class="nowrap" style="white-space: nowrap;">1 − <i>p</i></span>.</div>
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It is not hard to show that the statistical properties of the observable for the system prepared in such a mixed state are completely determined. However, there is no state vector <img alt=" |\xi\rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/2/6/526cbba1f251e3a248cc15bceea813ea.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />which determines this statistical behavior in the sense that the expectation value of <i>F</i>(<i>A</i>) is</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \langle \xi | F(A) | \xi \rangle \, . " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/b/0/5b094eefe5bec71d56542ef5005d957b.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Nevertheless, there <i>is</i> a unique operator <i>ρ</i> such that the expectation value of <i>F(A)</i> can be written as</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \operatorname{tr}[\rho F(A)]\, , " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/b/a/0ba49f7ef91bff0a8be88ef8b3d53d30.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where the operator <i>ρ</i> is the density operator of the mixed system. A simple calculation shows that the operator <i>ρ</i> for the above example is given by</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \rho = p | \psi\rangle \langle \psi | + (1-p) | \phi\rangle \langle \phi |\,. " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/8/6/c860f24021a626b4801a2e0317c38c33.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Density_matrix&action=edit&section=4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Formulation">edit</a>]</span><span class="mw-headline" id="Formulation">Formulation</span></h2>
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For a finite dimensional function space, the most general density operator is of the form</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \rho = \sum_j p_j |\psi_j \rang \lang \psi_j| " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/1/d/71d494438dcd981d6964c55e5328f837.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where the coefficients <i>p</i><sub style="line-height: 1em;"><i>j</i></sub> are non-negative and add up to one. This represents a statistical mixture of pure states. If the given system is closed, then one can think of a mixed state as representing a single system with an uncertain preparation history, as explicitly detailed above; <i>or</i> we can regard the mixed state as representing an <a href="http://en.wikipedia.org/wiki/Statistical_ensemble_(mathematical_physics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Statistical ensemble (mathematical physics)">ensemble</a> of systems, i.e. large number of copies of the system in question, where <i>p</i><sub style="line-height: 1em;"><i>j</i></sub> is the proportion of the ensemble being in the state <img alt="\textstyle |\psi_j \rang " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/a/c/7ac08775f2ab8da98a3fddcc8fa4037b.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. An ensemble is described by a pure state if every copy of the system in that ensemble is in the same state, i.e. it is a <i>pure ensemble</i>. If the system is not closed, however, then it is simply not correct to claim that it has some definite but unknown state vector, as the density operator may record physical entanglements to other systems.</div>
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Consider a quantum ensemble of size <i>N</i> with occupancy numbers <i>n</i><sub style="line-height: 1em;">1</sub>, <i>n</i><sub style="line-height: 1em;">2</sub>,...,<i>n<sub style="line-height: 1em;">k</sub></i> corresponding to the orthonormal states <img alt="\textstyle |1\rang,...,|k\rang" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/6/3/063e3a51b64358d5820bfa9f1ffc22e3.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, respectively, where <i>n</i><sub style="line-height: 1em;">1</sub>+...+<i>n<sub style="line-height: 1em;">k</sub></i> = <i>N</i>, and, thus, the coefficients <i>p<sub style="line-height: 1em;">j</sub></i> = <i>n<sub style="line-height: 1em;">j</sub></i> /<i>N</i>. For a pure ensemble, where all <i>N</i> particles are in state <img alt="\textstyle |i\rang " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/5/7/c57477e595fd64142442654f17f01ef9.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, we have <i>n<sub style="line-height: 1em;">j</sub></i> = 0, for all <i>j</i> ≠ <i>i</i>, from which we recover the corresponding density operator <img alt="\textstyle\rho = |i\rang\lang i|" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/6/7/f67e8d0544cd26b387383c3a298a21fc.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. However, the density operator of a mixed state does not capture all the information about a mixture; in particular, the coefficients <i>p</i><sub style="line-height: 1em;"><i>j</i></sub> and the kets ψ<sub style="line-height: 1em;"><i>j</i></sub> are not recoverable from the operator ρ without additional information. This non-uniqueness implies that different ensembles or mixtures may correspond to the same density operator. Such equivalent ensembles or mixtures cannot be distinguished by measurement of observables alone. This equivalence can be characterized precisely. Two ensembles ψ, ψ' define the same density operator <a href="http://en.wikipedia.org/wiki/If_and_only_if" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="If and only if">if and only if</a> there is a matrix U with</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" U^*U=I" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/7/2/e727ef01d33eebc497c93b1d58e339f1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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i.e., U is <a href="http://en.wikipedia.org/wiki/Unitary_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Unitary operator">unitary</a> and such that</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" | \psi_i'\rangle \sqrt {p_i'} = \sum_{j} u_{ij} | \psi_j\rangle \sqrt {p_j}." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/7/7/f77df1840156d3c635b23caf5b712862.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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This is simply a restatement of the following fact from linear algebra: for two square matrices <i>M</i> and <i>N</i>, <i>M M</i><sup style="line-height: 1em;">*</sup> = <i>N N</i><sup style="line-height: 1em;">*</sup> if and only if <i>M</i> = <i>NU</i> for some unitary <i>U</i>. (See <a href="http://en.wikipedia.org/wiki/Square_root_of_a_matrix" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Square root of a matrix">square root of a matrix</a> for more details.) Thus there is a unitary freedom in the ket mixture or ensemble that gives the same density operator. However if the kets in the mixture are <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Orthonormal" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Orthonormal">orthonormal</a> then the original probabilities <i>p</i><sub style="line-height: 1em;"><i>j</i></sub> are recoverable as the eigenvalues of the density matrix.</div>
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In operator language, a density operator is a <a href="http://en.wikipedia.org/wiki/Positive_semidefinite" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Positive semidefinite">positive semidefinite</a>, <a href="http://en.wikipedia.org/wiki/Hermitian" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermitian">hermitian</a> operator of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Trace_class_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Trace class operator">trace 1</a> acting on the state space. A density operator describes a <a href="http://en.wikipedia.org/wiki/Purity_(quantum_mechanics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Purity (quantum mechanics)">pure</a> state if it is a rank one projection. Equivalently, a density operator ρ is a <a href="http://en.wikipedia.org/wiki/Purity_(quantum_mechanics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Purity (quantum mechanics)">pure</a> state if and only if</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\; \rho = \rho^2" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/e/8/ce85a104c67909630a192dc80e6d1f21.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />,</dd></dl>
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i.e. the state is <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Idempotent" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Idempotent">idempotent</a>. This is true regardless of whether <i>H</i> is finite dimensional or not.</div>
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Geometrically, when the state is not expressible as a <a href="http://en.wikipedia.org/wiki/Convex_combination" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Convex combination">convex combination</a> of other states, it is a pure state. The family of mixed states is a convex set and a state is pure if it is an<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Extremal_point" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Extremal point">extremal point</a> of that set.</div>
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It follows from the <a href="http://en.wikipedia.org/wiki/Compact_operator_on_Hilbert_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Compact operator on Hilbert space">spectral theorem for compact self-adjoint operators</a> that every mixed state is an infinite convex combination of pure states. This representation is not unique. Furthermore, a theorem of <a href="http://en.wikipedia.org/wiki/Andrew_Gleason" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Andrew Gleason">Andrew Gleason</a> states that certain functions defined on the family of projections and taking values in [0,1] (which can be regarded as quantum analogues of probability measures) are determined by unique mixed states. See <a href="http://en.wikipedia.org/wiki/Quantum_logic#Statistical_structure" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum logic">quantum logic</a> for more details.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Density_matrix&action=edit&section=5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Measurement">edit</a>]</span><span class="mw-headline" id="Measurement">Measurement</span></h2>
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Let <i>A</i> be an <a href="http://en.wikipedia.org/wiki/Observable" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Observable">observable</a> of the system, and suppose the ensemble is in a mixed state such that each of the pure states <img alt="\textstyle |\psi_j\rang" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/a/c/7ac08775f2ab8da98a3fddcc8fa4037b.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> occurs with probability <i>p<sub style="line-height: 1em;">j</sub></i>. Then the corresponding density operator is:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\rho = \sum_j p_j |\psi_j \rang \lang \psi_j| ." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/e/d/6ed7cdfd576a96211ae699a38b78c816.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The <a href="http://en.wikipedia.org/wiki/Expectation_value_(quantum_mechanics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Expectation value (quantum mechanics)">expectation value</a> of the measurement can be calculated by extending from the case of pure states (see <a href="http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Measurement in quantum mechanics">Measurement in quantum mechanics</a>):</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \lang A \rang = \sum_j p_j \lang \psi_j|A|\psi_j \rang = \operatorname{tr}[\rho A]," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/4/b/54b895065c5f933fc2313b8ccdceead8.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where tr denotes <a href="http://en.wikipedia.org/wiki/Trace_(linear_algebra)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Trace (linear algebra)">trace</a>. Moreover, if <i>A</i> has spectral resolution</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="A = \sum_i a_i |a_i \rang \lang a_i| = \sum _i a_i P_i," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/2/8/c28abd6a257179d33251db53c6cd471d.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <img alt="P_i = |a_i \rang \lang a_i|" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/b/9/db937a08e539a8421fab4f1f00411222.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, the corresponding density operator after the measurement is given by:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\; \rho ^' = \sum_i P_i \rho P_i." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/6/e/76ee0109c6c81f15a213c1c6b6671ce7.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Note that the above density operator describes the full ensemble after measurement. The sub-ensemble for which the measurement result was the particular value <i>a<sub style="line-height: 1em;">i</sub></i> is described by the different density operator</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\rho_i' = \frac{P_i \rho P_i}{\operatorname{tr}[\rho P_i]}." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/a/f/eafe954839ed578d94ad55114bf7347e.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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This is true assuming that <img alt="\textstyle |a_i\rang" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/c/8/fc86fc283f133def086fa6c39dd2a757.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is the only eigenket (up to <a href="http://en.wikipedia.org/wiki/Phase_factor" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Phase factor">phase</a>) with <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Eigenvalue" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eigenvalue">eigenvalue</a> <i>a<sub style="line-height: 1em;">i</sub></i>; more generally, <i>P<sub style="line-height: 1em;">i</sub></i> in this expression would be replaced by the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Projection_operator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Projection operator">projection operator</a> into the<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Eigenspace" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eigenspace">eigen<i>space</i></a> corresponding to eigenvalue <i>a<sub style="line-height: 1em;">i</sub></i>.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Density_matrix&action=edit&section=6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Entropy">edit</a>]</span><span class="mw-headline" id="Entropy">Entropy</span></h2>
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The <a href="http://en.wikipedia.org/wiki/Von_Neumann_entropy" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Von Neumann entropy">von Neumann entropy</a> <img alt="S" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/d/b/5dbc98dcc983a70728bd082d1a47546e.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> of a mixture can be expressed in terms of the eigenvalues of <img alt="\rho" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/7/f/f7f177957cf064a93e9811df8fe65ed1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> or in terms of the <a href="http://en.wikipedia.org/wiki/Trace_(linear_algebra)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Trace (linear algebra)">trace</a> and <a href="http://en.wikipedia.org/wiki/Logarithm" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Logarithm">logarithm</a> of the density operator <img alt="\rho" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/7/f/f7f177957cf064a93e9811df8fe65ed1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. Since <img alt=" \rho " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/7/f/f7f177957cf064a93e9811df8fe65ed1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is a positive semi-definite operator, it has a spectral decomposition such that <img alt=" \rho= \sum_i \lambda_i |\varphi_i\rangle\langle\varphi_i| " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/d/8/ad8ab9e5b90c6ae0bbb6e128130de780.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> where <img alt=" |\varphi_i\rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/0/e/10e3a8b61b6c87fcf6114b5b122f1ce4.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> are orthonormal vectors. Therefore the entropy of a quantum system with density matrix <img alt=" \rho " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/7/f/f7f177957cf064a93e9811df8fe65ed1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="S = -\sum_i \lambda_i \ln \,\lambda_i = -\operatorname{tr}(\rho \ln \rho)\quad. " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/9/1/69179e3185c9e32d4f9ccfc50174db1a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Also it can be shown that</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="S\left(\rho=\sum_i p_i\rho_i\right)= H(p_i) + \sum_i p_iS(\rho_i)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/5/e/c5ea4d944eb5bbc138cf940e5c2b243b.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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when <img alt="\rho_i" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/5/c/c/5cc7498d7c9bb56d212889327004b54d.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> have orthogonal support, where <img alt="H(p)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/6/b/46bbb752c0f72ea66e5206edb12b5e78.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Shannon_entropy" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Shannon entropy">Shannon entropy</a>. This entropy can increase but never decrease with a projective measurement, however generalised measurements can decrease entropy <sup class="reference" id="cite_ref-5" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_note-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[6]</a></sup><sup class="reference" id="cite_ref-everett56_6-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_note-everett56-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[7]</a></sup>. The entropy of a pure state is zero, while that of a proper mixture always greater than zero. Therefore a pure state may be converted into a mixture by a measurement, but a proper mixture can <i>never</i> be converted into a pure state. Thus the act of measurement induces a fundamental <a href="http://en.wikipedia.org/wiki/Irreversible" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Irreversible">irreversible</a> change on the density matrix; this is analogous to the "collapse" of the state vector, or <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Wavefunction_collapse" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wavefunction collapse">wavefunction collapse</a>. Perhaps counterintuitively, the measurement actually <i>decreases</i> information by erasing quantum interference in the composite system—cf. <a href="http://en.wikipedia.org/wiki/Quantum_entanglement" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum entanglement">quantum entanglement</a> and <a href="http://en.wikipedia.org/wiki/Quantum_decoherence" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum decoherence">quantum decoherence</a>.</div>
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(A subsystem of a larger system can be turned from a mixed to a pure state, but only by increasing the von Neumann entropy elsewhere in the system. This is analogous to how the entropy of an object can be lowered by putting it in a refrigerator: The air outside the refrigerator's heat-exchanger warms up, gaining even more entropy than was lost by the object in the refrigerator. See <a href="http://en.wikipedia.org/wiki/Second_law_of_thermodynamics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Second law of thermodynamics">second law of thermodynamics</a>. See <a href="http://en.wikipedia.org/wiki/Entropy_in_thermodynamics_and_information_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Entropy in thermodynamics and information theory">Entropy in thermodynamics and information theory</a>.)</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Density_matrix&action=edit&section=7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: The Von Neumann equation for time evolution">edit</a>]</span><span class="mw-headline" id="The_Von_Neumann_equation_for_time_evolution">The Von Neumann equation for time evolution</span></h2>
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Just as the <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schrödinger equation">Schrödinger equation</a> describes how pure states evolve in time, the <i>von Neumann equation</i> (also known as Liouville-von Neumann equation) describes how a density operator evolves in time (in fact, the two equations are equivalent, in the sense that either can be derived from the other.) The von Neumann equation dictates that<sup class="reference" id="cite_ref-7" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_note-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[8]</a></sup><sup class="reference" id="cite_ref-8" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_note-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[9]</a></sup></div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" i \hbar \frac{\partial \rho}{\partial t} = [H,\rho]~, " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/a/4/ba406e6d3bbcdab2d096cbcd6e768cf8.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where the brackets denote a <a href="http://en.wikipedia.org/wiki/Commutator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Commutator">commutator</a>.</div>
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Note that this equation only holds when the density operator is taken to be in the <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_picture" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schrödinger picture">Schrödinger picture</a>, even though this equation seems at first look to emulate the Heisenberg equation of motion in the <a href="http://en.wikipedia.org/wiki/Heisenberg_picture" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Heisenberg picture">Heisenberg picture</a>, with a crucial sign difference:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \frac{dA^{(H)}}{dt}=-\frac{i}{\hbar}[A^{(H)},H] ~," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/6/7/067852fb376dafaf224d3708886e96e1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <img alt="A^{(H)}(t)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/5/f/d5f0e7d82701be4c561ad94d29111168.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is some <i>Heisenberg picture</i> operator; but in this picture the density matrix is <i>not time-dependent</i>, and the relative sign ensures that the time derivative of the expected value <img alt="\langle A \rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/a/b/fabf0d63904baffb0abd7e5b1b731b52.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> comes out <i>the same as in the Schrödinger picture</i>.</div>
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Taking the density operator to be in the Schrödinger picture makes sense, since it is composed of 'Schrödinger' kets and bras evolved in time, as per the Schrödinger picture. If the Hamiltonian is time-independent, this differential equation can be easily solved to yield</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\rho(t) = e^{-i H t/\hbar} \rho(0) e^{i H t/\hbar}." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/2/9/3296ca086859d5e5d689e1466b88714d.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Density_matrix&action=edit&section=8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: "Quantum Liouville", Moyal's equation">edit</a>]</span><span class="mw-headline" id=".22Quantum_Liouville.22.2C_Moyal.27s_equation">"Quantum Liouville", Moyal's equation</span></h2>
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The density matrix operator may also be realized in <a href="http://en.wikipedia.org/wiki/Phase_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Phase space">phase space</a>. Under the <a href="http://en.wikipedia.org/wiki/Wigner_quasi-probability_distribution#The_Wigner.E2.80.93Weyl_transformation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wigner quasi-probability distribution">Wigner map</a>, the density matrix transforms into the equivalent <a href="http://en.wikipedia.org/wiki/Wigner_quasi-probability_distribution" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wigner quasi-probability distribution">Wigner function</a>,</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" W(x,p)\stackrel{\mathrm{def}}{=}\frac{1}{\pi\hbar}\int_{-\infty}^\infty \psi^*(x+y)\psi(x-y)e^{2ipy/\hbar}\,dy ~." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/e/1/3e119d85dc34147f51a623dcb9fe2ee8.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The equation for the time-evolution of the Wigner function is then the Wigner-transform of the above von Neumann equation,</div>
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<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\frac{\partial W(q,p,t)}{\partial t} = -\{\{W(q,p,t) , H(q,p )\}\}~," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/4/c/c4c3a1aa35ac5ba01cdf7b9ecc35568b.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <i>H(q,p)</i> is the Hamiltonian, and { { •,• } } is the <a href="http://en.wikipedia.org/wiki/Moyal_bracket" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Moyal bracket">Moyal bracket</a>, the transform of the quantum <a href="http://en.wikipedia.org/wiki/Commutator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Commutator">commutator</a>.</div>
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The evolution equation for the Wigner function is then analogous to that of its classical limit, the <a href="http://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian)#Liouville_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Liouville's theorem (Hamiltonian)">Liouville equation</a> of <a href="http://en.wikipedia.org/wiki/Classical_physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Classical physics">classical physics</a>. In the limit of vanishing Planck's constant ħ,<i>W(q,p,t)</i> reduces to the classical Liouville probability density function in <a href="http://en.wikipedia.org/wiki/Phase_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Phase space">phase space</a>.</div>
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The classical Liouville equation can be solved using the <a href="http://en.wikipedia.org/wiki/Method_of_characteristics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Method of characteristics">method of characteristics</a> for partial differential equations, the characteristic equations being Hamilton's equations. The Moyal equation in quantum mechanics similarly admits formal solutions in terms of <a href="http://en.wikipedia.org/wiki/Method_of_quantum_characteristics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Method of quantum characteristics">quantum characteristics</a>, predicated on the <a href="http://en.wikipedia.org/wiki/Moyal_product" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Moyal product">∗−product</a> of phase space, although, in actual practice, solution-seeking follows different methods.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Density_matrix&action=edit&section=9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Composite Systems">edit</a>]</span><span class="mw-headline" id="Composite_Systems">Composite Systems</span></h2>
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The joint density matrix of a composite system of two systems A and B is described by <img alt=" \rho_{AB} " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/5/9/259ec796a26c40be46a28e92657227b0.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. Then the subsystems are described by their <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Reduced_density_matrix#Reduced_density_matrices" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Reduced density matrix">reduced density operator</a>.</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\rho_A=\operatorname{tr}_B\rho_{AB}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/f/d/1fdfcd48777be63de07c8d411e698b1d.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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<img alt="\operatorname{tr}_B" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/6/8/368a4facf84c034afcfa805b4dfcb1b4.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is called <i><a href="http://en.wikipedia.org/wiki/Partial_trace" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Partial trace">partial trace</a></i> over system B. If A and B are two distinct and independent systems then <img alt="\rho_{AB}=\rho_{A}\otimes\rho_{B}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/f/1/0f1254b1f947289c5a199d66dc07fe70.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> which is a <i>product state</i>.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Density_matrix&action=edit&section=10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: C*-algebraic formulation of states">edit</a>]</span><span class="mw-headline" id="C.2A-algebraic_formulation_of_states">C*-algebraic formulation of states</span></h2>
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It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable.<sup class="reference" id="cite_ref-9" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_note-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[10]</a></sup><sup class="reference" id="cite_ref-10" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_note-10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[11]</a></sup> For this reason, observables are identified to elements of an abstract <a href="http://en.wikipedia.org/wiki/C*-algebra" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="C*-algebra">C*-algebra</a> <i>A</i> (that is one without a distinguished representation as an algebra of operators) and <a href="http://en.wikipedia.org/wiki/State_(functional_analysis)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="State (functional analysis)">states</a> are positive <a href="http://en.wikipedia.org/wiki/Linear_functional" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear functional">linear functionals</a> on <i>A</i>. However, by using the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/GNS_construction" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="GNS construction">GNS construction</a>, we can recover Hilbert spaces which realize <i>A</i> as a subalgebra of operators.</div>
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Geometrically, a pure state on a C*-algebra <i>A</i> is a state which is an extreme point of the set of all states on <i>A</i>. By properties of the GNS construction these states correspond to<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Irreducible_representation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Irreducible representation">irreducible representations</a> of <i>A</i>.</div>
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The states of the C*-algebra of compact operators <i>K</i>(<i>H</i>) correspond exactly to the density operators and therefore the pure states of <i>K</i>(<i>H</i>) are exactly the pure states in the sense of quantum mechanics.</div>
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The C*-algebraic formulation can be seen to include both classical and quantum systems. When the system is classical, the algebra of observables become an abelian C*-algebra. In that case the states become probability measures, as noted in the introduction.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Density_matrix&action=edit&section=11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: See also">edit</a>]</span><span class="mw-headline" id="See_also">See also</span></h2>
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<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Fluctuation_theorem" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fluctuation theorem">Fluctuation theorem</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Density_functional_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Density functional theory">Density functional theory</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Dynamic_nuclear_polarisation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dynamic nuclear polarisation">Dynamic nuclear polarisation</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Linear_response_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear response function">Linear response function</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Green%27s_function_(many-body_theory)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Green's function (many-body theory)">Green's function (many-body theory)</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Lindblad_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lindblad equation">Lindblad equation</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Green%E2%80%93Kubo_relations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Green–Kubo relations">Green–Kubo relations</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Purification_of_quantum_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Purification of quantum state">Purification of quantum state</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/POVM" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="POVM">POVM</a> Generalized measurement of density states</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wigner_quasi-probability_distribution" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wigner quasi-probability distribution">Wigner quasi-probability distribution</a></li>
</ul>
</div>
<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Density_matrix&action=edit&section=12" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Notes and references">edit</a>]</span><span class="mw-headline" id="Notes_and_references">Notes and references</span></h2>
<div class="reflist" style="background-color: white; font-family: sans-serif; font-size: 12px; line-height: 20px; list-style-type: decimal; margin-bottom: 0.5em;">
<ol class="references" style="line-height: 1.5em; list-style-image: none; list-style-type: inherit; margin-bottom: 0.5em; margin-left: 3.2em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li id="cite_note-0" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_ref-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation" id="CITEREFFano1957" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Ugo_Fano" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Ugo Fano">Fano, Ugo</a> (1957), "Description of States in Quantum Mechanics by Density Matrix and Operator Techniques", <i>Reviews of Modern Physics</i> <b>29</b>: 74–93, <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1957RvMP...29...74F" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1957RvMP...29...74F</a>,<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FRevModPhys.29.74" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/RevModPhys.29.74</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Description+of+States+in+Quantum+Mechanics+by+Density+Matrix+and+Operator+Techniques&rft.jtitle=Reviews+of+Modern+Physics&rft.aulast=Fano&rft.aufirst=Ugo&rft.au=Fano%2C%26%2332%3BUgo&rft.date=1957&rft.volume=29&rft.pages=74%26ndash%3B93&rft_id=info:bibcode/1957RvMP...29...74F&rft_id=info:doi/10.1103%2FRevModPhys.29.74&rfr_id=info:sid/en.wikipedia.org:Density_matrix"></span></span></li>
<li id="cite_note-1" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_ref-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation" id="CITEREFvon_Neumann1927" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/John_von_Neumann" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John von Neumann">von Neumann, John</a> (1927), "Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik", <i>Göttinger Nachrichten</i> <b>1</b>: 245–272.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Wahrscheinlichkeitstheoretischer+Aufbau+der+Quantenmechanik&rft.jtitle=G%C3%B6ttinger+Nachrichten&rft.aulast=von+Neumann&rft.aufirst=John&rft.au=von+Neumann%2C%26%2332%3BJohn&rft.date=1927&rft.volume=1&rft.pages=245%26ndash%3B272&rfr_id=info:sid/en.wikipedia.org:Density_matrix"></span></span></li>
<li id="cite_note-2" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_ref-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation" id="CITEREFLandau1927" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Lev_Landau" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lev Landau">Landau, L. D.</a> (1927), "Das Dämpfungsproblem in der Wellenmechanik", <i>Zeitschrift für Physik</i> <b>45</b> (5–6): 430–441, <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1927ZPhy...45..430L" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1927ZPhy...45..430L</a>, <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1007%2FBF01343064" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1007/BF01343064</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Das+D%C3%A4mpfungsproblem+in+der+Wellenmechanik&rft.jtitle=Zeitschrift+f%C3%BCr+Physik&rft.aulast=Landau&rft.aufirst=L.+D.&rft.au=Landau%2C%26%2332%3BL.+D.&rft.date=1927&rft.volume=45&rft.issue=5%E2%80%936&rft.pages=430%E2%80%93441&rft_id=info:bibcode/1927ZPhy...45..430L&rft_id=info:doi/10.1007%2FBF01343064&rfr_id=info:sid/en.wikipedia.org:Density_matrix"></span></span></li>
<li id="cite_note-3" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_ref-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation" id="CITEREFLandau1977" style="word-wrap: break-word;">Landau, L. D., and Lifshitz, E. M. (1977), <i>Quantum Mechanics, Non-Relativistic Theory: Volume 3</i>, Oxford: Pergamon Press, pp. 41, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-08-017801-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-08-017801-4">0-08-017801-4</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics%2C+Non-Relativistic+Theory%3A+Volume+3&rft.aulast=Landau&rft.aufirst=L.+D.%2C+and+Lifshitz%2C+E.+M.&rft.au=Landau%2C%26%2332%3BL.+D.%2C+and+Lifshitz%2C+E.+M.&rft.date=1977&rft.pages=pp.%26nbsp%3B41&rft.place=Oxford&rft.pub=Pergamon+Press&rft.isbn=0-08-017801-4&rfr_id=info:sid/en.wikipedia.org:Density_matrix"></span></span></li>
<li id="cite_note-4" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_ref-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Technically, <i>F</i> must be a Borel function</span></li>
<li id="cite_note-5" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_ref-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation" id="CITEREFNielsenChuang2000" style="word-wrap: break-word;">Nielsen, Michael; Chuang, Isaac (2000), <i>Quantum Computation and Quantum Information</i>, <a href="http://en.wikipedia.org/wiki/Cambridge_University_Press" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cambridge University Press">Cambridge University Press</a>, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-521-63503-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-521-63503-5">978-0-521-63503-5</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computation+and+Quantum+Information&rft.aulast=Nielsen&rft.aufirst=Michael&rft.au=Nielsen%2C%26%2332%3BMichael&rft.au=Chuang%2C%26%2332%3BIsaac&rft.date=2000&rft.pub=%5B%5BCambridge+University+Press%5D%5D&rft.isbn=978-0-521-63503-5&rfr_id=info:sid/en.wikipedia.org:Density_matrix"></span>. Chapter 11: Entropy and information, Theorem 11.9, "Projective measurements cannot decrease entropy"</span></li>
<li id="cite_note-everett56-6" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_ref-everett56_6-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation" id="CITEREFEverett1973" style="word-wrap: break-word;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Hugh_Everett" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hugh Everett">Everett, Hugh</a> (1973), "The Theory of the Universal Wavefunction (1956) Appendix I. "Monotone decrease of information for stochastic processes"", <i>The Many-Worlds Interpretation of Quantum Mechanics</i>, Princeton Series in Physics, <a href="http://en.wikipedia.org/wiki/Princeton_University_Press" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Princeton University Press">Princeton University Press</a>, pp. 128–129, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-691-08131-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; color: #0b0080;" title="Special:BookSources/978-0-691-08131-1">978-0-691-08131-1</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=The+Theory+of+the+Universal+Wavefunction+%281956%29+Appendix+I.+%22Monotone+decrease+of+information+for+stochastic+processes%22&rft.atitle=The+Many-Worlds+Interpretation+of+Quantum+Mechanics&rft.aulast=Everett&rft.aufirst=Hugh&rft.au=Everett%2C%26%2332%3BHugh&rft.date=1973&rft.series=Princeton+Series+in+Physics&rft.pages=pp.%26nbsp%3B128%E2%80%93129&rft.pub=%5B%5BPrinceton+University+Press%5D%5D&rft.isbn=978-0-691-08131-1&rfr_id=info:sid/en.wikipedia.org:Density_matrix"></span></span></li>
<li id="cite_note-7" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_ref-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external text" href="http://books.google.com/books?id=0Yx5VzaMYm8C&pg=PA110" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;"><i>The theory of open quantum systems</i>, by Breuer and Petruccione, p110</a>.</span></li>
<li id="cite_note-8" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_ref-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external text" href="http://books.google.com/books?id=o-HyHvRZ4VcC&pg=PA16" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;"><i>Statistical mechanics</i>, by Schwabl, p16</a>.</span></li>
<li id="cite_note-9" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_ref-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">See appendix, <span class="citation" id="CITEREFMackey1963" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/George_Mackey" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="George Mackey">Mackey, George Whitelaw</a> (1963), <i>Mathematical Foundations of Quantum Mechanics</i>, Dover Books on Mathematics, New York: <a href="http://en.wikipedia.org/wiki/Dover_Publications" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dover Publications">Dover Publications</a>, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-486-43517-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-486-43517-6">978-0-486-43517-6</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Foundations+of+Quantum+Mechanics&rft.aulast=Mackey&rft.aufirst=George+Whitelaw&rft.au=Mackey%2C%26%2332%3BGeorge+Whitelaw&rft.date=1963&rft.series=Dover+Books+on+Mathematics&rft.place=New+York&rft.pub=%5B%5BDover+Publications%5D%5D&rft.isbn=978-0-486-43517-6&rfr_id=info:sid/en.wikipedia.org:Density_matrix"></span></span></li>
<li id="cite_note-10" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Density_matrix#cite_ref-10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation" id="CITEREFEmch1972" style="word-wrap: break-word;">Emch, Gerard G. (1972), <i>Algebraic methods in statistical mechanics and quantum field theory</i>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Wiley-Interscience" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wiley-Interscience">Wiley-Interscience</a>, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/978-0-471-23900-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/978-0-471-23900-0">978-0-471-23900-0</a></span></span></li>
</ol>
</div>
<span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px;"><br /><br /></span>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com1tag:blogger.com,1999:blog-5303246073824127471.post-57828386605493070982012-05-02T23:54:00.000-04:002012-05-02T23:54:08.935-04:00The Wavefunction<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><img alt="File:QuantumHarmonicOscillatorAnimation.gif" src="http://upload.wikimedia.org/wikipedia/commons/9/90/QuantumHarmonicOscillatorAnimation.gif" style="margin-left: auto; margin-right: auto;" /></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left;">Some trajectories of a </span><a href="http://en.wikipedia.org/wiki/Harmonic_oscillator" style="background-attachment: initial; background-clip: initial; background-color: #f9f9f9; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left; text-decoration: none;" title="Harmonic oscillator">harmonic oscillator</a><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left;"> (a ball attached to a </span><a href="http://en.wikipedia.org/wiki/Hooke%27s_law" style="background-attachment: initial; background-clip: initial; background-color: #f9f9f9; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left; text-decoration: none;" title="Hooke's law">spring</a><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left;">) in </span><a href="http://en.wikipedia.org/wiki/Classical_mechanics" style="background-attachment: initial; background-clip: initial; background-color: #f9f9f9; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left; text-decoration: none;" title="Classical mechanics">classical mechanics</a><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left;"> (A-B) and</span><a href="http://en.wikipedia.org/wiki/Quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: #f9f9f9; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left; text-decoration: none;" title="Quantum mechanics">quantum mechanics</a><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left;"> (C-H). In quantum mechanics (C-H), the ball has a </span><b style="background-color: #f9f9f9; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left;">wave function</b><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left;">, which is shown with </span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Real_part" style="background-attachment: initial; background-clip: initial; background-color: #f9f9f9; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left; text-decoration: none;" title="Real part">real part</a><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left;">in blue and </span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Imaginary_part" style="background-attachment: initial; background-clip: initial; background-color: #f9f9f9; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left; text-decoration: none;" title="Imaginary part">imaginary part</a><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left;"> in red. The trajectories C,D,E,F, (but not G or H) are examples of </span><a href="http://en.wikipedia.org/wiki/Standing_wave" style="background-attachment: initial; background-clip: initial; background-color: #f9f9f9; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left; text-decoration: none;" title="Standing wave">standing waves</a><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left;">, (or "</span><a href="http://en.wikipedia.org/wiki/Stationary_state" style="background-attachment: initial; background-clip: initial; background-color: #f9f9f9; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left; text-decoration: none;" title="Stationary state">stationary states</a><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left;">"). Each standing-wave frequency is proportional to a possible </span><a href="http://en.wikipedia.org/wiki/Energy_level" style="background-attachment: initial; background-clip: initial; background-color: #f9f9f9; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left; text-decoration: none;" title="Energy level">energy level</a><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left;"> of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have </span><i style="background-color: #f9f9f9; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left;">any</i><span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 11px; line-height: 16px; text-align: left;"> energy.</span>
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The Wavefunction is at the heart of quantum mechanics in Physics, therefore at the heart of understanding our Universe, as we live in a quantum mechanical universe. Developed by Ernest Schrodinger in the 1920's based on Louis DeBroglie's contribution that Matter can behave as waves, this equation has lead to more quantum "weirdness" as well as scientific advances than any other. While practicing physicists use density matrices in calculations, Physics undergrads study this important function, then square it (as the Wavefunction has 2 parts: Real and imaginary) to find probabilities. From Wikipedia:
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A <b>wave function</b> or <b>wavefunction</b> is a probability amplitude in <a href="http://en.wikipedia.org/wiki/Quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum mechanics">quantum mechanics</a> describing the <a href="http://en.wikipedia.org/wiki/Quantum_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum state">quantum state</a> of a particle and how it behaves. Typically, its values are <a href="http://en.wikipedia.org/wiki/Complex_number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Complex number">complex numbers</a> and, for a single particle, it is a <a href="http://en.wikipedia.org/wiki/Function_(mathematics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Function (mathematics)">function</a> of space and time. The laws of quantum mechanics (the <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schrödinger equation">Schrödinger equation</a>) describe how the wave function evolves over time. The wave function behaves qualitatively like other <a href="http://en.wikipedia.org/wiki/Wave" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wave">waves</a>, like <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Water_wave" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Water wave">water waves</a> or waves on a string, because the Schrödinger equation is mathematically a type of <a href="http://en.wikipedia.org/wiki/Wave_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wave equation">wave equation</a>. This explains the name "wave function", and gives rise to <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Wave-particle_duality" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wave-particle duality">wave-particle duality</a>.</div>
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The most common symbols for a wave function are <i>ψ</i> or <i>Ψ</i> (lower-case and capital <a href="http://en.wikipedia.org/wiki/Psi_(letter)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Psi (letter)">psi</a>).</div>
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Although <i>ψ</i> is a complex number, |<i>ψ</i>|<sup style="line-height: 1em;">2</sup> is real, and corresponds to the <a href="http://en.wikipedia.org/wiki/Probability_density_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability density function">probability density</a> of finding a particle in a given place at a given time, if the particle's position is <a href="http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Measurement in quantum mechanics">measured</a>.</div>
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The SI units for <i>ψ</i> depend on the system. For one particle in three dimensions, its units are m<sup style="line-height: 1em;">–3/2</sup>. These unusual units are required so that an integral of |<i>ψ</i>|<sup style="line-height: 1em;">2</sup> over a region of three-dimensional space is a unitless probability (i.e., the probability that the particle is in that region). For different numbers of particles and/or dimensions, the units may be different (though can be determined by <a href="http://en.wikipedia.org/wiki/Dimensional_analysis" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dimensional analysis">dimensional analysis</a>).</div>
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The wave function is absolutely central to quantum mechanics—it makes the subject what it is. It is also the source of the mysterious consequences and philosophical difficulties in what quantum mechanics means in nature, and even how nature itself behaves at the atomic scale and beyond—topics that continue to be debated today.</div>
<table class="toc" id="toc" style="background-color: #f9f9f9; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; border-image: initial; border-left-color: rgb(170, 170, 170); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(170, 170, 170); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(170, 170, 170); border-top-style: solid; border-top-width: 1px; color: black; font-family: sans-serif; font-size: 12px; line-height: 20px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;"><tbody>
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Contents</h2>
<span class="toctoggle" style="-webkit-user-select: none;"> [<a class="internal" href="http://en.wikipedia.org/wiki/Wavefunction#" id="togglelink" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">hide</a>] </span></div>
<ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;">
<li class="toclevel-1 tocsection-1" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Historical_background" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1</span> <span class="toctext">Historical background</span></a></li>
<li class="toclevel-1 tocsection-2" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Mathematical_introduction" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2</span> <span class="toctext">Mathematical introduction</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-3" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Wavefunctions_as_multi-variable_functions_-_analytical_calculus_formalism" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.1</span> <span class="toctext">Wavefunctions as multi-variable functions - analytical calculus formalism</span></a></li>
<li class="toclevel-2 tocsection-4" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Wave_functions_as_an_abstract_vector_space_-_linear.2Fabstract_algebra_formalism" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.2</span> <span class="toctext">Wave functions as an abstract vector space - linear/abstract algebra formalism</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-3 tocsection-5" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Introduction_to_vector_formalism" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.2.1</span> <span class="toctext">Introduction to vector formalism</span></a></li>
</ul>
</li>
<li class="toclevel-2 tocsection-6" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Requirements" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.3</span> <span class="toctext">Requirements</span></a></li>
<li class="toclevel-2 tocsection-7" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Information_about_quantum_systems" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.4</span> <span class="toctext">Information about quantum systems</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-8" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Definition_.28single_spin-0_particle_in_one_spatial_dimension.29" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3</span> <span class="toctext">Definition (single spin-0 particle in one spatial dimension)</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-9" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Position-space_wavefunction" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.1</span> <span class="toctext">Position-space wavefunction</span></a></li>
<li class="toclevel-2 tocsection-10" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Momentum-space_wavefunction" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.2</span> <span class="toctext">Momentum-space wavefunction</span></a></li>
<li class="toclevel-2 tocsection-11" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Relation_between_wavefunctions" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.3</span> <span class="toctext">Relation between wavefunctions</span></a></li>
<li class="toclevel-2 tocsection-12" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Example_of_normalization" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3.4</span> <span class="toctext">Example of normalization</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-13" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Definition_.28other_cases.29" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4</span> <span class="toctext">Definition (other cases)</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-14" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Many_spin-0_particles_in_one_spatial_dimension" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.1</span> <span class="toctext">Many spin-0 particles in one spatial dimension</span></a></li>
