Monday, February 21, 2011

Math Phys According to Wikipedia

Stephen Hawking, Babe Magnet
Wikipedia does a surprisingly very good job in giving a grand overview, in very few paragraphs, of what Mathematical Physics actually IS.

I say "surprisingly" because if you want to kill the passion in a young lad or lassie for a subject in Math or Physics, tell them to read a textbook on the subject. But if you REALLY want to destroy that passion with a large factor of safety, just send them to Wikipedia.

Not that any of these things are WRONG mind you, let's just say they're written by professionals in their fields writing moreso to other professional in their fields as opposed to say potential students, and in doing so they're really just showing off how big their "intellectual penis" is, so to speak. Like Sheldon.. ;-)

Technically, their entries are ... correct, just dry as a bone. EXAMPLES people, please, EXAMPLES! Well, at least Wiki has the modern miracle of "hyperlinks", which is good in one way because it allows you to surf the internet and get turned on to other subjects, papers, do great research, etc, but bad in another way as it often sends one to even dryer country. Well, whatever.

Anyway, here's a BETTER example of an entry at Wikipedia, with great links that should open up this seemingly "weird" world of ours, and does IMO the great job of explaining how Math and Physics have helped each other, with each returning the favor to the other, in an endless dance that IS ONGOING and gets deeper, every day.

Scope of the subject

The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories."[1]. There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods.

The theory of partial differential equations (and the related areas of variational calculus, Fourier analysis, potential theory, and vector analysis) are perhaps most closely associated with mathematical physics. These were developed intensively from the second half of the eighteenth century (by, for example, D'Alembert, Euler, and Lagrange) until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, elasticity theory, acoustics, thermodynamics, electricity, magnetism, and aerodynamics.

The theory of atomic spectra (and, later, quantum mechanics) developed almost concurrently with the mathematical fields of linear algebra, the spectral theory of operators, and more broadly, functional analysis. These constitute the mathematical basis of another branch of mathematical physics.

The special and general theories of relativity require a rather different type of mathematics. This was group theory: and it played an important role in both quantum field theory and differential geometry. This was, however, gradually supplemented by topology in the mathematical description of cosmological as well as quantum field theory phenomena.

Statistical mechanics forms a separate field, which is closely related with the more mathematical ergodic theory and some parts of probability theory.

There are increasing interactions between combinatorics and physics, in particular statistical physics.

The usage of the term 'Mathematical physics' is sometimes idiosyncratic. Certain parts of mathematics that initially arose from the development of physics are not considered parts of mathematical physics, while other closely related fields are. For example, ordinary differential equations and symplectic geometry are generally viewed as purely mathematical disciplines, whereas dynamical systems and Hamiltonian mechanics belong to mathematical physics.

 Prominent mathematical physicists

The seventeenth century English physicist and mathematician, Isaac Newton [1642–1727], developed a wealth of new mathematics (for example, calculus and several numerical methods (most notably Newton's method) to solve problems in physics. Other important mathematical physicists of the seventeenth century included the Dutchman Christiaan Huygens [1629–1695] (famous for suggesting the wave theory of light), and the German Johannes Kepler [1571–1630] (Tycho Brahe's assistant, and discoverer of the equations for planetary motion/orbit).

In the eighteenth century, two of the innovators of mathematical physics were Swiss: Daniel Bernoulli [1700–1782] (for contributions to fluid dynamics, and vibrating strings), and, more especially, Leonhard Euler [1707–1783], (for his work in variational calculus, dynamics, fluid dynamics, and many other things). Another notable contributor was the Italian-born Frenchman, Joseph-Louis Lagrange [1736–1813] (for his work in mechanics and variational methods).

In the late eighteenth and early nineteenth centuries, important French figures were Pierre-Simon Laplace [1749–1827] (in mathematical astronomy, potential theory, and mechanics) and Siméon Denis Poisson [1781–1840] (who also worked in mechanics and potential theory). In Germany, both Carl Friedrich Gauss [1777–1855] (in magnetism) and Carl Gustav Jacobi [1804–1851] (in the areas of dynamics and canonical transformations) made key contributions to the theoretical foundations of electricity, magnetism, mechanics, and fluid dynamics.

Gauss's contributions to non-Euclidean geometry laid the groundwork for the subsequent development of Riemannian geometry by Bernhard Riemann [1826–1866]. As we shall see later, this work is at the heart of general relativity.