<li class="toclevel-2 tocsection-15" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#One_spin-0_particle_in_three_spatial_dimensions" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.2</span> <span class="toctext">One spin-0 particle in three spatial dimensions</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-3 tocsection-16" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Position_space_wavefunction" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.2.1</span> <span class="toctext">Position space wavefunction</span></a></li>
<li class="toclevel-3 tocsection-17" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Momentum_space_wavefunction" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.2.2</span> <span class="toctext">Momentum space wavefunction</span></a></li>
<li class="toclevel-3 tocsection-18" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Relation_between_wavefunctions_2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.2.3</span> <span class="toctext">Relation between wavefunctions</span></a></li>
</ul>
</li>
<li class="toclevel-2 tocsection-19" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Many_spin-0_particles_in_three_spatial_dimensions" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.3</span> <span class="toctext">Many spin-0 particles in three spatial dimensions</span></a></li>
<li class="toclevel-2 tocsection-20" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#One_particle_with_spin_in_three_dimensions" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.4</span> <span class="toctext">One particle with spin in three dimensions</span></a></li>
<li class="toclevel-2 tocsection-21" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Many_particles_with_spin_in_three_dimensions" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.5</span> <span class="toctext">Many particles with spin in three dimensions</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-22" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Normalization_invariance" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5</span> <span class="toctext">Normalization invariance</span></a></li>
<li class="toclevel-1 tocsection-23" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Wavefunctions_as_vector_spaces" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6</span> <span class="toctext">Wavefunctions as vector spaces</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-2 tocsection-24" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Basis_representation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6.1</span> <span class="toctext">Basis representation</span></a></li>
<li class="toclevel-2 tocsection-25" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Finite_dimensional_basis_vectors" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6.2</span> <span class="toctext">Finite dimensional basis vectors</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li class="toclevel-3 tocsection-26" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Formalism" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6.2.1</span> <span class="toctext">Formalism</span></a></li>
</ul>
</li>
<li class="toclevel-2 tocsection-27" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Infinite_dimensional_basis_vectors" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6.3</span> <span class="toctext">Infinite dimensional basis vectors</span></a></li>
<li class="toclevel-2 tocsection-28" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Continuously_indexed_vectors" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6.4</span> <span class="toctext">Continuously indexed vectors</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-29" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Ontology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">7</span> <span class="toctext">Ontology</span></a></li>
<li class="toclevel-1 tocsection-30" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Examples" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">8</span> <span class="toctext">Examples</span></a></li>
<li class="toclevel-1 tocsection-31" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#See_also" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">9</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-32" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#References" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">10</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-33" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#Further_reading" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">11</span> <span class="toctext">Further reading</span></a></li>
<li class="toclevel-1 tocsection-34" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#External_links" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">12</span> <span class="toctext">External links</span></a></li>
</ul>
</td></tr>
</tbody></table>
<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Historical background">edit</a>]</span><span class="mw-headline" id="Historical_background">Historical background</span></h2>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
In the 1920s and 1930s, there were two divisions (so to speak) of <a href="http://en.wikipedia.org/wiki/Theoretical_physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theoretical physics">theoretical physicists</a> who simultaneously founded quantum mechanics: one for <a href="http://en.wikipedia.org/wiki/Calculus" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Calculus">calculus</a> and one for <a href="http://en.wikipedia.org/wiki/Linear_algebra" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear algebra">linear algebra</a>. Those who used the techniques of calculus included <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Louis-Victor_de_Broglie" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Louis-Victor de Broglie">Louis de Broglie</a>, <a href="http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Erwin Schrödinger">Erwin Schrödinger</a>, <a href="http://en.wikipedia.org/wiki/Paul_Dirac" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Paul Dirac">Paul Dirac</a>, <a href="http://en.wikipedia.org/wiki/Hermann_Weyl" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermann Weyl">Hermann Weyl</a>, <a href="http://en.wikipedia.org/wiki/Oskar_Klein" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Oskar Klein">Oskar Klein</a>, <a href="http://en.wikipedia.org/wiki/Walter_Gordon" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Walter Gordon">Walter Gordon</a>, <a href="http://en.wikipedia.org/wiki/Douglas_Hartree" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Douglas Hartree">Douglas Hartree</a> and<a href="http://en.wikipedia.org/wiki/Vladimir_Fock" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vladimir Fock">Vladimir Fock</a>. This hand of quantum mechanics became known as "<a href="http://en.wikipedia.org/wiki/Wave_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wave mechanics">wave mechanics</a>". Those who applied the methods of linear algebra included <a href="http://en.wikipedia.org/wiki/Werner_Heisenberg" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Werner Heisenberg">Werner Heisenberg</a>, <a href="http://en.wikipedia.org/wiki/Max_Born" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Max Born">Max Born</a>,<a href="http://en.wikipedia.org/wiki/Wolfgang_Pauli" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wolfgang Pauli">Wolfgang Pauli</a> and <a href="http://en.wikipedia.org/wiki/John_Slater" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John Slater">John Slater</a>. This other hand of quantum mechanics came to be called "matrix mechanics". Schrödinger was one who subsequently showed that the two approaches were equivalent.<sup class="reference" id="cite_ref-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[1]</a></sup> In each case, the wavefunction was at the centre of attention in two forms, giving quantum mechanics its unity.</div>
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De Broglie could be considered the founder of the wave model in 1925, due to his <a href="http://en.wikipedia.org/wiki/Symmetric_relation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Symmetric relation">symmetric relation</a> between <a href="http://en.wikipedia.org/wiki/Momentum" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Momentum">momentum</a> and <a href="http://en.wikipedia.org/wiki/Wavelength" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wavelength">wavelength</a>: the <a href="http://en.wikipedia.org/wiki/Matter_wave#The_de_Broglie_relations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Matter wave">De Broglie equation</a>. Schrödinger searched for an equation that would describe these waves, and was the first to construct and publish an equation for which the wave function satisfied in 1926, based on <a href="http://en.wikipedia.org/wiki/Classical_physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Classical physics">classical</a><a href="http://en.wikipedia.org/wiki/Energy_conservation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Energy conservation">energy conservation</a>. Indeed it is now called the <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schrödinger equation">Schrödinger equation</a>. However, <i>no-one</i>, even Schrödinger and De Broglie, were clear on <i>how to interpret it</i>. What did this function<i>mean</i>? <sup class="reference" id="cite_ref-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup> Around 1924–27, Born, Heisenberg, Bohr and others provided the perspective of <i><a href="http://en.wikipedia.org/wiki/Probability_amplitude" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability amplitude">probability amplitude</a></i>.<sup class="reference" id="cite_ref-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup> This is the <i><a href="http://en.wikipedia.org/wiki/Copenhagen_interpretation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Copenhagen interpretation">Copenhagen interpretation</a></i> of quantum mechanics. There are many other <a href="http://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Interpretations of quantum mechanics">interpretations of quantum mechanics</a>, but this is considered the most important - since quantum <i>calculations</i> can be understood.</div>
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In 1927, Hartree and Fock made the first step in an attempt to solve the <a href="http://en.wikipedia.org/wiki/Many-body_problem" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Many-body problem">N-body</a> wave function, and developed the <i>self-consistency cycle</i>: an <a href="http://en.wikipedia.org/wiki/Iteration" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Iteration">iterative</a> <a href="http://en.wikipedia.org/wiki/Algorithm" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Algorithm">algorithm</a> to approximate the solution. Now it is also known as the <a href="http://en.wikipedia.org/wiki/Hartree%E2%80%93Fock_method" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hartree–Fock method">Hartree–Fock method</a>.<sup class="reference" id="cite_ref-Quanta_1974_3-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-Quanta_1974-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[4]</a></sup> The <a href="http://en.wikipedia.org/wiki/Slater_determinant" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Slater determinant">Slater</a> <a href="http://en.wikipedia.org/wiki/Determinant" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Determinant">determinant</a> and <a href="http://en.wikipedia.org/wiki/Permanent" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Permanent">permanent</a> (of a <a href="http://en.wikipedia.org/wiki/Matrix_(mathematics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Matrix (mathematics)">matrix</a>) was part of the method, provided by Slater.</div>
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Interestingly, Schrödinger did encounter an equation for which the wave function satisfied <a href="http://en.wikipedia.org/wiki/Relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Relativity">relativistic</a> energy conservation <i>before</i> he published the non-relativistic one, but it lead to unacceptable consequences for that time so he discarded it.<sup class="reference" id="cite_ref-4" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[5]</a></sup> In 1927, Klein, Gorden and Fock also found it, but taking a step further: enmeshed the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Electromagnetic_force" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electromagnetic force">electromagnetic</a> <a href="http://en.wikipedia.org/wiki/Interaction" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Interaction">interaction</a>into it and proved it was <a href="http://en.wikipedia.org/wiki/Lorentz_covariance" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lorentz covariance">Lorentz-invariant</a>. De Broglie also arrived at exactly the same equation in 1928. This wave equation is now known most commonly as the <a href="http://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Klein–Gordon equation">Klein–Gordon equation</a>.<sup class="reference" id="cite_ref-5" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[6]</a></sup></div>
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By 1928 Dirac deduced his equation from the first successful unified combination of <a href="http://en.wikipedia.org/wiki/Special_relativity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special relativity">special relativity</a> and quantum mechanics to the <a href="http://en.wikipedia.org/wiki/Electron" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Electron">electron</a> - the <a href="http://en.wikipedia.org/wiki/Dirac_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac equation">Dirac equation</a>. He found an unusual character of the wavefunction for this equation: it was not a single complex number, but a <i><a href="http://en.wikipedia.org/wiki/Spinor" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spinor">spinor</a></i>.<sup class="reference" id="cite_ref-Quanta_1974_3-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-Quanta_1974-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[4]</a></sup> <a href="http://en.wikipedia.org/wiki/Spin" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin">Spin</a> automatically entered into the properties of the wavefunction. Although there were problems, Dirac was capable of resolving them. Around the same time Weyl also found his relativistic equation, which also had spinor solutions. Later other wave equations were developed: see <a href="http://en.wikipedia.org/wiki/Relativistic_wave_equations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Relativistic wave equations">Relativistic wave equations</a> for further information.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Mathematical introduction">edit</a>]</span><span class="mw-headline" id="Mathematical_introduction">Mathematical introduction</span></h2>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Wavefunctions as multi-variable functions - analytical calculus formalism">edit</a>]</span><span class="mw-headline" id="Wavefunctions_as_multi-variable_functions_-_analytical_calculus_formalism">Wavefunctions as multi-variable functions - analytical calculus formalism</span></h3>
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<a href="http://en.wikipedia.org/wiki/Multivariable_calculus" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Multivariable calculus">Multivariable calculus</a> and <a href="http://en.wikipedia.org/wiki/Mathematical_analysis" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mathematical analysis">analysis</a> (study of <a href="http://en.wikipedia.org/wiki/Function_(mathematics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Function (mathematics)">functions</a>, change etc.) can be used to <i><a href="http://en.wikipedia.org/wiki/Representation_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Representation theory">represent</a></i> the wavefunction in a number of situations. Superficially, this formalism is simple to understand for the following reasons.</div>
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<li style="margin-bottom: 0.1em;">It is more directly intuitive to have probability amplitudes as functions of space and time. At every position and time coordinate, the probability amplitude has a value by direct calculation.</li>
<li style="margin-bottom: 0.1em;">Functions can easily describe <a href="http://en.wikipedia.org/wiki/Wave" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wave">wave</a>-like motion, using <a href="http://en.wikipedia.org/wiki/Periodic_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Periodic function">periodic functions</a>, and <a href="http://en.wikipedia.org/wiki/Fourier_analysis" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fourier analysis">Fourier analysis</a> can be readily done.</li>
<li style="margin-bottom: 0.1em;">Functions are easy to produce, visualize and interpret, due to the pictorial nature of the <a href="http://en.wikipedia.org/wiki/Graph_of_a_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Graph of a function">graph of a function</a> (i.e. <a href="http://en.wikipedia.org/wiki/Curve" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Curve">curves</a>, <a href="http://en.wikipedia.org/wiki/Contour" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Contour">contours</a>, and <a href="http://en.wikipedia.org/wiki/Surface" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Surface">surfaces</a>). When the situation is in a high number of dimensions (say 3-d space) - it is possible to analyse the function in a lower dimensional slice (say a 2-d plane) or contour plots of the function to determine the behaviour of the system within that confined region.</li>
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Although these functions are <a href="http://en.wikipedia.org/wiki/Continuous_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Continuous function">continuous</a>, they are not <a href="http://en.wikipedia.org/wiki/Deterministic_system" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Deterministic system">deterministic</a>; rather, they are <a href="http://en.wikipedia.org/wiki/Probability_distribution" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability distribution">probability distributions</a>. Perhaps oddly, this approach is <i>not</i> the most general way to represent probability amplitudes. The more advanced techniques use <a href="http://en.wikipedia.org/wiki/Linear_algebra" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear algebra">linear algebra</a> (the study of <a href="http://en.wikipedia.org/wiki/Vector_space_model" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vector space model">vectors</a>, <a href="http://en.wikipedia.org/wiki/Matrix_(mathematics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Matrix (mathematics)">matrices</a>, etc.) and, more generally still, <a href="http://en.wikipedia.org/wiki/Abstract_algebra" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Abstract algebra">abstract algebra</a> (algebraic structures, generalizations of Euclidean spaces etc.).</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Wave functions as an abstract vector space - linear/abstract algebra formalism">edit</a>]</span><span class="mw-headline" id="Wave_functions_as_an_abstract_vector_space_-_linear.2Fabstract_algebra_formalism">Wave functions as an abstract vector space - linear/abstract algebra formalism</span></h3>
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The set of all possible wave functions (at any given time) forms an abstract mathematical <a href="http://en.wikipedia.org/wiki/Vector_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vector space">vector space</a>. Specifically, the <i>entire</i> wave function is treated as a <i>single</i> abstract vector:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\psi(\mathbf{r}) \leftrightarrow |\psi\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/1/3/813655bd593a2b695b72557687b97377.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <img alt="|\psi\rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/0/5/605b44d27aad3e8e841b1dd43053faa1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is a <a href="http://en.wikipedia.org/wiki/Column_vector" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Column vector">column vector</a> written in <a href="http://en.wikipedia.org/wiki/Bra-ket_notation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bra-ket notation">bra-ket notation</a>. The statement that "wave functions form an abstract vector space" simply means that it is possible to add together different wave functions, and multiply wave functions by complex numbers (see <a href="http://en.wikipedia.org/wiki/Vector_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vector space">vector space</a> for details). (Technically, because of the normalization condition, wave functions form a<a href="http://en.wikipedia.org/wiki/Projective_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Projective space">projective space</a> rather than an ordinary vector space.) This vector space is infinite-<a href="http://en.wikipedia.org/wiki/Dimension_(vector_space)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dimension (vector space)">dimensional</a>, because there is no finite set of functions which can be added together in various combinations to create every possible function. Also, it is a <a href="http://en.wikipedia.org/wiki/Hilbert_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hilbert space">Hilbert space</a>, because the inner product of wave functions <img alt="\Psi_1(x)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/4/3/343aa87511f1ecdc289b4b06dc196a41.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> and <img alt="\Psi_2(x)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/9/f/39f8376c312aba9b5736091fb78c9cd9.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> can be defined as</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\langle \Psi_1 | \Psi_2 \rangle \equiv \int\limits_{-\infty}^\infty \Psi_1^*(x)\Psi_2(x) \, \mathrm{d}x," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/5/f/35f9c613d5fa8d1c5b34c8cd55180722.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where * denotes <a href="http://en.wikipedia.org/wiki/Complex_conjugate" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Complex conjugate">complex conjugate</a>.</div>
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There are several advantages to understanding wave functions as elements of an abstract vector space:</div>
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<li style="margin-bottom: 0.1em;">All the powerful tools of <a href="http://en.wikipedia.org/wiki/Linear_algebra" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear algebra">linear algebra</a> can be used to manipulate and understand wave functions. For example:<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;">Linear algebra explains how a vector space can be given a <a href="http://en.wikipedia.org/wiki/Basis_(linear_algebra)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Basis (linear algebra)">basis</a>, and then any vector can be expressed in this basis. This explains the relationship between a wave function in position space and a wave function in momentum space, and suggests that there are other possibilities too.</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Bra-ket_notation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bra-ket notation">Bra-ket notation</a> can be used to manipulate wave functions.</li>
</ul>
</li>
<li style="margin-bottom: 0.1em;">The idea that <a href="http://en.wikipedia.org/wiki/Quantum_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum state">quantum states</a> are vectors in a Hilbert space is completely general in all aspects of quantum mechanics and <a href="http://en.wikipedia.org/wiki/Quantum_field_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum field theory">quantum field theory</a>, whereas the idea that quantum states are complex-valued "wave" functions of space is only true in certain situations.</li>
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<h4 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 15px; line-height: 20px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Introduction to vector formalism">edit</a>]</span><span class="mw-headline" id="Introduction_to_vector_formalism">Introduction to vector formalism</span></h4>
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Given an isolated physical system, the allowed states of this system (i.e. the states the system could occupy without violating the laws of physics) are part of a <a href="http://en.wikipedia.org/wiki/Hilbert_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hilbert space">Hilbert space</a> <i>H</i>. Some properties of such a space are</div>
<ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;">If <img alt="| \psi \rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/0/5/605b44d27aad3e8e841b1dd43053faa1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /> and <img alt="| \phi \rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/4/7/647e7b8a3252e5f6eac8022f9ca5de17.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /> are two allowed states, then <img alt="a | \psi \rangle + b | \phi \rangle\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/4/7/b4706d0924ebf9c9da3a413e14c34b3b.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /> is also an allowed state, provided <img alt="|a|^2+|b|^2=1\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/c/b/bcbd72980ba92ed3377d3340c3622179.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />. (This condition is due to normalisation, see below.)</li>
<li style="margin-bottom: 0.1em;">There is always an <a href="http://en.wikipedia.org/wiki/Orthonormal_basis" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Orthonormal basis">orthonormal basis</a> of allowed states of the vector space <i>H</i>.</li>
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Physically, the nature of the inner product is dependent on the basis in use, because the basis is chosen to reflect the quantum state of the system.</div>
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When the basis is a countable set <img alt="\{ | \phi_i \rangle \}\," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/6/8/7689ef3849e883f9aa8d3f915b04982c.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> and orthonormal, that is</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\langle \phi_i | \phi_j \rangle = \delta_{ij}," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/a/3/ea3b597ffea285b32d706b5fb139081c.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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then an arbitrary vector <img alt="| \psi \rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/0/5/605b44d27aad3e8e841b1dd43053faa1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> can be expressed as</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="| \psi \rangle = \sum_i c_i | \phi_i \rangle, " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/b/b/bbb27c5ba2cfef779b1a91773ab9ac31.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where the components are the (complex) numbers <img alt="c_i = \langle \phi_i | \psi \rangle." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/8/9/4899e03ce58faf4e3c7ea170e209ea35.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> This wave function is known as a <i>discrete spectrum</i>, since the bases are discrete.</div>
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When the basis is an uncountable set, the orthonormality condition holds similarly,</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\langle \phi | \phi_0 \rangle = \delta \left ( \phi - \phi_0 \right )," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/2/7/3271465aac186c5125b0d15897dfc6d6.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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then an arbitrary vector <img alt="| \psi \rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/0/5/605b44d27aad3e8e841b1dd43053faa1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> can be expressed as</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="| \psi \rangle = \int c(\phi) | \phi \rangle \mathrm{d} \phi. " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/2/b/32b8296b3504fa2d1f507ffe7dce863e.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where the components are the functions <img alt=" c(\phi) = \langle \phi | \psi \rangle." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/b/2/eb29ab9c2726ef5d1857c5fce5a5efd0.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> This wave function is known as a <i>continuous spectrum</i>, since the bases are continuous.</div>
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Paramount to the analysis is the <a href="http://en.wikipedia.org/wiki/Kronecker_delta" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Kronecker delta">Kronecker delta</a>, <img alt=" \delta_{ij}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/3/4/0345354c195713650c0b66ebdedd7a33.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, and the <a href="http://en.wikipedia.org/wiki/Dirac_delta_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Dirac delta function">Dirac delta function</a>, <img alt="\delta \left ( \phi - \phi_0 \right ) \," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/4/e/e4e241a08a810e0360eb580f9d83f814.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, since the bases used are orthonormal. More detailed discussion of wave functions as elements of vector spaces is below, following further definitions.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Requirements">edit</a>]</span><span class="mw-headline" id="Requirements">Requirements</span></h3>
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<a class="image" href="http://en.wikipedia.org/wiki/File:Wavefunction_continuity_space.svg" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><img alt="" class="thumbimage" height="503" src="http://upload.wikimedia.org/wikipedia/commons/thumb/1/11/Wavefunction_continuity_space.svg/300px-Wavefunction_continuity_space.svg.png" style="background-color: white; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; border-color: initial; border-image: initial; border-left-color: rgb(204, 204, 204); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(204, 204, 204); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(204, 204, 204); border-top-style: solid; border-top-width: 1px; border-width: initial; vertical-align: middle;" width="300" /></a><div class="thumbcaption" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; font-size: 11px; line-height: 1.4em; padding-bottom: 3px !important; padding-left: 3px !important; padding-right: 3px !important; padding-top: 3px !important; text-align: left;">
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Continuity of the wavefunction and its first spatial derivative (in the <i>x</i> direction, <i>y</i> and <i>z</i> coordinates not shown), at some time <i>t</i>.</div>
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The wavefunction must satisfy the following constraints for the calculations and physical interpretation to make sense:<sup class="reference" id="cite_ref-Atoms.2C_Molecules_1985_6-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-Atoms.2C_Molecules_1985-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[7]</a></sup></div>
<ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;">It must everywhere be finite.</li>
<li style="margin-bottom: 0.1em;">It must everywhere be a <a href="http://en.wikipedia.org/wiki/Continuous_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Continuous function">continuous function</a>, and <a href="http://en.wikipedia.org/wiki/Smooth_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Smooth function">continuously differentiable</a> (at least up to all possible first order derivatives).<ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;">As a corollary, the function would be single-valued, else multiple probabilities occur at the same position and time, again unphysical.</li>
</ul>
</li>
<li style="margin-bottom: 0.1em;">It must everywhere satisfy the relevant <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Normalizable_wave_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Normalizable wave function">normalization condition</a>, so that the particle/system of particles exists somewhere with 100% certainty.</li>
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If these requirements are not met, it's not possible to interpret the wavefunction as a probability amplitude; the values of the wavefunction and its first order derivatives may not be finite and definite (with exactly one value), i.e. probabilities can be<i>infinite</i> and <i>multiple-valued</i> at any one position and time - which is nonsense, as it does not satisfy the <a href="http://en.wikipedia.org/wiki/Probability_axioms" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability axioms">probability axioms</a>. Furthermore, when using the wavefunction to calculate a measurable observable of the quantum system without meeting these requirements, there will not be finite or definite values to calculate from - in this case the observable can take a number of values and can be infinite. This is unphysical and not observed when measuring in an experiment. Hence a wavefunction is meaningful only if these conditions are satisfied.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Information about quantum systems">edit</a>]</span><span class="mw-headline" id="Information_about_quantum_systems">Information about quantum systems</span></h3>
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Main articles: <a href="http://en.wikipedia.org/wiki/Quantum_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum state">Quantum state</a> and <a href="http://en.wikipedia.org/wiki/Operator_(physics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Operator (physics)">Operator (physics)</a></div>
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Although the wavefunction contains information, it is a <a href="http://en.wikipedia.org/wiki/Complex_number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Complex number">complex number</a> valued quantity; only its relative phase and relative magnitude can be measured. It does not directly tell anything about the magnitudes or directions of measurable observables. An operator extracts this information by acting on the wavefunction <i>ψ</i>. For details and examples on how quantum mechanical operators act on the wave function, commutation of operators, and expectation values of operators; see <a href="http://en.wikipedia.org/wiki/Operator_(physics)#Operators_in_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Operator (physics)">Operator (physics)</a>.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Definition (single spin-0 particle in one spatial dimension)">edit</a>]</span><span class="mw-headline" id="Definition_.28single_spin-0_particle_in_one_spatial_dimension.29">Definition (single spin-0 particle in one spatial dimension)</span></h2>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Position-space wavefunction">edit</a>]</span><span class="mw-headline" id="Position-space_wavefunction">Position-space wavefunction</span></h3>
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For now, consider the simple case of a single particle, without <a href="http://en.wikipedia.org/wiki/Spin_(physics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin (physics)">spin</a>, in one spatial dimension. (More general cases are discussed below). The state of such a particle is completely described by its wave function:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\Psi(x,t)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/6/e/96ee3a1fc2fc3bc89e210ba2bac750a8.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />,</dd></dl>
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where <i>x</i> is position and <i>t</i> is time. This function is <a href="http://en.wikipedia.org/wiki/Complex_number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Complex number">complex-valued</a>, meaning that <img alt="\Psi(x,t)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/6/e/96ee3a1fc2fc3bc89e210ba2bac750a8.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is a <a href="http://en.wikipedia.org/wiki/Complex_number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Complex number">complex number</a>.</div>
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If the particle's position is <a href="http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Measurement in quantum mechanics">measured</a>, its location is not deterministic, but is described by a <a href="http://en.wikipedia.org/wiki/Probability_distribution" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability distribution">probability distribution</a>. The probability that its position <i>x</i> will be in the interval [<i>a</i>, <i>b</i>] (meaning <i>a</i> ≤ <i>x</i> ≤ <i>b</i>) is:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="P_{a<x<b} = \int\limits_a^b |\Psi(x,t)|^2 \,\mathrm{d}x" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/b/7/7b75d569cfcc3d5c91205cb570bc2ab5.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <i>t</i> is the time at which the particle was measured. In other words, <img alt="|\Psi(x,t)|^2" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/6/6/96668f49082dfcf028cf4cdc44722dd6.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is the <a href="http://en.wikipedia.org/wiki/Probability_density_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Probability density function"><i>probability density</i></a> that the particle is at <i>x</i>, rather than some other location.</div>
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This leads to the <b>normalization condition</b>:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\int\limits_{-\infty}^\infty |\Psi(x,t)|^2\, \mathrm{d}x = 1" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/9/1/39158443d25589b67b07136d707cc884.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />,</dd></dl>
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because if the particle is measured, there is 100% probability that it will be <i>somewhere</i>.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Momentum-space wavefunction">edit</a>]</span><span class="mw-headline" id="Momentum-space_wavefunction">Momentum-space wavefunction</span></h3>
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Main article: <a href="http://en.wikipedia.org/wiki/Momentum_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Momentum space">Momentum space</a></div>
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The particle also has a wave function in <a href="http://en.wikipedia.org/wiki/Momentum_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Momentum space">momentum space</a>:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\Phi(p,t)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/d/d/bdd9dfddc47561efa383ab799033b9a4.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <i>p</i> is the <a href="http://en.wikipedia.org/wiki/Momentum" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Momentum">momentum</a> in one dimension, which can be any value from <img alt="-\infty" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/e/a/beab416080922c84a90ba092f7734fe5.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> to <img alt="+\infty" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/8/c/28cfe0a2608499ff5984a938e0d16d64.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, and <i>t</i> is time. If the particle's momentum is <a href="http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Measurement in quantum mechanics">measured</a>, the result is not deterministic, but is described by a probability distribution:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="P_{a<p<b} = \int\limits_a^b |\Phi(p,t)|^2 \mathrm{d}p" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/f/8/df87a14fc1bcd48b1b3c0a4d0fde34b2.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />,</dd></dl>
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analogous to the position case.</div>
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The normalization condition is also similar:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \int\limits_{-\infty}^{\infty} \left | \Phi \left ( p, t \right ) \right |^2 \mathrm{d}p = 1." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/d/b/6/db6b3ea9c04a850a54bd3e24a3a988c3.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Relation between wavefunctions">edit</a>]</span><span class="mw-headline" id="Relation_between_wavefunctions">Relation between wavefunctions</span></h3>
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The position-space and momentum-space wave functions are <a href="http://en.wikipedia.org/wiki/Fourier_transform" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fourier transform">Fourier transforms</a> of each other, therefore both contain the same information, and either one alone is sufficient to calculate any property of the particle. For one-dimension:<sup class="reference" id="cite_ref-7" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[8]</a></sup></div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\begin{align} \Phi(p,t) & = \frac{1}{\sqrt{2\pi\hbar}}\int\limits_{-\infty}^\infty e^{-ipx/\hbar} \Psi(x,t)\mathrm{d}x \\
&\upharpoonleft \downharpoonright\\
\Psi(x,t) & = \frac{1}{\sqrt{2\pi\hbar}}\int\limits_{-\infty}^\infty e^{ipx/\hbar} \Phi(p,t)\mathrm{d}p .
\end{align}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/a/a/6aaa6d2625b53b7dbc94cc7bf4b6e395.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Sometimes the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Wave-vector" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wave-vector">wave-vector</a> <i>k</i> is used in place of <a href="http://en.wikipedia.org/wiki/Momentum" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Momentum">momentum</a> <i>p</i>, since they are related by the <a href="http://en.wikipedia.org/wiki/Matter_wave#The_de_Broglie_relations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Matter wave">de Broglie relation</a></div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="p = \hbar k," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/6/c/b6c60acb37a0cf22ef6c3b8d58a469da.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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and the equivalent space is referred to as <a href="http://en.wikipedia.org/wiki/Momentum_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Momentum space">k-space</a>. Again it makes no difference which is used since <i>p</i> and <i>k</i> are equivalent - up to a constant. In practice, the position-space wavefunction is used much more often than the momentum-space wavefunction.</div>
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Wave-particles "stationary".</div>
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<a class="image" href="http://en.wikipedia.org/wiki/File:Quantum_mechanics_travelling_wavefunctions.svg" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wave-particles travelling."><img alt="" height="431" src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Quantum_mechanics_travelling_wavefunctions.svg/520px-Quantum_mechanics_travelling_wavefunctions.svg.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" width="520" /></a></div>
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Wave-particles travelling.</div>
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Interpretation of wave function for one spin-0 particle in one dimension. The wavefunctions shown are continuous, finite, single-valued and normalized. The colour opacity (%) of the particles corresponds to the probability density (which can measure in %) of finding the particle at the points on the x-axis.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=12" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Example of normalization">edit</a>]</span><span class="mw-headline" id="Example_of_normalization">Example of normalization</span></h3>
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A particle is restricted to a 1D region between <i>x</i> = 0 and <i>x</i> = L; its wave function is:</div>
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\Psi (x,t) & = Ae^{i(kx-\omega t)}, & x \in [0,L] \\
\Psi (x,t) & = 0, & x \notin [0,L] \\
\end{align} " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/e/3/be3b156e9bd6e88320098c44e49eea6c.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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and is zero elsewhere. To normalize the wave function we need to find the value of the arbitrary constant <i>A</i>; solved from</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \int\limits_{-\infty}^{\infty} |\Psi|^2 {\rm d}x = 1 . " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/9/d/99d67af8496a7d0d77c7cc00693d0584.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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From <i>Ψ</i>, we have |<i>Ψ</i>|<sup style="line-height: 1em;">2</sup>;</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" | \Psi | ^2 = A^2 e^{i(kx - \omega t)} e^{-i(kx - \omega t)} =A^2 , " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/2/9/a29cc6c944ec7dcbc7a7ce5b7d3abc06.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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so the integral becomes;</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \int\limits_{-\infty}^0 0 {\rm d}x + \int\limits_0^L A^2 {\rm d}x + \int\limits_L^\infty 0 {\rm d}x = 1 , " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/5/c/35c2ed72130c70b179727028d0dd31a1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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therefore the constant is;</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="A^2 L = 1 \rightarrow A = \frac{1}{\sqrt{L}} ." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/4/7/34742617cc86bece701c987935d21c3b.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The normalized wave function (in the region) is then given by;</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \Psi (x,t) = \frac{1}{\sqrt{L}} e^{i(kx-\omega t)}, \quad x\in[0,L]." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/d/6/cd6da4f7977aa566e6522c04f9ceda46.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=13" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Definition (other cases)">edit</a>]</span><span class="mw-headline" id="Definition_.28other_cases.29">Definition (other cases)</span></h2>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=14" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Many spin-0 particles in one spatial dimension">edit</a>]</span><span class="mw-headline" id="Many_spin-0_particles_in_one_spatial_dimension">Many spin-0 particles in one spatial dimension</span></h3>
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The previous wavefunction can be generalized to incorporate <i>N</i> particles in one dimension:</div>
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The probability that particle 1 is in an <i>x</i>-interval <i>R</i><sub style="line-height: 1em;">1</sub> = [<i>a</i><sub style="line-height: 1em;">1</sub>,<i>b</i><sub style="line-height: 1em;">1</sub>] <i>and</i> particle 2 in interval <i>R</i><sub style="line-height: 1em;">2</sub> = [<i>a</i><sub style="line-height: 1em;">2</sub>,<i>b</i><sub style="line-height: 1em;">2</sub>] etc., up to particle <i>N</i> in interval <i>R<sub style="line-height: 1em;">N</sub></i> = [<i>a<sub style="line-height: 1em;">N</sub></i>,<i>b<sub style="line-height: 1em;">N</sub></i>], all measured simultaneously at time <i>t</i>, is given by:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="P_{x_1\in R_1,x_2\in R_2 \cdots x_N\in R_N} = \int\limits_{a_1}^{b_1} \mathrm{d}x_1 \int\limits_{a_2}^{b_2} \mathrm{d}x_2 \cdots \int\limits_{a_N}^{b_N} \mathrm{d}x_N | \Psi(x_1 \cdots x_N,t)|^2 " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/2/c/f2c3325d4b3dea5eaa4dd5f554c7f1f7.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The normalization condition becomes:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\int\limits_{-\infty}^\infty \mathrm{d}x_1 \int\limits_{-\infty}^\infty \mathrm{d}x_2 \cdots \int\limits_{-\infty}^\infty \mathrm{d}x_N |\Psi(x_1 \cdots x_N,t)|^2 = 1" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/a/b/1aba66fa098ea2ad1e4c7ec8051dd4fa.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />.</dd></dl>
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In each case, there are <i>N</i> one-dimensional integrals, one for each particle.</div>
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Visulization of the wavefunction for two spin-0 particles, in one dimension. Above: The two particles interact, then recoil from each other with 100% certainty. This situation occurs in <a href="http://en.wikipedia.org/wiki/Quantum_entanglement" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum entanglement">quantum entanglement</a>. Below: The particles are simply travelling.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=15" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: One spin-0 particle in three spatial dimensions">edit</a>]</span><span class="mw-headline" id="One_spin-0_particle_in_three_spatial_dimensions">One spin-0 particle in three spatial dimensions</span></h3>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=16" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Position space wavefunction">edit</a>]</span><span class="mw-headline" id="Position_space_wavefunction">Position space wavefunction</span></h4>
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The electron probability density for the first few <a href="http://en.wikipedia.org/wiki/Hydrogen_atom" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hydrogen atom">hydrogen atom</a> electron<a href="http://en.wikipedia.org/wiki/Atomic_orbital" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Atomic orbital">orbitals</a> shown as cross-sections. These orbitals form an <a href="http://en.wikipedia.org/wiki/Orthonormal_basis" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Orthonormal basis">orthonormal basis</a> for the wave function of the electron. Different orbitals are depicted with different scale.</div>
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The position-space wave function of a single particle in three spatial dimensions is similar to the case of one spatial dimension above:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\Psi(\mathbf{r},t)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/2/2/322da38bacfd5d799a1ee38e7a886cd0.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <b>r</b> is the position in three-dimensional space (<b>r</b> is short for (<i>x</i>,<i>y</i>,<i>z</i>)), and <i>t</i> is time. As always <img alt="\Psi(\mathbf{r},t)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/2/2/322da38bacfd5d799a1ee38e7a886cd0.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />is a <a href="http://en.wikipedia.org/wiki/Complex_number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Complex number">complex number</a>. If the particle's position is measured at time <i>t</i>, the probability that it is in a region <i>R</i>is:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="P_{\mathbf{r}\in R} = \int\limits_R \left |\Psi(\mathbf{r},t) \right |^2 \mathrm{d}^3\mathbf{r}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/8/9/289ae8de1445ede35fb06db198d5868a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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(a three-dimensional integral over the region <i>R</i>, with differential volume element d<sup style="line-height: 1em;">3</sup><b>r</b>, also written "d<i>V</i>" or "d<i>x</i> d<i>y</i> d<i>z</i>"). The <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Normalisable_wave_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Normalisable wave function">normalization condition</a> is:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\int\limits_{{\rm all \, space}} \left | \Psi(\mathbf{r},t)\right |^2 \mathrm{d}^3\mathbf{r} = 1," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/d/a/bdadc53255003d33cee888e104da0a14.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where the integrals are taken over all of three-dimensional space (or 3d momentum space).</div>
<h4 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 15px; line-height: 20px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=17" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Momentum space wavefunction">edit</a>]</span><span class="mw-headline" id="Momentum_space_wavefunction">Momentum space wavefunction</span></h4>
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There is a corresponding momentum space wavefunction for three-dimensions also:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\Phi(\mathbf{p},t) " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/c/c/8ccaaddc1e15369e7e5da6d7bd8fdebf.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <b>p</b> is the momentum in 3-dimensional space, and <i>t</i> is time. This time there are three components of momentum which can have values <img alt="-\infty " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/e/a/beab416080922c84a90ba092f7734fe5.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> to <img alt="+ \infty " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/8/c/28cfe0a2608499ff5984a938e0d16d64.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> in each direction, in Cartesian coordinates <i>x</i>, <i>y</i>, <i>z</i>.</div>
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The probability of measuring the momentum components <i>p<sub style="line-height: 1em;">x</sub></i> between <i>a</i> and <i>b</i>, <i>p<sub style="line-height: 1em;">y</sub></i> between <i>c</i> and <i>d</i>, and <i>p<sub style="line-height: 1em;">z</sub></i> between <i>e</i> and <i>f</i>, is given by:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="P_{p_x\in[a,b],p_y\in[c,d],p_z\in[e,f]} = \int\limits_e^f \int\limits_c^d \int\limits_a^b \left | \Phi \left ( \mathbf{p}, t \right ) \right |^2 \mathrm{d}p_x \mathrm{d}p_y \mathrm{d}p_z ," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/d/1/1d170d1092c353499337061670ecfae6.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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hence the normalization:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \int\limits_{{\rm all \, space}} \left | \Phi \left ( \mathbf{p}, t \right ) \right |^2 \mathrm{d}^3\mathbf{p} = 1." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/e/3/1e3b191acb9dbb6765019dc2b40d71f5.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
analogous to space, d<sup style="line-height: 1em;">3</sup><b>p</b> = d<i>p<sub style="line-height: 1em;">x</sub></i>d<i>p<sub style="line-height: 1em;">y</sub></i>d<i>p<sub style="line-height: 1em;">z</sub></i> is a differential 3-momentum volume element in momentum space.</div>
<h4 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 15px; line-height: 20px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=18" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Relation between wavefunctions">edit</a>]</span><span class="mw-headline" id="Relation_between_wavefunctions_2">Relation between wavefunctions</span></h4>
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The generalization of the previous Fourier transform is <sup class="reference" id="cite_ref-8" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[9]</a></sup></div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\begin{align} \Phi(\mathbf{p},t) & = \frac{1}{\sqrt{\left(2\pi\hbar\right)^3}}\int\limits_{{\rm all \, space}} e^{-i \mathbf{r}\cdot \mathbf{p} /\hbar} \Psi(\mathbf{r},t)\mathrm{d}^3\mathbf{r} \\
&\upharpoonleft \downharpoonright\\
\Psi(\mathbf{r},t) & = \frac{1}{\sqrt{\left(2\pi\hbar\right)^3}}\int\limits_{{\rm all \, space}} e^{i \mathbf{r}\cdot \mathbf{p} /\hbar} \Phi(\mathbf{p},t)\mathrm{d}^3\mathbf{p}.
\end{align}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/8/9/c8959e152e2c7b702c3d75762ab30b96.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
<h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 18px; line-height: 20px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=19" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Many spin-0 particles in three spatial dimensions">edit</a>]</span><span class="mw-headline" id="Many_spin-0_particles_in_three_spatial_dimensions">Many spin-0 particles in three spatial dimensions</span></h3>
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When there are many particles, in general there is only one wave function, not a separate wave function for each particle. The fact that <i>one</i> wave function describes <i>many</i> particles is what makes <a href="http://en.wikipedia.org/wiki/Quantum_entanglement" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum entanglement">quantum entanglement</a> and the <a href="http://en.wikipedia.org/wiki/EPR_paradox" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="EPR paradox">EPR paradox</a> possible. The position-space wave function for <i>N</i> particles is written:<sup class="reference" id="cite_ref-Quanta_1974_3-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-Quanta_1974-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[4]</a></sup></div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\Psi(\mathbf{r}_1,\mathbf{r}_2 \cdots \mathbf{r}_N,t)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/2/e/a2ebf09aa3dfc7183adaed3daec63322.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <b>r</b><sub style="line-height: 1em;">i</sub> is the position of the <i>i</i>th particle in three-dimensional space, and <i>t</i> is time. If the particles' positions are all measured simultaneously at time <i>t</i>, the probability that particle 1 is in region <i>R</i><sub style="line-height: 1em;">1</sub> <i>and</i> particle 2 is in region <i>R</i><sub style="line-height: 1em;">2</sub> and so on is:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="P_{\mathbf{r}_1\in R_1,\mathbf{r}_2\in R_2 \cdots \mathbf{r}_N\in R_N} = \int\limits_{R_1} \mathrm{d}^3\mathbf{r}_1 \int\limits_{R_2} \mathrm{d}^3\mathbf{r}_2\cdots \int\limits_{R_N} \mathrm{d}^3\mathbf{r}_N |\Psi(\mathbf{r}_1 \cdots \mathbf{r}_N,t)|^2" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/3/c/83c378b122ba88f69a1e56ff6171c9b9.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The normalization condition is:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\int\limits_{{\rm all \, space}} \mathrm{d}^3\mathbf{r}_1 \int\limits_{{\rm all \, space}} \mathrm{d}^3\mathbf{r}_2\cdots \int\limits_{{\rm all \, space}} \mathrm{d}^3\mathbf{r}_N |\Psi(\mathbf{r}_1 \cdots \mathbf{r}_N,t)|^2 = 1" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/f/5/9f5c8847cca4c968cb603e1c5bbac1fb.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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(altogether, this is 3<i>N</i> one-dimensional integrals).</div>
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In quantum mechanics there is a fundamental distinction between <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Identical_particle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Identical particle">identical particles</a> and distinguishable particles. For example, any two electrons are fundamentally indistinguishable from each other; the laws of physics make it impossible to "stamp an identification number" on a certain electron to keep track of it.<sup class="reference" id="cite_ref-9" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[10]</a></sup> This translates to a requirement on the wavefunction: For example, if particles 1 and 2 are indistinguishable, then:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\Psi \left (\mathbf{r},\mathbf{r'},\mathbf{r}_3,\mathbf{r}_4,\cdots \right ) = \pm \Psi \left ( \mathbf{r'},\mathbf{r},\mathbf{r}_3,\mathbf{r}_4,\cdots \right )" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/4/3/7/437996145dd21820924e4162d7cc525a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where the + sign is required if the particles are <a href="http://en.wikipedia.org/wiki/Boson" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Boson">bosons</a>, and the – sign is required if they are <a href="http://en.wikipedia.org/wiki/Fermion" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermion">fermions</a>. More exactly stated:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\Psi \left ( \mathbf{r},\mathbf{r'},\mathbf{r}_3,\mathbf{r}_4,\cdots \right ) = \left ( -1 \right )^{2s} \Psi \left ( \mathbf{r'},\mathbf{r},\mathbf{r}_3,\mathbf{r}_4,\cdots \right )" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/5/f/b5f4e4023a4e1f8b60a12e84172cadcc.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
where <i>s</i> = spin quantum number,</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">integer for bosons: <img alt="s \in \left \{ \pm 1,\pm 2,\pm 3 \cdots \right \}, " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/9/d/e9d53339b9d049f6b915509747dbc9da.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">and half-integer for fermions: <img alt="s \in \left \{ \pm \frac{1}{2}, \pm \frac{3}{2} \cdots \right \} . " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/9/e/99eaf2907956292c2211312592f874ee.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The wavefunction is said to be <i>symmetric</i> (no sign change) under boson interchange and <i>antisymmetric</i> (sign changes) under fermion interchange. This feature of the wavefunction is known as the <a href="http://en.wikipedia.org/wiki/Pauli_exclusion_principle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pauli exclusion principle">Pauli principle</a>.</div>
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For <i>N</i> interacting particles, i.e. particles which interact mutually and constitute a many-body system, the wavefunction is a function of all positions of the particles and time, it can't be separated into the separate wavefunctions of the particles. However, for non-interacting particles, i.e. particles which do not interact mutually and move independently, in a time-independent potential, the wavefunction <i>can</i> be separated into the product of separate wavefunctions for each particle:<sup class="reference" id="cite_ref-Atoms.2C_Molecules_1985_6-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-Atoms.2C_Molecules_1985-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[7]</a></sup></div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\Psi = \phi(t)\prod_{i=1}^N\psi(\bold{r}_i) = \phi(t)\psi(\bold{r}_1)\psi(\bold{r}_2)\cdots\psi(\bold{r}_N)." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/3/0/f30900a774049aa791227abfa3a8e44f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=20" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: One particle with spin in three dimensions">edit</a>]</span><span class="mw-headline" id="One_particle_with_spin_in_three_dimensions">One particle with spin in three dimensions</span></h3>
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<a class="image" href="http://en.wikipedia.org/wiki/File:Spin-wavefunction.svg" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><img alt="" class="thumbimage" height="318" src="http://upload.wikimedia.org/wikipedia/commons/thumb/5/51/Spin-wavefunction.svg/300px-Spin-wavefunction.svg.png" style="background-color: white; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; border-color: initial; border-image: initial; border-left-color: rgb(204, 204, 204); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(204, 204, 204); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(204, 204, 204); border-top-style: solid; border-top-width: 1px; border-width: initial; vertical-align: middle;" width="300" /></a><div class="thumbcaption" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; font-size: 11px; line-height: 1.4em; padding-bottom: 3px !important; padding-left: 3px !important; padding-right: 3px !important; padding-top: 3px !important; text-align: left;">
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<a class="internal" href="http://en.wikipedia.org/wiki/File:Spin-wavefunction.svg" style="background-attachment: initial !important; background-clip: initial !important; background-color: initial !important; background-image: none !important; background-origin: initial !important; background-position: initial initial !important; background-repeat: initial initial !important; border-bottom-style: none !important; border-color: initial !important; border-image: initial !important; border-left-style: none !important; border-right-style: none !important; border-top-style: none !important; border-width: initial !important; color: #0b0080; display: block; text-decoration: none;" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.20wmf1/common/images/magnify-clip.png" style="background-attachment: initial !important; background-clip: initial !important; background-color: initial !important; background-image: none !important; background-origin: initial !important; background-position: initial initial !important; background-repeat: initial initial !important; border-bottom-style: none !important; border-color: initial !important; border-image: initial !important; border-left-style: none !important; border-right-style: none !important; border-top-style: none !important; border-width: initial !important; display: block; vertical-align: middle;" width="15" /></a></div>
Visulization of the wavefunction for a spin-1/2 particle, in one dimension. The spin orientations are shown in full opacity, with common notations for each value. Particles do not literally spin about their axes, this is just a representation.</div>
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For a particle with <a href="http://en.wikipedia.org/wiki/Spin_(physics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin (physics)">spin</a>, the wave function can be written in "position-spin-space" as:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\Psi(\mathbf{r},s_z,t)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/f/3/7f37bdbda0fe2fe13d18c73e5b25b2a0.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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where <b>r</b> is a position in three-dimensional space, <i>t</i> is time, and <i>s</i><sub style="line-height: 1em;">z</sub> is the <a href="http://en.wikipedia.org/wiki/Spin_(physics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Spin (physics)">spin projection quantum number</a> along the <i>z</i> axis. (The <i>z</i> axis is an arbitrary choice; other axes can be used instead if the wave function is transformed appropriately, see below.) The <i>s<sub style="line-height: 1em;">z</sub></i> parameter, unlike <b>r</b> and <i>t</i>, is a <i>discrete variable</i>. For example, for a spin-1/2 particle, <i>s</i><sub style="line-height: 1em;">z</sub> can only be +1/2 or -1/2, and not any other value. (In general, for spin <i>s</i>, <i>s</i><sub style="line-height: 1em;">z</sub> can be s, s–1,...,–s.) If the particle's position and spin is measured simultaneously at time <i>t</i>, the probability that its position is in <i>R</i><sub style="line-height: 1em;">1</sub> <i>and</i> its spin projection quantum number is a certain value <i>m</i>is:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="P_{\mathbf{r}\in R,s_z=m} = \int\limits_{R} \mathrm{d}^3\mathbf{r} |\Psi(\mathbf{r},t,m)|^2" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/a/5/2a54b7bac6849fb6a5d94d98a158e2be.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The normalization condition is:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\sum_{\mathrm{all\, }s_z} \int\limits_{{\rm all \, space}} |\Psi(\mathbf{r},t,s_z)|^2 \mathrm{d}^3\mathbf{r} = 1" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/c/b/2cb54d4375a6145a9321f8768d3bc6b1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />.</dd></dl>
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Since the spin quantum number has discrete values, it must be written as a sum rather than an integral, taken over all possible values.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=21" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Many particles with spin in three dimensions">edit</a>]</span><span class="mw-headline" id="Many_particles_with_spin_in_three_dimensions">Many particles with spin in three dimensions</span></h3>
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Likewise, the wavefunction for <i>N</i> particles each with spin is:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\Psi(\mathbf{r}_1, \mathbf{r}_2 \cdots \mathbf{r}_N, s_{z\,1}, s_{z\,2} \cdots s_{z\,N}, t)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/a/7/ea718b152523ef2473bda401fb497f6f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The probability that particle 1 is in region <i>R</i><sub style="line-height: 1em;">1</sub> with spin <i>s</i><sub style="line-height: 1em;"><i>z</i>1</sub> = <i>m</i><sub style="line-height: 1em;">1</sub> <i>and</i> particle 2 is in region <i>R</i><sub style="line-height: 1em;">2</sub> with spin <i>s</i><sub style="line-height: 1em;"><i>z</i>2</sub> = <i>m</i><sub style="line-height: 1em;">2</sub> etc. reads (probability subscripts now removed due to their great length):</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="P = \int\limits_{R_1} \mathrm{d}^3\mathbf{r}_1 \int\limits_{R_2} \mathrm{d}^3\mathbf{r}_2\cdots \int\limits_{R_N} \mathrm{d}^3\mathbf{r}_N \left | \Psi\left (\mathbf{r}_1 \cdots \mathbf{r}_N,m_1\cdots m_N,t \right ) \right |^2" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/9/0/9902787809736d102c9c57971edee926.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The normalization condition is:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \sum_{s_{z\,N}} \cdots \sum_{s_{z\,2}} \sum_{s_{z\,1}} \int\limits_{{\rm all \, space}} \mathrm{d}^3\mathbf{r}_1 \int\limits_{{\rm all \, space}} \mathrm{d}^3\mathbf{r}_2\cdots \int\limits_{{\rm all \, space}} \mathrm{d}^3 \mathbf{r}_N \left | \Psi \left (\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) \right |^2 = 1" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/f/d/bfdc497e02d95cc1b6902105c69a0c20.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Now there are 3<i>N</i> one-dimensional integrals followed by <i>N</i> sums.</div>
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Again, for non-interacting particles in a time-independent potential the wavefunction is the product of separate wavefunctions for each particle:<sup class="reference" id="cite_ref-Atoms.2C_Molecules_1985_6-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-Atoms.2C_Molecules_1985-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[7]</a></sup></div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\Psi = \phi(t)\prod_{i=1}^N\psi(\bold{r}_i,s_{z\,i}) = \phi(t)\psi(\bold{r}_1,s_{z\,1})\psi(\bold{r}_2,s_{z\,2})\cdots\psi(\bold{r}_N,s_{z\,N})." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/c/9/9c9ec3419af4ff923bd3cdaf953552cf.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=22" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Normalization invariance">edit</a>]</span><span class="mw-headline" id="Normalization_invariance">Normalization invariance</span></h2>
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It is important that the properties associated with the wave function are <i>invariant under normalization</i>. If normalization of a wave function changed the properties, the process becomes pointless as we still cannot yield any information about the particle associated with the non-normalized wave function.</div>
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All properties of the particle, such as momentum, energy, expectation value of position, associated probability distributions etc., are solved from the <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schrödinger equation">Schrödinger equation</a> (or other<a href="http://en.wikipedia.org/wiki/Relativistic_wave_equations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Relativistic wave equations">relativistic wave equations</a>). The <a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schrödinger equation">Schrödinger equation</a> is a <a href="http://en.wikipedia.org/wiki/Linear_differential_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear differential equation">linear differential equation</a>, so if <i>Ψ</i> is normalized and becomes <i>AΨ</i> (<i>A</i> is the normalization constant), then the equation reads:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \hat{H} (A\Psi) = i\hbar\frac{\partial }{\partial t}(A\Psi) \rightarrow \hat{H} \Psi = i\hbar\frac{\partial }{\partial t}\Psi " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/3/a/a/3aa473cac74e8785beabf0a59e7b4aef.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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which is the original Schrödinger equation. That is to say, the Schrödinger equation is <a href="http://en.wikipedia.org/wiki/Invariant_(mathematics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Invariant (mathematics)">invariant</a> under normalization, and consequently associated properties are unchanged.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=23" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Wavefunctions as vector spaces">edit</a>]</span><span class="mw-headline" id="Wavefunctions_as_vector_spaces">Wavefunctions as vector spaces</span></h2>
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Main article: <a href="http://en.wikipedia.org/wiki/Quantum_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum state">Quantum state</a></div>
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As explained above, <a href="http://en.wikipedia.org/wiki/Quantum_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum state">quantum states</a> are always vectors in an abstract vector space (technically, a complex <a href="http://en.wikipedia.org/wiki/Projective_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Projective space">projective</a> <a href="http://en.wikipedia.org/wiki/Hilbert_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hilbert space">Hilbert space</a>). For the wave functions above, the Hilbert space usually has not only infinite dimensions, but <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Uncountable" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Uncountable">uncountably</a></i> infinitely many dimensions. However, <a href="http://en.wikipedia.org/wiki/Linear_algebra" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear algebra">linear algebra</a> is much simpler for finite-dimensional vector spaces. Therefore it is helpful to look at an example where the Hilbert space of wave functions is finite dimensional.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=24" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Basis representation">edit</a>]</span><span class="mw-headline" id="Basis_representation">Basis representation</span></h3>
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A wave function describes the state of a physical system <img alt=" \left | \psi \right \rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/f/d/afd4dec0777186d8fc68768f51a9b847.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, by expanding it in terms of other possible states of the same system - collectively referred to as a <i>basis</i> or<i>representation</i> <img alt=" \left | \phi_i \right \rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/f/e/2fea404aa121dfd101dfd70466f62baf.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. In what follows, all wave functions are assumed to be normalized.</div>
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An element of a <a href="http://en.wikipedia.org/wiki/Vector_space" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Vector space">vector space</a> can be expressed in different <a href="http://en.wikipedia.org/wiki/Basis_(linear_algebra)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Basis (linear algebra)">bases elements</a>; and so the same applies to wave functions. The components of a wave function describing the same physical state take different <a href="http://en.wikipedia.org/wiki/Complex_number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Complex number">complex</a> values depending on the basis being used; however, just like elements of a vector space, the wave function itself is independent on the basis chosen. Choosing a new coordinate system does not change the vector itself, only the <i>representation</i> of the vector with respect to the new coordinate frame, since the <i>components</i>will be different but the linear combination of them still equals the vector.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=25" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Finite dimensional basis vectors">edit</a>]</span><span class="mw-headline" id="Finite_dimensional_basis_vectors">Finite dimensional basis vectors</span></h3>
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To start, consider the finite basis representation. A wave function <img alt="\vec \psi" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/d/2/1d2416aeb5659aa4d21f7e14b30a7d3d.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> with <i>n</i> components describes how to express the state of the physical system <img alt="| \psi \rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/0/5/605b44d27aad3e8e841b1dd43053faa1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> as the <a href="http://en.wikipedia.org/wiki/Linear_combination" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Linear combination">linear combination</a> of <i>n</i>basis elements <img alt="| \phi_i \rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/a/0/2a0663a3edc837468423a869cce63e3f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, (<i>i</i> = 1, 2...<i>n</i>). The following is a breakdown of the used formalism.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=26" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Formalism">edit</a>]</span><span class="mw-headline" id="Formalism">Formalism</span></h4>
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<b>Conventional vector: <i>Ψ</i> and conventional notation</b></div>
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As a column vector or column matrix:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix}. " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/f/4/ff42870cedb5b01f69a9f402571e8733.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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<b>State vector: <i>Ψ</i> and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Bra-ket" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bra-ket">bra-ket</a> notation</b></div>
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Equivalently in bra-ket notation, the <i>state</i> of a particle with wave function <i>Ψ</i> can be written as a ket;</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \left | \psi \right \rangle
= \sum_{i = 1}^n c_i \left | \phi_i \right \rangle
= c_1 \left | \phi_1 \right \rangle + c_2 \left | \phi_2 \right \rangle + \cdots c_n \left | \phi_n \right \rangle
= \begin{bmatrix} \left \langle \phi_1 \vert \psi \right \rangle \\ \vdots \\ \left \langle \phi_n \vert \psi \right \rangle \end{bmatrix}
= \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix} ." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/2/d/e2d0195ca8b10284a7ebafad464e8e1f.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The corresponding bra is the complex conjugate of the transposed matrix (into a row matrix/row vector):</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \begin{align}
\langle \psi | = | \psi \rangle^{*} & = \begin{bmatrix} \langle \phi_1 | \psi \rangle & \cdots & \langle \phi_n | \psi \rangle \end{bmatrix}^{*} = \begin{bmatrix} \langle \phi_1 | \psi \rangle^{*} & \cdots & \langle \phi_n | \psi \rangle^{*} \end{bmatrix} \\
& = \begin{bmatrix} c_1 & \cdots & c_n \end{bmatrix}^{*} = \begin{bmatrix} c_1^{*} & \cdots & c_n^{*} \end{bmatrix}
\end{align}
\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/a/e/bae95e882b70d3593257629a3eef7de9.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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By "the state of a particle with wavefunction <i>Ψ</i>", written as <img alt=" | \psi \rangle \,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/8/0/6801320d4f8324c427d202f0f2a461c1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />, this means the variables which characterize the system, with respect to the wavefunction. The wave function associated with a particular state may be seen as an expansion of the state in a basis of <img alt="H" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/1/d/c1d9f50f86825a1a2302ec2449c17196.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. For example, a basis could be for a free particle travelling in one dimension, with momentum eigenstates <i>ψ</i><sub style="line-height: 1em;">±</sub> corresponding to the ±<i>x</i> direction:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" | \psi \rangle = \psi_+ | p_{+} \rangle + \psi_{-} | p_{-} \rangle . " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/d/e/ede899c88562b49cf84f2525d5df5d7d.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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Another example is the superposition of two energy eigenstates for a particle trapped in a 1-d box (these states are <a href="http://en.wikipedia.org/wiki/Stationary_state" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Stationary state">stationary state</a>):</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" | \psi \rangle = \psi_1 | E_1 \rangle + \psi_2 | E_2 \rangle. " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/e/6/f/e6f6daed95b6a25655c6452c7ad90d26.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The most characteristic example is a particle in a spin up or down configuration:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" | \psi \rangle = \psi_+ | \uparrow_z \rangle + \psi_{-} | \downarrow_z \rangle , " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/c/f/0cf5c9f3a04e3dce515296c49ac06edf.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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(see below for details of this frequent case). Notice how kets are not completely analogous to the ordinary notion of vectors - rather they are labels for a state of a wavefunction, which are used in a similar way. In all of the above examples, the particle is not in any one definite or preferred state, but rather <i>in both at the same time</i> - hence the term<i>superposition</i>. The free particle could be have momentum in the +<i>x</i> <i>or</i> –<i>x</i> direction simultaneously, the trapped particle in the 1-d potential well can be in the energy eigenstates corresponding to eigenvalues <i>E</i><sub style="line-height: 1em;">1</sub> and <i>E</i><sub style="line-height: 1em;">2</sub> at the same time, the particle with spin could be in spin up or down orientation at any instant of time. The relative chance of which state occurs is related to the (moduli squares of the) coefficients.</div>
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Ket <i>Ψ</i>, ket bases, and orthonormality<a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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Ket <i>Ψ</i> and its components, the collapse postulate<a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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Ket <i>Ψ</i> and function <i>Ψ</i><a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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Inner product of two ket vectors <i>ψ</i> and <i>χ</i><a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle12" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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Outer product of two ket vectors <i>ψ</i> and <i>χ</i> and the closure relation<a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle13" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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Ket <i>Ψ</i> normalization<a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle14" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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<big>Application to one spin-½ particle (neglect spatial freedom)</big><a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle15" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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<h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 18px; line-height: 20px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=27" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Infinite dimensional basis vectors">edit</a>]</span><span class="mw-headline" id="Infinite_dimensional_basis_vectors">Infinite dimensional basis vectors</span></h3>
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The case of a countably infinite vector, with a discrete index, is treated and interpreted in the same manner as a finite vector, except the sum is extended over an infinite number of basis elements.</div>
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<b>Conventional vector: <i>Ψ</i> and conventional notation</b></div>
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As a column vector or column matrix, there are infinitely many entries:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \\ \vdots \end{bmatrix}. " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/2/b/e/2be7be5bdc7e42034cc5855633c0ca3c.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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<b>State vector: <i>Ψ</i> and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Bra-ket" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bra-ket">bra-ket</a> notation</b></div>
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In bra-ket notation;</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \left | \psi \right \rangle
= \sum_{i = 1}^\infty c_i \left | \phi_i \right \rangle
= c_1 \left | \phi_1 \right \rangle + c_2 \left | \phi_2 \right \rangle + \cdots
= \begin{bmatrix} \left \langle \phi_1 \vert \psi \right \rangle \\ \vdots \\ \left \langle \phi_n \vert \psi \right \rangle \\ \vdots \end{bmatrix}
= \begin{bmatrix} c_1 \\ \vdots \\ c_n \\ \vdots \end{bmatrix} ." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/0/9/b09e291714b5321c1bc621814aa033cf.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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The corresponding bra is as before:</div>
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\langle \psi | = | \psi \rangle^{*} & = \begin{bmatrix} \langle \phi_1 | \psi \rangle & \cdots & \langle \phi_n | \psi \rangle & \cdots \end{bmatrix}^{*} = \begin{bmatrix} \langle \phi_1 | \psi \rangle^{*} & \cdots & \langle \phi_n | \psi \rangle^{*} & \cdots \end{bmatrix} \\
& = \begin{bmatrix} c_1 & \cdots & c_n & \cdots \end{bmatrix}^{*} = \begin{bmatrix} c_1^{*} & \cdots & c_n^{*} & \cdots \end{bmatrix}
\end{align}
\,\!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/e/6/fe61035a8cf229a1ce9643e03dae290a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=28" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Continuously indexed vectors">edit</a>]</span><span class="mw-headline" id="Continuously_indexed_vectors">Continuously indexed vectors</span></h3>
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Now consider an uncountably infinite number of components of the physical state of the particle, <img alt=" \left | \psi \right \rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/f/d/afd4dec0777186d8fc68768f51a9b847.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" />. For this reason the collection of all states <img alt=" \left | \psi \right \rangle" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/f/d/afd4dec0777186d8fc68768f51a9b847.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is known as a <i>continuum</i> or<i>spectrum</i> of states. Finite or countably infinite basis vectors are summed over a discrete index - for a continuous basis the integral is over the continuous index, replacing the sum.</div>
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<b>Continuously indexed vector: <i>Ψ</i> and bra-ket notation</b></div>
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As usual <img alt=" | \psi \rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/0/5/605b44d27aad3e8e841b1dd43053faa1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is the physical state of the particle. The sum for a superposition of states now becomes an integral. In what follows, all integrals are with respect to the basis variable <i>ϕ</i>, over the required range. Usually this is just the real line or subset intervals of it. The state <img alt=" | \psi \rangle " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/6/0/5/605b44d27aad3e8e841b1dd43053faa1.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /> is given by:</div>
<dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" | \psi \rangle = \int | \phi \rangle \langle \phi | \psi \rangle \mathrm{d}\phi = \int | \phi \rangle c(\phi) \mathrm{d}\phi = \int| \phi \rangle \psi(\phi) \mathrm{d}\phi ." class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/b/f/c/bfcb1b03aacf0b0e0ca26031992bc5d6.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl>
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See below for more on notation of basis and components.</div>
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Ket <i>Ψ</i>, ket bases and orthonormality<a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle16" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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Ket <i>Ψ</i> and its components<a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle17" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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Inner product of two ket vectors <i>ψ</i> and <i>χ</i><a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle18" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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Outer product of two ket vectors <i>ψ</i> and <i>χ</i> and the closure relation<a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle19" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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Ket <i>Ψ</i> normalization<a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle20" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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<big>Application to one spin-0 particle in one spatial dimension</big><a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle21" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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<big>Application to one spin-0 particle in three spatial dimensions</big><a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle22" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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<big>Application to one spin particle in three spatial dimensions</big><a class="NavToggle" href="http://en.wikipedia.org/wiki/Wavefunction" id="NavToggle23" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; font-size: 13px; font-weight: normal; position: absolute; right: 3px; text-decoration: none; top: 0px;">[show]</a></div>
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<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=29" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Ontology">edit</a>]</span><span class="mw-headline" id="Ontology">Ontology</span></h2>
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Main article: <a href="http://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Interpretations of quantum mechanics">Interpretations of quantum mechanics</a></div>
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Whether the wave function really exists, and what it represents, are major questions in the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Interpretation_of_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Interpretation of quantum mechanics">interpretation of quantum mechanics</a>. Many famous physicists of a previous generation puzzled over this problem, such as <a href="http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Erwin Schrödinger">Schrödinger</a>, <a href="http://en.wikipedia.org/wiki/Albert_Einstein" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Albert Einstein">Einstein</a> and <a href="http://en.wikipedia.org/wiki/Niels_Bohr" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Niels Bohr">Bohr</a>. Some advocate formulations or variants of the <a href="http://en.wikipedia.org/wiki/Copenhagen_interpretation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Copenhagen interpretation">Copenhagen interpretation</a> (e.g. Bohr, <a href="http://en.wikipedia.org/wiki/Eugene_Wigner" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Eugene Wigner">Wigner</a> and <a href="http://en.wikipedia.org/wiki/John_von_Neumann" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John von Neumann">von Neumann</a>) while others, such as <a href="http://en.wikipedia.org/wiki/John_Archibald_Wheeler" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John Archibald Wheeler">Wheeler</a> or <a href="http://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edwin Thompson Jaynes">Jaynes</a>, take the more classical approach<sup class="reference" id="cite_ref-13" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-13" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[14]</a></sup> and regard the wave function as representing information in the mind of the observer, i.e. a measure of our knowledge of reality. Some, ranging from Schrödinger, Einstein, <a href="http://en.wikipedia.org/wiki/David_Bohm" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="David Bohm">Bohm</a> and <a href="http://en.wikipedia.org/wiki/Hugh_Everett_III" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hugh Everett III">Everett</a> and others, argued that the wave function must have an objective, physical existence. The later argument was recently supported by the demonstration (not peer reviewed) of a theorem stating the physical reality of the quantum state.<sup class="reference" id="cite_ref-pusey_14-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_note-pusey-14" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[15]</a></sup> For more on this topic, see<a href="http://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Interpretations of quantum mechanics">Interpretations of quantum mechanics</a>.</div>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=30" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Examples">edit</a>]</span><span class="mw-headline" id="Examples">Examples</span></h2>
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Here are examples of wavefunctions for specific applications:</div>
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<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Free_particle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Free particle">Free particle</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Particle_in_a_box" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Particle in a box">Particle in a box</a></li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Finite_square_well" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Finite square well">Finite square well</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Delta_potential" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Delta potential">Delta potential</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Quantum harmonic oscillator">Quantum harmonic oscillator</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Hydrogen_atom#Mathematical_summary_of_eigenstates_of_hydrogen_atom" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hydrogen atom">Hydrogen atom</a> and <a href="http://en.wikipedia.org/wiki/Hydrogen-like_atom" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hydrogen-like atom">Hydrogen-like atom</a></li>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=31" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: See also">edit</a>]</span><span class="mw-headline" id="See_also">See also</span></h2>
<ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Boson" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Boson">Boson</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Double-slit_experiment" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Double-slit experiment">Double-slit experiment</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Faraday_wave" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Faraday wave">Faraday wave</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Fermion" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Fermion">Fermion</a></li>
<li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Normalisable_wave_function" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Normalisable wave function">Normalisable wave function</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schrödinger equation">Schrödinger equation</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wave_function_collapse" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wave function collapse">Wave function collapse</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Wave_packet" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wave packet">Wave packet</a></li>
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<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=32" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: References">edit</a>]</span><span class="mw-headline" id="References">References</span></h2>
<div class="reflist" style="background-color: white; font-family: sans-serif; font-size: 12px; line-height: 20px; list-style-type: decimal; margin-bottom: 0.5em;">
<ol class="references" style="line-height: 1.5em; list-style-image: none; list-style-type: inherit; margin-bottom: 0.5em; margin-left: 3.2em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li id="cite_note-0" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation" id="CITEREFHanle1977" style="word-wrap: break-word;">Hanle, P.A. (December 1977), "Erwin Schrodinger's Reaction to Louis de Broglie's Thesis on the Quantum Theory.", <i>Isis</i> <b>68</b> (4): 606–609, <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1086%2F351880" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1086/351880</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Erwin+Schrodinger%27s+Reaction+to+Louis+de+Broglie%27s+Thesis+on+the+Quantum+Theory.&rft.jtitle=Isis&rft.aulast=Hanle&rft.aufirst=P.A.&rft.au=Hanle%2C%26%2332%3BP.A.&rft.date=December+1977&rft.volume=68&rft.issue=4&rft.pages=606%E2%80%93609&rft_id=info:doi/10.1086%2F351880&rfr_id=info:sid/en.wikipedia.org:Wave_function"></span></span></li>
<li id="cite_note-1" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/0716789647" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 0-7167-8964-7</a></span></li>
<li id="cite_note-2" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Sears' and Zemansky's University Physics, Young and Freedman (12th edition), Pearson Ed. & Addison-Wesley Inc., 2008, <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/9780321501301" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 978-0-321-50130-1</a></span></li>
<li id="cite_note-Quanta_1974-3" style="margin-bottom: 0.1em;">^ <a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-Quanta_1974_3-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-Quanta_1974_3-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-Quanta_1974_3-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>c</b></i></sup></a> <span class="reference-text">Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/0198554931" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 0-19-855493-1</a></span></li>
<li id="cite_note-4" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/9780071543828" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 978-0-07-154382-8</a></span></li>
<li id="cite_note-5" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Particle Physics (3rd Edition), B.R. Martin, G. Shaw, Manchester Physics Series, John Wiley & Sons, 2008, <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/9780470032947" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 978-0-470-03294-7</a></span></li>
<li id="cite_note-Atoms.2C_Molecules_1985-6" style="margin-bottom: 0.1em;">^ <a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-Atoms.2C_Molecules_1985_6-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-Atoms.2C_Molecules_1985_6-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-Atoms.2C_Molecules_1985_6-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>c</b></i></sup></a> <span class="reference-text">Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/9780471873730" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 978-0-471-87373-0</a></span></li>
<li id="cite_note-7" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Griffiths, page 107 of the first edition</span></li>
<li id="cite_note-8" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Quantum Mechanics (3rd Edition), Eugen Merzbacher, 1998, John Wiley & Sons, <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/0471887021" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 0-471-88702-1</a></span></li>
<li id="cite_note-9" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Griffiths, p179 of the first edition</span></li>
<li id="cite_note-10" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external free" href="http://www.users.csbsju.edu/~frioux/dirac/dirac.pdf" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif); background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 18px; text-decoration: none;">http://www.users.csbsju.edu/~frioux/dirac/dirac.pdf</a> , F. Rioux</span></li>
<li id="cite_note-11" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/9780131461000" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 978-0-13-146100-0</a></span></li>
<li id="cite_note-12" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-12" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external free" href="http://www.nyu.edu/classes/tuckerman/adv.chem/lectures/lecture_9/node2.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://www.nyu.edu/classes/tuckerman/adv.chem/lectures/lecture_9/node2.html</a></span></li>
<li id="cite_note-13" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-13" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">E. T. Jaynes. <i>Probability Theory: The Logic of Science</i>, <a href="http://en.wikipedia.org/wiki/Cambridge_University_Press" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cambridge University Press">Cambridge University Press</a> (2003),</span></li>
<li id="cite_note-pusey-14" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Wavefunction#cite_ref-pusey_14-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">Pusey, Matthew F.; Jonathan Barrett, Terry Rudolph (14). <a class="external text" href="http://arxiv.org/abs/1111.3328" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">"The quantum state cannot be interpreted statistically"</a>. <i>arXiv.org</i>: arxiv:1111.3328v1.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+quantum+state+cannot+be+interpreted+statistically&rft.jtitle=arXiv.org&rft.aulast=Pusey&rft.aufirst=Matthew+F.&rft.au=Pusey%2C%26%2332%3BMatthew+F.&rft.date=14&rft.pages=arxiv%3A1111.3328v1&rft_id=http%3A%2F%2Farxiv.org%2Fabs%2F1111.3328&rfr_id=info:sid/en.wikipedia.org:Wave_function"></span></span></li>
</ol>
</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; margin-bottom: 0.5em; margin-top: 0.4em;">
2.Quantum Mechanics(Non-Relativistic Theory), L.D. Landau and E.M. Lifshitz, <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/0080209408" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">ISBN 0-08-020940-8</a></div>
<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=33" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Further reading">edit</a>]</span><span class="mw-headline" id="Further_reading">Further reading</span></h2>
<ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;"><span class="citation book" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/David_Griffiths_(physicist)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="David Griffiths (physicist)">Griffiths, David J.</a> (2004). <i>Introduction to Quantum Mechanics (2nd ed.)</i>. Prentice Hall. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-13-111892-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-13-111892-7">0-13-111892-7</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Quantum+Mechanics+%282nd+ed.%29&rft.aulast=Griffiths%2C+David+J.&rft.au=Griffiths%2C+David+J.&rft.date=2004&rft.pub=Prentice+Hall&rft.isbn=0-13-111892-7&rfr_id=info:sid/en.wikipedia.org:Wave_function"></span></li>
<li style="margin-bottom: 0.1em;"><span class="citation Journal" style="word-wrap: break-word;">Yong-Ki Kim (September 2, 2000). <a class="external text" href="http://amods.kaeri.re.kr/mcdf/lectnote.pdf" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif); background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 18px; text-decoration: none;">"Practical Atomic Physics"</a>. <i>National Institute of Standards and Technology</i> (Maryland): 1 (55 pages)<span class="reference-accessdate">. Retrieved 2010-08-17</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Practical+Atomic+Physics&rft.jtitle=National+Institute+of+Standards+and+Technology&rft.aulast=Yong-Ki+Kim&rft.au=Yong-Ki+Kim&rft.date=September+2%2C+2000&rft.pages=1+%2855+pages%29&rft.place=Maryland&rft_id=http%3A%2F%2Famods.kaeri.re.kr%2Fmcdf%2Flectnote.pdf&rfr_id=info:sid/en.wikipedia.org:Wave_function"></span></li>
</ul>
<h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 20px; font-weight: normal; line-height: 20px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;">
<span class="editsection" style="-webkit-user-select: none; float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Wave_function&action=edit&section=34" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: External links">edit</a>]</span><span class="mw-headline" id="External_links">External links</span></h2>
<ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 20px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">
<li style="margin-bottom: 0.1em;"><a class="external autonumber" href="http://www.eng.fsu.edu/~dommelen/quantum/style_a/complexs.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">[1]</a>, <a class="external autonumber" href="http://www.nyu.edu/classes/tuckerman/adv.chem/lectures/lecture_9/node2.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">[2]</a>, <a class="external autonumber" href="http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/IdenticalParticlesRevisited.htm" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">[3]</a>, <a class="external autonumber" href="http://vergil.chemistry.gatech.edu/notes/quantrev/node34.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">[4]</a></li>
<li style="margin-bottom: 0.1em;"><a class="external autonumber" href="http://cat.middlebury.edu/~chem/chemistry/class/physical/quantum/help/normalize/normalize.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">[5]</a> Normalization.</li>
</ul>
<br />
<br />Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com2tag:blogger.com,1999:blog-5303246073824127471.post-68163782418219963092012-05-01T00:00:00.003-04:002012-05-01T00:03:14.753-04:00The Quantum Twenty-Five
<br /><br /> These are the twenty-five men who gave us the world we have today. Thank you, gents. <br />
<br />
They can be roughly split between the First Quantum Eleven (sometimes called "Ten" because they always leave out Weyl) and The Second Quantum Fifteen (which adds up to 26, I know, but Paul Dirac is part of both).<br />
<br />
In mostly chronological order:<br />
<br />
1) Maxwell Planck<br />
<br />
2) Albert Einstein <br />
<br />
3) Niels Bohr<br />
<br />
4) Louis de Broglie<br />
<br />
5) Hermann Weyl<br />
<br />
6) Wolfgang Pauli<br />
<br />
7) Werner Heisenberg<br />
<br />
8) Maxwell Born<br />
<br />
9) Pascual Jordan<br />
<br />
10) Erwin Schrödinger<br />
<br />
11) PAUL ADRIEN MAURICE (P.A.M.) DIRAC<br />
<br />
T12) Richard Phillips Feynman<br />
<br />
T12) Julian Schwinger<br />
<br />
T12) Sin-Itiro Tomonaga<br />
<br />
T15) Murray Gell-Mann<br />
<br />
T15) George Zweig <br />
<br />
T17) Sheldon Glashow<br />
<br />
T17) Abdus Salaam<br />
<br />
T17) Steven Weinberg<br />
<br />
T20) Martinus Veltman<br />
<br />
T20) Gerardus 't Hooft<br />
<br />
T22) Sidney Coleman<br />
<br />
T22) Hugh David Politzer<br />
<br />
T22) David Gross<br />
<br />
T22) Frank Wilczek<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgHEcFVHgIzaTfF-FA8szU4LpvBk-zaJIHfc59gx4UNragHRpHOS1wr7bx35vvKOCk8itPe9J4_TDbw6c9RD1v7M96Ym0k55AQCR8fGeqUYlbJTNcEkljvhAI9Sjo0HGmeTY716loQkw/s1600/Frank.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="352" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgHEcFVHgIzaTfF-FA8szU4LpvBk-zaJIHfc59gx4UNragHRpHOS1wr7bx35vvKOCk8itPe9J4_TDbw6c9RD1v7M96Ym0k55AQCR8fGeqUYlbJTNcEkljvhAI9Sjo0HGmeTY716loQkw/s640/Frank.jpg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Frank Wilczek ... the youngest of the bunch.</td></tr>
</tbody></table>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com2tag:blogger.com,1999:blog-5303246073824127471.post-75009136372550201882012-03-25T15:05:00.000-04:002012-03-25T15:05:59.835-04:00The Scale of The Universe 2<div style="background-color: white; border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #333333; font-family: Georgia, Century, Times, serif; font-size: 16px; line-height: 21px; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 15px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;"><a href="http://htwins.net/scale2/">CLICK HERE TO SEE IT</a><br />
<br />
You could call it <a href="http://powersof10.com/film" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #4e677b; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: none; outline-width: initial; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-decoration: none;" target="_hplink">"Powers of Ten"</a> for the Millennial Generation.</div><div style="background-color: white; border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #333333; font-family: Georgia, Century, Times, serif; font-size: 16px; line-height: 21px; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 15px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;"><a href="http://htwins.net/" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #4e677b; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: none; outline-width: initial; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-decoration: none;" target="_hplink">Cary and Michael Huang</a>, the minds behind the 2010 hit "<a href="http://scaleofuniverse.com/" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #4e677b; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: none; outline-width: initial; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-decoration: none;" target="_hplink">Scale of the Universe</a>," have <a href="http://htwins.net/scale2/" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #4e677b; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: none; outline-width: initial; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-decoration: none;" target="_hplink">released a follow-up</a> to their original interactive visualization.</div><div style="background-color: white; border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #333333; font-family: Georgia, Century, Times, serif; font-size: 16px; line-height: 21px; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 15px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;">Using objects varying in size from 10<sup style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">-35</sup> meters to 9.3 x 10<sup style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;">28</sup> meters, the tool attempts to help viewers grasp of the size of the universe in relation to familiar (and unfamiliar) objects, land masses, planets and the like.</div><div style="background-color: white; border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #333333; font-family: Georgia, Century, Times, serif; font-size: 16px; line-height: 21px; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 15px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;">But unlike the visualization that was released a couple of years ago, each object -- and there are hundreds -- is clickable and comes with its own explanation.</div><div style="background-color: white; border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #333333; font-family: Georgia, Century, Times, serif; font-size: 16px; line-height: 21px; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 15px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;">Want to know about the <a href="http://www.nasa.gov/multimedia/imagegallery/image_feature_89.html" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #4e677b; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: none; outline-width: initial; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-decoration: none;" target="_hplink">Horsehead Nebula</a>? <a href="http://en.wikipedia.org/wiki/Quantum_foam" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #4e677b; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: none; outline-width: initial; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-decoration: none;" target="_hplink">Quantum foam</a>? The <a href="http://www.usbr.gov/lc/hooverdam/" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #4e677b; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: none; outline-width: initial; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-decoration: none;" target="_hplink">Hoover Dam</a>? Just click on the icons as you travel between<a href="http://en.wikipedia.org/wiki/Yoctometre" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #4e677b; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: none; outline-width: initial; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-decoration: none;" target="_hplink">yoctometers</a> and <a href="http://en.wikipedia.org/wiki/Parsec" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #4e677b; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: none; outline-width: initial; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-decoration: none;" target="_hplink">parsecs</a>.</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqJ-dozV-REHs3p8fS3sSJ4wMdLKqCP21m6_jx7ePz57HzVwtuoakNssRzc7h0610aARDn6A57g0iz3Qc2svxtIm6ZcXxu-I32fqpT3lramwyY2E3oFT0Yyo0dkd5DaRj1KM6ZonaWSg/s1600/SaturnNewRing.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="314" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqJ-dozV-REHs3p8fS3sSJ4wMdLKqCP21m6_jx7ePz57HzVwtuoakNssRzc7h0610aARDn6A57g0iz3Qc2svxtIm6ZcXxu-I32fqpT3lramwyY2E3oFT0Yyo0dkd5DaRj1KM6ZonaWSg/s640/SaturnNewRing.jpg" width="640" /></a></div><div style="background-color: white; border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #333333; font-family: Georgia, Century, Times, serif; font-size: 16px; line-height: 21px; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 15px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;">The folks at JayIsGames.com <a href="http://jayisgames.com/archives/2012/02/scale_of_the_universe_2.php" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #4e677b; list-style-image: initial; list-style-position: initial; list-style-type: none; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; outline-color: initial; outline-style: none; outline-width: initial; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-decoration: none;" target="_hplink">say it only takes about 20 minutes</a> to go through and read the descriptions of all of the objects, but we've been playing with this for quite a bit longe.<br />
<br />
by The Huffington Post SCIENCE<br />
<br />
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<br />
</div>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com0tag:blogger.com,1999:blog-5303246073824127471.post-6618880162063943352012-03-16T08:36:00.000-04:002012-03-16T08:36:44.425-04:00Mag Tegmark and the Mathematical Universe Hypothesis<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5L2UuMV5PGxcBvbIvw5LbzF4BlABn4OXncoUVaIdBgOXP3xR4sZDubZAALwF2cj9RfEZF9kjUtPceCuqJ-PWn0hkd1KvGQkOMtzdpC5Uce7xyLwCVP7FGcMROFpwkwNYQU0ejiU0hyw/s1600/Max_Tegmark.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="315" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5L2UuMV5PGxcBvbIvw5LbzF4BlABn4OXncoUVaIdBgOXP3xR4sZDubZAALwF2cj9RfEZF9kjUtPceCuqJ-PWn0hkd1KvGQkOMtzdpC5Uce7xyLwCVP7FGcMROFpwkwNYQU0ejiU0hyw/s320/Max_Tegmark.jpg" width="320" /></a></div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;"><b><br />
<br />