The nineteenth century also saw the Scot, James Clerk Maxwell [1831–1879], win renown for his four equations of electromagnetism, and his countryman, Lord Kelvin [1824–1907] make substantial discoveries in thermodynamics. Among the English physics community, Lord Rayleigh [1842–1919] worked on sound; and George Gabriel Stokes [1819–1903] was a leader in optics and fluid dynamics; while the Irishman William Rowan Hamilton [1805–1865] was noted for his work in dynamics. The German Hermann von Helmholtz [1821–1894] is best remembered for his work in the areas of electromagnetism, waves, fluids, and sound. In the U.S.A., the pioneering work of Josiah Willard Gibbs [1839–1903] became the basis for statistical mechanics. Together, these men laid the foundations of electromagnetic theory, fluid dynamics and statistical mechanics.

The late nineteenth and the early twentieth centuries saw the birth of special relativity. This had been anticipated in the works of the Dutchman, Hendrik Lorentz [1853–1928], with important insights from Jules-Henri Poincaré [1854–1912], but which were brought to full clarity by Albert Einstein [1879–1955].

Einstein then developed the invariant approach further to arrive at the remarkable geometrical approach to gravitational physics embodied in general relativity. This was based on the non-Euclidean geometry created by Gauss and Riemann in the previous century.

Einstein's special relativity replaced the Galilean transformations of space and time with Lorentz transformations in four dimensional Minkowski space-time. His general theory of relativity replaced the flat Euclidean geometry with that of a Riemannian manifold, whose curvature is determined by the distribution of gravitational matter. This replaced Newton's vector gravitational force by the Riemann curvature tensor.

Another revolutionary development of the twentieth century has been quantum theory, which emerged from the seminal contributions of Max Planck [1856–1947] (on black body radiation) and Einstein's work on the photoelectric effect. This was, at first, followed by a heuristic framework devised by Arnold Sommerfeld [1868–1951] and Niels Bohr [1885–1962], but this was soon replaced by the quantum mechanics developed by Max Born [1882–1970], Werner Heisenberg [1901–1976], Paul Dirac [1902–1984], Erwin Schrödinger [1887–1961], and Wolfgang Pauli [1900–1958].

This revolutionary theoretical framework is based on a probabilistic interpretation of states, and evolution and measurements in terms of self-adjoint operators on an infinite dimensional vector space (Hilbert space, introduced by David Hilbert [1862–1943]).

Paul Dirac, for example, used algebraic constructions to produce a relativistic model for the electron, predicting its magnetic moment and the existence of its antiparticle, the positron.

Later important contributors to twentieth century mathematical physics include Satyendra Nath Bose [1894–1974], Julian Schwinger [1918–1994], Sin-Itiro Tomonaga [1906–1979], Richard Feynman [1918–1988], Freeman Dyson [1923– ], Hideki Yukawa [1907–1981], Roger Penrose [1931– ], Stephen Hawking [1942– ], Edward Witten [1951– ] and Rudolf Haag [1922– ]

 Mathematically rigorous physics

The term 'mathematical' physics is also sometimes used in a special sense, to denote research aimed at studying and solving problems inspired by physics within a mathematically rigorous framework. Mathematical physics in this sense covers a very broad area of topics with the common feature that they blend pure mathematics and physics. Although related to theoretical physics, 'mathematical' physics in this sense emphasizes the mathematical rigour of the same type as found in mathematics.

On the other hand, theoretical physics emphasizes the links to observations and experimental physics which often requires theoretical physicists (and mathematical physicists in the more general sense) to use heuristic, intuitive, and approximate arguments. Such arguments are not considered rigorous by mathematicians. Arguably, rigorous mathematical physics is closer to mathematics, and theoretical physics is closer to physics.

Such mathematical physicists primarily expand and elucidate physical theories. Because of the required rigor, these researchers often deal with questions that theoretical physicists have considered to already be solved. However, they can sometimes show (but neither commonly nor easily) that the previous solution was incorrect.[example needed]

The field has concentrated in three main areas: (1) quantum field theory, especially the precise construction of models; (2) statistical mechanics, especially the theory of phase transitions; and (3) nonrelativistic quantum mechanics (Schrödinger operators), including the connections to atomic and molecular physics.

The effort to put physical theories on a mathematically rigorous footing has inspired many mathematical developments. For example, the development of quantum mechanics and some aspects of functional analysis parallel each other in many ways. The mathematical study of quantum statistical mechanics has motivated results in operator algebras. The attempt to construct a rigorous quantum field theory has brought about progress in fields such as representation theory. Use of geometry and topology plays an important role in string theory.