Max Tegmark</b> (born 5 May 1967) is a <a href="http://en.wikipedia.org/wiki/Sweden" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sweden">Swedish</a>-<a href="http://en.wikipedia.org/wiki/United_States" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="United States">American</a> <a href="http://en.wikipedia.org/wiki/Physical_cosmology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physical cosmology">cosmologist</a>. Tegmark is a <a href="http://en.wikipedia.org/wiki/Professor" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Professor">professor</a> at the <a href="http://en.wikipedia.org/wiki/Massachusetts_Institute_of_Technology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Massachusetts Institute of Technology">Massachusetts Institute of Technology</a> and belongs to the scientific directorate of the <a href="http://en.wikipedia.org/wiki/Foundational_Questions_Institute" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Foundational Questions Institute">Foundational Questions Institute</a>.</div><table class="toc" id="toc" style="background-color: white; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; border-image: initial; border-left-color: rgb(170, 170, 170); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(170, 170, 170); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(170, 170, 170); border-top-style: solid; border-top-width: 1px; font-family: sans-serif; font-size: 12px; line-height: 19px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;"><tbody>
<tr><td><div id="toctitle" style="direction: ltr; text-align: center;"><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-style: none; border-color: initial; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; border-width: initial; display: inline; font-size: 12px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: auto;">Contents</h2> <span class="toctoggle" style="font-size: 11px;"> </span></div><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;"><li class="toclevel-1 tocsection-1" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark#Biography" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1</span> <span class="toctext">Biography</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li class="toclevel-2 tocsection-2" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark#Early_life" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1.1</span> <span class="toctext">Early life</span></a></li>
<li class="toclevel-2 tocsection-3" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark#Career" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1.2</span> <span class="toctext">Career</span></a></li>
<li class="toclevel-2 tocsection-4" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark#Personal_life" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1.3</span> <span class="toctext">Personal life</span></a></li>
</ul></li>
<li class="toclevel-1 tocsection-5" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark#In_the_media" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2</span> <span class="toctext">In the media</span></a></li>
<li class="toclevel-1 tocsection-6" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark#Notes" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3</span> <span class="toctext">Notes</span></a></li>
<li class="toclevel-1 tocsection-7" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark#External_links" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4</span> <span class="toctext">External links</span></a></li>
</ul></td></tr>
</tbody></table><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Max_Tegmark&action=edit&section=1&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Biography">edit</a>]</span><span class="mw-headline" id="Biography">Biography</span></h2><h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 17px; line-height: 19px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Max_Tegmark&action=edit&section=2&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Early life">edit</a>]</span><span class="mw-headline" id="Early_life">Early life</span></h3><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">Tegmark was born as Max Shapiro in Sweden, son of Karin Tegmark and <a href="http://en.wikipedia.org/wiki/Harold_S._Shapiro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Harold S. Shapiro">Harold S. Shapiro</a>, studied at the <a href="http://en.wikipedia.org/wiki/Royal_Institute_of_Technology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Royal Institute of Technology">Royal Institute of Technology</a> in <a href="http://en.wikipedia.org/wiki/Stockholm" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Stockholm">Stockholm</a>, and later received his Ph.D. from the <a href="http://en.wikipedia.org/wiki/University_of_California,_Berkeley" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="University of California, Berkeley">University of California, Berkeley</a>. After having worked at the <a href="http://en.wikipedia.org/wiki/University_of_Pennsylvania" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="University of Pennsylvania">University of Pennsylvania</a>, he is now at the <a href="http://en.wikipedia.org/wiki/Massachusetts_Institute_of_Technology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Massachusetts Institute of Technology">Massachusetts Institute of Technology</a>. While still in high-school, Max wrote, and sold commercially, together with school buddy Magnus Bodin a word processor written in pure machine code <sup class="reference" id="cite_ref-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark#cite_note-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[1]</a></sup> for the Swedish 8-bit computer <a class="mw-redirect" href="http://en.wikipedia.org/wiki/ABC80" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ABC80">ABC80</a>.</div><h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 17px; line-height: 19px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Max_Tegmark&action=edit&section=3&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Career">edit</a>]</span><span class="mw-headline" id="Career">Career</span></h3><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">His research has focused on <a href="http://en.wikipedia.org/wiki/Cosmology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cosmology">cosmology</a>, combining theoretical work with new measurements to place constraints on cosmological models and their free parameters, often in collaboration with experimentalists. He has over 200 publications, of which 9 have been cited over 500 times.<sup class="reference" id="cite_ref-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark#cite_note-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup> He has developed data analysis tools based on information theory and applied them to <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Cosmic_Microwave_Background" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cosmic Microwave Background">Cosmic Microwave Background</a> experiments such as <a class="mw-redirect" href="http://en.wikipedia.org/wiki/COBE" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="COBE">COBE</a>, <a href="http://en.wikipedia.org/wiki/QMAP" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="QMAP">QMAP</a>, and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/WMAP" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="WMAP">WMAP</a>, and to galaxy redshift surveys such as the <a href="http://en.wikipedia.org/wiki/Las_Campanas_Redshift_Survey" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Las Campanas Redshift Survey">Las Campanas Redshift Survey</a>, the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/2dF" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="2dF">2dF</a> Survey and the <a href="http://en.wikipedia.org/wiki/Sloan_Digital_Sky_Survey" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sloan Digital Sky Survey">Sloan Digital Sky Survey</a>.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">With Daniel Eisenstein and Wayne Hu, he introduced the idea of using <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Baryon_Acoustic_Oscillations" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Baryon Acoustic Oscillations">Baryon Acoustic Oscillations</a> as a <a href="http://en.wikipedia.org/wiki/Standard_ruler" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Standard ruler">Standard Ruler</a>.<sup class="reference" id="cite_ref-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark#cite_note-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup><sup class="noprint Template-Fact" style="line-height: 1em; white-space: nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:No_original_research#Primary.2C_secondary_and_tertiary_sources" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:No original research"><span title="This claim needs references to reliable secondary sources. from January 2012">non-primary</span></a> source <a href="http://en.wikipedia.org/wiki/Wikipedia:Verifiability" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:Verifiability">needed</a></i>]</sup></div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">With Angelica de Oliveira-Costa and Andrew Hamilton, he discovered the anomalous multipole alignment in the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/WMAP" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="WMAP">WMAP</a> data sometimes referred to as the "axis of evil".<sup class="reference" id="cite_ref-3" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark#cite_note-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[4]</a></sup><sup class="noprint Template-Fact" style="line-height: 1em; white-space: nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:No_original_research#Primary.2C_secondary_and_tertiary_sources" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:No original research"><span title="This claim needs references to reliable secondary sources. from January 2012">non-primary</span></a> source <a href="http://en.wikipedia.org/wiki/Wikipedia:Verifiability" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:Verifiability">needed</a></i>]</sup></div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">Tegmark has also formulated the "Ultimate ensemble theory of everything", whose only postulate is that "all structures that exist mathematically exist also physically". This simple theory, with no free parameters at all, suggests that in those structures complex enough to contain self-aware substructures (SASs), these SASs will subjectively perceive themselves as existing in a physically "real" world. This idea is formalized as the "<a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mathematical universe hypothesis">Mathematical universe hypothesis</a>" <sup class="reference" id="cite_ref-4" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark#cite_note-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[5]</a></sup></div><h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 17px; line-height: 19px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Max_Tegmark&action=edit&section=4&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Personal life">edit</a>]</span><span class="mw-headline" id="Personal_life">Personal life</span></h3><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">He was married to astrophysicist Angelica de Oliveira-Costa in 1997, and divorced in 2009. They have two sons, Philip and Alexander.<sup class="reference" id="cite_ref-5" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark#cite_note-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[6]</a></sup></div><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Max_Tegmark&action=edit&section=5&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: In the media">edit</a>]</span><span class="mw-headline" id="In_the_media">In the media</span></h2><ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;">In 2006, Tegmark was one of fifty scientists interviewed by <i><a href="http://en.wikipedia.org/wiki/New_Scientist" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="New Scientist">New Scientist</a></i> about their predictions for the future. His prediction: "In 50 years, you may be able to buy T-shirts on which are printed equations describing the unified laws of our universes."<sup class="reference" id="cite_ref-6" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark#cite_note-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[7]</a></sup></li>
<li style="margin-bottom: 0.1em;">Tegmark appears in the documentary <i><a href="http://en.wikipedia.org/wiki/Parallel_Worlds,_Parallel_Lives" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Parallel Worlds, Parallel Lives">Parallel Worlds, Parallel Lives</a></i> in which he is interviewed by <a href="http://en.wikipedia.org/wiki/Mark_Oliver_Everett" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mark Oliver Everett">Mark Oliver Everett</a>, son of the founder of the <a href="http://en.wikipedia.org/wiki/Many-worlds_interpretation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Many-worlds interpretation">many-worlds interpretation</a> of quantum mechanics,<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Hugh_Everett" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hugh Everett">Hugh Everett</a>.</li>
<li style="margin-bottom: 0.1em;">Tegmark also appears in "Who's afraid of a big black hole?","What time is it?", "To Infinity and Beyond", "Is Everything We Know About The Universe Wrong?" and "What is Reality?", all part of the BBC's <i><a href="http://en.wikipedia.org/wiki/Horizon_(BBC_TV_series)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Horizon (BBC TV series)">Horizon</a></i> scientific series of programmes.</li>
</ul><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Max_Tegmark&action=edit&section=6&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Notes">edit</a>]</span><span class="mw-headline" id="Notes">Notes</span></h2><div class="reflist" style="background-color: white; font-family: sans-serif; font-size: 12px; line-height: 19px; list-style-type: decimal; margin-bottom: 0.5em;"><ol class="references" style="line-height: 1.5em; list-style-image: none; list-style-type: inherit; margin-bottom: 0.5em; margin-left: 3.2em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li id="cite_note-0" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Max_Tegmark#cite_ref-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external autonumber" href="http://x42.com/teddy/" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">[1]</a></span></li>
<li id="cite_note-1" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Max_Tegmark#cite_ref-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external autonumber" href="http://www-spires.slac.stanford.edu/spires/find/hep/wwwcitesummary?rawcmd=FIND+a+tegmark" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">[2]</a></span></li>
<li id="cite_note-2" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Max_Tegmark#cite_ref-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">Eisenstein, Daniel J.; Hu, Wayne, Tegmark, Max. "Cosmic Complementarity: <img alt="H_0" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/c/5/7c5081abe6c2100f0e44396b6ac51661.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /> and <img alt="\Omega_m" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/9/d/7/9d70b5b3c87e464d74c3a6bb31ec8eb6.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /> from Combining Cosmic Microwave Background Experiments and Redshift Surveys". <i>The Astrophysical Journal</i><b>504</b> (2): L57–L60. <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/astro-ph/9805239" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">astro-ph/9805239</a>. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1998ApJ...504L..57E" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1998ApJ...504L..57E</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1086%2F311582" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1086/311582</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Cosmic+Complementarity%3A+%7FUNIQ6748e568c300b4-math-00000018-QINU%7F+and+%7FUNIQ6748e568c300b4-math-00000019-QINU%7F+from+Combining+Cosmic+Microwave+Background+Experiments+and+Redshift+Surveys&rft.jtitle=The+Astrophysical+Journal&rft.aulast=Eisenstein&rft.aufirst=Daniel+J.&rft.au=Eisenstein%2C%26%2332%3BDaniel+J.&rft.volume=504&rft.issue=2&rft.pages=L57%E2%80%93L60&rft_id=info:arxiv/astro-ph%2F9805239&rft_id=info:bibcode/1998ApJ...504L..57E&rft_id=info:doi/10.1086%2F311582&rfr_id=info:sid/en.wikipedia.org:Max_Tegmark"></span></span></li>
<li id="cite_note-3" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Max_Tegmark#cite_ref-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">Tegmark, Max; de Oliveira-Costa, Angélica, Hamilton, Andrew (1 December 2003). "High resolution foreground cleaned CMB map from WMAP". <i>Physical Review D</i> <b>68</b> (12). <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/astro-ph/0302496" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">astro-ph/0302496</a>.<a href="http://en.wikipedia.org/wiki/Bibcode" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/2003PhRvD..68l3523T" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">2003PhRvD..68l3523T</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1103%2FPhysRevD.68.123523" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1103/PhysRevD.68.123523</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=High+resolution+foreground+cleaned+CMB+map+from+WMAP&rft.jtitle=Physical+Review+D&rft.aulast=Tegmark&rft.aufirst=Max&rft.au=Tegmark%2C%26%2332%3BMax&rft.date=1+December+2003&rft.volume=68&rft.issue=12&rft_id=info:arxiv/astro-ph%2F0302496&rft_id=info:bibcode/2003PhRvD..68l3523T&rft_id=info:doi/10.1103%2FPhysRevD.68.123523&rfr_id=info:sid/en.wikipedia.org:Max_Tegmark"></span></span></li>
<li id="cite_note-4" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Max_Tegmark#cite_ref-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">Tegmark, Max. "The Mathematical Universe". <i>Foundations of Physics</i> <b>38</b> (2): 101–150. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/2008FoPh...38..101T" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">2008FoPh...38..101T</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1007%2Fs10701-007-9186-9" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1007/s10701-007-9186-9</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Mathematical+Universe&rft.jtitle=Foundations+of+Physics&rft.aulast=Tegmark&rft.aufirst=Max&rft.au=Tegmark%2C%26%2332%3BMax&rft.volume=38&rft.issue=2&rft.pages=101%E2%80%93150&rft_id=info:bibcode/2008FoPh...38..101T&rft_id=info:doi/10.1007%2Fs10701-007-9186-9&rfr_id=info:sid/en.wikipedia.org:Max_Tegmark"></span> a short version of which is available at <a class="external text" href="http://arxiv.org/abs/0709.4024" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;"><i>Shut up and calculate.</i></a> (in reference to David Mermin's famous quote "shut up and calculate" <a class="external autonumber" href="http://physicstoday.org/journals/doc/PHTOAD-ft/vol_57/iss_5/10_1.shtml" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">[3]</a>.</span></li>
<li id="cite_note-5" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Max_Tegmark#cite_ref-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external autonumber" href="http://space.mit.edu/home/tegmark/personal.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">[4]</a></span></li>
<li id="cite_note-6" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Max_Tegmark#cite_ref-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external autonumber" href="http://www.newscientist.com/article/mg19225780.084-max-tegmark-forecasts-the-future.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">[5]</a></span></li>
</ol></div><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Max_Tegmark&action=edit&section=7&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: External links">edit</a>]</span><span class="mw-headline" id="External_links">External links</span></h2><table class="metadata mbox-small plainlinks" style="background-color: #f9f9f9; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; border-image: initial; border-left-color: rgb(170, 170, 170); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(170, 170, 170); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(170, 170, 170); border-top-style: solid; border-top-width: 1px; clear: right; float: right; font-family: sans-serif; font-size: 11px; line-height: 1.25em; margin-bottom: 4px; margin-left: 1em; margin-right: 0px; margin-top: 4px; width: 238px;"><tbody>
<tr><td class="mbox-image" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; padding-bottom: 2px; padding-left: 0.9em; padding-right: 0px; padding-top: 2px; text-align: center;"><img alt="" height="40" src="http://upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Wikiquote-logo-en.svg/40px-Wikiquote-logo-en.svg.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" width="40" /></td><td class="mbox-text" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; padding-bottom: 0.25em; padding-left: 0.9em; padding-right: 0.9em; padding-top: 0.25em; width: 160px;">Wikiquote has a collection of quotations related to: <i><b><a class="external text" href="http://en.wikiquote.org/wiki/Special:Search/Max_Tegmark" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 0% 0%; background-repeat: no-repeat no-repeat; color: #663366; padding-bottom: 0px !important; padding-left: 0px !important; padding-right: 13px; padding-top: 0px !important; text-decoration: none;">Max Tegmark</a></b></i></td></tr>
</tbody></table><ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;"><a class="external text" href="http://space.mit.edu/home/tegmark/" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Max Tegmark's website</a></li>
</ul><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;"><br />
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In <a href="http://en.wikipedia.org/wiki/Physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physics">physics</a> and <a href="http://en.wikipedia.org/wiki/Cosmology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cosmology">cosmology</a>, the <b>mathematical universe hypothesis</b> (<b>MUH</b>), also known as the <b>Ultimate Ensemble</b>, is a speculative "theory of everything" (TOE) proposed by the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Theoretical_physicist" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theoretical physicist">theoretical physicist</a>, <a href="http://en.wikipedia.org/wiki/Max_Tegmark" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Max Tegmark">Max Tegmark</a>.<sup class="reference" id="cite_ref-Tegmark1998_0-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-Tegmark1998-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[1]</a></sup></div><table class="toc" id="toc" style="background-color: white; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; border-image: initial; border-left-color: rgb(170, 170, 170); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(170, 170, 170); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(170, 170, 170); border-top-style: solid; border-top-width: 1px; font-family: sans-serif; font-size: 12px; line-height: 19px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;"><tbody>
<tr><td><div id="toctitle" style="direction: ltr; text-align: center;"><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-style: none; border-color: initial; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; border-width: initial; display: inline; font-size: 12px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: auto;">Contents</h2> <span class="toctoggle" style="font-size: 11px;"> [<a class="internal" href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#" id="togglelink" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">hide</a>] </span></div><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;"><li class="toclevel-1 tocsection-1" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#Description" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1</span> <span class="toctext">Description</span></a></li>
<li class="toclevel-1 tocsection-2" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#Criticisms_and_responses" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2</span> <span class="toctext">Criticisms and responses</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li class="toclevel-2 tocsection-3" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#Definition_of_the_Ensemble" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.1</span> <span class="toctext">Definition of the Ensemble</span></a></li>
<li class="toclevel-2 tocsection-4" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#Consistency_with_G.C3.B6del.27s_theorem" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.2</span> <span class="toctext">Consistency with Gödel's theorem</span></a></li>
<li class="toclevel-2 tocsection-5" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#Observability" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.3</span> <span class="toctext">Observability</span></a></li>
<li class="toclevel-2 tocsection-6" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#Plausibility_of_Radical_Platonism" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.4</span> <span class="toctext">Plausibility of Radical Platonism</span></a></li>
<li class="toclevel-2 tocsection-7" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#Coexistence_of_all_mathematical_structures" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.5</span> <span class="toctext">Coexistence of all mathematical structures</span></a></li>
<li class="toclevel-2 tocsection-8" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#Consistency_with_our_.22simple_universe.22" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2.6</span> <span class="toctext">Consistency with our "simple universe"</span></a></li>
</ul></li>
<li class="toclevel-1 tocsection-9" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#See_also" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-10" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#References" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-11" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#Further_reading" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5</span> <span class="toctext">Further reading</span></a></li>
<li class="toclevel-1 tocsection-12" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#External_links" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6</span> <span class="toctext">External links</span></a></li>
</ul></td></tr>
</tbody></table><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_universe_hypothesis&action=edit&section=1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Description">edit</a>]</span><span class="mw-headline" id="Description">Description</span></h2><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">Tegmark's mathematical universe hypothesis (MUH) is: <i>Our external physical reality is a mathematical structure</i>. That is, the universe <i>is</i> mathematics in a well-defined sense, and that "in those [worlds] complex enough to contain self-aware substructures [they] will subjectively perceive themselves as existing in a physically 'real' world".<sup class="reference" id="cite_ref-Tegmark2008_1-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-Tegmark2008-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup><sup class="reference" id="cite_ref-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup> The hypothesis suggests that worlds corresponding to different sets of initial conditions, physical constants, or altogether different equations may be considered equally real. Tegmark elaborates the MUH into the Computable Universe Hypothesis (CUH), which posits that all computable mathematical structures exist.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">The theory can be considered a form of <a href="http://en.wikipedia.org/wiki/Platonism" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Platonism">Platonism</a> in that it posits the existence of mathematical entities, can be considered a <a href="http://en.wikipedia.org/wiki/Philosophy_of_mathematics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Philosophy of mathematics">mathematical monism</a> in that it denies that anything exists except mathematical objects, and can be considered a formal expression of <a href="http://en.wikipedia.org/wiki/Scientific_realism#Structural_realism" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Scientific realism">ontic structural realism</a>.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">Tegmark claims that the hypothesis has no free parameters and is not observationally ruled out. Thus, he reasons, it is preferred over other theories-of-everything by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Occam%27s_Razor" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Occam's Razor">Occam's Razor</a>. He suggests conscious experience would take the form of mathematical "self-aware substructures" that exist in a physically "'real'" world.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">The hypothesis is related to the <a href="http://en.wikipedia.org/wiki/Anthropic_principle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Anthropic principle">anthropic principle</a> and to Tegmark's categorization of theories of the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Multiverse_(science)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Multiverse (science)">multiverse</a>.<sup class="reference" id="cite_ref-Tegmark2003_3-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-Tegmark2003-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[4]</a></sup></div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">Andreas Albrecht of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Imperial_College" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Imperial College">Imperial College</a> in London called it a "provocative" solution to one of the central problems facing physics. Although he "wouldn't dare" go so far as to say he believes it, he noted that "it's actually quite difficult to construct a theory where everything we see is all there is".<sup class="reference" id="cite_ref-4" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[5]</a></sup></div><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_universe_hypothesis&action=edit&section=2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Criticisms and responses">edit</a>]</span><span class="mw-headline" id="Criticisms_and_responses">Criticisms and responses</span></h2><h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 17px; line-height: 19px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_universe_hypothesis&action=edit&section=3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Definition of the Ensemble">edit</a>]</span><span class="mw-headline" id="Definition_of_the_Ensemble">Definition of the Ensemble</span></h3><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;"><a href="http://en.wikipedia.org/wiki/J%C3%BCrgen_Schmidhuber" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Jürgen Schmidhuber">Jürgen Schmidhuber</a> <sup class="reference" id="cite_ref-5" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[6]</a></sup> argues that "Although Tegmark suggests that '... all mathematical structures are a priori given equal statistical weight', there is no way of assigning equal nonvanishing probability to all (infinitely many) mathematical structures". Schmidhuber puts forward a more restricted ensemble which admits only universe representations describable by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Constructive_mathematics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Constructive mathematics">constructive mathematics</a>, that is, <a href="http://en.wikipedia.org/wiki/Computer_program" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Computer program">computer programs</a>. He explicitly includes universe representations describable by non-<a href="http://en.wikipedia.org/wiki/Halting_problem" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Halting problem">halting programs</a> whose output bits converge after finite time, although the convergence time itself may not be predictable by a halting program, due to <a href="http://en.wikipedia.org/wiki/Kurt_G%C3%B6del" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Kurt Gödel">Kurt Gödel</a>'s limitations.<sup class="reference" id="cite_ref-6" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[7]</a></sup> In response, Tegmark notes <sup class="reference" id="cite_ref-Tegmark2008_1-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-Tegmark2008-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup> (sec. V.E) that the measure over all universes has not yet been constructed for the <a href="http://en.wikipedia.org/wiki/String_theory_landscape" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="String theory landscape">String theory landscape</a> either, so this should not be regarded as a "show-stopper".</div><h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 17px; line-height: 19px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_universe_hypothesis&action=edit&section=4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Consistency with Gödel's theorem">edit</a>]</span><span class="mw-headline" id="Consistency_with_G.C3.B6del.27s_theorem">Consistency with Gödel's theorem</span></h3><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">It has also been suggested that the MUH is inconsistent with <a class="mw-redirect" href="http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorem" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gödel's incompleteness theorem">Gödel's incompleteness theorem</a>. In a three-way debate between Tegmark and fellow physicists <a href="http://en.wikipedia.org/wiki/Piet_Hut" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Piet Hut">Piet Hut</a> and <a class="new" href="http://en.wikipedia.org/w/index.php?title=Mark_Alford&action=edit&redlink=1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Mark Alford (page does not exist)">Mark Alford</a>,<sup class="reference" id="cite_ref-HAT_7-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-HAT-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[8]</a></sup> the "secularist" (Alford) states that "the methods allowed by formalists cannot prove all the theorems in a sufficiently powerful system... The idea that math is "out there" is incompatible with the idea that it consists of formal systems." Tegmark's response in <sup class="reference" id="cite_ref-HAT_7-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-HAT-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[8]</a></sup> (sec VI.A.1) is to offer a new hypothesis "that only Godel-complete (fully decidable) mathematical structures have physical existence. This drastically shrinks the Level IV multiverse, essentially placing an upper limit on complexity, and may have the attractive side effect of explaining the relative simplicity of our universe." Tegmark goes on to note that although conventional theories in physics are Godel-undecidable, the actual mathematical structure describing our world could still be Godel-complete, and "could in principle contain observers capable of thinking about Godel-incomplete mathematics, just as finite-state digital computers can prove certain theorems about Godel-incomplete formal systems like Peano arithmetic." In<sup class="reference" id="cite_ref-Tegmark2008_1-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-Tegmark2008-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup> (sec. VII) he gives a more detailed response, proposing as an alternative to MUH the more restricted "Computable Universe Hypothesis" (CUH) which only includes mathematical structures that are simple enough that Gödel's theorem does not require them to contain any undecidable/uncomputable theorems. Tegmark admits that this approach faces "serious challeges", including (a) it excludes much of the mathematical landscape; (b) the measure on the space of allowed theories may itself be uncomputable; and (c) "virtually all historically successful theories of physics violate the CUH".</div><h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 17px; line-height: 19px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_universe_hypothesis&action=edit&section=5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Observability">edit</a>]</span><span class="mw-headline" id="Observability">Observability</span></h3><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">Stoeger, Ellis, and Kircher <sup class="reference" id="cite_ref-8" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[9]</a></sup> (sec. 7) note that in a true multiverse theory, "the universes are then completely disjoint and nothing that happens in any one of them is causally linked to what happens in any other one. This lack of any causal connection in such multiverses really places them beyond any scientific support". Ellis <sup class="reference" id="cite_ref-9" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[10]</a></sup> (p29) specifically criticizes the MUH, stating that an infinite ensemble of completely disconnected universes is "completely untestable, despite hopeful remarks sometimes made, see, e.g., Tegmark (1998)." Tegmark maintains that MUH is testable, stating that it predicts (a) that "physics research will uncover mathematical regularities in nature", and (b) by assuming that we occupy a typical member of the multiverse of mathematical structures, one could "start testing multiverse predictions by assessing how typical our universe is" (,<sup class="reference" id="cite_ref-Tegmark2008_1-3" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-Tegmark2008-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup> sec. VIII.C).</div><h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 17px; line-height: 19px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_universe_hypothesis&action=edit&section=6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Plausibility of Radical Platonism">edit</a>]</span><span class="mw-headline" id="Plausibility_of_Radical_Platonism">Plausibility of Radical Platonism</span></h3><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">The MUH is based on the Radical Platonist view that math is an external reality (<sup class="reference" id="cite_ref-Tegmark2008_1-4" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-Tegmark2008-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup> sec V.C). However, Jannes <sup class="reference" id="cite_ref-Jannes2009_10-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-Jannes2009-10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[11]</a></sup> argues that "mathematics is at least in part a human construction", on the basis that if it is an external reality then "non-human intelligent beings should exist that understand the language of advanced mathematics. However, none of the non-human intelligent beings that we know of confirm the status of (advanced) mathematics as an objective language." In <sup class="reference" id="cite_ref-HAT_7-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-HAT-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[8]</a></sup> the secularist argues (sec. VI.A) that math is evolving over time, there is "no reason to think it is converging to a definite structure, with fixed questions and established ways to address them", and also that "The Radical Platonist position is just another metaphysical theory like solipsism... In the end the metaphysics just demands that we use a different language for saying what we already knew." Tegmark responds (sec VI.A.1) that "The notion of a mathematical structure is rigorously defined in any book on Model Theory", and that non-human mathematics would only differ from our own "because we are uncovering a different part of what is in fact a consistent and unified picture, so math is converging in this sense."</div><h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 17px; line-height: 19px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_universe_hypothesis&action=edit&section=7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Coexistence of all mathematical structures">edit</a>]</span><span class="mw-headline" id="Coexistence_of_all_mathematical_structures">Coexistence of all mathematical structures</span></h3><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;"><a href="http://en.wikipedia.org/wiki/Don_Page_(physicist)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Don Page (physicist)">Don Page</a> has argued <sup class="reference" id="cite_ref-11" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[12]</a></sup> (sec 4) that "At the ultimate level, there can be only one world and, if mathematical structures are broad enough to include all possible worlds or at least our own, there must be one unique mathematical structure that describes ultimate reality. So I think it is logical nonsense to talk of Level 4 in the sense of the co-existence of all mathematical structures." Tegmark responds (,<sup class="reference" id="cite_ref-Tegmark2008_1-5" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-Tegmark2008-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup> sec. V.E) that "this is less inconsistent with Level IV than it may sound, since many mathematical structures decompose into unrelated substructures, and separate ones can be unified."</div><h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 17px; line-height: 19px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_universe_hypothesis&action=edit&section=8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Consistency with our "simple universe"">edit</a>]</span><span class="mw-headline" id="Consistency_with_our_.22simple_universe.22">Consistency with our "simple universe"</span></h3><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;"><a href="http://en.wikipedia.org/wiki/Alexander_Vilenkin" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Alexander Vilenkin">Alexander Vilenkin</a> comments <sup class="reference" id="cite_ref-Vilenkin2006_12-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-Vilenkin2006-12" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[13]</a></sup> (Ch.19, p203) that "the number of mathematical structures increases with increasing complexity, suggesting that 'typical' structures should be horrendously large and cumbersome. This seems to be in conflict with the beauty and simplicity of the theories describing our world". He goes on to note (footnote 8, p222) that Tegmark's solution to this problem, the assigning of lower "weights" to the more complex structures (<sup class="reference" id="cite_ref-Tegmark2003_3-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_note-Tegmark2003-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[4]</a></sup> sec. V.B) seems arbitrary ("Who determines the weights?") and may not be logically consistent ("It seems to introduce an additional mathematical structure, but all of them are supposed to be already included in the set").</div><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_universe_hypothesis&action=edit&section=9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: See also">edit</a>]</span><span class="mw-headline" id="See_also">See also</span></h2><ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Cosmology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cosmology">Cosmology</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Digital_physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital physics">Digital physics</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Impossible_world" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Impossible world">Impossible world</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Modal_realism" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Modal realism">Modal realism</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Multiverse" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Multiverse">Multiverse</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Ontology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Ontology">Ontology</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/String_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="String theory">String theory</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Theory_of_everything" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theory of everything">Theory of everything</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="The Unreasonable Effectiveness of Mathematics in the Natural Sciences">The Unreasonable Effectiveness of Mathematics in the Natural Sciences</a></li>
</ul><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_universe_hypothesis&action=edit&section=10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: References">edit</a>]</span><span class="mw-headline" id="References">References</span></h2><div class="reflist references-column-width" style="-webkit-column-width: 30em; background-color: white; font-family: sans-serif; font-size: 12px; line-height: 19px; list-style-type: decimal; margin-bottom: 0.5em;"><ol class="references" style="line-height: 1.5em; list-style-image: none; list-style-type: inherit; margin-bottom: 0.5em; margin-left: 3.2em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li id="cite_note-Tegmark1998-0" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-Tegmark1998_0-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">Tegmark, Max (November 1998). "Is "the Theory of Everything" Merely the Ultimate Ensemble Theory?". <i>Annals of Physics</i> <b>270</b> (1): 1–51. <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/gr-qc/9704009" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">gr-qc/9704009</a>. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a><a class="external text" href="http://adsabs.harvard.edu/abs/1998AnPhy.270....1T" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1998AnPhy.270....1T</a>. <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1006%2Faphy.1998.5855" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1006/aphy.1998.5855</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Is+%22the+Theory+of+Everything%22+Merely+the+Ultimate+Ensemble+Theory%3F&rft.jtitle=Annals+of+Physics&rft.aulast=Tegmark&rft.aufirst=Max&rft.au=Tegmark%2C%26%2332%3BMax&rft.date=November+1998&rft.volume=270&rft.issue=1&rft.pages=1%E2%80%9351&rft_id=info:arxiv/gr-qc%2F9704009&rft_id=info:bibcode/1998AnPhy.270....1T&rft_id=info:doi/10.1006%2Faphy.1998.5855&rfr_id=info:sid/en.wikipedia.org:Mathematical_universe_hypothesis"></span></span></li>
<li id="cite_note-Tegmark2008-1" style="margin-bottom: 0.1em;">^ <a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-Tegmark2008_1-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-Tegmark2008_1-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-Tegmark2008_1-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>c</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-Tegmark2008_1-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>d</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-Tegmark2008_1-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>e</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-Tegmark2008_1-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>f</b></i></sup></a> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">Tegmark, Max (February 2008). "The Mathematical Universe". <i>Foundations of Physics</i> <b>38</b> (2): 101–150.<a href="http://en.wikipedia.org/wiki/ArXiv" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/0704.0646" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">0704.0646</a>. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/2008FoPh...38..101T" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">2008FoPh...38..101T</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1007%2Fs10701-007-9186-9" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1007/s10701-007-9186-9</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Mathematical+Universe&rft.jtitle=Foundations+of+Physics&rft.aulast=Tegmark&rft.aufirst=Max&rft.au=Tegmark%2C%26%2332%3BMax&rft.date=February+2008&rft.volume=38&rft.issue=2&rft.pages=101%E2%80%93150&rft_id=info:arxiv/0704.0646&rft_id=info:bibcode/2008FoPh...38..101T&rft_id=info:doi/10.1007%2Fs10701-007-9186-9&rfr_id=info:sid/en.wikipedia.org:Mathematical_universe_hypothesis"></span></span></li>
<li id="cite_note-2" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Tegmark (1998), p. 1.</span></li>
<li id="cite_note-Tegmark2003-3" style="margin-bottom: 0.1em;">^ <a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-Tegmark2003_3-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-Tegmark2003_3-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;">Tegmark, Max (2003). "Parallel Universes". In Barrow, J.D.; Davies, P.C.W.' & Harper, C.L.. <i>"Science and Ultimate Reality: From Quantum to Cosmos" honoring John Wheeler's 90th birthday</i>. Cambridge University Press. <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/astro-ph/0302131" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">astro-ph/0302131</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Parallel+Universes&rft.atitle=%22Science+and+Ultimate+Reality%3A+From+Quantum+to+Cosmos%22+honoring+John+Wheeler%27s+90th+birthday&rft.aulast=Tegmark&rft.aufirst=Max&rft.au=Tegmark%2C%26%2332%3BMax&rft.date=2003&rft.pub=Cambridge+University+Press&rft_id=info:arxiv/astro-ph%2F0302131&rfr_id=info:sid/en.wikipedia.org:Mathematical_universe_hypothesis"></span></span></li>
<li id="cite_note-4" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">Chown, Markus (June 1998). "Anything goes". <i>New Scientist</i><b>158</b> (2157 url=<a class="external free" href="http://space.mit.edu/home/tegmark/toe_press.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://space.mit.edu/home/tegmark/toe_press.html</a>).</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Anything+goes&rft.jtitle=New+Scientist&rft.aulast=Chown&rft.aufirst=Markus&rft.au=Chown%2C%26%2332%3BMarkus&rft.date=June+1998&rft.volume=158&rft.issue=2157+url%3Dhttp%3A%2F%2Fspace.mit.edu%2Fhome%2Ftegmark%2Ftoe_press.html&rfr_id=info:sid/en.wikipedia.org:Mathematical_universe_hypothesis"></span></span></li>
<li id="cite_note-5" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a href="http://en.wikipedia.org/wiki/J%C3%BCrgen_Schmidhuber" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Jürgen Schmidhuber">J. Schmidhuber</a> (2000) "<a class="external text" href="http://arxiv.org/abs/quant-ph/0011122" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Algorithmic Theories of Everything.</a>"</span></li>
<li id="cite_note-6" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/J%C3%BCrgen_Schmidhuber" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Jürgen Schmidhuber">Schmidhuber, J.</a> (2002). <a class="external text" href="http://www.idsia.ch/~juergen/kolmogorov.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">"Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit"</a>. <i>International Journal of Foundations of Computer Science</i> <b>13</b> (4): 587–612.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1142%2FS0129054102001291" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1142/S0129054102001291</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Hierarchies+of+generalized+Kolmogorov+complexities+and+nonenumerable+universal+measures+computable+in+the+limit&rft.jtitle=International+Journal+of+Foundations+of+Computer+Science&rft.aulast=Schmidhuber&rft.aufirst=J.&rft.au=Schmidhuber%2C%26%2332%3BJ.&rft.date=2002&rft.volume=13&rft.issue=4&rft.pages=587%E2%80%93612&rft_id=info:doi/10.1142%2FS0129054102001291&rft_id=http%3A%2F%2Fwww.idsia.ch%2F%7Ejuergen%2Fkolmogorov.html&rfr_id=info:sid/en.wikipedia.org:Mathematical_universe_hypothesis"></span></span></li>
<li id="cite_note-HAT-7" style="margin-bottom: 0.1em;">^ <a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-HAT_7-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-HAT_7-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>b</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-HAT_7-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><sup style="line-height: 1em;"><i><b>c</b></i></sup></a> <span class="reference-text"><span class="citation Journal" style="word-wrap: break-word;">Hut, P.; Alford, M.; Tegmark, M. (2006). "On Math, Matter and Mind". <i>Foundations of Physics</i> <b>36</b>: 765–94.<a href="http://en.wikipedia.org/wiki/ArXiv" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/physics/0510188" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">physics/0510188</a>. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/2006FoPh...36..765H" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">2006FoPh...36..765H</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1007%2Fs10701-006-9048-x" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1007/s10701-006-9048-x</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+Math%2C+Matter+and+Mind&rft.jtitle=Foundations+of+Physics&rft.aulast=Hut&rft.aufirst=P.&rft.au=Hut%2C%26%2332%3BP.&rft.au=Alford%2C%26%2332%3BM.&rft.au=Tegmark%2C%26%2332%3BM.&rft.date=2006&rft.volume=36&rft.pages=765%E2%80%9394&rft_id=info:arxiv/physics%2F0510188&rft_id=info:bibcode/2006FoPh...36..765H&rft_id=info:doi/10.1007%2Fs10701-006-9048-x&rfr_id=info:sid/en.wikipedia.org:Mathematical_universe_hypothesis"></span></span></li>
<li id="cite_note-8" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">W. R. Stoeger, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/G._F._R._Ellis" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="G. F. R. Ellis">G. F. R. Ellis</a>, U. Kirchner (2006) "<a class="external text" href="http://arxiv.org/abs/astro-ph/0407329" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Multiverses and Cosmology: Philosophical Issues.</a>"</span></li>
<li id="cite_note-9" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">G.F.R. Ellis, "83 years of general relativity and cosmology: Progress and problems", Class. Quant. Grav. 16, A37-A75, 1999</span></li>
<li id="cite_note-Jannes2009-10" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-Jannes2009_10-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Gil Jannes, "Some comments on 'The Mathematical Universe'", Found. Phys. 39, 397-406, 2009 <a class="external text" href="http://arxiv.org/abs/0904.0867" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">arXiv:0904.0867</a></span></li>
<li id="cite_note-11" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">D. Page, "<a class="external text" href="http://arxiv.org/abs/hep-th/0610101" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Predictions and Tests of Multiverse Theories.</a>"</span></li>
<li id="cite_note-Vilenkin2006-12" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis#cite_ref-Vilenkin2006_12-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">A. Vilenkin (2006) <i>Many Worlds in One: The Search for Other Universes</i>. Hill and Wang, New York.</span></li>
</ol></div><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_universe_hypothesis&action=edit&section=11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Further reading">edit</a>]</span><span class="mw-headline" id="Further_reading">Further reading</span></h2><ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/J%C3%BCrgen_Schmidhuber" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Jürgen Schmidhuber">Jürgen Schmidhuber</a> (1997) "<a class="external text" href="http://www.idsia.ch/~juergen/everything/" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">A Computer Scientist's View of Life, the Universe, and Everything</a>" in C. Freksa, ed., <i>Foundations of Computer Science: Potential - Theory - Cognition</i>. Lecture Notes in Computer Science. Springer: 201-08.</li>
<li style="margin-bottom: 0.1em;"><span class="citation Journal" style="word-wrap: break-word;"><a href="http://en.wikipedia.org/wiki/Max_Tegmark" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Max Tegmark">Tegmark, Max</a> (1998). "Is the 'theory of everything' merely the ultimate ensemble theory?". <i>Annals of Physics</i> <b>270</b>: 1–51. <a href="http://en.wikipedia.org/wiki/ArXiv" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="ArXiv">arXiv</a>:<a class="external text" href="http://arxiv.org/abs/gr-qc/9704009" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">gr-qc/9704009</a>. <a href="http://en.wikipedia.org/wiki/Bibcode" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bibcode">Bibcode</a> <a class="external text" href="http://adsabs.harvard.edu/abs/1998AnPhy.270....1T" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">1998AnPhy.270....1T</a>.<a href="http://en.wikipedia.org/wiki/Digital_object_identifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Digital object identifier">doi</a>:<a class="external text" href="http://dx.doi.org/10.1006%2Faphy.1998.5855" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">10.1006/aphy.1998.5855</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Is+the+%27theory+of+everything%27+merely+the+ultimate+ensemble+theory%3F&rft.jtitle=Annals+of+Physics&rft.aulast=Tegmark&rft.aufirst=Max&rft.au=Tegmark%2C%26%2332%3BMax&rft.date=1998&rft.volume=270&rft.pages=1%E2%80%9351&rft_id=info:arxiv/gr-qc%2F9704009&rft_id=info:bibcode/1998AnPhy.270....1T&rft_id=info:doi/10.1006%2Faphy.1998.5855&rfr_id=info:sid/en.wikipedia.org:Mathematical_universe_hypothesis"></span></li>
<li style="margin-bottom: 0.1em;">-------- (2008) "<a class="external text" href="http://arxiv.org/abs/0704.0646" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">The Mathematical Universe,</a>" <i>Foundations of Physics</i> 38: 101-50.</li>
</ul><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Mathematical_universe_hypothesis&action=edit&section=12" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: External links">edit</a>]</span><span class="mw-headline" id="External_links">External links</span></h2><ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/J%C3%BCrgen_Schmidhuber" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Jürgen Schmidhuber">Jürgen Schmidhuber</a> "<a class="external text" href="http://www.idsia.ch/~juergen/computeruniverse.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">The ensemble of universes describable by constructive mathematics.</a>"</li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://space.mit.edu/home/tegmark/toe_frames.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Page maintained by Max Tegmark</a> with links to his technical and popular writings.</li>
<li style="margin-bottom: 0.1em;">"<a class="external text" href="http://www.weidai.com/everything.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">The 'Everything' mailing list</a>" (and archives). Discusses the idea that all possible universes exist.</li>
<li style="margin-bottom: 0.1em;">"<a class="external text" href="http://discovermagazine.com/2008/jul/16-is-the-universe-actually-made-of-math" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Is the universe actually made of math?</a>" Interview with Max Tegmark in <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Discover_Magazine" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Discover Magazine">Discover Magazine</a></i>.</li>
</ul>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com2tag:blogger.com,1999:blog-5303246073824127471.post-90427020162893852652012-03-16T08:21:00.003-04:002012-03-16T08:25:23.699-04:00Fred Kavli and His Institutes"The curiosity of the human being is what has brought us where we are today, and I have complete confidence that it will take us where we need to be in the future. " ... Fred Kavli<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgN2zteN9HqFiuWkfR_tiAjzbvX4A8fzKw33X2nAuVer7VpjuKJ9XVAIBM9UuY8Dd70zuSGAB0ranFBT2kGt64i87fxNUYGXgCjcQOgxMV4Va8gvt1J4zMgfbh1h6BjmAsqA3VyDvYwZQ/s1600/0309121332.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgN2zteN9HqFiuWkfR_tiAjzbvX4A8fzKw33X2nAuVer7VpjuKJ9XVAIBM9UuY8Dd70zuSGAB0ranFBT2kGt64i87fxNUYGXgCjcQOgxMV4Va8gvt1J4zMgfbh1h6BjmAsqA3VyDvYwZQ/s400/0309121332.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Poster at The M.I.T. Kavli Institute for Astrophysics and Space Science in Cambridge, MA</td></tr>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0No5bb6nGMPAJgJqRLVxsJMplg9kuSZTY8TxOuLZvBhwgNYRe2n28fSTWtcnOk-Df8VHLH6tpBplKLmw0VhTKbKc8flmjLW1WJhcJ2sU5AOguRS5ps3hZJvIxiAUAYtYluMk6N128xw/s1600/0309121343b.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0No5bb6nGMPAJgJqRLVxsJMplg9kuSZTY8TxOuLZvBhwgNYRe2n28fSTWtcnOk-Df8VHLH6tpBplKLmw0VhTKbKc8flmjLW1WJhcJ2sU5AOguRS5ps3hZJvIxiAUAYtYluMk6N128xw/s400/0309121343b.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Poster at The M.I.T. Kavli Institute for Astrophysics and Space Science in Cambridge, MA </td></tr>
</tbody></table><div style="text-align: center;"><b style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto;">Fred Kavli</b><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto;"> (born 1927) is a Norwegian and naturalized American physicist, business leader, inventor, and </span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Philanthropist" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto; text-decoration: none;" title="Philanthropist">philanthropist</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto;">. He was born in the village of Eresfjord, </span><a href="http://en.wikipedia.org/wiki/Nesset" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto; text-decoration: none;" title="Nesset">Nesset</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto;"> municipality in </span><a href="http://en.wikipedia.org/wiki/M%C3%B8re_og_Romsdal" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto; text-decoration: none;" title="Møre og Romsdal">Møre og Romsdal</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto;"> county, </span><a href="http://en.wikipedia.org/wiki/Norway" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto; text-decoration: none;" title="Norway">Norway</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto;">. Today Kavli lives in the city of </span><a href="http://en.wikipedia.org/wiki/Santa_Barbara,_California" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto; text-decoration: none;" title="Santa Barbara, California">Santa Barbara</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto;">, </span><a href="http://en.wikipedia.org/wiki/California" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto; text-decoration: none;" title="California">California</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto;"> in the United States. He established </span><a href="http://en.wikipedia.org/wiki/The_Kavli_Foundation" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; color: #0b0080; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto;" title="The Kavli Foundation">The Kavli Foundation</a><span style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; text-align: -webkit-auto;"> in the year 2000 to support basic scientific research. He has only recently appeared in the mainstream media for his work, primarily his philanthropic efforts. He is divorced and has two grown children. An avid art collector, Kavli has gathered a large collection of Norwegian oil paintings.</span></div><br />
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<table class="toc" id="toc" style="background-color: white; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; border-image: initial; border-left-color: rgb(170, 170, 170); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(170, 170, 170); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(170, 170, 170); border-top-style: solid; border-top-width: 1px; font-family: sans-serif; font-size: 12px; line-height: 19px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;"><tbody>
<tr><td><div id="toctitle" style="direction: ltr; text-align: center;"><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-style: none; border-color: initial; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; border-width: initial; display: inline; font-size: 12px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: auto;">Contents</h2> <span class="toctoggle" style="font-size: 11px;"> </span></div><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;"><li class="toclevel-1 tocsection-1" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#Kavli.27s_life" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1</span> <span class="toctext">Kavli's life</span></a></li>
<li class="toclevel-1 tocsection-2" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#Kavli_Prizes" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2</span> <span class="toctext">Kavli Prizes</span></a></li>
<li class="toclevel-1 tocsection-3" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#The_Kavli_Foundation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3</span> <span class="toctext">The Kavli Foundation</span></a></li>
<li class="toclevel-1 tocsection-4" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#Kavli_Institutes" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4</span> <span class="toctext">Kavli Institutes</span></a><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0px; margin-left: 2em; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li class="toclevel-2 tocsection-5" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#Astrophysics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.1</span> <span class="toctext">Astrophysics</span></a></li>
<li class="toclevel-2 tocsection-6" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#Nanoscience" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.2</span> <span class="toctext">Nanoscience</span></a></li>
<li class="toclevel-2 tocsection-7" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#Neuroscience" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.3</span> <span class="toctext">Neuroscience</span></a></li>
<li class="toclevel-2 tocsection-8" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#Theoretical_physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4.4</span> <span class="toctext">Theoretical physics</span></a></li>
</ul></li>
<li class="toclevel-1 tocsection-9" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#Quotes" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5</span> <span class="toctext">Quotes</span></a></li>
<li class="toclevel-1 tocsection-10" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#References" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-11" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#External_links" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">7</span> <span class="toctext">External links</span></a></li>
</ul></td></tr>
</tbody></table><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Fred_Kavli&action=edit&section=1&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Kavli's life">edit</a>]</span><span class="mw-headline" id="Kavli.27s_life">Kavli's life</span></h2><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">Kavli grew up on the family farm in the tiny Norwegian village of Eresfjord (pop. 450).</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">At 14, together with his brother Aslak, he began his first enterprise creating wood pellet fuel for cars. This was during the <a href="http://en.wikipedia.org/wiki/World_War_II" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="World War II">Second World War</a> and the <a href="http://en.wikipedia.org/wiki/Occupation_of_Norway_by_Nazi_Germany" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Occupation of Norway by Nazi Germany">Nazi occupation of Norway</a>.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">Inspired by his father's 13 years in San Francisco the young Kavli wanted to move to the US. Three days after he received his engineering <a href="http://en.wikipedia.org/wiki/Physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Physics">physics</a> degree from the <a href="http://en.wikipedia.org/wiki/Norwegian_Institute_of_Technology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Norwegian Institute of Technology">Norwegian Institute of Technology</a>(NTH) in <a href="http://en.wikipedia.org/wiki/Trondheim" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Trondheim">Trondheim</a> he left for America on the SS <i>Stavangerfjord</i>.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">Having no job or sponsor waiting for him, his visa application was initially rejected, and so in 1955 he immigrated to <a href="http://en.wikipedia.org/wiki/Montreal" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Montreal">Montreal</a>, <a href="http://en.wikipedia.org/wiki/Canada" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Canada">Canada</a> instead. The following year his visa was approved and he moved to the <a href="http://en.wikipedia.org/wiki/United_States" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="United States">United States</a>. He found work as an <a href="http://en.wikipedia.org/wiki/Engineer" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Engineer">engineer</a> for a Los Angeles business that developed feedback flight controls for <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Atlas_(missile)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Atlas (missile)">Atlas missiles</a>. He would rise to the position of Chief Engineer here.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">Looking to start his own business he advertised in the <a href="http://en.wikipedia.org/wiki/Los_Angeles_Times" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Los Angeles Times">Los Angeles Times</a> newspaper soliciting financial backers with the simple but effective text "Engineer seeking financial backing to start own business".</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">Two years later he had founded the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Kavlico_Corporation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Kavlico Corporation">Kavlico Corporation</a>, located in <a href="http://en.wikipedia.org/wiki/Moorpark,_California" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Moorpark, California">Moorpark</a>, <a href="http://en.wikipedia.org/wiki/California" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="California">California</a>. Under his leadership, the company became one of the world's largest suppliers of <a href="http://en.wikipedia.org/wiki/Sensor" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sensor">sensors</a> for aeronautic, automotive, and industrial applications supplying amongst others General Electric and the Ford Motor Company. In 2000 he sold Kavlico for $345 million to C-Mac Industries Inc. Kavlico is today owned by the French company <a href="http://en.wikipedia.org/wiki/Schneider_Electric" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Schneider Electric">Schneider Electric</a>. Much of Kavli's wealth is a result of his real estate investments in Southern California. As a philanthropist, Kavli subsequently established The Kavli Foundation and has dedicated much of his wealth to funding research institutions and programs worldwide.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">On June 19, 2006, he was appointed Grand Officer, Commander with Star, of the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/The_Royal_Norwegian_Order_of_St._Olav" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="The Royal Norwegian Order of St. Olav">Royal Norwegian Order of Merit</a> by King <a href="http://en.wikipedia.org/wiki/Harald_V_of_Norway" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Harald V of Norway">Harald V of Norway</a> <sup class="reference" id="cite_ref-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[1]</a></sup> in recognition of his work on behalf of <a href="http://en.wikipedia.org/wiki/Norway" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Norway">Norway</a> and humanity. In 2008, he was also awarded an honorary doctorate, Doctor Honoris Causa, by the Norwegian University of Science and Technology, in recognition of his work to the benefit and advancement of science and research.<sup class="reference" id="cite_ref-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup> Kavli is a Fellow of the American Academy of Arts and Sciences.<sup class="reference" id="cite_ref-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup> He is also a former member of the U.S. President’s Council of Advisors on Science and Technology, and former member of the University of California President’s Board on Science and Innovation. In 2009, Mr. Kavli received an honorary Doctor of Science degree from Northwestern University.<sup class="reference" id="cite_ref-3" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[4]</a></sup> In 2011 he received the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Bower_Award_for_Business_Leadership" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bower Award for Business Leadership">Bower Award for Business Leadership</a> from the <a href="http://en.wikipedia.org/wiki/Franklin_Institute" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Franklin Institute">Franklin Institute</a>,<sup class="reference" id="cite_ref-4" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[5]</a></sup> one of the oldest science education centers in the United States, and the Carnegie Medal of Philanthropy, which is given every biennially to one or more individuals who, like Andrew Carnegie, have dedicated their private wealth to public good, and who have sustained impressive careers as philanthropists.<sup class="reference" id="cite_ref-5" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[6]</a></sup> In 2011, Mr. Kavli was also conferred the degree of <i>doctor philosopliae honoris causa</i> by the University of Oslo.<sup class="reference" id="cite_ref-6" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[7]</a></sup></div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">A Trustee of the University of California, Santa Barbara (UCSB) Foundation, in addition to supporting scientific research and education, his philanthropic activities include the Fred Kavli Theatre for Performing Arts at the Thousand Oaks Civic Arts Plaza, California, as well as other projects.</div><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Fred_Kavli&action=edit&section=2&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Kavli Prizes">edit</a>]</span><span class="mw-headline" id="Kavli_Prizes">Kavli Prizes</span></h2><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">Through <a href="http://en.wikipedia.org/wiki/The_Kavli_Foundation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="The Kavli Foundation">The Kavli Foundation</a>, Kavli established scientific prizes in the fields of <a href="http://en.wikipedia.org/wiki/Astrophysics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Astrophysics">Astrophysics</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Nanoscience" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nanoscience">Nanoscience</a>, and <a href="http://en.wikipedia.org/wiki/Neuroscience" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Neuroscience">Neuroscience</a>. The Kavli Prizes are presented in cooperation with the <a href="http://en.wikipedia.org/wiki/Norwegian_Academy_of_Science_and_Letters" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Norwegian Academy of Science and Letters">Norwegian Academy of Science and Letters</a> and the Norwegian Ministry of Education and Research, and have been awarded biennially at a ceremony in <a href="http://en.wikipedia.org/wiki/Oslo" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Oslo">Oslo</a> since 2008.<sup class="reference" id="cite_ref-7" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[8]</a></sup> Each prize consists of a scroll, gold medal, and $1,000,000 cash.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">Kavli chose to focus on these three areas of interest – "from the biggest, to the smallest, to the most complex" – because he thinks these fields are the most exciting scientific fields for the 21st century with potentially great benefits.<sup class="reference" id="cite_ref-8" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[9]</a></sup> Kavli has also noted his intent that the Prizes distinguish themselves from the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Nobel_prize" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nobel prize">Nobel</a> prizes in science.<sup class="reference" id="cite_ref-9" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[10]</a></sup> Consequently, one key distinction between the prizes: Kavli Prize laureates are selected by committees composed of distinguished international scientists. These committee members are recommended by the Chinese Academy of Sciences, the French Academy of Sciences, the Max Planck Society, the U.S. National Academy of Sciences and The Royal Society, with committee chairs chosen by the Norwegian Academy of Science and Letters.<sup class="reference" id="cite_ref-10" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[11]</a></sup></div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">The first Kavli Prize winners were announced on May 28, 2008, simultaneously in Oslo and at the opening of the <a href="http://en.wikipedia.org/wiki/World_Science_Festival" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="World Science Festival">World Science Festival</a> in <a href="http://en.wikipedia.org/wiki/New_York_City" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="New York City">New York City</a>. The first Kavli Prize for astrophysics was awarded to <a href="http://en.wikipedia.org/wiki/Maarten_Schmidt" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Maarten Schmidt">Maarten Schmidt</a> and <a href="http://en.wikipedia.org/wiki/Donald_Lynden-Bell" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Donald Lynden-Bell">Donald Lynden-Bell</a>. <a href="http://en.wikipedia.org/wiki/Louis_E._Brus" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Louis E. Brus">Louis E. Brus</a> and <a href="http://en.wikipedia.org/wiki/Sumio_Iijima" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sumio Iijima">Sumio Iijima</a> shared the nanoscience prize, while <a href="http://en.wikipedia.org/wiki/Pasko_Rakic" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Pasko Rakic">Pasko Rakic</a>, <a href="http://en.wikipedia.org/wiki/Thomas_Jessell" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Thomas Jessell">Thomas Jessell</a> and <a href="http://en.wikipedia.org/wiki/Sten_Grillner" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Sten Grillner">Sten Grillner</a> were awarded the neuroscience prize.<sup class="reference" id="cite_ref-11" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[12]</a></sup> The four US winners of the Kavli Prize were honored by President <a href="http://en.wikipedia.org/wiki/George_W._Bush" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="George W. Bush">George W. Bush</a> and Science Advisor, Dr. <a href="http://en.wikipedia.org/wiki/John_Marburger" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John Marburger">John Marburger</a>, at an <a href="http://en.wikipedia.org/wiki/Oval_Office" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Oval Office">Oval Office</a> reception in the <a href="http://en.wikipedia.org/wiki/White_House" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="White House">White House</a> on November 12, 2008.<sup class="reference" id="cite_ref-12" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-12" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[13]</a></sup> (See <a href="http://en.wikipedia.org/wiki/Kavli_Prize" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Kavli Prize">Kavli Prize</a> for laureates in subsequent years.)</div><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Fred_Kavli&action=edit&section=3&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: The Kavli Foundation">edit</a>]</span><span class="mw-headline" id="The_Kavli_Foundation">The Kavli Foundation</span></h2><div class="rellink relarticle mainarticle" style="background-color: white; font-family: sans-serif; font-size: 13px; font-style: italic; line-height: 19px; margin-bottom: 0.5em; padding-left: 1.6em;">Main article: <a href="http://en.wikipedia.org/wiki/The_Kavli_Foundation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="The Kavli Foundation">The Kavli Foundation</a></div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;"><a class="external text" href="http://www.kavlifoundation.org/" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">The Kavli Foundation</a>, based in Oxnard, California, is dedicated to the goals of advancing science for the benefit of humanity and promoting increased public understanding and support for scientists and their work.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">The Foundation's mission is implemented through an international program of research institutes, professorships, and symposia in the fields of astrophysics, nanoscience, neuroscience, and theoretical physics as well as prizes in the fields of astrophysics, nanoscience, and neuroscience.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">The Kavli Foundation was established in December 2000 by its founder and benefactor, Fred Kavli, a prominent California business leader and noted philanthropist whose foundation is currently actively involved in establishing major research institutes at leading universities and institutions in the United States, Europe and Asia.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">The Kavli Foundation has made grants to establish Kavli Institutes on the campuses of the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/University_of_California_Santa_Barbara" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="University of California Santa Barbara">University of California Santa Barbara</a>, <a href="http://en.wikipedia.org/wiki/Stanford_University" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Stanford University">Stanford University</a>, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/The_California_Institute_of_Technology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="The California Institute of Technology">the California Institute of Technology</a>, the<a href="http://en.wikipedia.org/wiki/University_of_Chicago" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="University of Chicago">University of Chicago</a>, <a href="http://en.wikipedia.org/wiki/Columbia_University" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Columbia University">Columbia University</a>, <a href="http://en.wikipedia.org/wiki/Yale_University" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Yale University">Yale University</a>, <a href="http://en.wikipedia.org/wiki/New_York_University" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="New York University">New York University</a>, <a href="http://en.wikipedia.org/wiki/Cornell_University" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Cornell University">Cornell University</a>, the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/University_of_California_San_Diego" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="University of California San Diego">University of California San Diego</a>, <a href="http://en.wikipedia.org/wiki/Delft_University_of_Technology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Delft University of Technology">Delft University of Technology</a> in The Netherlands, the Massachusetts Institute of Technology, <a href="http://en.wikipedia.org/wiki/Peking_University" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Peking University">Peking University</a>, <a href="http://en.wikipedia.org/wiki/Chinese_Academy_of_Sciences" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Chinese Academy of Sciences">Chinese Academy of Sciences</a>, <a href="http://en.wikipedia.org/wiki/Harvard_University" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Harvard University">Harvard University</a>, <a href="http://en.wikipedia.org/wiki/University_of_Cambridge" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="University of Cambridge">University of Cambridge</a> and the Norwegian University of Science and Technology. These institutions are the beneficiaries of the Kavli Foundation as on date, and the list is bound to grow in the future.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">In addition to the Kavli Institutes, six Kavli professorships have been established: two at University of California Santa Barbara, one at University of California Los Angeles, one at University of California Irvine, one at Columbia University, and one at California Institute of Technology.</div><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Fred_Kavli&action=edit&section=4&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Kavli Institutes">edit</a>]</span><span class="mw-headline" id="Kavli_Institutes">Kavli Institutes</span></h2><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">The Kavli Foundation has established research institutes at leading universities worldwide. Consistent with its business-like approach, Kavli requires each partner University to match the average $7.5 million donation. The institutes are not required to focus on any specific subject but are free to do any basic research they see fit.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">Three researchers associated with the Kavli institutes have been awarded Nobel prizes: <a href="http://en.wikipedia.org/wiki/David_Gross" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="David Gross">David Gross</a>, <a href="http://en.wikipedia.org/wiki/Frank_Wilczek" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Frank Wilczek">Frank Wilczek</a> and <a href="http://en.wikipedia.org/wiki/Richard_Axel" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Richard Axel">Richard Axel</a>.</div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">As of March 2008, there are 15 institutes in the <a href="http://en.wikipedia.org/wiki/United_States" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="United States">United States</a>, 2 in <a class="mw-redirect" href="http://en.wikipedia.org/wiki/People%27s_Republic_of_China" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="People's Republic of China">China</a>, 1 in the <a href="http://en.wikipedia.org/wiki/Netherlands" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Netherlands">Netherlands</a>, 1 in Norway and 1 in the <a href="http://en.wikipedia.org/wiki/United_Kingdom" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="United Kingdom">United Kingdom</a>.<sup class="reference" id="cite_ref-13" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-13" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[14]</a></sup> According to the Foundation eventually there might be as many 20 centres. The Institute for Physics and Mathematics of the Universe in Tokyo has also received an endowment to setup a Kavli institute from April 1st 2012 <sup class="reference" id="cite_ref-14" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-14" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[15]</a></sup></div><div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">The fifteen Kavli Institutes are:</div><h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 17px; line-height: 19px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Fred_Kavli&action=edit&section=5&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Astrophysics">edit</a>]</span><span class="mw-headline" id="Astrophysics">Astrophysics</span></h3><ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Kavli_Institute_for_Particle_Astrophysics_and_Cosmology" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Kavli Institute for Particle Astrophysics and Cosmology">Kavli Institute for Particle Astrophysics and Cosmology</a> at Stanford University</li>
<li style="margin-bottom: 0.1em;">Kavli Institute for Cosmological Physics at the University of Chicago</li>
<li style="margin-bottom: 0.1em;">Kavli Institute for Astrophysics and Space Research at the Massachusetts Institute of Technology</li>
<li style="margin-bottom: 0.1em;">Kavli Institute for Cosmology at the University of Cambridge</li>
<li style="margin-bottom: 0.1em;">Kavli Institute for Astronomy and Astrophysics at Peking University in China</li>
</ul><h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 17px; line-height: 19px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Fred_Kavli&action=edit&section=6&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Nanoscience">edit</a>]</span><span class="mw-headline" id="Nanoscience">Nanoscience</span></h3><ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;">Kavli Nanoscience Institute at Caltech</li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.kavlifoundation.org/cornell-university" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Kavli Institute at Cornell for Nanoscale Science</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Kavli_Institute_of_Nanoscience" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Kavli Institute of Nanoscience">Kavli Institute of Nanoscience</a> at Delft University of Technology in The Netherlands</li>
<li style="margin-bottom: 0.1em;">Kavli Institute for Bionano Science and Technology at Harvard University</li>
</ul><h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 17px; line-height: 19px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Fred_Kavli&action=edit&section=7&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Neuroscience">edit</a>]</span><span class="mw-headline" id="Neuroscience">Neuroscience</span></h3><ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.kavli.columbia.edu/" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Kavli Institute for Brain Science</a> at Columbia University</li>
<li style="margin-bottom: 0.1em;">Kavli Institute for Brain and Mind at the University of California, San Diego</li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Kavli_Institute_for_Systems_Neuroscience_and_Centre_for_the_Biology_of_Memory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Kavli Institute for Systems Neuroscience and Centre for the Biology of Memory">Kavli Institute for Systems Neuroscience</a> at the Norwegian Institute of Technology</li>
<li style="margin-bottom: 0.1em;">Kavli Institute for Neuroscience at Yale University</li>
</ul><h3 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: initial; border-bottom-style: none; border-bottom-width: initial; font-family: sans-serif; font-size: 17px; line-height: 19px; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; font-weight: normal; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Fred_Kavli&action=edit&section=8&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Theoretical physics">edit</a>]</span><span class="mw-headline" id="Theoretical_physics">Theoretical physics</span></h3><ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Kavli_Institute_for_Theoretical_Physics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Kavli Institute for Theoretical Physics">Kavli Institute for Theoretical Physics</a> at the University of California, Santa Barbara</li>
<li style="margin-bottom: 0.1em;">Kavli Institute for Theoretical Physics China at the Chinese Academy of Sciences</li>
<li style="margin-bottom: 0.1em;">Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU) <sup class="reference" id="cite_ref-15" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_note-15" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[16]</a></sup></li>
</ul><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Fred_Kavli&action=edit&section=9&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Quotes">edit</a>]</span><span class="mw-headline" id="Quotes">Quotes</span></h2><dl style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><i>The curiosity of the human being is what has brought us where we are today, and I have complete confidence that it will take us where we need to be in the future.</i></dd></dl><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Fred_Kavli&action=edit&section=10&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: References">edit</a>]</span><span class="mw-headline" id="References">References</span></h2><div class="reflist" style="background-color: white; font-family: sans-serif; font-size: 12px; line-height: 19px; list-style-type: decimal; margin-bottom: 0.5em;"><ol class="references" style="line-height: 1.5em; list-style-image: none; list-style-type: inherit; margin-bottom: 0.5em; margin-left: 3.2em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li id="cite_note-0" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external text" href="http://www.kongehuset.no/default.asp" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Kongehuset.no (Official site)</a></span></li>
<li id="cite_note-1" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external text" href="http://www.ntnu.no/cbm/events/fred_kavli_receives_honorary_doctorate" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Norwegian University of Science and Technology</a></span></li>
<li id="cite_note-2" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external text" href="http://www.amacad.org/news/alpha2006.aspx" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">American Academy of Arts and Sciences</a></span></li>
<li id="cite_note-3" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external text" href="http://www.northwestern.edu/newscenter/stories/2009/05/honorarydegrees.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Northwestern University</a></span></li>
<li id="cite_note-4" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation web" style="word-wrap: break-word;"><a class="external text" href="http://www.fi.edu/franklinawards/11/bowerbus.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">"Bower Award for Business Leadership"</a>. Franklin Institute. 2011<span class="reference-accessdate">. Retrieved December 23, 2011</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Bower+Award+for+Business+Leadership&rft.atitle=&rft.date=2011&rft.pub=Franklin+Institute&rft_id=http%3A%2F%2Fwww.fi.edu%2Ffranklinawards%2F11%2Fbowerbus.html&rfr_id=info:sid/en.wikipedia.org:Fred_Kavli"></span></span></li>
<li id="cite_note-5" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external text" href="http://www.carnegiemedals.org/index.php?option=com_content&view=article&id=47&Itemid=120" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Carnegie Medal of Philanthropy</a></span></li>
<li id="cite_note-6" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external text" href="http://www.uio.no/om/tall-og-fakta/aresdoktorer/eresdoktorer-2011.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">University of Oslo</a></span></li>
<li id="cite_note-7" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external text" href="http://www.kavliprize.no/" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Kavli Prize official website</a></span></li>
<li id="cite_note-8" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Andrew Farrell for <a class="new" href="http://en.wikipedia.org/w/index.php?title=Forbes_Magazine_Online&action=edit&redlink=1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #a55858; text-decoration: none;" title="Forbes Magazine Online (page does not exist)">Forbes Magazine Online</a>: <a class="external text" href="http://www.forbes.com/businessbillionaires/2008/05/16/wealth-science-kavli-biz-billies-cx_af_0516kavli.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">The Man With The Million-Dollar Prize</a>. May 16, 2008. Retrieved on 2008-05-16.</span></li>
<li id="cite_note-9" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Time_Magazine" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Time Magazine">Time Magazine</a>: <a class="external text" href="http://www.time.com/time/magazine/article/0,9171,1649308,00.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">The Next Nobel?</a></span></li>
<li id="cite_note-10" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external text" href="http://www.kavliprize.no/" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Kavli Prize official website</a></span></li>
<li id="cite_note-11" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external text" href="http://www.worldsciencefestival.com/media-resources/Kavli-Press-Release-080528.pdf" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif); background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 18px; text-decoration: none;">Kavli Foundation Press Release, May 28, 2008</a></span></li>
<li id="cite_note-12" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-12" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation news" style="word-wrap: break-word;"><a class="external text" href="http://www.norway.org/restech/researchnews/BushKavli.htm" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">"President Bush honors U.S. Kavli Prize recipients"</a>. <i><a href="http://en.wikipedia.org/wiki/Executive_Office_of_the_President_of_the_United_States" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Executive Office of the President of the United States">Executive Office of the President of the United States</a>/Royal Norwegian Embassy in Washington D.C.</i> (<a href="http://en.wikipedia.org/wiki/Ministry_of_Foreign_Affairs_(Norway)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Ministry of Foreign Affairs (Norway)">Norwegian Ministry of Foreign Affairs</a>). November 17, 2008<span class="reference-accessdate">. Retrieved 20 November 2008</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=President+Bush+honors+U.S.+Kavli+Prize+recipients&rft.jtitle=%5B%5BExecutive+Office+of+the+President+of+the+United+States%5D%5D%2FRoyal+Norwegian+Embassy+in+Washington+D.C.&rft.date=November+17%2C+2008&rft.pub=%5B%5BMinistry+of+Foreign+Affairs+%28Norway%29%7CNorwegian+Ministry+of+Foreign+Affairs%5D%5D&rft_id=http%3A%2F%2Fwww.norway.org%2Frestech%2Fresearchnews%2FBushKavli.htm&rfr_id=info:sid/en.wikipedia.org:Fred_Kavli"></span></span></li>
<li id="cite_note-13" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-13" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external free" href="http://www.admin.cam.ac.uk/reporter/current/weekly/6051/12.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://www.admin.cam.ac.uk/reporter/current/weekly/6051/12.html</a></span></li>
<li id="cite_note-14" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-14" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external free" href="http://www.ipmu.jp/node/1231" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://www.ipmu.jp/node/1231</a></span></li>
<li id="cite_note-15" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/Fred_Kavli#cite_ref-15" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="external free" href="http://physicsworld.com/cws/article/news/48597" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">http://physicsworld.com/cws/article/news/48597</a></span></li>
</ol></div><h2 style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 19px; font-weight: normal; line-height: 19px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=Fred_Kavli&action=edit&section=11&editintro=Template:BLP_editintro" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: External links">edit</a>]</span><span class="mw-headline" id="External_links">External links</span></h2><ul style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.kavlifoundation.org/fred-kavli" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Concise Fred Kavli biography</a> – From <a class="external text" href="http://www.kavlifoundation.org/" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">The Kavli Foundation</a></li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.nytimes.com/2005/04/19/science/19prof.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Kavli Strives to Leave Mark on Science</a> – New York Times article</li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.time.com/time/magazine/article/0,9171,1649308,00.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">The Next Nobel?</a> Time Magazine profile</li>
<li style="margin-bottom: 0.1em;"><a class="external text" href="http://www.usatoday.com/tech/science/2006-11-13-kavli-philanthropy_x.htm" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Kavli strives to leave mark on science</a> USA Today</li>
</ul>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com0tag:blogger.com,1999:blog-5303246073824127471.post-90320181957612198692012-03-05T08:12:00.001-05:002012-03-05T08:15:37.677-05:00The History of Entropy<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhELFqvseCLKKC7_mYeNstNkbIHUOehFucy-dZtEC6GXGNwEBAiyD2LJVxnACgFNQgY3YZplDJS5BUhuOWGOqNxANTUDKhASQrZQYH1ZhiyF4KB2tI9zIUsYck3d0vDFd-2UtEQfOBoA/s1600/entropy.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="552" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhELFqvseCLKKC7_mYeNstNkbIHUOehFucy-dZtEC6GXGNwEBAiyD2LJVxnACgFNQgY3YZplDJS5BUhuOWGOqNxANTUDKhASQrZQYH1ZhiyF4KB2tI9zIUsYck3d0vDFd-2UtEQfOBoA/s640/entropy.jpg" width="640" /></a></div><h1 class="firstHeading" id="firstHeading" style="background-attachment: initial; background-clip: initial; background-color: white; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-family: sans-serif; font-size: 1.6em; font-weight: normal; line-height: 1.2em; margin-bottom: 0.1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0px; padding-top: 0px; width: auto;"><span dir="auto"><br />
<div class="mw-content-ltr" dir="ltr" lang="en" style="direction: ltr; font-size: 13px; line-height: 19px;"><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">The concept of <b><a href="http://en.wikipedia.org/wiki/Entropy" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Entropy">entropy</a></b> developed in response to the observation that a certain amount of functional energy released from <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Combustion_reactions" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Combustion reactions">combustion reactions</a> is always lost to dissipation or friction and is thus not transformed into <a href="http://en.wikipedia.org/wiki/Work_(thermodynamics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; color: #0b0080;" title="Work (thermodynamics)">useful work</a> .<sup class="Template-Fact" style="line-height: 1em; white-space: nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources from September 2010">citation needed</span></a></i>]</sup> Early heat-powered engines such as <a href="http://en.wikipedia.org/wiki/Thomas_Savery" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Thomas Savery">Thomas Savery</a>'s (1698), the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Newcomen_engine" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Newcomen engine">Newcomen engine</a> (1712) and the Cugnot <a href="http://en.wikipedia.org/wiki/Steam_tricycle" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Steam tricycle">steam tricycle</a> (1769) were inefficient, converting less than two percent of the input energy into useful <a href="http://en.wikipedia.org/wiki/Work_output" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Work output">work output</a>; a great deal of useful energy was dissipated or lost into what seemed like a state of immeasurable randomness.<sup class="noprint Inline-Template" style="line-height: 1em; white-space: nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:No_original_research" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:No original research"><span title="The material in the vicinity of this tag may be based upon unreliable original research from September 2010">original research?</span></a></i>]</sup> Over the next two centuries, physicists investigated this puzzle of lost energy; the result was the concept of <a href="http://en.wikipedia.org/wiki/Entropy" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Entropy">entropy</a>.</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">In the early 1850s, <a href="http://en.wikipedia.org/wiki/Rudolf_Clausius" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Rudolf Clausius">Rudolf Clausius</a> set forth the concept of the <a href="http://en.wikipedia.org/wiki/Thermodynamic_system" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Thermodynamic system">thermodynamic system</a> and posited the argument that in any irreversible process a small amount of heat energy <i>δQ</i> is incrementally dissipated across the system boundary. Clausius continued to develop his ideas of lost energy, and coined the term <i>entropy</i>.</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">Since the mid-20th century the concept of entropy has found application in the analogous field of data loss in information transmission systems.<sup class="Template-Fact" style="line-height: 1em; white-space: nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources from September 2010">citation needed</span></a></i>]</sup></div><table class="toc" id="toc" style="background-color: #f9f9f9; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; border-image: initial; border-left-color: rgb(170, 170, 170); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(170, 170, 170); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(170, 170, 170); border-top-style: solid; border-top-width: 1px; font-size: 12px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; padding-top: 5px;"><tbody>
<tr><td><div id="toctitle" style="direction: ltr; text-align: center;"><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-style: none; border-color: initial; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; border-width: initial; display: inline; font-size: 12px; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; width: auto;">Contents</h2> <span class="toctoggle" style="font-size: 11px;"> [<a class="internal" href="http://en.wikipedia.org/wiki/History_of_entropy#" id="togglelink" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">hide</a>] </span></div><ul style="line-height: 1.5em; list-style-image: none; list-style-type: none; margin-bottom: 0.3em; margin-left: 0px; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: left;"><li class="toclevel-1 tocsection-1" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#Classical_thermodynamic_views" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">1</span> <span class="toctext">Classical thermodynamic views</span></a></li>
<li class="toclevel-1 tocsection-2" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#1854_definition" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">2</span> <span class="toctext">1854 definition</span></a></li>
<li class="toclevel-1 tocsection-3" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#1856_definition" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">3</span> <span class="toctext">1856 definition</span></a></li>
<li class="toclevel-1 tocsection-4" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#1862_definition" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">4</span> <span class="toctext">1862 definition</span></a></li>
<li class="toclevel-1 tocsection-5" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#1865_definition" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">5</span> <span class="toctext">1865 definition</span></a></li>
<li class="toclevel-1 tocsection-6" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#Later_developments" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">6</span> <span class="toctext">Later developments</span></a></li>
<li class="toclevel-1 tocsection-7" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#Statistical_thermodynamic_views" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">7</span> <span class="toctext">Statistical thermodynamic views</span></a></li>
<li class="toclevel-1 tocsection-8" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#Information_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">8</span> <span class="toctext">Information theory</span></a></li>
<li class="toclevel-1 tocsection-9" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#Popular_use" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">9</span> <span class="toctext">Popular use</span></a></li>
<li class="toclevel-1 tocsection-10" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#Terminology_overlap" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">10</span> <span class="toctext">Terminology overlap</span></a></li>
<li class="toclevel-1 tocsection-11" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#See_also" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">11</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-12" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#References" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">12</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-13" style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#External_links" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><span class="tocnumber">13</span> <span class="toctext">External links</span></a></li>
</ul></td></tr>
</tbody></table><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 19px; font-weight: normal; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=History_of_entropy&action=edit&section=1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Classical thermodynamic views">edit</a>]</span><span class="mw-headline" id="Classical_thermodynamic_views">Classical thermodynamic views</span></h2><div class="rellink relarticle mainarticle" style="font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em;">Main article: <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Classical_thermodynamics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Classical thermodynamics">classical thermodynamics</a></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">In 1803, mathematician <a href="http://en.wikipedia.org/wiki/Lazare_Carnot" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Lazare Carnot">Lazare Carnot</a> published a work entitled <i>Fundamental Principles of Equilibrium and Movement</i>. This work includes a discussion on the efficiency of fundamental machines, i.e. pulleys and inclined planes. Lazare Carnot saw through all the details of the mechanisms to develop a general discussion on the conservation of mechanical energy. Over the next three decades, Lazare Carnot’s theorem was taken as a statement that in any machine the accelerations and shocks of the moving parts all represent losses of <i>moment of activity</i>, i.e. the <a href="http://en.wikipedia.org/wiki/Work_(thermodynamics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Work (thermodynamics)">useful work</a> done. From this Lazare drew the inference that perpetual motion was impossible. This <i>loss of moment of activity</i> was the first-ever rudimentary statement of the <a href="http://en.wikipedia.org/wiki/Second_law_of_thermodynamics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Second law of thermodynamics">second law of thermodynamics</a> and the concept of 'transformation-energy' or <i>entropy</i>, i.e. energy lost to dissipation and friction.<sup class="reference" id="cite_ref-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_note-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[1]</a></sup></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">Lazare Carnot died in exile in 1823. During the following year Lazare’s son <a href="http://en.wikipedia.org/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nicolas Léonard Sadi Carnot">Sadi Carnot</a>, having graduated from the <a href="http://en.wikipedia.org/wiki/%C3%89cole_Polytechnique" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="École Polytechnique">École Polytechnique</a> training school for engineers, but now living on half-pay with his brother Hippolyte in a small apartment in Paris, wrote the <i>Reflections on the Motive Power of Fire</i>. In this paper, Sadi visualized an ideal engine in which the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Thermodynamic_reversibility" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Thermodynamic reversibility">thermodynamic reversibility</a>. Building on his father's work, Sadi postulated the concept that “some caloric is always lost”, not being converted to mechanical work. Hence any real heat engine could not realize the Carnot cycle's reversibility and was condemned to be less efficient. This lost caloric was a precursory form of entropy loss as we now know it. Though formulated in terms of caloric, rather than entropy, this was an early insight into the <a href="http://en.wikipedia.org/wiki/Second_law_of_thermodynamics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Second law of thermodynamics">second law of thermodynamics</a>.</div><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 19px; font-weight: normal; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=History_of_entropy&action=edit&section=2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: 1854 definition">edit</a>]</span><span class="mw-headline" id="1854_definition">1854 definition</span></h2><div class="thumb tright" style="background-color: transparent; clear: right; float: right; margin-bottom: 1.3em; margin-left: 1.4em; margin-right: 0px; margin-top: 0.5em; width: auto;"><div class="thumbinner" style="background-color: #f9f9f9; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; border-image: initial; border-left-color: rgb(204, 204, 204); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(204, 204, 204); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(204, 204, 204); border-top-style: solid; border-top-width: 1px; font-size: 12px; min-width: 100px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 3px !important; padding-left: 3px !important; padding-right: 3px !important; padding-top: 3px !important; text-align: center; width: 227px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Clausius.jpg" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><img alt="" class="thumbimage" height="267" src="http://upload.wikimedia.org/wikipedia/commons/thumb/4/40/Clausius.jpg/225px-Clausius.jpg" style="background-color: white; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; border-color: initial; border-image: initial; border-left-color: rgb(204, 204, 204); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(204, 204, 204); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(204, 204, 204); border-top-style: solid; border-top-width: 1px; border-width: initial; vertical-align: middle;" width="225" /></a><div class="thumbcaption" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; font-size: 11px; line-height: 1.4em; padding-bottom: 3px !important; padding-left: 3px !important; padding-right: 3px !important; padding-top: 3px !important; text-align: left;"><div class="magnify" style="background-attachment: initial !important; background-clip: initial !important; background-color: initial !important; background-image: none !important; background-origin: initial !important; background-position: initial initial !important; background-repeat: initial initial !important; border-bottom-style: none !important; border-color: initial !important; border-image: initial !important; border-left-style: none !important; border-right-style: none !important; border-top-style: none !important; border-width: initial !important; float: right;"><a class="internal" href="http://en.wikipedia.org/wiki/File:Clausius.jpg" style="background-attachment: initial !important; background-clip: initial !important; background-color: initial !important; background-image: none !important; background-origin: initial !important; background-position: initial initial !important; background-repeat: initial initial !important; border-bottom-style: none !important; border-color: initial !important; border-image: initial !important; border-left-style: none !important; border-right-style: none !important; border-top-style: none !important; border-width: initial !important; color: #0b0080; display: block; text-decoration: none;" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.19/common/images/magnify-clip.png" style="background-attachment: initial !important; background-clip: initial !important; background-color: initial !important; background-image: none !important; background-origin: initial !important; background-position: initial initial !important; background-repeat: initial initial !important; border-bottom-style: none !important; border-color: initial !important; border-color: initial; border-image: initial !important; border-left-style: none !important; border-right-style: none !important; border-top-style: none !important; border-width: initial !important; border-width: initial; display: block; vertical-align: middle;" width="15" /></a></div><a href="http://en.wikipedia.org/wiki/Rudolf_Clausius" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Rudolf Clausius">Rudolf Clausius</a> - originator of the concept of <b>"entropy"</b></div></div></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">In his 1854 memoir, Clausius first develops the concepts of <i>interior work</i>, i.e. that "which the atoms of the body exert upon each other", and <i>exterior work</i>, i.e. that "which arise from foreign influences [to] which the body may be exposed", which may act on a working body of fluid or gas, typically functioning to work a piston. He then discusses the three categories into which heat <i>Q</i> may be divided:</div><ol style="line-height: 1.5em; list-style-image: none; margin-bottom: 0px; margin-left: 3.2em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;">Heat employed in increasing the heat actually existing in the body.</li>
<li style="margin-bottom: 0.1em;">Heat employed in producing the interior work.</li>
<li style="margin-bottom: 0.1em;">Heat employed in producing the exterior work.</li>
</ol><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">Building on this logic, and following a mathematical presentation of the <i>first fundamental theorem</i>, Clausius then presented the first-ever mathematical formulation of entropy, although at this point in the development of his theories he called it "equivalence-value", perhaps referring to the concept of the<a href="http://en.wikipedia.org/wiki/Mechanical_equivalent_of_heat" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mechanical equivalent of heat">mechanical equivalent of heat</a> which was developing at the time rather than entropy, a term which was to come into use later.<sup class="reference" id="cite_ref-1" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_note-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[2]</a></sup> He stated:<sup class="reference" id="cite_ref-2" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_note-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[3]</a></sup></div><blockquote><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">the <i>second fundamental theorem</i> in the mechanical <a href="http://en.wikipedia.org/wiki/Theory_of_heat" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theory of heat">theory of heat</a> may thus be enunciated: If two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generations of the quantity of <a href="http://en.wikipedia.org/wiki/Heat" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Heat">heat</a> <i>Q</i> from <a href="http://en.wikipedia.org/wiki/Work_(thermodynamics)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Work (thermodynamics)">work</a> at the temperature <i>T</i> , has the <i>equivalence-value</i>:</div><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \frac {Q}{T}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/3/5/035b6a76ed538cd7c9465e5f52b012c5.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl></dd></dl>and the passage of the quantity of heat <i>Q</i> from the <a href="http://en.wikipedia.org/wiki/Temperature" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Temperature">temperature</a> <i>T<sub style="line-height: 1em;">1</sub></i> to the temperature <i>T<sub style="line-height: 1em;">2</sub></i>, has the equivalence-value:<br />
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">
<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" Q \left( \frac {1}{T_2} - \frac {1}{T_1}\right)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/a/5/0a5a69456445bd3359b82ee22abaa145.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl></dd></dl>wherein <i>T</i> is a function of the temperature, independent of the nature of the process by which the transformation is effected.</blockquote><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">In modern terminology, we think of this equivalence-value as "entropy", symbolized by <i>S</i>. Thus, using the above description, we can calculate the entropy change Δ<i>S</i> for the passage of the quantity of<a href="http://en.wikipedia.org/wiki/Heat" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Heat">heat</a> <i>Q</i> from the <a href="http://en.wikipedia.org/wiki/Temperature" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Temperature">temperature</a> <i>T<sub style="line-height: 1em;">1</sub></i>, through the "working body" of fluid (see <a href="http://en.wikipedia.org/wiki/Heat_engine" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Heat engine">heat engine</a>), which was typically a body of steam, to the temperature <i>T<sub style="line-height: 1em;">2</sub></i> as shown below:</div><div class="thumb tright" style="background-color: transparent; clear: right; float: right; margin-bottom: 1.3em; margin-left: 1.4em; margin-right: 0px; margin-top: 0.5em; width: auto;"><div class="thumbinner" style="background-color: #f9f9f9; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; border-image: initial; border-left-color: rgb(204, 204, 204); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(204, 204, 204); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(204, 204, 204); border-top-style: solid; border-top-width: 1px; font-size: 12px; min-width: 100px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 3px !important; padding-left: 3px !important; padding-right: 3px !important; padding-top: 3px !important; text-align: center; width: 327px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Entropy-diagram.png" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;"><img alt="" class="thumbimage" height="214" src="http://upload.wikimedia.org/wikipedia/en/thumb/d/dc/Entropy-diagram.png/325px-Entropy-diagram.png" style="background-color: white; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; border-color: initial; border-image: initial; border-left-color: rgb(204, 204, 204); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(204, 204, 204); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(204, 204, 204); border-top-style: solid; border-top-width: 1px; border-width: initial; vertical-align: middle;" width="325" /></a><div class="thumbcaption" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; font-size: 11px; line-height: 1.4em; padding-bottom: 3px !important; padding-left: 3px !important; padding-right: 3px !important; padding-top: 3px !important; text-align: left;"><div class="magnify" style="background-attachment: initial !important; background-clip: initial !important; background-color: initial !important; background-image: none !important; background-origin: initial !important; background-position: initial initial !important; background-repeat: initial initial !important; border-bottom-style: none !important; border-color: initial !important; border-image: initial !important; border-left-style: none !important; border-right-style: none !important; border-top-style: none !important; border-width: initial !important; float: right;"><a class="internal" href="http://en.wikipedia.org/wiki/File:Entropy-diagram.png" style="background-attachment: initial !important; background-clip: initial !important; background-color: initial !important; background-image: none !important; background-origin: initial !important; background-position: initial initial !important; background-repeat: initial initial !important; border-bottom-style: none !important; border-color: initial !important; border-image: initial !important; border-left-style: none !important; border-right-style: none !important; border-top-style: none !important; border-width: initial !important; color: #0b0080; display: block; text-decoration: none;" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.19/common/images/magnify-clip.png" style="background-attachment: initial !important; background-clip: initial !important; background-color: initial !important; background-image: none !important; background-origin: initial !important; background-position: initial initial !important; background-repeat: initial initial !important; border-bottom-style: none !important; border-color: initial !important; border-color: initial; border-image: initial !important; border-left-style: none !important; border-right-style: none !important; border-top-style: none !important; border-width: initial !important; border-width: initial; display: block; vertical-align: middle;" width="15" /></a></div>Diagram of Sadi Carnot's <a href="http://en.wikipedia.org/wiki/Heat_engine" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Heat engine">heat engine</a>, 1824</div></div></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">If we make the assignment:</div><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" S= \frac {Q}{T}" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/e/4/8e465ecb58dcade915a306733330cb9a.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">Then, the entropy change or "equivalence-value" for this transformation is:</div><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \Delta S = S_{\rm final} - S_{\rm initial} \, " class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/f/5/1f59a80e91ce19edd2a9a32a8336393e.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">which equals:</div><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \Delta S = \left(\frac {Q}{T_2} - \frac {Q}{T_1}\right)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/f/0/2/f02796ac59b47e024d1edab0fdce2be2.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">and by factoring out Q, we have the following form, as was derived by Clausius:</div><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt=" \Delta S = Q\left(\frac {1}{T_2} - \frac {1}{T_1}\right)" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/1/d/a/1da26002a27bc094ea9bebd3f4364fea.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 19px; font-weight: normal; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=History_of_entropy&action=edit&section=3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: 1856 definition">edit</a>]</span><span class="mw-headline" id="1856_definition">1856 definition</span></h2><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">In 1856, Clausius stated what he called the "second fundamental theorem in the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Mechanical_theory_of_heat" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Mechanical theory of heat">mechanical theory of heat</a>" in the following form:</div><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\int \frac{\delta Q}{T} = -N" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/0/a/b/0abef3292f6afbd6dede3a2155ae1b45.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">where <i>N</i> is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. This equivalence-value was a precursory formulation of entropy.<sup class="reference" id="cite_ref-3" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_note-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[4]</a></sup></div><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 19px; font-weight: normal; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=History_of_entropy&action=edit&section=4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: 1862 definition">edit</a>]</span><span class="mw-headline" id="1862_definition">1862 definition</span></h2><div class="rellink relarticle mainarticle" style="font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em;">Main article: <a href="http://en.wikipedia.org/wiki/Disgregation" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Disgregation">disgregation</a></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">In 1862, Clausius stated what he calls the “theorem respecting the equivalence-values of the transformations” or what is now known as the <a href="http://en.wikipedia.org/wiki/Second_law_of_thermodynamics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Second law of thermodynamics">second law of thermodynamics</a>, as such:</div><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><i>The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to nothing.</i></dd></dl><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">Quantitatively, Clausius states the mathematical expression for this theorem is as follows. Let <i>δQ</i> be an element of the heat given up by the body to any reservoir of heat during its own changes, heat which it may absorb from a reservoir being here reckoned as negative, and <i>T</i> the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Absolute_temperature" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Absolute temperature">absolute temperature</a> of the body at the moment of giving up this heat, then the equation:</div><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\int \frac{\delta Q}{T} = 0" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/8/f/4/8f4eac9ecf7c7d35d5b348b86fa249de.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">must be true for every reversible cyclical process, and the relation:</div><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="\int \frac{\delta Q}{T} \ge 0" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/c/4/a/c4aa908ad7660cc85cf570ba3df6b842.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">must hold good for every cyclical process which is in any way possible. This was an early formulation of the second law and one of the original forms of the concept of entropy.</div><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 19px; font-weight: normal; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=History_of_entropy&action=edit&section=5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: 1865 definition">edit</a>]</span><span class="mw-headline" id="1865_definition">1865 definition</span></h2><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">In 1865, Clausius gave irreversible heat loss, or what he had previously been calling "equivalence-value", a name:<sup class="reference" id="cite_ref-4" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_note-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[5]</a></sup><sup class="reference" id="cite_ref-5" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_note-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[6]</a></sup></div><table class="cquote" style="background-color: transparent; border-bottom-style: none; border-collapse: collapse; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; font-size: 13px; margin-bottom: auto; margin-left: auto; margin-right: auto; margin-top: auto; width: auto;"><tbody>
<tr><td style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #b2b7f2; font-family: 'Times New Roman', serif; font-size: 35px; font-weight: bold; padding-bottom: 10px; padding-left: 10px; padding-right: 10px; padding-top: 10px; text-align: left;" valign="top" width="20">“</td><td style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; padding-bottom: 4px; padding-left: 10px; padding-right: 10px; padding-top: 4px;" valign="top">I propose to name the quantity <i>S</i> the entropy of the system, after the Greek word [τροπη <i>trope</i>], the transformation. I have deliberately chosen the word entropy to be as similar as possible to the word energy: the two quantities to be named by these words are so closely related in physical significance that a certain similarity in their names appears to be appropriate.</td><td style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #b2b7f2; font-family: 'Times New Roman', serif; font-size: 35px; font-weight: bold; padding-bottom: 10px; padding-left: 10px; padding-right: 10px; padding-top: 10px; text-align: right;" valign="bottom" width="20">”</td></tr>
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Although Clausius did not specify why he chose the symbol "S" to represent entropy, it is arguable that Clausius chose "S" in honor of <a href="http://en.wikipedia.org/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Nicolas Léonard Sadi Carnot">Sadi Carnot</a>, to whose 1824 article Clausius devoted over 15 years of work and research. On the first page of his original 1850 article "On the Motive Power of Heat, and on the Laws which can be Deduced from it for the Theory of Heat", Clausius calls Carnot the most important of the researchers in the <a href="http://en.wikipedia.org/wiki/Theory_of_heat" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Theory of heat">theory of heat</a>.<sup class="reference" id="cite_ref-Clausius_6-0" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_note-Clausius-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[7]</a></sup></div><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 19px; font-weight: normal; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=History_of_entropy&action=edit&section=6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Later developments">edit</a>]</span><span class="mw-headline" id="Later_developments">Later developments</span></h2><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">In 1876, physicist <a class="mw-redirect" href="http://en.wikipedia.org/wiki/J._Willard_Gibbs" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="J. Willard Gibbs">J. Willard Gibbs</a>, building on the work of Clausius, <a href="http://en.wikipedia.org/wiki/Hermann_von_Helmholtz" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Hermann von Helmholtz">Hermann von Helmholtz</a> and others, proposed that the measurement of "available energy" Δ<i>G</i> in a thermodynamic system could be mathematically accounted for by subtracting the "energy loss" <i>T</i>Δ<i>S</i> from total energy change of the system Δ<i>H</i>. These concepts were further developed by <a href="http://en.wikipedia.org/wiki/James_Clerk_Maxwell" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="James Clerk Maxwell">James Clerk Maxwell</a> [1871] and <a href="http://en.wikipedia.org/wiki/Max_Planck" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Max Planck">Max Planck</a>[1903].</div><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 19px; font-weight: normal; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=History_of_entropy&action=edit&section=7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Statistical thermodynamic views">edit</a>]</span><span class="mw-headline" id="Statistical_thermodynamic_views">Statistical thermodynamic views</span></h2><div class="rellink relarticle mainarticle" style="font-style: italic; margin-bottom: 0.5em; padding-left: 1.6em;">Main article: <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Statistical_thermodynamics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Statistical thermodynamics">statistical thermodynamics</a></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">In 1877, <a href="http://en.wikipedia.org/wiki/Ludwig_Boltzmann" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Ludwig Boltzmann">Ludwig Boltzmann</a> formulated the alternative definition of entropy <i>S</i> defined as:</div><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="S = k_{\rm B} \ln \Omega \!" class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/a/4/d/a4df765afa539fe865ce994f2584a921.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">where</div><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><i>k</i><sub style="line-height: 1em;">B</sub> is <a href="http://en.wikipedia.org/wiki/Boltzmann_constant" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Boltzmann constant">Boltzmann's constant</a> and</dd><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><i>Ω</i> is the number of microstates consistent with the given macrostate.</dd></dl><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">Boltzmann saw entropy as a measure of statistical "mixedupness" or disorder. This concept was soon refined by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/J._Willard_Gibbs" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="J. Willard Gibbs">J. Willard Gibbs</a>, and is now regarded as one of the cornerstones of the theory of<a href="http://en.wikipedia.org/wiki/Statistical_mechanics" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Statistical mechanics">statistical mechanics</a>.</div><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 19px; font-weight: normal; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=History_of_entropy&action=edit&section=8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Information theory">edit</a>]</span><span class="mw-headline" id="Information_theory">Information theory</span></h2><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">An analog to <i>thermodynamic entropy</i> is <b>information entropy</b>. In 1948, while working at <a href="http://en.wikipedia.org/wiki/Bell_Telephone" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bell Telephone">Bell Telephone</a> Laboratories electrical engineer <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Claude_Elwood_Shannon" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Claude Elwood Shannon">Claude Shannon</a> set out to mathematically quantify the statistical nature of “lost information” in phone-line signals. To do this, Shannon developed the very general concept of <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Information_entropy" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Information entropy">information entropy</a>, a fundamental cornerstone of <a href="http://en.wikipedia.org/wiki/Information_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Information theory">information theory</a>. Although the story varies, initially it seems that Shannon was not particularly aware of the close similarity between his new quantity and earlier work in thermodynamics. In 1949, however, when Shannon had been working on his equations for some time, he happened to visit the mathematician <a href="http://en.wikipedia.org/wiki/John_von_Neumann" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="John von Neumann">John von Neumann</a>. During their discussions, regarding what Shannon should call the “measure of uncertainty” or attenuation in phone-line signals with reference to his new information theory, according to one source:<sup class="reference" id="cite_ref-7" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_note-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[8]</a></sup></div><table class="cquote" style="background-color: transparent; border-bottom-style: none; border-collapse: collapse; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; font-size: 13px; margin-bottom: auto; margin-left: auto; margin-right: auto; margin-top: auto; width: auto;"><tbody>
<tr><td style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #b2b7f2; font-family: 'Times New Roman', serif; font-size: 35px; font-weight: bold; padding-bottom: 10px; padding-left: 10px; padding-right: 10px; padding-top: 10px; text-align: left;" valign="top" width="20">“</td><td style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; padding-bottom: 4px; padding-left: 10px; padding-right: 10px; padding-top: 4px;" valign="top">My greatest concern was what to call it. I thought of calling it ‘information’, but the word was overly used, so I decided to call it ‘uncertainty’. When I discussed it with John von Neumann, he had a better idea. Von Neumann told me, ‘You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage.</td><td style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #b2b7f2; font-family: 'Times New Roman', serif; font-size: 35px; font-weight: bold; padding-bottom: 10px; padding-left: 10px; padding-right: 10px; padding-top: 10px; text-align: right;" valign="bottom" width="20">”</td></tr>
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According to another source, when von Neumann asked him how he was getting on with his information theory, Shannon replied:<sup class="reference" id="cite_ref-8" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_note-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[9]</a></sup></div><table class="cquote" style="background-color: transparent; border-bottom-style: none; border-collapse: collapse; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; font-size: 13px; margin-bottom: auto; margin-left: auto; margin-right: auto; margin-top: auto; width: auto;"><tbody>
<tr><td style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #b2b7f2; font-family: 'Times New Roman', serif; font-size: 35px; font-weight: bold; padding-bottom: 10px; padding-left: 10px; padding-right: 10px; padding-top: 10px; text-align: left;" valign="top" width="20">“</td><td style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; padding-bottom: 4px; padding-left: 10px; padding-right: 10px; padding-top: 4px;" valign="top">The theory was in excellent shape, except that he needed a good name for “missing information”. “Why don’t you call it entropy”, von Neumann suggested. “In the first place, a mathematical development very much like yours already exists in Boltzmann’s statistical mechanics, and in the second place, no one understands entropy very well, so in any discussion you will be in a position of advantage.</td><td style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; color: #b2b7f2; font-family: 'Times New Roman', serif; font-size: 35px; font-weight: bold; padding-bottom: 10px; padding-left: 10px; padding-right: 10px; padding-top: 10px; text-align: right;" valign="bottom" width="20">”</td></tr>
</tbody></table><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;"><br />
In 1948 Shannon published his famous paper <i>A Mathematical Theory of Communication</i>, in which he devoted a section to what he calls Choice, Uncertainty, and Entropy.<sup class="reference" id="cite_ref-9" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_note-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[10]</a></sup> In this section, Shannon introduces an <i>H function</i> of the following form:</div><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="line-height: 1.5em; margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><img alt="H = -K\sum_{i=1}^k p(i) \log p(i)," class="tex" src="http://upload.wikimedia.org/wikipedia/en/math/7/c/8/7c8e3c5cb430461ced0090347964a592.png" style="border-bottom-style: none; border-color: initial; border-image: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" /></dd></dl><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">where <i>K</i> is a positive constant. Shannon then states that “any quantity of this form, where <i>K</i> merely amounts to a choice of a unit of measurement, plays a central role in information theory as measures of information, choice, and uncertainty.” Then, as an example of how this expression applies in a number of different fields, he references R.C. Tolman’s 1938 <i>Principles of Statistical Mechanics</i>, stating that “the form of <i>H</i> will be recognized as that of entropy as defined in certain formulations of statistical mechanics where <i>p<sub style="line-height: 1em;">i</sub></i> is the probability of a system being in cell <i>i</i> of its phase space… <i>H</i> is then, for example, the <i>H</i> in Boltzmann’s famous <a href="http://en.wikipedia.org/wiki/H-theorem" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="H-theorem">H theorem</a>.” As such, over the last fifty years, ever since this statement was made, people have been overlapping the two concepts or even stating that they are exactly the same.</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">Shannon's information entropy is a much more general concept than statistical thermodynamic entropy. Information entropy is present whenever there are unknown quantities that can be described only by a probability distribution. In a series of papers by <a class="mw-redirect" href="http://en.wikipedia.org/wiki/E._T._Jaynes" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="E. T. Jaynes">E. T. Jaynes</a> starting in 1957,<sup class="reference" id="cite_ref-10" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_note-10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[11]</a></sup><sup class="reference" id="cite_ref-11" style="line-height: 1em;"><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_note-11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none; white-space: nowrap;">[12]</a></sup> the statistical thermodynamic entropy can be seen as just a particular application of Shannon's information entropy to the probabilities of particular microstates of a system occurring in order to produce a particular macrostate.</div><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 19px; font-weight: normal; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=History_of_entropy&action=edit&section=9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Popular use">edit</a>]</span><span class="mw-headline" id="Popular_use">Popular use</span></h2><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">The term entropy is often used in popular language to denote a variety of unrelated phenomena. One example is the concept of <b>corporate entropy</b> as put forward somewhat humorously by authors Tom DeMarco and Timothy Lister in their 1987 classic publication <i>Peopleware</i>, a book on growing and managing productive teams and successful software projects. Here, they view energy waste as red tape and business team inefficiency as a form of entropy, i.e. energy lost to waste. This concept has caught on and is now common jargon in business schools.</div><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 19px; font-weight: normal; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=History_of_entropy&action=edit&section=10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: Terminology overlap">edit</a>]</span><span class="mw-headline" id="Terminology_overlap">Terminology overlap</span></h2><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-top: 0.4em;">When necessary, to disambiguate between the statistical thermodynamic concept of entropy, and entropy-like formulae put forward by different researchers, the statistical thermodynamic entropy is most properly referred to as the <b><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Gibbs_entropy" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Gibbs entropy">Gibbs entropy</a></b>. The terms <i>Boltzmann-Gibbs entropy</i> or <i>BG entropy</i>, and <i>Boltzmann-Gibbs-Shannon entropy</i> or <i>BGS entropy</i> are also seen in the literature.</div><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 19px; font-weight: normal; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=History_of_entropy&action=edit&section=11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: See also">edit</a>]</span><span class="mw-headline" id="See_also">See also</span></h2><ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Entropy" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Entropy">Entropy</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Enthalpy" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Enthalpy">Enthalpy</a></li>
<li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Thermodynamic_free_energy" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Thermodynamic free energy">Thermodynamic free energy</a></li>
</ul><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 19px; font-weight: normal; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=History_of_entropy&action=edit&section=12" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: References">edit</a>]</span><span class="mw-headline" id="References">References</span></h2><div class="reflist" style="font-size: 12px; list-style-type: decimal; margin-bottom: 0.5em;"><ol class="references" style="line-height: 1.5em; list-style-image: none; list-style-type: inherit; margin-bottom: 0.5em; margin-left: 3.2em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li id="cite_note-0" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_ref-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;">Mendoza, E. (1988). <i>Reflections on the Motive Power of Fire – and other Papers on the Second Law of Thermodynamics by E. Clapeyron and R. Clausius</i>. New York: Dover Publications. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-486-44641-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-486-44641-7">0-486-44641-7</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Reflections+on+the+Motive+Power+of+Fire+%E2%80%93+and+other+Papers+on+the+Second+Law+of+Thermodynamics+by+E.+Clapeyron+and+R.+Clausius&rft.aulast=Mendoza%2C+E.&rft.au=Mendoza%2C+E.&rft.date=1988&rft.place=New+York&rft.pub=Dover+Publications&rft.isbn=0-486-44641-7&rfr_id=info:sid/en.wikipedia.org:History_of_entropy"></span></span></li>
<li id="cite_note-1" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_ref-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><i>Mechanical Theory of Heat</i>, by <a href="http://en.wikipedia.org/wiki/Rudolf_Clausius" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Rudolf Clausius">Rudolf Clausius</a>, 1850-1865</span></li>
<li id="cite_note-2" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_ref-2" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Published in Poggendoff’s Annalen, December 1854, vol. xciii. p. 481; translated in the Journal de Mathematiques, vol. xx. Paris, 1855, and in the Philosophical Magazine, August 1856, s. 4. vol. xii, p. 81</span></li>
<li id="cite_note-3" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_ref-3" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">Clausius, Rudolf. (1856). "<i>On the Application of the Mechanical theory of Heat to the Steam-Engine</i>." as found in: Clausius, R. (1865). <a class="external text" href="http://books.google.com/books?id=8LIEAAAAYAAJ" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies</a>. London: John van Voorst, 1 Paternoster Row. MDCCCLXVII.</span></li>
<li id="cite_note-4" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_ref-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;">Laidler, Keith J. (1995). <i>The Physical World of Chemistry</i>. Oxford University Press. pp. 104–105. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-19-855919-4" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-19-855919-4">0-19-855919-4</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Physical+World+of+Chemistry&rft.aulast=Laidler&rft.aufirst=Keith+J.&rft.au=Laidler%2C%26%2332%3BKeith+J.&rft.date=1995&rft.pages=pp.%26nbsp%3B104%E2%80%93105&rft.pub=Oxford+University+Press&rft.isbn=0-19-855919-4&rfr_id=info:sid/en.wikipedia.org:History_of_entropy"></span></span></li>
<li id="cite_note-5" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_ref-5" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><a class="mw-redirect" href="http://en.wikipedia.org/wiki/OED" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="OED">OED</a>, Second Edition, 1989, "<i>Clausius (Pogg. Ann. CXXV. 390), assuming (unhistorically) the etymological sense of energy to be ‘work-contents’ (werk-inhalt), devised the term entropy as a corresponding designation for the ‘transformation-contents’ (verwandlungsinhalt) of a system"</i></span></li>
<li id="cite_note-Clausius-6" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_ref-Clausius_6-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;">Clausius, Rudolf (1850). <i>On the Motive Power of Heat, and on the Laws which can be deduced from it for the Theory of Heat</i>. Poggendorff's <i>Annalen der Physick</i>, LXXIX (Dover Reprint). <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-486-59065-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/0-486-59065-8">0-486-59065-8</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+the+Motive+Power+of+Heat%2C+and+on+the+Laws+which+can+be+deduced+from+it+for+the+Theory+of+Heat&rft.aulast=Clausius&rft.aufirst=Rudolf&rft.au=Clausius%2C%26%2332%3BRudolf&rft.date=1850&rft.pub=Poggendorff%27s+%27%27Annalen+der+Physick%27%27%2C+LXXIX+%28Dover+Reprint%29&rft.isbn=0-486-59065-8&rfr_id=info:sid/en.wikipedia.org:History_of_entropy"></span></span></li>
<li id="cite_note-7" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_ref-7" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">M. Tribus, E.C. McIrvine, “Energy and information”, Scientific American, 224 (September 1971).</span></li>
<li id="cite_note-8" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_ref-8" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text"><span class="citation book" style="word-wrap: break-word;">Avery, John (2003). <i>Information Theory and Evolution</i>. World Scientific. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/981-238-400-6" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Special:BookSources/981-238-400-6">981-238-400-6</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Information+Theory+and+Evolution&rft.aulast=Avery%2C+John&rft.au=Avery%2C+John&rft.date=2003&rft.pub=World+Scientific&rft.isbn=981-238-400-6&rfr_id=info:sid/en.wikipedia.org:History_of_entropy"></span></span></li>
<li id="cite_note-9" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_ref-9" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">C.E. Shannon, "A Mathematical Theory of Communication", <i><a href="http://en.wikipedia.org/wiki/Bell_System_Technical_Journal" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Bell System Technical Journal">Bell System Technical Journal</a></i>, vol. 27, pp. 379-423, 623-656, July, October, 1948, <a class="external text" href="http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;">Eprint</a>, <a class="external text" href="http://cm.bell-labs.com/cm/ms/what/shannonday/shannon1948.pdf" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif); background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 18px; text-decoration: none;">PDF</a></span></li>
<li id="cite_note-10" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_ref-10" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">E. T. Jaynes (1957) <a class="external text" href="http://bayes.wustl.edu/etj/articles/theory.1.pdf" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif); background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 18px; text-decoration: none;">Information theory and statistical mechanics</a>, <i>Physical Review</i> <b>106</b>:620</span></li>
<li id="cite_note-11" style="margin-bottom: 0.1em;"><b><a href="http://en.wikipedia.org/wiki/History_of_entropy#cite_ref-11" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;">^</a></b> <span class="reference-text">E. T. Jaynes (1957) <a class="external text" href="http://bayes.wustl.edu/etj/articles/theory.2.pdf" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(http://upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif); background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 18px; text-decoration: none;">Information theory and statistical mechanics II</a>, <i>Physical Review</i> <b>108</b>:171</span></li>
</ol></div><h2 style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 19px; font-weight: normal; margin-bottom: 0.6em; margin-left: 0px; margin-right: 0px; margin-top: 0px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 0.17em; padding-top: 0.5em; width: auto;"><span class="editsection" style="float: right; font-size: 13px; margin-left: 5px;">[<a href="http://en.wikipedia.org/w/index.php?title=History_of_entropy&action=edit&section=13" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Edit section: External links">edit</a>]</span><span class="mw-headline" id="External_links">External links</span></h2><ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;"><a href="http://en.wikipedia.org/wiki/Max_Jammer" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #0b0080; text-decoration: none;" title="Max Jammer">Max Jammer</a> (1973). <a class="external text" href="http://etext.lib.virginia.edu/cgi-local/DHI/dhi.cgi?id=dv2-12" rel="nofollow" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(data:image/png; background-origin: initial; background-position: 100% 50%; background-repeat: no-repeat no-repeat; color: #663366; padding-right: 13px; text-decoration: none;"><i>Dictionary of the History of Ideas</i>: Entropy</a></li>
</ul></div><div class="articleFeedback" id="mw-articlefeedback" style="display: inline-block; font-size: 13px; line-height: 19px; margin-top: 1em; position: relative;"><div class="articleFeedback-panel" style="background-color: #f9f9f9; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; border-image: initial; border-left-color: rgb(204, 204, 204); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(204, 204, 204); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(204, 204, 204); border-top-style: solid; border-top-width: 1px; float: left; padding-bottom: 1px;"><div class="articleFeedback-buffer articleFeedback-ui" style="padding-bottom: 0.75em; padding-left: 1em; padding-right: 1em; padding-top: 0.75em;"><br class="Apple-interchange-newline" /></div></div></div>History of entrop</span></h1><div id="bodyContent" style="background-color: white; font-family: sans-serif; font-size: 0.8em; line-height: 1.5em; position: relative; width: 1157px;"><div class="mw-content-ltr" dir="ltr" lang="en" style="direction: ltr;"><ul style="line-height: 1.5em; list-style-image: url(data:image/png; list-style-type: square; margin-bottom: 0px; margin-left: 1.6em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"></ul></div></div>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com0tag:blogger.com,1999:blog-5303246073824127471.post-59690783809283661512012-02-28T10:40:00.001-05:002012-02-28T10:44:11.751-05:00The Universe in a Glass of WineFor Billy Stiltner of Virgina on his 30th Birthday with thanks for his work on Fractals, and Sheila Whitaker of Tennessee for keeping the Astrophysics and Solid-State Physics alive, and overall great sense of humor.<br />
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"A poet once said 'The whole universe is in a glass of wine.'<br />
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We will probably never know in what sense he meant that, for poets do not write to be understood. But it is true that if we look at a glass closely enough we see the entire universe.<br />
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There are the things of physics: the twisting liquid which evaporates depending on the wind and weather, the reflections in the glass, and our imagination adds the atoms. The glass is a distillation of the earth's rocks, and in its composition we see the secret of the universe's age, and the evolution of the stars.<br />
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What strange array of chemicals are there in the wine? How did they come to be?<br />
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There are the ferments, the enzymes, the substrates, and the products.<br />
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There in wine is found the great generalization: all life is fermentation. Nobody can discover the chemistry of wine without discovering, as did Louis Pasteur, the cause of much disease.<br />
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How vivid is the claret, pressing its existence into the consciousness that watches it! If our small minds, for some convenience, divide this glass of wine, this universe, into parts - physics, biology, geology, astronomy, psychology, and so on - remember that Nature does not know it!<br />
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So let us put it all back together, not forgetting ultimately what it is for. Let it give us one more final pleasure: drink it and forget it all!"<br />
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... Richard Feynman<br />
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Feynman Flowchart (What Would Richard Feynman Do?) :<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIaGtQ9YI_V835mdVf_QHqa2OGAVijCwkvTv7jQTH36e-5YsaEf-xlzn1qAlqbJG-NT-w-5-hLyFPVGK_CUkXJfS-9ltl5_6WaKYWoZ5eEBC_ktUzpDCOnh6bPk9d1H0bLumDXpfx3sg/s1600/richard-feynman-do.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="480" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIaGtQ9YI_V835mdVf_QHqa2OGAVijCwkvTv7jQTH36e-5YsaEf-xlzn1qAlqbJG-NT-w-5-hLyFPVGK_CUkXJfS-9ltl5_6WaKYWoZ5eEBC_ktUzpDCOnh6bPk9d1H0bLumDXpfx3sg/s640/richard-feynman-do.png" width="640" /></a>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com0tag:blogger.com,1999:blog-5303246073824127471.post-20592989340249776622012-02-25T10:02:00.001-05:002012-02-25T10:03:21.408-05:00EULER !<a href="http://en.wikipedia.org/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a> called Euler's formula "our jewel"<sup class="reference" id="cite_ref-1"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#cite_note-1">[2]</a></sup> and "one of the most remarkable, almost astounding, formulas in all of mathematics."<sup class="reference" id="cite_ref-2"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#cite_note-2">[3]</a></sup><br />
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<tr><td class="tr-caption" style="text-align: center;"><a href="http://www.titaniumringsforever.com/search/node/euler%20type:ring">You can buy the ring above for a mere $245 here.</a><br />
One equation to bring them all and in the darkness bind them. In the land of Euler (that would be Switzerland, although Euler did most of his work and lived in St. Petersberg and Prussia).</td></tr>
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<a href="http://www.amazon.com/Dr-Eulers-Fabulous-Formula-Mathematical/dp/0691150370/ref=sr_1_1?ie=UTF8&qid=1303594437&sr=8-1">Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills</a> by Paul J. Nahin, Professor Emeritus, University of New Hampshire, PhDEE<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBz3_ZOLYX443tnFvULhx_vgEozXojtezM_udaRS4pWnfeXcjl8IQzgAvl9SC-ZKPz_MCYpqs7fsgobRiOO4bBqZyYpG1iXQLPkq4lLbDoREWdXPqq708i19ZUSrGw4t3RSdPSbqKT0Q/s1600/cover.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBz3_ZOLYX443tnFvULhx_vgEozXojtezM_udaRS4pWnfeXcjl8IQzgAvl9SC-ZKPz_MCYpqs7fsgobRiOO4bBqZyYpG1iXQLPkq4lLbDoREWdXPqq708i19ZUSrGw4t3RSdPSbqKT0Q/s400/cover.jpg" width="265" /></a></div><br />
From <a href="http://plus.maths.org/content/dr-eulers-fabulous-formula">+ Plus magazine ... Living Mathematics</a> Review by Lewis Dartnell<br />
<br />
<br />
<div style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 14px; line-height: 18px; margin-bottom: 1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0em;">The hero of this book is Euler's formula:</div><div style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 14px; line-height: 18px; margin-bottom: 1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0em;"><i>e</i><sup>i<i>π</i></sup> + 1 = 0</div><div style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 14px; line-height: 18px; margin-bottom: 1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0em;">This simple equation has been widely considered through the last two centuries to be one of the most beautiful formulae of mathematics, and Nahin tells us why.</div><div style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 14px; line-height: 18px; margin-bottom: 1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0em;">The constant <i>e</i> is the base of the natural logarithm (and you might have used it in calculations on radioactive decay in physics lessons, for example), whereas <i>&pi</i>, as we all know, is the ratio of a circle's circumference to its diameter. Both<i>e</i> and <i>&pi</i> are irrational numbers, that is, you could never write down all of their decimal places as they can be proven to continue for ever. The symbol <i>i</i> denotes the square root of -1, a value that does not even exist on the standard number line. Each of these three quantities is therefore curious in its own right, and they were originally developed within very different areas of maths. So how on Earth does <i>e</i><sup>i<i>&pi</i></sup> equal <i>exactly</i> -1? It seems like the biggest fluke in the Universe — and this is part the formula's exquisite beauty. It is also fairly easy to derive the formula yourself, and the proof can be found in any textbook on complex numbers. And as Nahin's book shows, it is also one of the most influential formulae in the history of mathematics.</div><div style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 14px; line-height: 18px; margin-bottom: 1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0em;">The book starts off gently enough, with an Introduction leading the reader through a few examples of the most fundamental skill in mathematics; constructing a proof. We see, for example, why √2 must be irrational through a simple proof by contradiction. The process of proving things is mostly ignored at GCSE and A-level, but really does give an eye-opening insight into how real mathematics is often like solving a intriguing puzzle, rather than slogging through homework exercises to practice the basic skills.</div><div style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 14px; line-height: 18px; margin-bottom: 1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0em;">Through the following six chapters of the book, Nahin shows us many of the applications of imaginary numbers, Euler's formula, and other mathematical formulae and techniques that have been built on this eighteenth century foundation, both in pure and applied maths. Regular <i>Plus</i>readers might already know a little bit about some of these, and the sections in the book include: drawing regular prime number polygons, such as the 17-gon, using only a ruler and a compass; how to deconstruct any continuous function, such as a sound wave, into a set of sine waves — a technique that is crucial to modern gadgets like mp3 players. Nahin also shows us the maths of complex numbers lying behind the uncertainty principle of quantum mechanics, listening to the radio, and even how to build a telephone voice scrambler from simple electronics.</div><div style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 14px; line-height: 18px; margin-bottom: 1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; text-indent: 0em;">The book is well-written and illustrated with numerous diagrams and graphs. For those wanting a little more detail, or to follow up on the bibliography, the book is usefully cross-referenced and has hordes of end notes. But I do have one major warning for <i>Plus</i> readers. Although this book is written to be more easy-reading and popularist than a text book, it is still very heavy-going. More importantly it is pitched at the level of university undergraduates. However, if you're enjoying maths A-level, then this book has a lot to offer, even if you don't understand everything. Every chapter begins with an interesting introduction on the history of a particular problem and the lives of the mathematicians involved, before heading into streams of equations and derivations. The final section of the book gives a detailed biography of the genius behind all of this mathematics through the ages, <a href="http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Euler.html" style="color: #005689; text-decoration: none;">Leonhard Euler</a>. If you have a class project to write an essay on an influential mathematician, then Euler would certainly be an inspirational choice, and this final chapter a good place to start your research.</div><br />
We will be discussing two equations. The ring illustrated above is Euler's Identity, which we will discuss second. Feynman's quote refers to Euler's formula, which we will discuss first. From Wikipedia:<br />
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<b>Euler's formula</b>, named after <a href="http://en.wikipedia.org/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, is a <a href="http://en.wikipedia.org/wiki/Mathematics" title="Mathematics">mathematical</a> <a href="http://en.wikipedia.org/wiki/Formula" title="Formula">formula</a> in <a href="http://en.wikipedia.org/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> that establishes the deep relationship between the <a href="http://en.wikipedia.org/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> and the <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex</a> <a href="http://en.wikipedia.org/wiki/Exponential_function" title="Exponential function">exponential function</a>. Euler's formula states that, for any <a href="http://en.wikipedia.org/wiki/Real_number" title="Real number">real number</a> <i>x</i>,<br />
<dl><dd><img alt="e^{ix} = \cos x + i\sin x \ " class="tex" src="http://upload.wikimedia.org/math/c/9/f/c9f2055dadfb49853eff822a453d9ceb.png" /></dd></dl>where <i>e</i> is the <a href="http://en.wikipedia.org/wiki/E_%28mathematical_constant%29" title="E (mathematical constant)">base of the natural logarithm</a>, <i>i</i> is the <a href="http://en.wikipedia.org/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>, and cos and sin are the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Trigonometric_function" title="Trigonometric function">trigonometric functions</a> cosine and sine respectively, with the argument <i>x</i> given in <a href="http://en.wikipedia.org/wiki/Radian" title="Radian">radians</a>. This complex exponential function is sometimes called <b>cis</b>(<i>x</i>). The formula is still valid if <i>x</i> is a <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex number</a>, and so some authors refer to the more general complex version as Euler's formula.<sup class="reference" id="cite_ref-0"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#cite_note-0">[1]</a></sup><br />
<table class="toc tochidden" id="toc"><tbody>
<tr> <td><ul style="display: none;"><li class="toclevel-1 tocsection-1"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#History"><span class="tocnumber">1</span> <span class="toctext">History</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#Applications_in_complex_number_theory"><span class="tocnumber">2</span> <span class="toctext">Applications in complex number theory</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#Relationship_to_trigonometry"><span class="tocnumber">3</span> <span class="toctext">Relationship to trigonometry</span></a></li>
<li class="toclevel-1 tocsection-4"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#Other_applications"><span class="tocnumber">4</span> <span class="toctext">Other applications</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#Definitions_of_complex_exponentiation"><span class="tocnumber">5</span> <span class="toctext">Definitions of complex exponentiation</span></a> <ul><li class="toclevel-2 tocsection-6"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#Power_series_definition"><span class="tocnumber">5.1</span> <span class="toctext">Power series definition</span></a></li>
<li class="toclevel-2 tocsection-7"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#Limit_definition"><span class="tocnumber">5.2</span> <span class="toctext">Limit definition</span></a></li>
</ul></li>
<li class="toclevel-1 tocsection-8"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#Proofs"><span class="tocnumber">6</span> <span class="toctext">Proofs</span></a> <ul><li class="toclevel-2 tocsection-9"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#Using_power_series"><span class="tocnumber">6.1</span> <span class="toctext">Using power series</span></a></li>
<li class="toclevel-2 tocsection-10"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#Using_calculus"><span class="tocnumber">6.2</span> <span class="toctext">Using calculus</span></a></li>
<li class="toclevel-2 tocsection-11"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#Using_differential_equations"><span class="tocnumber">6.3</span> <span class="toctext">Using differential equations</span></a></li>
</ul></li>
<li class="toclevel-1 tocsection-12"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#See_also"><span class="tocnumber">7</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-13"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#References"><span class="tocnumber">8</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-14"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#External_links"><span class="tocnumber">9</span> <span class="toctext">External links</span></a></li>
</ul></td> </tr>
</tbody></table><h2><span class="editsection"></span><span class="mw-headline" id="History">History</span></h2>It was <a href="http://en.wikipedia.org/wiki/Bernoulli" title="Bernoulli">Bernoulli</a> [1702] who noted that<br />
<dl><dd><img alt="\frac{1}{1+x^2}=\frac{1}{2} \left(\frac{1}{1-ix}+\frac{1}{1+ix} \right) \ ." class="tex" src="http://upload.wikimedia.org/math/d/b/c/dbcba9dc5aac67049709f3f5c8ed5b3a.png" /></dd></dl>And since<br />
<dl><dd><img alt="\int \frac{dx}{1+ax}=\frac{1}{a}\ln(1+ax)+C \ ," class="tex" src="http://upload.wikimedia.org/math/9/d/f/9dfad84bb86db69d2e6dd74b303a304c.png" /></dd></dl>the above equation tells us something about complex logarithms. Bernoulli, however, did not evaluate the integral. His correspondence with Euler (who also knew the above equation) shows that he didn't fully understand logarithms. Euler also suggested that the complex logarithms can have infinitely many values.<br />
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Meanwhile, <a href="http://en.wikipedia.org/wiki/Roger_Cotes" title="Roger Cotes">Roger Cotes</a>, in 1714, discovered<br />
<dl><dd><img alt=" \ln(\cos x + i\sin x)=ix \ " class="tex" src="http://upload.wikimedia.org/math/d/d/1/dd10cb1ac2c564b31c1e55638e25d209.png" /></dd></dl>(where "ln" means <a href="http://en.wikipedia.org/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>, i.e. log with base <i>e</i>).<sup class="reference" id="cite_ref-3"><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#cite_note-3">[4]</a></sup> We now know that the above equation is only true <a href="http://en.wikipedia.org/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> integer multiples of <span class="texhtml">2π<i>i</i></span>, but Cotes missed the fact that a complex logarithm can have infinitely many values which owes to the periodicity of the trigonometric functions.<br />
<br />
It was Euler (presumably around 1740) who turned his attention to the exponential function instead of logarithms, and obtained the correct formula now coined after his name. It was published in 1748, and his proof was based on the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Infinite_series" title="Infinite series">infinite series</a> of both sides being equal. Neither of these men saw the geometrical interpretation of the formula: the view of complex numbers as points in the <a href="http://en.wikipedia.org/wiki/Complex_plane" title="Complex plane">complex plane</a> arose only some 50 years later (see <a href="http://en.wikipedia.org/wiki/Caspar_Wessel" title="Caspar Wessel">Caspar Wessel</a>).<br />
<h2><span class="editsection"></span><span class="mw-headline" id="Applications_in_complex_number_theory">Applications in complex number theory</span></h2><div class="thumb tright"><div class="thumbinner" style="width: 222px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Euler%27s_formula.svg"><img alt="Euler's formula.svg" class="thumbimage" height="217" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/220px-Euler%27s_formula.svg.png" width="220" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:Euler%27s_formula.svg" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.5/common/images/magnify-clip.png" width="15" /></a></div></div></div></div><div class="thumb tright"><div class="thumbinner" style="width: 332px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Euler%27s_Formula_c.png"><img alt="" class="thumbimage" height="330" src="http://upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Euler%27s_Formula_c.png/330px-Euler%27s_Formula_c.png" width="330" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:Euler%27s_Formula_c.png" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.5/common/images/magnify-clip.png" width="15" /></a></div>Three-dimensional visualization of Euler's formula. See also <a href="http://en.wikipedia.org/wiki/Circular_polarization" title="Circular polarization">circular polarization</a>.</div></div></div><br />
This formula can be interpreted as saying that the function <i>e</i><sup><i>ix</i></sup> traces out the <a href="http://en.wikipedia.org/wiki/Unit_circle" title="Unit circle">unit circle</a> in the <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex number</a> plane as <i>x</i> ranges through the real numbers. Here, <i>x</i> is the <a href="http://en.wikipedia.org/wiki/Angle" title="Angle">angle</a> that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in <a href="http://en.wikipedia.org/wiki/Radian" title="Radian">radians</a>.<br />
<br />
The original proof is based on the <a href="http://en.wikipedia.org/wiki/Taylor_series" title="Taylor series">Taylor series</a> expansions of the <a href="http://en.wikipedia.org/wiki/Exponential_function" title="Exponential function">exponential function</a> <i>e</i><sup><i>z</i></sup> (where <i>z</i> is a complex number) and of sin <i>x</i> and cos <i>x</i> for real numbers <i>x</i> (see below). In fact, the same proof shows that Euler's formula is even valid for all <i>complex</i> numbers <i>z</i>.<br />
<br />
A point in the <a href="http://en.wikipedia.org/wiki/Complex_plane" title="Complex plane">complex plane</a> can be represented by a complex number written in <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Coordinates_%28elementary_mathematics%29#Cartesian_coordinates" title="Coordinates (elementary mathematics)">cartesian coordinates</a>. Euler's formula provides a means of conversion between cartesian coordinates and <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Coordinates_%28elementary_mathematics%29#Polar_coordinates" title="Coordinates (elementary mathematics)">polar coordinates</a>. The polar form reduces the number of <a href="http://en.wikipedia.org/wiki/Term_%28mathematics%29" title="Term (mathematics)">terms</a> from two to one, which simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number <i>z</i> = <i>x</i> + <i>iy</i> can be written as<br />
<dl><dd><img alt=" z = x + iy = |z| (\cos \phi + i\sin \phi ) = r e^{i \phi} \ " class="tex" src="http://upload.wikimedia.org/math/7/8/1/7810bad1ee2b17aa0dd72367a6bedb76.png" /></dd><dd><img alt=" \bar{z} = x - iy = |z| (\cos \phi - i\sin \phi ) = r e^{-i \phi} \ " class="tex" src="http://upload.wikimedia.org/math/f/3/9/f39844364cb655f5a4dc04011241bc9e.png" /></dd></dl>where<br />
<dl><dd><img alt=" x = \mathrm{Re}\{z\} \," class="tex" src="http://upload.wikimedia.org/math/0/4/b/04bfad7c61a5331cbec6c2f135acbbf4.png" /> the real part</dd><dd><img alt=" y = \mathrm{Im}\{z\} \," class="tex" src="http://upload.wikimedia.org/math/5/c/8/5c883511f99eec9c89974418a5f43fbb.png" /> the imaginary part</dd><dd><img alt=" r = |z| = \sqrt{x^2+y^2}" class="tex" src="http://upload.wikimedia.org/math/8/7/8/878a3519f2c6c1e9965dbeb479a11b9e.png" /> the <a href="http://en.wikipedia.org/wiki/Magnitude_%28mathematics%29" title="Magnitude (mathematics)">magnitude</a> of <i>z</i></dd><dd><img alt="\phi = \arg z = \," class="tex" src="http://upload.wikimedia.org/math/f/1/9/f19b678734154067129c05ae4008bf0f.png" /> <a href="http://en.wikipedia.org/wiki/Atan2" title="Atan2">atan2</a>(<i>y</i>, <i>x</i>) .</dd></dl><img alt="\phi \," class="tex" src="http://upload.wikimedia.org/math/c/d/0/cd014731964c742c274df08d7cc238fb.png" /> is the <i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Arg_%28mathematics%29" title="Arg (mathematics)">argument</a></i> of <i>z</i>—i.e., the angle between the <i>x</i> axis and the vector <i>z</i> measured counterclockwise and in <a href="http://en.wikipedia.org/wiki/Radian" title="Radian">radians</a>—which is defined <a href="http://en.wikipedia.org/wiki/Up_to" title="Up to">up to</a> addition of 2π. Many texts write tan<sup>-1</sup>(<i>y</i>/<i>x</i>) instead of atan2(<i>y</i>,<i>x</i>) but this needs adjustment when <i>x</i> ≤ 0.<br />
<br />
Now, taking this derived formula, we can use Euler's formula to define the <a href="http://en.wikipedia.org/wiki/Logarithm" title="Logarithm">logarithm</a> of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation) that<br />
<dl><dd><img alt="a = e^{\ln (a)} \ " class="tex" src="http://upload.wikimedia.org/math/6/9/1/691bfeaecf6b3a2e6e11801eae5cc292.png" /></dd></dl>and that<br />
<dl><dd><img alt="e^a e^b = e^{a + b} \ " class="tex" src="http://upload.wikimedia.org/math/1/b/6/1b630f30ff74d1eee33a0d84b7c7aeb8.png" /></dd></dl>both valid for any complex numbers <i>a</i> and <i>b</i>.<br />
<br />
Therefore, one can write:<br />
<dl><dd><img alt=" z = |z| e^{i \phi} = e^{\ln |z|} e^{i \phi} = e^{\ln |z| + i \phi} \ " class="tex" src="http://upload.wikimedia.org/math/1/8/0/1804780755af2618fa7175f62e271d33.png" /></dd></dl>for any <i>z</i> ≠ 0. Taking the logarithm of both sides shows that:<br />
<dl><dd><img alt="\ln z= \ln |z| + i \phi \ ." class="tex" src="http://upload.wikimedia.org/math/b/8/5/b852751e5a5e429b52a18d04019c4334.png" /></dd></dl>and in fact this can be used as the definition for the <a href="http://en.wikipedia.org/wiki/Complex_logarithm" title="Complex logarithm">complex logarithm</a>. The logarithm of a complex number is thus a <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Multi-valued_function" title="Multi-valued function">multi-valued function</a>, because <span class="texhtml">φ</span> is multi-valued.<br />
<br />
Finally, the other exponential law<br />
<dl><dd><img alt="(e^a)^k = e^{a k} \ ," class="tex" src="http://upload.wikimedia.org/math/8/7/7/87739e76a47aeaa7c26559fd2cd505c9.png" /></dd></dl>which can be seen to hold for all integers <i>k</i>, together with Euler's formula, implies several <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Trigonometric_identity" title="Trigonometric identity">trigonometric identities</a> as well as <a href="http://en.wikipedia.org/wiki/De_Moivre%27s_formula" title="De Moivre's formula">de Moivre's formula</a>.<br />
<h2><span class="editsection"></span><span class="mw-headline" id="Relationship_to_trigonometry">Relationship to trigonometry</span></h2>Euler's formula provides a powerful connection between <a href="http://en.wikipedia.org/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a> and <a href="http://en.wikipedia.org/wiki/Trigonometry" title="Trigonometry">trigonometry</a>, and provides an interpretation of the sine and cosine functions as <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Weighted_sum" title="Weighted sum">weighted sums</a> of the exponential function<b>:</b><br />
<dl><dd><img alt="\cos x = \mathrm{Re}\{e^{ix}\} ={e^{ix} + e^{-ix} \over 2}" class="tex" src="http://upload.wikimedia.org/math/4/b/3/4b331f79eaadfba81f95d40a6e80b88a.png" /></dd><dd><img alt="\sin x = \mathrm{Im}\{e^{ix}\} ={e^{ix} - e^{-ix} \over 2i} \ ." class="tex" src="http://upload.wikimedia.org/math/6/d/f/6dfbf63c384cff19f602cbc3e0a4d2df.png" /></dd></dl>The two equations above can be derived by adding or subtracting Euler's formulas<b>:</b><br />
<dl><dd><img alt="e^{ix} = \cos x + i \sin x \;" class="tex" src="http://upload.wikimedia.org/math/9/1/f/91fc7a888736b55389bdbeed2593ee84.png" /></dd></dl><dl><dd><img alt="e^{-ix} = \cos(- x) + i \sin(- x) = \cos x - i \sin x \;" class="tex" src="http://upload.wikimedia.org/math/4/4/e/44ea7e9ee99fd1db849f55a024988795.png" /></dd></dl>and solving for either cosine or sine.<br />
<br />
These formulas can even serve as the definition of the trigonometric functions for complex arguments <i>x</i>. For example, letting <i>x</i> = <i>iy</i>, we have<b>:</b><br />
<dl><dd><img alt=" \cos(iy) = {e^{-y} + e^{y} \over 2} = \cosh(y) " class="tex" src="http://upload.wikimedia.org/math/2/8/7/2877d21c421a2d0df43cddfc2c3b44dc.png" /></dd></dl><dl><dd><img alt=" \sin(iy) = {e^{-y} - e^{y} \over 2i} = -{1 \over i} {e^{y} - e^{-y} \over 2} = i\sinh(y) \ . " class="tex" src="http://upload.wikimedia.org/math/e/a/0/ea06f104531ce3a80c58d177bfa5173a.png" /></dd></dl>Complex exponentials can simplify trigonometry, because they are easier to manipulate than their sinusoidal components. One technique is simply to convert sinusoids into equivalent expressions in terms of exponentials.<br />
<br />
After the manipulations, the simplified result is still real-valued. For example<b>:</b><br />
<dl><dd><img alt="\begin{align}
\cos x\cdot \cos y & = \frac{(e^{ix}+e^{-ix})}{2} \cdot \frac{(e^{iy}+e^{-iy})}{2} \\
& = \frac{1}{2}\cdot \frac{e^{i(x+y)}+e^{i(x-y)}+e^{i(-x+y)}+e^{i(-x-y)}}{2} \\
& = \frac{1}{2} \left[ \underbrace{ \frac{e^{i(x+y)} + e^{-i(x+y)}}{2} }_{\cos(x+y)} + \underbrace{ \frac{e^{i(x-y)} + e^{-i(x-y)}}{2} }_{\cos(x-y)} \right] \ .
\end{align}" class="tex" src="http://upload.wikimedia.org/math/4/5/3/453afc84086fb72dbe8dc2bd17eed023.png" /></dd></dl>Another technique is to represent the sinusoids in terms of the <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Real_part" title="Real part">real part</a> of a more complex expression, and perform the manipulations on the complex expression. For example<b>:</b><br />
<dl><dd><img alt="\begin{align}
\cos(nx) & = \mathrm{Re} \{\ e^{inx}\ \}
= \mathrm{Re} \{\ e^{i(n-1)x}\cdot e^{ix}\ \} \\
& = \mathrm{Re} \{\ e^{i(n-1)x}\cdot (e^{ix} + e^{-ix} - e^{-ix})\ \} \\
& = \mathrm{Re} \{\ e^{i(n-1)x}\cdot \underbrace{(e^{ix} + e^{-ix})}_{2\cos(x)} - e^{i(n-2)x}\ \} \\
& = \cos[(n-1)x]\cdot 2 \cos(x) - \cos[(n-2)x] \ .
\end{align}" class="tex" src="http://upload.wikimedia.org/math/a/7/e/a7e33560e9ae009ce7eb56b99babb19d.png" /></dd></dl>This formula is used for recursive generation of cos(<i>nx</i>) for integer values of <i>n</i> and arbitrary <i>x</i> (in radians).<br />
<h2><span class="editsection"></span><span class="mw-headline" id="Other_applications">Other applications</span></h2>In <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Differential_equations" title="Differential equations">differential equations</a>, the function <i>e</i><sup><i>ix</i></sup> is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. The reason for this is that the complex exponential is the <a href="http://en.wikipedia.org/wiki/Eigenfunction" title="Eigenfunction">eigenfunction</a> of differentiation. <a href="http://en.wikipedia.org/wiki/Euler%27s_identity" title="Euler's identity">Euler's identity</a> is an easy consequence of Euler's formula.<br />
<br />
In <a href="http://en.wikipedia.org/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a> and other fields, signals that vary periodically over time are often described as a combination of sine and cosine functions (see <a href="http://en.wikipedia.org/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>), and these are more conveniently expressed as the real part of exponential functions with <a href="http://en.wikipedia.org/wiki/Imaginary_number" title="Imaginary number">imaginary</a> exponents, using Euler's formula. Also, <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Phasor_analysis" title="Phasor analysis">phasor analysis</a> of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor.<br />
<h2><span class="editsection"></span><span class="mw-headline" id="Definitions_of_complex_exponentiation">Definitions of complex exponentiation</span></h2><div class="rellink relarticle mainarticle">Main articles: <a href="http://en.wikipedia.org/wiki/Exponentiation" title="Exponentiation">Exponentiation</a> and <a href="http://en.wikipedia.org/wiki/Exponential_function" title="Exponential function">Exponential function</a></div><br />
The exponential function <i>e<sup>x</sup></i> for real values of <i>x</i> may be defined in a few different equivalent ways (see <a href="http://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function" title="Characterizations of the exponential function">Characterizations of the exponential function</a>). Several of these methods may be directly extended to give definitions of <i>e<sup>z</sup></i> for complex values of <i>z</i> simply by substituting <i>z</i> in place of <i>x</i> and using the complex algebraic operations. In particular we may use either of the two following definitions which are equivalent. From a more advanced perspective, each of these definitions may be interpreted as giving the unique <a href="http://en.wikipedia.org/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a> of <i>e<sup>x</sup></i> to the complex plane.<br />
<h3><span class="editsection"></span><span class="mw-headline" id="Power_series_definition">Power series definition</span></h3>For complex <i>z</i><br />
<dl><dd><img alt="e^z = 1 + \frac{z}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots = \sum_{n=0}^{\infty} \frac{z^n}{n!} ~." class="tex" src="http://upload.wikimedia.org/math/2/b/8/2b8f73649758a53d36c2ed52cc30bbdd.png" /></dd></dl>Using the <a href="http://en.wikipedia.org/wiki/Ratio_test" title="Ratio test">ratio test</a> it is possible to show that this <a href="http://en.wikipedia.org/wiki/Power_series" title="Power series">power series</a> has an infinite <a href="http://en.wikipedia.org/wiki/Radius_of_convergence" title="Radius of convergence">radius of convergence</a>, and so defines <i>e<sup>z</sup></i> for all complex <i>z</i>.<br />
<h3><span class="editsection"></span><span class="mw-headline" id="Limit_definition">Limit definition</span></h3>For complex <i>z</i><br />
<dl><dd><img alt="e^z = \lim_{n \rightarrow \infty} \left(1+\frac{z}{n}\right)^n ~." class="tex" src="http://upload.wikimedia.org/math/2/9/a/29aac5e101f209b343e5c367da7e54bb.png" /></dd></dl><h2><span class="editsection"></span><span class="mw-headline" id="Proofs">Proofs</span></h2>Various proofs of the formula are possible.<br />
<h3><span class="editsection"></span><span class="mw-headline" id="Using_power_series">Using power series</span></h3>Here is a proof of Euler's formula using power series expansions as well as basic facts about the powers of <i>i</i>:<br />
<dl><dd><img alt="\begin{align}
i^0 &{}= 1, \quad &
i^1 &{}= i, \quad &
i^2 &{}= -1, \quad &
i^3 &{}= -i, \\
i^4 &={} 1, \quad &
i^5 &={} i, \quad &
i^6 &{}= -1, \quad &
i^7 &{}= -i,
\end{align}" class="tex" src="http://upload.wikimedia.org/math/c/d/4/cd45d828d312cb20cde8395dd35444a5.png" /></dd></dl>and so on. Using now the power series definition from above we see that for real values of <i>x</i><br />
<dl><dd><img alt="\begin{align}
e^{ix} &{}= 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \frac{(ix)^6}{6!} + \frac{(ix)^7}{7!} + \frac{(ix)^8}{8!} + \cdots \\[8pt]
&{}= 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \frac{x^6}{6!} - \frac{ix^7}{7!} + \frac{x^8}{8!} + \cdots \\[8pt]
&{}= \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots \right) + i\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \right) \\[8pt]
&{}= \cos x + i\sin x \ .
\end{align}" class="tex" src="http://upload.wikimedia.org/math/8/7/9/8799ab90dd91d47cf82ea7b449556230.png" /></dd></dl>In the last step we have simply recognized the <a href="http://en.wikipedia.org/wiki/Taylor_series" title="Taylor series">Taylor series</a> for <i>sin(x)</i> and <i>cos(x)</i>. The rearrangement of terms is justified because each series is <a href="http://en.wikipedia.org/wiki/Absolute_convergence" title="Absolute convergence">absolutely convergent</a>.<br />
<h3><span class="editsection"></span><span class="mw-headline" id="Using_calculus">Using calculus</span></h3>Treating <i>i</i> as a constant, albeit an imaginary constant, note that<br />
<dl><dd><img alt=" \frac{d}{dx} e^{ix} = i e^{ix} \ ." class="tex" src="http://upload.wikimedia.org/math/1/3/0/13057c339c5223c96488e6ae8b15a644.png" /></dd></dl>Then define the function<br />
<dl><dd><img alt=" f(x) = (\cos x - i \sin x) \cdot e^{ix} \ ." class="tex" src="http://upload.wikimedia.org/math/8/a/b/8abff6c16dfddfb08c11f4cebac5972c.png" /></dd></dl>Because the product rule holds for complex valued functions of a real variable for the same reason as in the real case, the <a href="http://en.wikipedia.org/wiki/Derivative" title="Derivative">derivative</a> of ƒ(<i>x</i>) according to the <a href="http://en.wikipedia.org/wiki/Product_rule" title="Product rule">product rule</a> is:<br />
<dl><dd><img alt="\begin{align}
\frac{d}{dx}f(x) &= (\cos x - i\sin x)\cdot\frac{d}{dx}e^{ix} + \frac{d}{dx}(\cos x - i\sin x)\cdot e^{ix} \\
&= (\cos x - i\sin x)(i e^{ix}) + (-\sin x - i\cos x)\cdot e^{ix} \\
&= (i\cos x + \sin x - \sin x - i\cos x)\cdot e^{ix} \\
&= 0 \ .
\end{align} " class="tex" src="http://upload.wikimedia.org/math/a/1/e/a1e4da551eb52234beed2db00c0a0149.png" /></dd></dl>Therefore, ƒ(<i>x</i>) must be a <a href="http://en.wikipedia.org/wiki/Constant_function" title="Constant function">constant function</a> in <i>x</i>. Because ƒ(0) = 1 by inspection, ƒ(<i>x</i>) = 1, giving<br />
<dl><dd><img alt="1 = (\cos x - i \sin x) \cdot e^{ix} \ ." class="tex" src="http://upload.wikimedia.org/math/a/9/d/a9da593fe102f9742474defbdf5417c1.png" /></dd></dl>Multiplying both sides by cos <i>x</i> + <i>i</i> sin <i>x</i>, we obtain<br />
<dl><dd><img alt="\begin{align}
\cos x + i \sin x &= (\cos x + i \sin x)(\cos x - i \sin x) \cdot e^{ix} \\
&= (\cos^2 x -(i \sin x)^2) \cdot e^{ix} = (\cos^2 x + \sin^2 x) \cdot e^{ix} = e^{ix} \ .
\end{align}" class="tex" src="http://upload.wikimedia.org/math/4/f/7/4f7d62d0ce399d24550441cd79a84e61.png" /></dd></dl><h3><span class="editsection"></span><span class="mw-headline" id="Using_differential_equations">Using differential equations</span></h3>Here is another proof that follows from the differential identity above. Define a new function ƒ(<i>x</i>) of the real variable <i>x</i> as<br />
<dl><dd><img alt=" f(x) = \cos x + i \sin x \ ." class="tex" src="http://upload.wikimedia.org/math/c/0/b/c0b25f72c3aa79c50bdacec7e38c110c.png" /></dd></dl>Then we may check that<br />
<dl><dd><img alt="\begin{align}
\frac{d}{dx}f(x) &= -\sin x + i \cos x \\
&= i f(x) \ .
\end{align}" class="tex" src="http://upload.wikimedia.org/math/1/3/5/13525ce24f9de61028f31cd452a968b7.png" /></dd></dl>Thus ƒ(<i>x</i>) and <i>e<sup>ix</sup></i> satisfy the same first-order <a href="http://en.wikipedia.org/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> (here the complex values are considered as points in the plane ℝ<sup>2</sup>). Note also that both functions are equal to 1 at <i>x</i> = 0, then by the uniqueness of solutions to ordinary differential equations they must be equal everywhere (see <a href="http://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem</a> and note the comments concerning global uniqueness in the proof section there).<br />
<h2><span class="editsection"></span><span class="mw-headline" id="See_also">See also</span></h2><ul><li><a href="http://en.wikipedia.org/wiki/File:Sine_and_Cosine_fundamental_relationship_to_Circle_%28and_Helix%29.gif" title="File:Sine and Cosine fundamental relationship to Circle (and Helix).gif">Movie of Euler's formula</a></li>
<li><a href="http://en.wikipedia.org/wiki/Euler%27s_identity" title="Euler's identity">Euler's identity</a></li>
<li><a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">Complex number</a></li>
<li><a href="http://en.wikipedia.org/wiki/Integration_using_Euler%27s_formula" title="Integration using Euler's formula">Integration using Euler's formula</a></li>
</ul><h2><span class="editsection"></span><span class="mw-headline" id="References">References</span></h2><div class="references"><ol><li id="cite_note-0"><b><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#cite_ref-0">^</a></b> <span class="citation book">Moskowitz, Martin A. (2002). <i>A Course in Complex Analysis in One Variable</i>. World Scientific Publishing Co.. pp. 7. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/981-02-4780-X" title="Special:BookSources/981-02-4780-X">981-02-4780-X</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Complex+Analysis+in+One+Variable&rft.aulast=Moskowitz&rft.aufirst=Martin+A.&rft.au=Moskowitz%2C%26%2332%3BMartin+A.&rft.date=2002&rft.pages=pp.%26nbsp%3B7&rft.pub=World+Scientific+Publishing+Co.&rft.isbn=981-02-4780-X&rfr_id=info:sid/en.wikipedia.org:Euler%27s_formula"><span style="display: none;"> </span></span></li>
<li id="cite_note-1"><b><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#cite_ref-1">^</a></b> <span class="citation book">Feynman, Richard P. (1977). <i>The Feynman Lectures on Physics, vol. I</i>. Addison-Wesley. pp. 22–10. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-201-02010-6" title="Special:BookSources/0-201-02010-6">0-201-02010-6</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Feynman+Lectures+on+Physics%2C+vol.+I&rft.aulast=Feynman&rft.aufirst=Richard+P.&rft.au=Feynman%2C%26%2332%3BRichard+P.&rft.date=1977&rft.pages=pp.%26nbsp%3B22%26ndash%3B10&rft.pub=Addison-Wesley&rft.isbn=0-201-02010-6&rfr_id=info:sid/en.wikipedia.org:Euler%27s_formula"><span style="display: none;"> </span></span></li>
<li id="cite_note-2"><b><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#cite_ref-2">^</a></b> <span class="citation book">Feynman, Richard P. (1977). <i>The Feynman Lectures on Physics, vol. I</i>. Addison-Wesley. pp. 22–1. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0-201-02010-6" title="Special:BookSources/0-201-02010-6">0-201-02010-6</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Feynman+Lectures+on+Physics%2C+vol.+I&rft.aulast=Feynman&rft.aufirst=Richard+P.&rft.au=Feynman%2C%26%2332%3BRichard+P.&rft.date=1977&rft.pages=pp.%26nbsp%3B22%26ndash%3B1&rft.pub=Addison-Wesley&rft.isbn=0-201-02010-6&rfr_id=info:sid/en.wikipedia.org:Euler%27s_formula"><span style="display: none;"> </span></span></li>
<li id="cite_note-3"><b><a href="http://en.wikipedia.org/wiki/Euler%27s_formula#cite_ref-3">^</a></b> <span class="citation book">John Stillwell (2002). <i>Mathematics and Its History</i>. Springer.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+Its+History&rft.aulast=John+Stillwell&rft.au=John+Stillwell&rft.date=2002&rft.pub=Springer&rfr_id=info:sid/en.wikipedia.org:Euler%27s_formula"><span style="display: none;"> </span></span></li>
</ol></div><h2><span class="editsection"><a href="http://en.wikipedia.org/w/index.php?title=Euler%27s_formula&action=edit&section=14" title="Edit section: External links"></a></span><span class="mw-headline" id="External_links"></span><span class="mw-headline" id="External_links"></span><span class="mw-headline" id="External_links">External links</span></h2><ul><li><a class="external text" href="http://ccrma-www.stanford.edu/~jos/mdft/Proof_Euler_s_Identity.html" rel="nofollow">Proof of Euler's Formula</a> by Julius O. Smith III</li>
<li><a class="external text" href="http://fermatslasttheorem.blogspot.com/2006/02/eulers-formula.html" rel="nofollow">Euler's Formula and Fermat's Last Theorem</a></li>
<li><a class="external text" href="http://math.fullerton.edu/mathews/c2003/ComplexFunExponentialMod.html" rel="nofollow">Complex Exponential Function Module by John H. Mathews</a></li>
<li><a class="external text" href="http://web.mat.bham.ac.uk/C.J.Sangwin/euler/" rel="nofollow">Elements of Algebra</a></li>
<li><a class="external text" href="http://resonanceswavesandfields.blogspot.com/2007/08/eulers-equation-and-complex-numbers.html" rel="nofollow">Visual Representation of Euler's Formula</a></li>
</ul><div class="printfooter">Retrieved from "<a href="http://en.wikipedia.org/wiki/Euler%27s_formula">http://en.wikipedia.org/wiki/Euler%27s_formula</a>"</div><div class="catlinks" id="catlinks"><div id="mw-normal-catlinks"><a href="http://en.wikipedia.org/wiki/Special:Categories" title="Special:Categories">Categories</a>: <span dir="ltr"><a href="http://en.wikipedia.org/wiki/Category:Complex_analysis" title="Category:Complex analysis">Complex analysis</a></span> | <span dir="ltr"><a href="http://en.wikipedia.org/wiki/Category:Mathematical_theorems" title="Category:Mathematical theorems">Mathematical theorems</a></span> | <span dir="ltr"><a href="http://en.wikipedia.org/wiki/Category:Trigonometry" title="Category:Trigonometry">Trigonometry</a></span> | <span dir="ltr"><a href="http://en.wikipedia.org/wiki/Category:Articles_containing_proofs" title="Category:Articles containing proofs">Articles containing proofs</a></span><br />
<br />
</div><div id="mw-normal-catlinks"></div><div id="mw-normal-catlinks"><span dir="ltr"><b> EULER'S IDENTITY</b></span><br />
<br />
</div><div id="mw-normal-catlinks"></div></div><div class="thumb tright"><div class="thumbinner" style="width: 302px;"><a class="image" href="http://en.wikipedia.org/wiki/File:ExpIPi.gif"><img alt="" class="thumbimage" height="269" src="http://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/ExpIPi.gif/300px-ExpIPi.gif" width="300" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:ExpIPi.gif" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.5/common/images/magnify-clip.png" width="15" /></a></div>The <a href="http://en.wikipedia.org/wiki/Exponential_function" title="Exponential function">exponential function</a> <i>e</i><sup><i>z</i></sup> can be defined as the <a href="http://en.wikipedia.org/wiki/Limit_of_a_sequence" title="Limit of a sequence">limit</a> of <span style="white-space: nowrap;">(1 + <i>z</i>/<i>N</i>)<sup><i>N</i></sup></span>, as <i>N</i> approaches infinity, and thus <i>e</i><sup><i>iπ</i></sup> is the limit of <span style="white-space: nowrap;">(1 + <i>iπ/N</i>)<sup><i>N</i></sup></span>. In this animation <i>N</i> takes various increasing values from 1 to 100. The computation of <span style="white-space: nowrap;">(1 + <i>iπ/N</i>)<sup><i>N</i></sup></span> is displayed as the combined effect of <i>N</i> repeated multiplications in the <a href="http://en.wikipedia.org/wiki/Complex_plane" title="Complex plane">complex plane</a>, with the final point being the actual value of <span style="white-space: nowrap;">(1 + <i>iπ/N</i>)<sup><i>N</i></sup></span>. It can be seen that as <i>N</i> gets larger <span style="white-space: nowrap;">(1 + <i>iπ/N</i>)<sup><i>N</i></sup></span> approaches a limit of −1.</div></div></div><table cellpadding="0" cellspacing="0" class="infobox" style="clear: right; float: right;"><tbody>
<tr> <td style="padding: 0px;"><table cellpadding="0" cellspacing="0" style="background: none repeat scroll 0% 0% transparent; font-size: 11px; margin: 0.25em 0.25em 0em; text-align: center; width: 210px;"><tbody>
<tr> <td style="padding-left: 5px;"><small><span style="white-space: nowrap;"> </span>Part of a series of articles on</small><br />
<span style="font-size: 175%;">The mathematical constant <b><a href="http://en.wikipedia.org/wiki/E_%28mathematical_constant%29" title="E (mathematical constant)">e</a></b></span></td> </tr>
<tr> <td style="padding-bottom: 5px; padding-left: 12px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Euler%27s_formula.svg"><img alt="Euler's formula.svg" height="178" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/180px-Euler%27s_formula.svg.png" width="180" /></a></td> </tr>
<tr> <td style="border-top: 1px solid rgb(170, 170, 170); padding: 0px;"><a href="http://en.wikipedia.org/wiki/Natural_logarithm" title="Natural logarithm">Natural logarithm</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Exponential_function" title="Exponential function">Exponential function</a></td> </tr>
<tr> <td style="border-top: 1px solid rgb(170, 170, 170); padding: 0px;"><b>Applications in:</b> <a href="http://en.wikipedia.org/wiki/Compound_interest" title="Compound interest">compound interest</a> <span style="font-weight: bold;">·</span> <b class="selflink">Euler's identity</b> & <a href="http://en.wikipedia.org/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Half-life" title="Half-life">half-lives</a> & exponential <a href="http://en.wikipedia.org/wiki/Exponential_growth" title="Exponential growth">growth</a>/<a href="http://en.wikipedia.org/wiki/Exponential_decay" title="Exponential decay">decay</a></td> </tr>
<tr> <td style="border-top: 1px solid rgb(170, 170, 170); padding: 0px;"><b>Defining e:</b> <a href="http://en.wikipedia.org/wiki/Proof_that_e_is_irrational" title="Proof that e is irrational">proof that e is irrational</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Representations_of_e" title="Representations of e">representations of e</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem" title="Lindemann–Weierstrass theorem">Lindemann–Weierstrass theorem</a></td> </tr>
<tr> <td style="border-top: 1px solid rgb(170, 170, 170); padding: 0px;"><b>People</b> <a href="http://en.wikipedia.org/wiki/John_Napier" title="John Napier">John Napier</a> <span style="font-weight: bold;">·</span> <a href="http://en.wikipedia.org/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></td> </tr>
<tr> <td style="border-top: 1px solid rgb(170, 170, 170); padding: 0px;"><a href="http://en.wikipedia.org/wiki/Schanuel%27s_conjecture" title="Schanuel's conjecture">Schanuel's conjecture</a></td> </tr>
</tbody></table></td> </tr>
</tbody></table><br />
In <a href="http://en.wikipedia.org/wiki/Mathematical_analysis" title="Mathematical analysis">analytical mathematics</a>, <b>Euler's Identity</b>, named for the Swiss-German <a href="http://en.wikipedia.org/wiki/Mathematician" title="Mathematician">mathematician</a> <a href="http://en.wikipedia.org/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, is the equality<br />
<dl><dd><img alt="e^{i \pi} + 1 = 0\,\!" class="tex" src="http://upload.wikimedia.org/math/9/e/9/9e9a547076c6820b95e439dd1a5d6a32.png" /></dd></dl>where<br />
<dl><dd><img alt="e\,\!" class="tex" src="http://upload.wikimedia.org/math/4/6/0/460a1940ceddf45878d2e095af31128a.png" /> is <a href="http://en.wikipedia.org/wiki/E_%28mathematical_constant%29" title="E (mathematical constant)">Euler's number</a>, the base of <a href="http://en.wikipedia.org/wiki/Natural_logarithm" title="Natural logarithm">natural logarithms</a>,</dd><dd><img alt="i\,\!" class="tex" src="http://upload.wikimedia.org/math/a/7/9/a796b40d92e81ae190a1e4f4e2a2c3ed.png" /> is the <a href="http://en.wikipedia.org/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>, which satisfies <i>i</i><sup>2</sup> = −1, and</dd><dd><img alt="\pi\,\!" class="tex" src="http://upload.wikimedia.org/math/e/0/6/e06a022b86322d6ee502b6760a0ed4ec.png" /> is <a href="http://en.wikipedia.org/wiki/Pi" title="Pi">pi</a>, the <a href="http://en.wikipedia.org/wiki/Ratio" title="Ratio">ratio</a> of the circumference of a circle to its diameter.</dd></dl>Euler's Identity is also sometimes called <b>Euler's Equation</b>.<br />
<table class="toc tochidden" id="toc"><tbody>
<tr> <td><ul style="display: none;"><li class="toclevel-1 tocsection-1"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#Beauty"><span class="tocnumber">1</span> <span class="toctext">Beauty</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#Derivation"><span class="tocnumber">2</span> <span class="toctext">Derivation</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#Generalizations"><span class="tocnumber">3</span> <span class="toctext">Generalizations</span></a></li>
<li class="toclevel-1 tocsection-4"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#Attribution"><span class="tocnumber">4</span> <span class="toctext">Attribution</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#See_also"><span class="tocnumber">5</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-6"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#Notes"><span class="tocnumber">6</span> <span class="toctext">Notes</span></a></li>
<li class="toclevel-1 tocsection-7"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#References"><span class="tocnumber">7</span> <span class="toctext">References</span></a></li>
</ul></td> </tr>
</tbody></table><h2><span class="editsection"></span><span class="mw-headline" id="Beauty">Beauty</span></h2>Euler's identity is considered by many to be remarkable for its <a href="http://en.wikipedia.org/wiki/Mathematical_beauty" title="Mathematical beauty">mathematical beauty</a>. These three basic <a href="http://en.wikipedia.org/wiki/Arithmetic" title="Arithmetic">arithmetic</a> operations occur exactly once each: <a href="http://en.wikipedia.org/wiki/Addition" title="Addition">addition</a>, <a href="http://en.wikipedia.org/wiki/Multiplication" title="Multiplication">multiplication</a>, and <a href="http://en.wikipedia.org/wiki/Exponentiation" title="Exponentiation">exponentiation</a>. The identity also links five fundamental <a href="http://en.wikipedia.org/wiki/Mathematical_constant" title="Mathematical constant">mathematical constants</a>:<br />
<ul><li>The <a href="http://en.wikipedia.org/wiki/0_%28number%29" title="0 (number)">number 0</a>, the additive identity.</li>
<li>The <a href="http://en.wikipedia.org/wiki/1_%28number%29" title="1 (number)">number 1</a>, the multiplicative identity.</li>
<li>The <a href="http://en.wikipedia.org/wiki/Pi" title="Pi">number <i>π</i></a>, which is ubiquitous in <a href="http://en.wikipedia.org/wiki/Trigonometry" title="Trigonometry">trigonometry</a>, the geometry of <a href="http://en.wikipedia.org/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, and <a href="http://en.wikipedia.org/wiki/Mathematical_analysis" title="Mathematical analysis">analytical mathematics</a> (π = 3.14159265...)</li>
<li>The <a href="http://en.wikipedia.org/wiki/E_%28mathematical_constant%29" title="E (mathematical constant)">number <i>e</i></a>, the base of <a href="http://en.wikipedia.org/wiki/Natural_logarithm" title="Natural logarithm">natural logarithms</a>, which occurs widely in mathematical and scientific analysis (e = 2.718281828...). Both π and e are <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Transcendental_numbers" title="Transcendental numbers">transcendental numbers</a>.</li>
<li>The <a href="http://en.wikipedia.org/wiki/Imaginary_unit" title="Imaginary unit">number <i>i</i></a>, the imaginary unit of the <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex numbers</a>, a <a href="http://en.wikipedia.org/wiki/Field_%28mathematics%29" title="Field (mathematics)">field of numbers</a> that contains the roots of all polynomials (that are not constants), and whose study leads to deeper insights into many areas of <a href="http://en.wikipedia.org/wiki/Algebra" title="Algebra">algebra</a> and <a href="http://en.wikipedia.org/wiki/Calculus" title="Calculus">calculus</a>, such as <a href="http://en.wikipedia.org/wiki/Integral" title="Integral">integration in calculus</a>.</li>
</ul>Furthermore, in <a href="http://en.wikipedia.org/wiki/Algebra" title="Algebra">algebra</a> and other areas of mathematics, <a href="http://en.wikipedia.org/wiki/Equation" title="Equation">equations</a> are commonly written with zero on one side of the <a href="http://en.wikipedia.org/wiki/Equals_sign" title="Equals sign">equals sign</a>.<br />
<br />
A poll of readers conducted by <i><a href="http://en.wikipedia.org/wiki/The_Mathematical_Intelligencer" title="The Mathematical Intelligencer">The Mathematical Intelligencer</a></i> <a href="http://en.wikipedia.org/wiki/Magazine" title="Magazine">magazine</a> named <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Euler%27s_Identity" title="Euler's Identity">Euler's Identity</a> as the "most beautiful theorem in mathematics".<sup class="reference" id="cite_ref-0"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-0">[1]</a></sup> Another poll of readers that was conducted by <i><a href="http://en.wikipedia.org/wiki/Physics_World" title="Physics World">Physics World</a></i> magazine, in 2004, chose Euler's Identity tied with <a href="http://en.wikipedia.org/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell equations</a> (of <a href="http://en.wikipedia.org/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>) as the "greatest equation ever".<sup class="reference" id="cite_ref-1"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-1">[2]</a></sup><br />
<br />
An entire 400-page mathematics book, <i>Dr. Euler's Fabulous Formula</i> (published in 2006), written by Dr. Paul Nahin (a <a href="http://en.wikipedia.org/wiki/Professor" title="Professor">Professor</a> Emeritus at the <a href="http://en.wikipedia.org/wiki/University_of_New_Hampshire" title="University of New Hampshire">University of New Hampshire</a>), is devoted to Euler's Identity. This monograph states that Euler's Identity sets "the gold standard for mathematical beauty."<sup class="reference" id="cite_ref-2"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-2">[3]</a></sup><br />
<br />
<a href="http://en.wikipedia.org/wiki/Constance_Reid" title="Constance Reid">Constance Reid</a> claimed that Euler's Identity was "the most famous formula in all mathematics."<sup class="reference" id="cite_ref-3"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-3">[4]</a></sup><br />
The <a href="http://en.wikipedia.org/wiki/Mathematician" title="Mathematician">mathematician</a> <a href="http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> was reported to have commented that if this formula was not immediately apparent to a student upon being told it, that student would never become a first-class mathematician.<sup class="reference" id="cite_ref-4"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-4">[5]</a></sup><br />
<br />
After proving Euler's Identity during a lecture, <a href="http://en.wikipedia.org/wiki/Benjamin_Peirce" title="Benjamin Peirce">Benjamin Peirce</a>, a noted American 19th century <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Philosopher" title="Philosopher">philosopher</a>/mathematician and a professor at <a href="http://en.wikipedia.org/wiki/Harvard_University" title="Harvard University">Harvard University</a>, stated that "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth." <sup class="reference" id="cite_ref-5"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-5">[6]</a></sup><br />
<br />
The <a href="http://en.wikipedia.org/wiki/Stanford_University" title="Stanford University">Stanford University</a> mathematics professor, Dr. <a href="http://en.wikipedia.org/wiki/Keith_Devlin" title="Keith Devlin">Keith Devlin</a>, said, "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's Equation reaches down into the very depths of existence."<sup class="reference" id="cite_ref-6"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-6">[7]</a></sup><br />
<h2><span class="editsection"></span><span class="mw-headline" id="Derivation">Derivation</span></h2><div class="thumb tright"><div class="thumbinner" style="width: 302px;"><a class="image" href="http://en.wikipedia.org/wiki/File:Euler%27s_formula.svg"><img alt="" class="thumbimage" height="296" src="http://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/300px-Euler%27s_formula.svg.png" width="300" /></a> <br />
<div class="thumbcaption"><div class="magnify"><a class="internal" href="http://en.wikipedia.org/wiki/File:Euler%27s_formula.svg" title="Enlarge"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.5/common/images/magnify-clip.png" width="15" /></a></div>Euler's formula for a general angle</div></div></div><br />
The identity is a special case of <a href="http://en.wikipedia.org/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a> from <a href="http://en.wikipedia.org/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, which states that<br />
<dl><dd><img alt="e^{ix} = \cos x + i \sin x \,\!" class="tex" src="http://upload.wikimedia.org/math/c/6/7/c67d19a30b34c87a92e27e1458f21630.png" /></dd></dl>for any <a href="http://en.wikipedia.org/wiki/Real_number" title="Real number">real number</a> <i>x</i>. (Note that the arguments to the <a href="http://en.wikipedia.org/wiki/Trigonometry" title="Trigonometry">trigonometric</a> functions <i>sine</i> and <i>cosine</i> are taken to be in <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Radians" title="Radians">radians</a>, and not in degrees.) In particular,<br />
<dl><dd><img alt="e^{i \pi} = \cos \pi + i \sin \pi.\,\!" class="tex" src="http://upload.wikimedia.org/math/a/8/e/a8ea600cf1fad24caf2844a34ce3929a.png" /></dd></dl>Since<br />
<dl><dd><img alt="\cos \pi = -1 \, \! " class="tex" src="http://upload.wikimedia.org/math/e/0/f/e0fead469bf4c84bd856b6bb05f283b7.png" /></dd></dl>and<br />
<dl><dd><img alt="\sin \pi = 0,\,\!" class="tex" src="http://upload.wikimedia.org/math/6/2/2/6226f53433617628310b550a95965995.png" /></dd></dl>it follows that<br />
<dl><dd><img alt="e^{i \pi} = -1,\,\!" class="tex" src="http://upload.wikimedia.org/math/f/1/c/f1cd6e9c1e708549bf56fa80038cdd2f.png" /></dd></dl>which gives the identity<br />
<dl><dd><img alt="e^{i \pi} +1 = 0.\,\!" class="tex" src="http://upload.wikimedia.org/math/9/b/0/9b0db59874cc7c1cc97abd52402520fe.png" /></dd></dl><h2><span class="editsection"></span><span class="mw-headline" id="Generalizations">Generalizations</span></h2>Euler's Identity is actually a special case of the more general identity that the <i>n</i>th <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Roots_of_unity" title="Roots of unity">roots of unity</a>, for <i>n</i> > 1, add up to 0:<br />
<dl><dd><img alt="\sum_{k=0}^{n-1} e^{2 \pi i k/n} = 0 ." class="tex" src="http://upload.wikimedia.org/math/f/1/6/f1663f7e15c033562e3c4f95a1bd359a.png" /></dd></dl>Euler's identity is the case where <i>n</i> = 2.<br />
<br />
In another field of mathematics, by using <a href="http://en.wikipedia.org/wiki/Quaternion" title="Quaternion">quaternion</a> exponentiation, one can show that a similar identity also applies to quaternions:<br />
<dl><dd><img alt="e^{\frac{(i+j+k)}{\sqrt 3}\pi} + 1 = 0. \,\!" class="tex" src="http://upload.wikimedia.org/math/8/5/2/852949d5d16eba7d07445ff4a72e8e58.png" /></dd></dl><h2><span class="editsection"></span><span class="mw-headline" id="Attribution">Attribution</span></h2>While Euler wrote about his formula that relates <i>e</i> with <i>cosine</i> and <i>sine</i> terms, in the field of complex numbers, there is no known record of Euler's actually stating or deriving the simplified identity equation itself.<br />
<br />
Furthermore, Euler's formula was probably known before the life of Euler.<sup class="reference" id="cite_ref-7"><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_note-7">[8]</a></sup> (If so, then this usage would be an example of <a href="http://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy" title="Stigler's law of eponymy">Stigler's law of eponymy</a>.) Thus, the question of whether or not this identity should be attributed to Euler is unanswerable.<br />
<h2><span class="editsection"></span><span class="mw-headline" id="See_also">See also</span></h2><ul><li><a href="http://en.wikipedia.org/wiki/De_Moivre%27s_formula" title="De Moivre's formula">De Moivre's formula</a></li>
<li><a href="http://en.wikipedia.org/wiki/Exponential_function" title="Exponential function">Exponential function</a></li>
<li><a href="http://en.wikipedia.org/wiki/Gelfond%27s_constant" title="Gelfond's constant">Gelfond's constant</a></li>
</ul><h2><span class="editsection"></span><span class="mw-headline" id="Notes">Notes</span></h2><div class="reflist references-small"><div class="references"><ol><li id="cite_note-0"><b><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-0">^</a></b> Nahin, 2006, p.2–3 (poll published in the summer 1990 issue of the magazine).</li>
<li id="cite_note-1"><b><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-1">^</a></b> Crease, 2004.</li>
<li id="cite_note-2"><b><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-2">^</a></b> Cited in Crease, 2007.</li>
<li id="cite_note-3"><b><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-3">^</a></b> Reid.</li>
<li id="cite_note-4"><b><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-4">^</a></b> Derbyshire p.210.</li>
<li id="cite_note-5"><b><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-5">^</a></b> Maor p.160 and Kasner & Newman p.103–104.</li>
<li id="cite_note-6"><b><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-6">^</a></b> Nahin, 2006, p.1.</li>
<li id="cite_note-7"><b><a href="http://en.wikipedia.org/wiki/Euler%27s_identity#cite_ref-7">^</a></b> Sandifer.</li>
</ol></div></div><h2><span class="editsection"></span><span class="mw-headline" id="References">References</span></h2><ul><li>Crease, Robert P., "<a class="external text" href="http://physicsweb.org/articles/world/17/10/2" rel="nofollow">The greatest equations ever</a>", PhysicsWeb, October 2004 (registration required).</li>
<li>Crease, Robert P. "<a class="external text" href="http://physicsweb.org/articles/world/20/3/3/1" rel="nofollow">Equations as icons</a>," PhysicsWeb, March 2007 (registration required).</li>
<li>Derbyshire, J. <i>Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics</i> (New York: Penguin, 2004).</li>
<li>Kasner, E., and Newman, J., <i>Mathematics and the Imagination</i> (Bell and Sons, 1949).</li>
<li>Maor, Eli, <i>e: The Story of a number</i> (Princeton University Press, 1998), <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/0691058547">ISBN 0-691-05854-7</a></li>
<li>Nahin, Paul J., <i>Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills</i> (Princeton University Press, 2006), <a class="internal mw-magiclink-isbn" href="http://en.wikipedia.org/wiki/Special:BookSources/9780691118222">ISBN 978-0691118222</a></li>
<li>Reid, Constance, <i>From Zero to Infinity</i> (Mathematical Association of America, various editions).</li>
<li>Sandifer, Ed, "<a class="external text" href="http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2040%20Greatest%20Hits.pdf" rel="nofollow">Euler's Greatest Hits</a>", MAA Online, February 2007.</li>
</ul><div class="printfooter">Retrieved from "<a href="http://en.wikipedia.org/wiki/Euler%27s_identity">http://en.wikipedia.org/wiki/Euler%27s_identity</a>"</div><div class="catlinks" id="catlinks"><div id="mw-normal-catlinks"><a href="http://en.wikipedia.org/wiki/Special:Categories" title="Special:Categories">Categories</a>: <span dir="ltr"><a href="http://en.wikipedia.org/wiki/Category:Complex_analysis" title="Category:Complex analysis">Complex analysis</a></span> | <span dir="ltr"><a href="http://en.wikipedia.org/wiki/Category:Exponentials" title="Category:Exponentials">Exponentials</a></span> | <span dir="ltr"><a href="http://en.wikipedia.org/wiki/Category:Mathematical_identities" title="Category:Mathematical identities">Mathematical identities</a></span></div></div>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com21tag:blogger.com,1999:blog-5303246073824127471.post-64228107105129240382012-02-19T07:48:00.001-05:002012-02-19T07:49:08.844-05:00Are YOU an Ostrich?<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9dB_QOmInXFm6kUNtUEs9-4hRv8IBMH_Crfw6kRdr1h9UQEM5d3JUWhXNYZKA8RVAUMG0sjx4A69XZoOVymSoL3x_P1QqlMRh7ewbfO865iomvkDipin2orZeIL-yjDBbJvrdLelEEA/s1600/Occupy+Mind.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9dB_QOmInXFm6kUNtUEs9-4hRv8IBMH_Crfw6kRdr1h9UQEM5d3JUWhXNYZKA8RVAUMG0sjx4A69XZoOVymSoL3x_P1QqlMRh7ewbfO865iomvkDipin2orZeIL-yjDBbJvrdLelEEA/s320/Occupy+Mind.jpg" width="308" /></a></div><h6 class="uiStreamMessage" data-ft="{"type":1}"><span class="messageBody"><br />
<br />
Read this first about the oversupply of specialists in STEM:<br />
<br />
<a href="http://www.washingtonmonthly.com/college_guide/blog/do_we_really_have_a_scientist.php">Do We Really Have a Scientist Shortage?<br />
by Daniel Luzer</a><br />
<br />
Are YOU an Ostrich?<br />
by Steven H. Colyer<br />
<br />
"It’s not that America’s science and engineering firms don’t have enough applicants. They have plenty. They just want to ensure a steady oversupply of trained STEM professionals so they can continue to employ them cheaply.<br />
<br />
"Of course that makes a lot of sense for such firms, but let’s be careful about using a few companies’ human resource desires to formulate national education policy. "<br />
<br />
I had a dog and his name was .... BINGO !!<br />
<br />
Damn straight. The T & E in STEM (Scientists, Technologists, Engineers, and Mathematicians) are the most important to commerce and the government (military). America is an aggressive, military-industrial nation. Oh, you don't like that? Well too frigging bad then and no soup for you, because that's the way it is. Next!<br />
<br />
A few weeks ago, while I was walking around Thomas Jefferson's home in Virginia, Monticello, one of several personal Meccas for me and which was a damned fine spiritual experience as well since it was my first time there, hearing of Jefferson's love of all things Voltaire and Locke and freedom of speech and of the right to worship (or not) as one pleases, seeing his University of Virginia laid out like a map below his mountaintop home and his love of Education, reminded me that we used to stand for something else. Something we don't anymore.<br />
<br />
A few years ago I went to Plymouth, MA, and I went to Jamestown, VA the day before Monticello, also for the first time. Those visits drove home in me that which our country was built on: Co-operation among us, people helping people, in order to survive, against all odds. <br />
<br />
Back then the forces against us were Nature. Today it's human nature: unchecked greed by the monopolists who will crush, and do crush, all competition. The 1%-ers won. They're still winning.<br />
<br />
And what are any of you doing about it, other than bitching? Or worse, you can be like my wife, who is most common and typical of most Americans, and be a human ostrich, and spend your evenings allowing Alex Trebeck, Pat Sajek and Vanna White, and Simon Cowell and Cee-lo take your mind off your troubles.<br />
<br />
And that ain't a life folks, that's just putting in time until the Grim Reaper comes to call. That's want the 1% want you to do. <br />
<br />
Oh, and don't forget to vote Republican, the party of the One-Percent that control 99% of American Wealth! Pfft, hell no. What jerks.<br />
<br />
"Every man dies. Not every man lives."<br />
.... William Wallace<br />
<br />
Amen.<br />
<br />
P.S. Thanks to Neil Bates of Newport News, VA, for the link and his continued inspiration. <br />
<br />
Now excuse me, while I go out and actually DO SOMETHING today.</span></h6>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com0tag:blogger.com,1999:blog-5303246073824127471.post-86727596396049405182012-02-17T17:22:00.000-05:002012-02-17T17:22:24.258-05:00The Public Ivies<b>Public Ivy</b> is a term coined by Richard Moll in his 1985 book <i>Public Ivies: A Guide to America's best public undergraduate colleges and universities</i> to refer to universities which "provide an <a href="http://en.wikipedia.org/wiki/Ivy_League" title="Ivy League">Ivy League</a> collegiate experience at a public school price."<sup class="reference" id="cite_ref-PublicIvys_0-0"><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_note-PublicIvys-0"><span>[</span>1<span>]</span></a></sup> Public Ivies are considered, according to the <i>Journal of Blacks in Higher Education</i>, to be capable of "successfully competing with the Ivy League schools in academic rigor... attracting superstar faculty and in competing for the best and brightest students of all races."<sup class="reference" id="cite_ref-JBHE_1-0"><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_note-JBHE-1"><span>[</span>2<span>]</span></a></sup><br />
<table class="toc" id="toc"><tbody>
<tr> <td> <div id="toctitle"> <h2>Contents</h2><span class="toctoggle"></span></div><ul><li class="toclevel-1 tocsection-1"><a href="http://en.wikipedia.org/wiki/Public_Ivies#Origins_of_the_term"><span class="tocnumber">1</span> <span class="toctext">Origins of the term</span></a> <ul><li class="toclevel-2 tocsection-2"><a href="http://en.wikipedia.org/wiki/Public_Ivies#The_original_eight_Public_Ivies"><span class="tocnumber">1.1</span> <span class="toctext">The original eight Public Ivies</span></a></li>
<li class="toclevel-2 tocsection-3"><a href="http://en.wikipedia.org/wiki/Public_Ivies#The_worthy_runners-up"><span class="tocnumber">1.2</span> <span class="toctext">The worthy runners-up</span></a></li>
</ul></li>
<li class="toclevel-1 tocsection-4"><a href="http://en.wikipedia.org/wiki/Public_Ivies#Greenes.27_Guides"><span class="tocnumber">2</span> <span class="toctext">Greenes' Guides</span></a> <ul><li class="toclevel-2 tocsection-5"><a href="http://en.wikipedia.org/wiki/Public_Ivies#The_Public_Ivies_according_to_Greene.27s_Guides"><span class="tocnumber">2.1</span> <span class="toctext">The Public Ivies according to Greene's Guides</span></a></li>
<li class="toclevel-2 tocsection-6"><a href="http://en.wikipedia.org/wiki/Public_Ivies#Eastern"><span class="tocnumber">2.2</span> <span class="toctext">Eastern</span></a></li>
<li class="toclevel-2 tocsection-7"><a href="http://en.wikipedia.org/wiki/Public_Ivies#Western"><span class="tocnumber">2.3</span> <span class="toctext">Western</span></a></li>
<li class="toclevel-2 tocsection-8"><a href="http://en.wikipedia.org/wiki/Public_Ivies#Great_Lakes_.26_Midwest"><span class="tocnumber">2.4</span> <span class="toctext">Great Lakes & Midwest</span></a></li>
<li class="toclevel-2 tocsection-9"><a href="http://en.wikipedia.org/wiki/Public_Ivies#Southern"><span class="tocnumber">2.5</span> <span class="toctext">Southern</span></a></li>
</ul></li>
<li class="toclevel-1 tocsection-10"><a href="http://en.wikipedia.org/wiki/Public_Ivies#Institutional_comparisons"><span class="tocnumber">3</span> <span class="toctext">Institutional comparisons</span></a> <ul><li class="toclevel-2 tocsection-11"><a href="http://en.wikipedia.org/wiki/Public_Ivies#Academic_comparisons_and_rankings"><span class="tocnumber">3.1</span> <span class="toctext">Academic comparisons and rankings</span></a></li>
<li class="toclevel-2 tocsection-12"><a href="http://en.wikipedia.org/wiki/Public_Ivies#Athletic_comparisons"><span class="tocnumber">3.2</span> <span class="toctext">Athletic comparisons</span></a></li>
</ul></li>
<li class="toclevel-1 tocsection-13"><a href="http://en.wikipedia.org/wiki/Public_Ivies#See_also"><span class="tocnumber">4</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-14"><a href="http://en.wikipedia.org/wiki/Public_Ivies#References_and_other_resources"><span class="tocnumber">5</span> <span class="toctext">References and other resources</span></a> <ul><li class="toclevel-2 tocsection-15"><a href="http://en.wikipedia.org/wiki/Public_Ivies#Citations"><span class="tocnumber">5.1</span> <span class="toctext">Citations</span></a></li>
<li class="toclevel-2 tocsection-16"><a href="http://en.wikipedia.org/wiki/Public_Ivies#Books"><span class="tocnumber">5.2</span> <span class="toctext">Books</span></a></li>
</ul></li>
</ul></td> </tr>
</tbody></table><h2> <span class="mw-headline" id="Origins_of_the_term">Origins of the term</span></h2>Moll, who earned his <a href="http://en.wikipedia.org/wiki/Master_of_Divinity" title="Master of Divinity">Master of Divinity</a> degree from <a href="http://en.wikipedia.org/wiki/Yale_University" title="Yale University">Yale University</a> in 1959,<sup class="reference" id="cite_ref-YaleMagazine_2-0"><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_note-YaleMagazine-2"><span>[</span>3<span>]</span></a></sup> was an admissions officer at <a href="http://en.wikipedia.org/wiki/Yale_University" title="Yale University">Yale</a>, and the director of admissions at <a href="http://en.wikipedia.org/wiki/Bowdoin_College" title="Bowdoin College">Bowdoin College</a>, <a href="http://en.wikipedia.org/wiki/University_of_California,_Santa_Cruz" title="University of California, Santa Cruz">University of California, Santa Cruz</a>, and <a href="http://en.wikipedia.org/wiki/Vassar_College" title="Vassar College">Vassar College</a>.<sup class="reference" id="cite_ref-YaleMagazine_2-1"><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_note-YaleMagazine-2"><span>[</span>3<span>]</span></a></sup><sup class="reference" id="cite_ref-3"><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_note-3"><span>[</span>4<span>]</span></a></sup><sup class="reference" id="cite_ref-4"><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_note-4"><span>[</span>5<span>]</span></a></sup> He traveled the nation examining higher education and in particular, identified eight public institutions (the same as the number of Ivy League members) which he thought had the look and feel of an Ivy League university. In addition to academic excellence, other factors considered by Moll include visual appearance, age, and school traditions as well as certain other Ivy League characteristics.<br />
<h3> <span class="mw-headline" id="The_original_eight_Public_Ivies">The original eight Public Ivies</span></h3>The original eight Public Ivies as they were listed by Moll in 1985:<sup class="reference" id="cite_ref-JBHE_1-1"><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_note-JBHE-1"><span>[</span>2<span>]</span></a></sup><br />
<ul><li><a href="http://en.wikipedia.org/wiki/The_College_of_William_%26_Mary" title="The College of William & Mary">College of William & Mary</a> (<a href="http://en.wikipedia.org/wiki/Williamsburg,_Virginia" title="Williamsburg, Virginia">Williamsburg</a>, <a href="http://en.wikipedia.org/wiki/Virginia" title="Virginia">Virginia</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/Miami_University" title="Miami University">Miami University</a> (<a href="http://en.wikipedia.org/wiki/Oxford,_Ohio" title="Oxford, Ohio">Oxford</a>, <a href="http://en.wikipedia.org/wiki/Ohio" title="Ohio">Ohio</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/University_of_California" title="University of California">University of California</a> (campuses as of 1985)<sup class="reference" id="cite_ref-5"><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_note-5"><span>[</span>6<span>]</span></a></sup></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Michigan" title="University of Michigan">University of Michigan</a> (<a href="http://en.wikipedia.org/wiki/Ann_Arbor,_Michigan" title="Ann Arbor, Michigan">Ann Arbor</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/University_of_North_Carolina_at_Chapel_Hill" title="University of North Carolina at Chapel Hill">University of North Carolina at Chapel Hill</a></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Texas_at_Austin" title="University of Texas at Austin">University of Texas at Austin</a></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Vermont" title="University of Vermont">University of Vermont</a> (<a href="http://en.wikipedia.org/wiki/Burlington,_Vermont" title="Burlington, Vermont">Burlington</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Virginia" title="University of Virginia">University of Virginia</a> (<a href="http://en.wikipedia.org/wiki/Charlottesville,_Virginia" title="Charlottesville, Virginia">Charlottesville</a>)</li>
</ul><h3> <span class="mw-headline" id="The_worthy_runners-up">The worthy runners-up</span></h3>Moll also offered in the same book "a list of worthy runners-up" and brief summaries of them:<sup class="reference" id="cite_ref-6"><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_note-6"><span>[</span>7<span>]</span></a></sup><br />
<ul><li><a href="http://en.wikipedia.org/wiki/University_of_Colorado_at_Boulder" title="University of Colorado at Boulder">University of Colorado at Boulder</a></li>
<li><a href="http://en.wikipedia.org/wiki/Georgia_Institute_of_Technology" title="Georgia Institute of Technology">Georgia Institute of Technology</a></li>
<li><a class="mw-redirect" href="http://en.wikipedia.org/wiki/University_of_Illinois_at_Urbana-Champaign" title="University of Illinois at Urbana-Champaign">University of Illinois at Urbana-Champaign</a></li>
<li><a href="http://en.wikipedia.org/wiki/New_College_of_Florida" title="New College of Florida">New College of the University of South Florida</a> (now New College of Florida)</li>
<li><a href="http://en.wikipedia.org/wiki/Pennsylvania_State_University" title="Pennsylvania State University">Pennsylvania State University at University Park</a></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Pittsburgh" title="University of Pittsburgh">University of Pittsburgh</a></li>
<li><a href="http://en.wikipedia.org/wiki/Binghamton_University" title="Binghamton University">State University of New York at Binghamton</a> (also called <a href="http://en.wikipedia.org/wiki/Binghamton_University" title="Binghamton University">Binghamton University</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Washington" title="University of Washington">University of Washington</a></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Wisconsin%E2%80%93Madison" title="University of Wisconsin–Madison">University of Wisconsin–Madison</a></li>
</ul><h2> <span class="mw-headline" id="Greenes.27_Guides">Greenes' Guides</span></h2>The more recent and expansive Greene's list (including a list of approximately 30 schools) had one focus alone: public schools with academic quality comparable to an Ivy League institution.<br />
<h3> <span class="mw-headline" id="The_Public_Ivies_according_to_Greene.27s_Guides">The Public Ivies according to Greene's Guides</span></h3>A later book titled <i>The Public Ivies: America's Flagship Public Universities</i> (2001) by Howard and Matthew Greene of <i>Greene's Guides</i> expanded upon the first list (<i>italicized</i> below) to include 30 colleges and universities.<sup class="reference" id="cite_ref-7"><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_note-7"><span>[</span>8<span>]</span></a></sup> The table below is organized by region, and colleges are listed in alphabetical order.<br />
<table cellpadding="0" cellspacing="0" class="multicol" style="background: transparent; width: 100%;"><tbody>
<tr> <td align="left" valign="top"> <h3> <span class="mw-headline" id="Eastern">Eastern</span></h3><ul><li><a href="http://en.wikipedia.org/wiki/Pennsylvania_State_University" title="Pennsylvania State University">Pennsylvania State University</a> (<a href="http://en.wikipedia.org/wiki/University_Park,_Pennsylvania" title="University Park, Pennsylvania">University Park</a>)</li>
<li><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Rutgers,_The_State_University_of_New_Jersey" title="Rutgers, The State University of New Jersey">Rutgers, The State University of New Jersey</a> (<a href="http://en.wikipedia.org/wiki/New_Brunswick,_New_Jersey" title="New Brunswick, New Jersey">New Brunswick, New Jersey</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/Binghamton_University" title="Binghamton University">State University of New York at Binghamton</a></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Connecticut" title="University of Connecticut">University of Connecticut</a> (<a href="http://en.wikipedia.org/wiki/Storrs,_Connecticut" title="Storrs, Connecticut">Storrs</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Delaware" title="University of Delaware">University of Delaware</a> (<a href="http://en.wikipedia.org/wiki/Newark,_Delaware" title="Newark, Delaware">Newark</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Maryland,_College_Park" title="University of Maryland, College Park">University of Maryland</a> (<a href="http://en.wikipedia.org/wiki/College_Park,_Maryland" title="College Park, Maryland">College Park</a>)</li>
</ul><h3> <span class="mw-headline" id="Western">Western</span></h3><ul><li><a href="http://en.wikipedia.org/wiki/University_of_Arizona" title="University of Arizona">University of Arizona</a> (<a href="http://en.wikipedia.org/wiki/Tucson,_Arizona" title="Tucson, Arizona">Tucson</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/University_of_California,_Berkeley" title="University of California, Berkeley">University of California, Berkeley</a></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_California,_Los_Angeles" title="University of California, Los Angeles">University of California, Los Angeles</a></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_California,_Davis" title="University of California, Davis">University of California, Davis</a></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_California,_Irvine" title="University of California, Irvine">University of California, Irvine</a></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_California,_San_Diego" title="University of California, San Diego">University of California, San Diego</a></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_California,_Santa_Barbara" title="University of California, Santa Barbara">University of California, Santa Barbara</a></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Colorado_at_Boulder" title="University of Colorado at Boulder">University of Colorado at Boulder</a></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Washington" title="University of Washington">University of Washington</a> (<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Seattle,_Washington" title="Seattle, Washington">Seattle</a>)</li>
</ul></td> <td align="left" valign="top"> <h3> <span class="mw-headline" id="Great_Lakes_.26_Midwest">Great Lakes & Midwest</span></h3><ul><li><a href="http://en.wikipedia.org/wiki/Indiana_University_Bloomington" title="Indiana University Bloomington">Indiana University</a> (<a href="http://en.wikipedia.org/wiki/Bloomington,_Indiana" title="Bloomington, Indiana">Bloomington</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/Miami_University" title="Miami University">Miami University</a> (<a href="http://en.wikipedia.org/wiki/Oxford,_Ohio" title="Oxford, Ohio">Oxford</a>, <a href="http://en.wikipedia.org/wiki/Ohio" title="Ohio">Ohio</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/Michigan_State_University" title="Michigan State University">Michigan State University</a> (<a href="http://en.wikipedia.org/wiki/East_Lansing,_Michigan" title="East Lansing, Michigan">East Lansing</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/Ohio_State_University" title="Ohio State University">Ohio State University</a> (<a href="http://en.wikipedia.org/wiki/Columbus,_Ohio" title="Columbus, Ohio">Columbus</a>)</li>
<li><a class="mw-redirect" href="http://en.wikipedia.org/wiki/University_of_Illinois_at_Urbana-Champaign" title="University of Illinois at Urbana-Champaign">University of Illinois</a> (<a href="http://en.wikipedia.org/wiki/Urbana,_Illinois" title="Urbana, Illinois">Urbana</a>-<a href="http://en.wikipedia.org/wiki/Champaign,_Illinois" title="Champaign, Illinois">Champaign</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Iowa" title="University of Iowa">University of Iowa</a> (<a href="http://en.wikipedia.org/wiki/Iowa_City,_Iowa" title="Iowa City, Iowa">Iowa City</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Michigan" title="University of Michigan">University of Michigan</a> (<a href="http://en.wikipedia.org/wiki/Ann_Arbor,_Michigan" title="Ann Arbor, Michigan">Ann Arbor</a>)</li>
<li><a class="mw-redirect" href="http://en.wikipedia.org/wiki/University_of_Minnesota,_Twin_Cities" title="University of Minnesota, Twin Cities">University of Minnesota</a> (<a class="mw-redirect" href="http://en.wikipedia.org/wiki/Minneapolis-Saint_Paul" title="Minneapolis-Saint Paul">Minneapolis-Saint Paul</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Wisconsin%E2%80%93Madison" title="University of Wisconsin–Madison">University of Wisconsin</a> (<a href="http://en.wikipedia.org/wiki/Madison,_Wisconsin" title="Madison, Wisconsin">Madison</a>)</li>
</ul><h3> <span class="mw-headline" id="Southern">Southern</span></h3><ul><li><a href="http://en.wikipedia.org/wiki/The_College_of_William_%26_Mary" title="The College of William & Mary">College of William & Mary</a> (<a href="http://en.wikipedia.org/wiki/Williamsburg,_Virginia" title="Williamsburg, Virginia">Williamsburg</a>, <a href="http://en.wikipedia.org/wiki/Virginia" title="Virginia">Virginia</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Florida" title="University of Florida">University of Florida</a> (<a href="http://en.wikipedia.org/wiki/Gainesville,_Florida" title="Gainesville, Florida">Gainesville</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Georgia" title="University of Georgia">University of Georgia</a> (<a href="http://en.wikipedia.org/wiki/Athens,_Georgia" title="Athens, Georgia">Athens</a>)</li>
<li><a href="http://en.wikipedia.org/wiki/University_of_North_Carolina_at_Chapel_Hill" title="University of North Carolina at Chapel Hill">University of North Carolina at Chapel Hill</a></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Texas_at_Austin" title="University of Texas at Austin">University of Texas at Austin</a></li>
<li><a href="http://en.wikipedia.org/wiki/University_of_Virginia" title="University of Virginia">University of Virginia</a> (<a href="http://en.wikipedia.org/wiki/Charlottesville,_Virginia" title="Charlottesville, Virginia">Charlottesville</a>)</li>
</ul></td> </tr>
</tbody></table><h2> <span class="mw-headline" id="Institutional_comparisons">Institutional comparisons</span></h2><h3> <span class="mw-headline" id="Academic_comparisons_and_rankings">Academic comparisons and rankings</span></h3>Several schools considered as "Public Ivies" are consistently ranked among the top schools in the multitude of surveys on American colleges and universities undertaken by <i><a href="http://en.wikipedia.org/wiki/U.S._News_%26_World_Report" title="U.S. News & World Report">U.S. News & World Report</a></i>. For instance, half of the top 12 ranked national universities for undergraduate teaching in <i>U.S. News and World Report</i> are of the original Public Ivies listed by Moll.<sup class="reference" id="cite_ref-8"><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_note-8"><span>[</span>9<span>]</span></a></sup> Public Ivies can be found in the top ten ranked graduate schools in business, education, engineering, law, and medicine.<sup class="reference" id="cite_ref-9"><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_note-9"><span>[</span>10<span>]</span></a></sup><br />
<h3> <span class="mw-headline" id="Athletic_comparisons">Athletic comparisons</span></h3>One sharp distinction between the <a href="http://en.wikipedia.org/wiki/Ivy_League" title="Ivy League">Ivy League</a> and most "Public Ivies" is their participation in intercollegiate athletics. One of the <a href="http://en.wikipedia.org/wiki/Ivy_League" title="Ivy League">Ivy League</a>'s distinguishing characteristics is its prohibition on the awarding of <a href="http://en.wikipedia.org/wiki/Athletic_scholarship" title="Athletic scholarship">athletic scholarships</a> (athletes may only receive the same <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Student_financial_aid" title="Student financial aid">financial aid</a> to which they would be entitled even if they did not play a sport). In contrast, many of the "Public Ivies" participate in major athletic conferences such as the <a href="http://en.wikipedia.org/wiki/Big_East_Conference" title="Big East Conference">Big East</a>, <a href="http://en.wikipedia.org/wiki/Big_Ten_Conference" title="Big Ten Conference">Big Ten</a>, <a href="http://en.wikipedia.org/wiki/Big_12_Conference" title="Big 12 Conference">Big 12</a>, <a href="http://en.wikipedia.org/wiki/Atlantic_Coast_Conference" title="Atlantic Coast Conference">ACC</a>, <a href="http://en.wikipedia.org/wiki/Southeastern_Conference" title="Southeastern Conference">SEC</a>, or <a href="http://en.wikipedia.org/wiki/Pacific-12_Conference" title="Pacific-12 Conference">Pac-12</a>, and award athletic scholarships. These schools sometimes rely on profits, if any, from large-scale <a href="http://en.wikipedia.org/wiki/American_football" title="American football">football</a> and men's <a href="http://en.wikipedia.org/wiki/Basketball" title="Basketball">basketball</a> programs to support the athletic department as a whole.<br />
<h2> <span class="mw-headline" id="See_also">See also</span></h2><ul><li><a href="http://en.wikipedia.org/wiki/Black_Ivy_League" title="Black Ivy League">Black Ivy League</a></li>
<li><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Colonial_colleges" title="Colonial colleges">Colonial colleges</a></li>
<li><a href="http://en.wikipedia.org/wiki/Flagship#Education" title="Flagship">Flagship university</a></li>
<li><i><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Hidden_Ivies:_Thirty_Colleges_of_Excellence" title="Hidden Ivies: Thirty Colleges of Excellence">Hidden Ivies: Thirty Colleges of Excellence</a></i></li>
<li><a href="http://en.wikipedia.org/wiki/Ivy_League" title="Ivy League">Ivy League</a></li>
<li><a href="http://en.wikipedia.org/wiki/Little_Three" title="Little Three">Little Three</a></li>
<li><a href="http://en.wikipedia.org/wiki/Little_Ivies" title="Little Ivies">Little Ivies</a></li>
<li><a href="http://en.wikipedia.org/wiki/Seven_Sisters_%28colleges%29" title="Seven Sisters (colleges)">Seven Sisters</a></li>
<li><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Southern_Ivies" title="Southern Ivies">Southern Ivies</a></li>
</ul><h2> <span class="mw-headline" id="References_and_other_resources">References and other resources</span></h2><h3> <span class="mw-headline" id="Citations">Citations</span></h3><div class="reflist" style="list-style-type: decimal;"> <ol class="references"><li id="cite_note-PublicIvys-0"><b><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_ref-PublicIvys_0-0">^</a></b> <a class="new" href="http://en.wikipedia.org/w/index.php?title=Richard_Moll_%28Author%29&action=edit&redlink=1" title="Richard Moll (Author) (page does not exist)">Richard Moll</a> in his book <i>Public Ivys: A Guide to America's best public undergraduate colleges and universities</i> (1985)</li>
<li id="cite_note-JBHE-1">^ <a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_ref-JBHE_1-0"><sup><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_ref-JBHE_1-1"><sup><i><b>b</b></i></sup></a> <span class="citation Journal"><a class="external text" href="http://www.jbhe.com/news_views/49_blackenrollment_publicivies.html" rel="nofollow">"Comparing Black Enrollments at the Public Ivies"</a>. <i>Journal of Blacks in Higher Education</i>. Autumn 2005<span class="reference-accessdate">. Retrieved 2006-09-03</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Comparing+Black+Enrollments+at+the+Public+Ivies&rft.jtitle=Journal+of+Blacks+in+Higher+Education&rft.date=Autumn+2005&rft_id=http%3A%2F%2Fwww.jbhe.com%2Fnews_views%2F49_blackenrollment_publicivies.html&rfr_id=info:sid/en.wikipedia.org:Public_Ivy"></span></li>
<li id="cite_note-YaleMagazine-2">^ <a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_ref-YaleMagazine_2-0"><sup><i><b>a</b></i></sup></a> <a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_ref-YaleMagazine_2-1"><sup><i><b>b</b></i></sup></a> <span class="citation Journal"><a class="new" href="http://en.wikipedia.org/w/index.php?title=Mark_Alden_Branch&action=edit&redlink=1" title="Mark Alden Branch (page does not exist)">Branch, Mark Alden</a> (November 2000). <a class="external text" href="http://www.yalealumnimagazine.com/issues/00_11/admissions.html" rel="nofollow">"Deciphering the Admissions Map"</a>. <i>Yale Alumni Magazine</i> <b>109</b> (11)<span class="reference-accessdate">. Retrieved 2008-02-09</span>. "¶16: But Richard Moll '59MDiv, a former Yale admissions officer who later oversaw admissions at Bowdoin and Vassar, thinks Yale still is not as visible as it should be. "Yale has not had the presence at grassroots admissions and counseling conferences that Harvard and Stanford have," says Moll, author of Playing the Selective College Admissions Game."</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Deciphering+the+Admissions+Map&rft.jtitle=Yale+Alumni+Magazine&rft.aulast=Branch&rft.aufirst=Mark+Alden&rft.au=Branch%2C%26%2332%3BMark+Alden&rft.date=November+2000&rft.volume=109&rft.issue=11&rft_id=http%3A%2F%2Fwww.yalealumnimagazine.com%2Fissues%2F00_11%2Fadmissions.html&rfr_id=info:sid/en.wikipedia.org:Public_Ivy"></span></li>
<li id="cite_note-3"><b><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_ref-3">^</a></b> <span class="citation Journal"><a href="http://en.wikipedia.org/wiki/Kenneth_M._Pierce" title="Kenneth M. Pierce">Pierce, Kenneth M.</a> (24 November 1980). <a class="external text" href="http://www.time.com/time/magazine/article/0,9171,952854-1,00.html" rel="nofollow">"Dr. Fix-It Goes to Santa Cruz"</a>. <i><a href="http://en.wikipedia.org/wiki/Time_%28magazine%29" title="Time (magazine)">Time</a></i><span class="reference-accessdate">. Retrieved 2008-02-09</span>. "Trouble in paradise as "the touchy-feely school" sings the blues – <a href="http://en.wikipedia.org/wiki/Richard_Moll" title="Richard Moll">Richard Moll</a>, 45, a tweedy graduate of <a href="http://en.wikipedia.org/wiki/Yale_Divinity_School" title="Yale Divinity School">Yale's Divinity School</a>, has become a Dr. Fix-It for colleges that complain of sagging enrollment."</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Dr.+Fix-It+Goes+to+Santa+Cruz&rft.jtitle=%5B%5BTime+%28magazine%29%7CTime%5D%5D&rft.aulast=Pierce&rft.aufirst=Kenneth+M.&rft.au=Pierce%2C%26%2332%3BKenneth+M.&rft.date=24+November+1980&rft_id=http%3A%2F%2Fwww.time.com%2Ftime%2Fmagazine%2Farticle%2F0%2C9171%2C952854-1%2C00.html&rfr_id=info:sid/en.wikipedia.org:Public_Ivy"></span></li>
<li id="cite_note-4"><b><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_ref-4">^</a></b> <span class="citation web">Paul Marthers, Dean of Admission. <a class="external text" href="http://web.reed.edu/apply/news_and_articles/admission_messages.html" rel="nofollow">"Admissions Messages vs. Admissions Realities"</a>. <i>Office of Admissions</i>. <a href="http://en.wikipedia.org/wiki/Reed_College" title="Reed College">Reed College</a><span class="reference-accessdate">. Retrieved 2008-02-09</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Admissions+Messages+vs.+Admissions+Realities&rft.atitle=Office+of+Admissions&rft.aulast=Paul+Marthers%2C+Dean+of+Admission&rft.au=Paul+Marthers%2C+Dean+of+Admission&rft.pub=%5B%5BReed+College%5D%5D&rft_id=http%3A%2F%2Fweb.reed.edu%2Fapply%2Fnews_and_articles%2Fadmission_messages.html&rfr_id=info:sid/en.wikipedia.org:Public_Ivy"></span></li>
<li id="cite_note-5"><b><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_ref-5">^</a></b> In Moll's book, he refers to the entire <a href="http://en.wikipedia.org/wiki/University_of_California" title="University of California">UC</a> system</li>
<li id="cite_note-6"><b><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_ref-6">^</a></b> Moll, Richard (1985). <i>The Public Ivys: A Guide to America's Best Undergraduate Colleges and Universities</i>. Viking Penguin Inc. p. xxvi. <a class="external text" href="http://en.wikipedia.org/w/index.php?title=Special%3ABooksources&isbn=0670582050%7CISBN">0-670-58205-0</a></li>
<li id="cite_note-7"><b><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_ref-7">^</a></b> <a class="external text" href="http://www.amazon.com/gp/reader/006093459X/" rel="nofollow">Greenes' Guides: The Public Ivies</a> (accessed on May 16, 2007); see also <a class="external autonumber" href="http://www.jbhe.com/news_views/49_blackenrollment_publicivies.html" rel="nofollow">[1]</a>.</li>
<li id="cite_note-8"><b><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_ref-8">^</a></b> <span class="citation web">U.S. News and World Report. <a class="external text" href="http://colleges.usnews.rankingsandreviews.com/best-colleges/national-ut-rank" rel="nofollow">"Best Colleges: Undergraduate Teaching at National Universities"</a><span class="reference-accessdate">. Retrieved 2 August 2010</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Best+Colleges%3A+Undergraduate+Teaching+at+National+Universities&rft.atitle=&rft.aulast=U.S.+News+and+World+Report&rft.au=U.S.+News+and+World+Report&rft_id=http%3A%2F%2Fcolleges.usnews.rankingsandreviews.com%2Fbest-colleges%2Fnational-ut-rank&rfr_id=info:sid/en.wikipedia.org:Public_Ivy"></span></li>
<li id="cite_note-9"><b><a href="http://en.wikipedia.org/wiki/Public_Ivies#cite_ref-9">^</a></b> <span class="citation web">U.S. News and World Report. <a class="external text" href="http://grad-schools.usnews.rankingsandreviews.com/best-graduate-schools" rel="nofollow">"Best Graduate Schools"</a><span class="reference-accessdate">. Retrieved 2 August 2010</span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Best+Graduate+Schools&rft.atitle=&rft.aulast=U.S.+News+and+World+Report&rft.au=U.S.+News+and+World+Report&rft_id=http%3A%2F%2Fgrad-schools.usnews.rankingsandreviews.com%2Fbest-graduate-schools&rfr_id=info:sid/en.wikipedia.org:Public_Ivy"></span></li>
</ol></div><h3> <span class="mw-headline" id="Books">Books</span></h3><ul><li><span class="citation book">Greene, Howard; Matthew Greene (2001). <i>The Public Ivies: America's Flagship Public Universities</i>. New York: HarperCollins. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/006093459X" title="Special:BookSources/006093459X">006093459X</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Public+Ivies%3A+America%27s+Flagship+Public+Universities&rft.aulast=Greene&rft.aufirst=Howard&rft.au=Greene%2C%26%2332%3BHoward&rft.date=2001&rft.place=New+York&rft.pub=HarperCollins&rft.isbn=006093459X&rfr_id=info:sid/en.wikipedia.org:Public_Ivy"></span></li>
<li><span class="citation book">Greene, Howard; Matthew Greene (2000). <i>Hidden Ivies: Thirty Colleges of Excellence</i>. New York: HarperCollins. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0060953624" title="Special:BookSources/0060953624">0060953624</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hidden+Ivies%3A+Thirty+Colleges+of+Excellence&rft.aulast=Greene&rft.aufirst=Howard&rft.au=Greene%2C%26%2332%3BHoward&rft.date=2000&rft.place=New+York&rft.pub=HarperCollins&rft.isbn=0060953624&rfr_id=info:sid/en.wikipedia.org:Public_Ivy"></span></li>
<li><span class="citation book">Moll, Richard (1985). <i>The Public Ivies: A Guide to America's best public undergraduate colleges and universities</i>. New York: Penguin (Viking). <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0140093842" title="Special:BookSources/0140093842">0140093842</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Public+Ivies%3A+A+Guide+to+America%27s+best+public+undergraduate+colleges+and+universities&rft.aulast=Moll&rft.aufirst=Richard&rft.au=Moll%2C%26%2332%3BRichard&rft.date=1985&rft.place=New+York&rft.pub=Penguin+%28Viking%29&rft.isbn=0140093842&rfr_id=info:sid/en.wikipedia.org:Public_Ivy"></span></li>
<li><span class="citation book">Robert Franek ... (2006). <i>The Best 361 Colleges, 2007 Edition</i>. Princeton, New Jersey: <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Princeton_Review" title="Princeton Review">Princeton Review</a>. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="http://en.wikipedia.org/wiki/Special:BookSources/0375765581" title="Special:BookSources/0375765581">0375765581</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Best+361+Colleges%2C+2007+Edition&rft.aulast=Robert+Franek+...&rft.au=Robert+Franek+...&rft.date=2006&rft.place=Princeton%2C+New+Jersey&rft.pub=%5B%5BPrinceton+Review%5D%5D&rft.isbn=0375765581&rfr_id=info:sid/en.wikipedia.org:Public_Ivy"></span></li>
</ul><table cellspacing="0" class="navbox" style="border-spacing: 0;"><tbody>
<tr> <td style="padding: 2px;"> <table cellspacing="0" class="nowraplinks collapsible autocollapse navbox-inner" id="collapsibleTable0" style="background: transparent; border-spacing: 0; color: inherit;"><tbody>
<tr> <th class="navbox-title" colspan="2" scope="col"><span class="collapseButton">[<a href="http://en.wikipedia.org/wiki/Public_Ivies#" id="collapseButton0">hide</a>]</span> <div class="noprint plainlinks hlist navbar"> <ul><li class="nv-view"><a href="http://en.wikipedia.org/wiki/Template:Public_Ivy" title="Template:Public Ivy"><span style="background: none transparent; border: none;" title="View this template">v</span></a></li>
<li class="nv-talk"><a href="http://en.wikipedia.org/wiki/Template_talk:Public_Ivy" title="Template talk:Public Ivy"><span style="background: none transparent; border: none;" title="Discuss this template">d</span></a></li>
<li class="nv-edit"><a class="external text" href="http://en.wikipedia.org/w/index.php?title=Template:Public_Ivy&action=edit"><span style="background: none transparent; border: none;" title="Edit this template">e</span></a></li>
</ul></div><div class="" style="font-size: 110%;"><strong class="selflink">Public Ivy</strong> universities</div></th> </tr>
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</td> </tr>
<tr> <td class="navbox-list navbox-odd" colspan="2" style="padding: 0px; width: 100%;"> <div style="padding: 0em 0.25em;"><b>Richard Moll's 1985 list</b></div></td> </tr>
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</td> </tr>
<tr> <th class="navbox-group" scope="row">Original Eight</th> <td class="navbox-list navbox-even" style="border-left-style: solid; border-left-width: 2px; padding: 0px; text-align: left; width: 100%;"> <div style="padding: 0em 0.25em;"><a href="http://en.wikipedia.org/wiki/The_College_of_William_%26_Mary" title="The College of William & Mary">College of William & Mary</a> • <a href="http://en.wikipedia.org/wiki/Miami_University" title="Miami University">Miami University</a> • <a href="http://en.wikipedia.org/wiki/University_of_California" title="University of California">University of California</a> • <a href="http://en.wikipedia.org/wiki/University_of_Michigan" title="University of Michigan">University of Michigan</a> • <a href="http://en.wikipedia.org/wiki/University_of_North_Carolina_at_Chapel_Hill" title="University of North Carolina at Chapel Hill">University of North Carolina at Chapel Hill</a> • <a href="http://en.wikipedia.org/wiki/University_of_Texas_at_Austin" title="University of Texas at Austin">University of Texas at Austin</a> • <a href="http://en.wikipedia.org/wiki/University_of_Vermont" title="University of Vermont">University of Vermont</a> • <a href="http://en.wikipedia.org/wiki/University_of_Virginia" title="University of Virginia">University of Virginia</a></div></td> </tr>
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</td> </tr>
<tr> <th class="navbox-group" scope="row">"Worthy Runners-Up"</th> <td class="navbox-list navbox-odd" style="border-left-style: solid; border-left-width: 2px; padding: 0px; text-align: left; width: 100%;"> <div style="padding: 0em 0.25em;"><a href="http://en.wikipedia.org/wiki/Binghamton_University" title="Binghamton University">Binghamton University, State University of New York</a> • <a href="http://en.wikipedia.org/wiki/University_of_Colorado_at_Boulder" title="University of Colorado at Boulder">University of Colorado at Boulder</a> • <a href="http://en.wikipedia.org/wiki/Georgia_Institute_of_Technology" title="Georgia Institute of Technology">Georgia Institute of Technology</a> • <a href="http://en.wikipedia.org/wiki/University_of_Illinois_at_Urbana%E2%80%93Champaign" title="University of Illinois at Urbana–Champaign">University of Illinois at Urbana–Champaign</a> • <a href="http://en.wikipedia.org/wiki/New_College_of_Florida" title="New College of Florida">New College of Florida</a> • <a href="http://en.wikipedia.org/wiki/Pennsylvania_State_University" title="Pennsylvania State University">Pennsylvania State University</a> • <a href="http://en.wikipedia.org/wiki/University_of_Pittsburgh" title="University of Pittsburgh">University of Pittsburgh</a> • <a href="http://en.wikipedia.org/wiki/University_of_Washington" title="University of Washington">University of Washington at Seattle</a> • <a href="http://en.wikipedia.org/wiki/University_of_Wisconsin%E2%80%93Madison" title="University of Wisconsin–Madison">University of Wisconsin–Madison</a></div></td> </tr>
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</td> </tr>
<tr> <td class="navbox-list navbox-even" colspan="2" style="padding: 0px; width: 100%;"> <div style="padding: 0em 0.25em;"><b>Greenes' Guides 2001 list</b></div></td> </tr>
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</td> </tr>
<tr> <th class="navbox-group" scope="row">Eastern</th> <td class="navbox-list navbox-odd" style="border-left-style: solid; border-left-width: 2px; padding: 0px; text-align: left; width: 100%;"> <div style="padding: 0em 0.25em;"><a href="http://en.wikipedia.org/wiki/Binghamton_University" title="Binghamton University">Binghamton University, State University of New York</a> • <a href="http://en.wikipedia.org/wiki/The_College_of_William_%26_Mary" title="The College of William & Mary">College of William & Mary</a> • <a href="http://en.wikipedia.org/wiki/Pennsylvania_State_University" title="Pennsylvania State University">Pennsylvania State University</a> • <a class="mw-redirect" href="http://en.wikipedia.org/wiki/Rutgers,_The_State_University_of_New_Jersey" title="Rutgers, The State University of New Jersey">Rutgers, The State University of New Jersey</a> • <a href="http://en.wikipedia.org/wiki/University_of_Connecticut" title="University of Connecticut">University of Connecticut</a> • <a href="http://en.wikipedia.org/wiki/University_of_Delaware" title="University of Delaware">University of Delaware</a> • <a href="http://en.wikipedia.org/wiki/University_of_Maryland,_College_Park" title="University of Maryland, College Park">University of Maryland</a> • <a href="http://en.wikipedia.org/wiki/University_of_North_Carolina_at_Chapel_Hill" title="University of North Carolina at Chapel Hill">University of North Carolina at Chapel Hill</a> • <a href="http://en.wikipedia.org/wiki/University_of_Virginia" title="University of Virginia">University of Virginia</a> • <a href="http://en.wikipedia.org/wiki/University_of_Vermont" title="University of Vermont">University of Vermont</a></div></td> </tr>
<tr style="height: 2px;"> <td><br />
</td> </tr>
<tr> <th class="navbox-group" scope="row">Western</th> <td class="navbox-list navbox-even" style="border-left-style: solid; border-left-width: 2px; padding: 0px; text-align: left; width: 100%;"> <div style="padding: 0em 0.25em;"><a href="http://en.wikipedia.org/wiki/University_of_Arizona" title="University of Arizona">University of Arizona</a> • <a href="http://en.wikipedia.org/wiki/University_of_California" title="University of California">University of California</a>: (<small><a href="http://en.wikipedia.org/wiki/University_of_California,_Berkeley" title="University of California, Berkeley">Berkeley</a>, <a href="http://en.wikipedia.org/wiki/University_of_California,_Davis" title="University of California, Davis">Davis</a>, <a href="http://en.wikipedia.org/wiki/University_of_California,_Irvine" title="University of California, Irvine">Irvine</a>, <a href="http://en.wikipedia.org/wiki/University_of_California,_Los_Angeles" title="University of California, Los Angeles">Los Angeles</a>, <a href="http://en.wikipedia.org/wiki/University_of_California,_San_Diego" title="University of California, San Diego">San Diego</a>, and <a href="http://en.wikipedia.org/wiki/University_of_California,_Santa_Barbara" title="University of California, Santa Barbara">Santa Barbara</a></small>) • <a href="http://en.wikipedia.org/wiki/University_of_Colorado_at_Boulder" title="University of Colorado at Boulder">University of Colorado at Boulder</a> • <a href="http://en.wikipedia.org/wiki/University_of_Washington" title="University of Washington">University of Washington</a></div></td> </tr>
<tr style="height: 2px;"> <td><br />
</td> </tr>
<tr> <th class="navbox-group" scope="row">Great Lakes & Midwest</th> <td class="navbox-list navbox-odd" style="border-left-style: solid; border-left-width: 2px; padding: 0px; text-align: left; width: 100%;"> <div style="padding: 0em 0.25em;"><a href="http://en.wikipedia.org/wiki/Indiana_University_Bloomington" title="Indiana University Bloomington">Indiana University</a> • <a href="http://en.wikipedia.org/wiki/Miami_University" title="Miami University">Miami University</a> • <a href="http://en.wikipedia.org/wiki/Ohio_State_University" title="Ohio State University">Ohio State University</a> • <a href="http://en.wikipedia.org/wiki/University_of_Illinois_at_Urbana%E2%80%93Champaign" title="University of Illinois at Urbana–Champaign">University of Illinois</a> • <a href="http://en.wikipedia.org/wiki/University_of_Iowa" title="University of Iowa">University of Iowa</a> • <a href="http://en.wikipedia.org/wiki/University_of_Michigan" title="University of Michigan">University of Michigan</a> • <a href="http://en.wikipedia.org/wiki/Michigan_State_University" title="Michigan State University">Michigan State University</a> • <a href="http://en.wikipedia.org/wiki/University_of_Minnesota" title="University of Minnesota">University of Minnesota</a> • <a href="http://en.wikipedia.org/wiki/University_of_Wisconsin%E2%80%93Madison" title="University of Wisconsin–Madison">University of Wisconsin</a></div></td> </tr>
<tr style="height: 2px;"> <td><br />
</td> </tr>
<tr> <th class="navbox-group" scope="row">Southern</th> <td class="navbox-list navbox-even" style="border-left-style: solid; border-left-width: 2px; padding: 0px; text-align: left; width: 100%;"> <div style="padding: 0em 0.25em;"><a href="http://en.wikipedia.org/wiki/University_of_Florida" title="University of Florida">University of Florida</a> • <a href="http://en.wikipedia.org/wiki/University_of_Georgia" title="University of Georgia">University of Georgia</a> • <a href="http://en.wikipedia.org/wiki/University_of_Texas_at_Austin" title="University of Texas at Austin">University of Texas at Austin</a></div></td> </tr>
</tbody></table></td> </tr>
</tbody></table>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com1tag:blogger.com,1999:blog-5303246073824127471.post-35622181921749744642012-02-14T22:48:00.001-05:002012-02-14T22:49:51.285-05:00Hearts<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXmdrIwU_f3c7rBXwRrp6qcQu5wFV7GkXGZp-cgPoTXt6ioUeQVkUKg5GBLMenkTF4LWM_CgRUBNVDLeQ8kKLRANvR3chErueRAKSQUvy_BPDA8K9oadcAIUaCZvW_7Dcm2_wZ-Nd7SA/s1600/heart+graph.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="394" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXmdrIwU_f3c7rBXwRrp6qcQu5wFV7GkXGZp-cgPoTXt6ioUeQVkUKg5GBLMenkTF4LWM_CgRUBNVDLeQ8kKLRANvR3chErueRAKSQUvy_BPDA8K9oadcAIUaCZvW_7Dcm2_wZ-Nd7SA/s400/heart+graph.jpg" width="400" /></a></div><br />
Thanks to Pat Ballew of Pat's Blog for <a href="http://pballew.blogspot.com/2012/02/on-this-day-in-math-feb-14.html">the Mathematical Heart</a>.<br />
<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQ_hhFZZkPW5iGOI0XB_kZtayI5o-0tGSrWByf6gXRfqoRsgWqJ9ZPxyfxHXaDS1j8k4A4VxIsQ6aMu2xsMu95MNm2LAuShnnnZGe2tcYjm-PRuF4DOj_SPJXfmI_Qb6nxtgYaac5f6w/s1600/heart4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQ_hhFZZkPW5iGOI0XB_kZtayI5o-0tGSrWByf6gXRfqoRsgWqJ9ZPxyfxHXaDS1j8k4A4VxIsQ6aMu2xsMu95MNm2LAuShnnnZGe2tcYjm-PRuF4DOj_SPJXfmI_Qb6nxtgYaac5f6w/s320/heart4.jpg" width="213" /></a></div>Steven Colyerhttp://www.blogger.com/profile/10435759210177642257noreply@blogger.com0