 See also

 Notes

  1. ^ Definition from the Journal of Mathematical Physics. http://jmp.aip.org/jmp/staff.jsp

 References

 Further reading

 The Classics

 Textbooks for undergraduate studies

  • Arfken, George B.; Weber, Hans J. (1995), 'Mathematical methods for physicists' (4th ed.), San Diego, [CA.]: Academic Press, ISBN 0-120-59816-7  (pbk.)
  • Boas, Mary L. (2006), 'Mathematical Methods in the Physical Sciences' (3rd ed.), Hoboken, [NJ.]: John Wiley & Sons, ISBN 9780471198260 
  • Butkov, Eugene (1968), 'Mathematical physics', Reading, [Mass.]: Addison-Wesley 
  • Jeffreys, Harold; Swirles Jeffreys, Bertha (1956), 'Methods of mathematical physics' (3rd rev. ed.), Cambridge, [England]: Cambridge University Press 
  • Kusse, Bruce R. (2006), 'Mathematical Physics: Applied Mathematics for Scientists and Engineers' (2nd ed.), [Germany]: Wiley-VCH, ISBN 3-527-40672-7 
  • Joos, Georg; Freeman, Ira M. (1987), Theoretical Physics, Dover Publications, ISBN 0-486-65227-0 
  • Mathews, Jon; Walker, Robert L. (1970), 'Mathematical methods of physics' (2nd ed.), New York, [NY.]: W. A. Benjamin, ISBN 0-8053-7002-1 
  • Menzel, Donald Howard (1961), Mathematical Physics, Dover Publications, ISBN 0-486-60056-4 
  • Stakgold, Ivar (c.2000), 'Boundary value problems of mathematical physics (2 vol.)', Philadelphia, [PA.]: Society for Industrial and Applied Mathematics, ISBN 0-898-71456-7  (set : pbk.)

Textbooks for graduate studies

  • Hassani, Sadri (1999), 'Mathematical Physics: A Modern Introduction to Its Foundations', Berlin, [Germany]: Springer-Verlag, ISBN 0387985794 
  • Reed,Michael; Simon,Barry Methods of Modern Mathematical Physics (Academic Press).[Four volumes]

 Other specialised subareas

8 comments:

Jérôme CHAUVET said...

You are so right, Steven. Many people criticize Wikipedia for being "written by unprofessional people" though using it anyway. Furthermore, it is wrong to say Wikipedia must be overwhelmed with uncorrect knowledge. 99% of the information put in Wikipedia can be retrieved in articles of specialists without the need to deny what you first read in Wikipedia. Its articles are the very starting point of any of my queries when I am in search for something on a subject. I cannot say there's only gibberish in it, it would not be honest to say so.

I once in the past made a correction of the french article which deals with computer viruses. There was a sentence which compared them with biological viruses which didn't make sense, clumsily written (clearly the guy was rather of the computing side of science) so I made a free correction of the sentence. I am a molecular biologist, so I could recognize where the awkward thing is. What if a non specialist gazes into the text ? He may surely either find it boring and leave or simply read and believe what he reads. Those who write and make corrections in Wikipedia articles know the meaning of things so we should not distrust their writings that much.

Best,

Steven Colyer said...

The BEST thing about Wiki Jérôme may end up being its articles re Mathematics, Physics, and Computer Science, and those things in the other Sciences that are non-controversial and therefore KNOWN to be true, via established experimental proof.

The WORST thing about about Wiki are those things that are open for debate, to whit virtually all things Philosophy, Corporate, and Biographical. We just have NO idea what we read there as to what is true, unless we do further research, because the debatable stuff is too often and unfortunately presented at Wikipedia as "fact."

"The Net of a Million Lies" is where we live, virtually. It behooves each person to use their intelligence in a skeptical fashion to discern the truth from that which is falsely presented in a satanic Third Reich-ian truth-mixed-with-lies manner, that is to say: utter shit.

Steven Colyer said...

Although I very much love this entry, I must complain that Louis de Broglie and Pascual Jordan were not mentioned, and most especially Hermann Weyl.

Jérôme CHAUVET said...

Sorry, I read your article too fast, thought you were on the side of Wikipedia. Re-reading it, I found you're rather skeptical :)

Well, that's your opinion. I just use Wikipedia as a base for all my searches, nothing more.

Steven Colyer said...

Again, I DO like Wikipedia for mathematics and the hard sciences, and as you say, a "base" for other subjects.

I like Wiki for those things that are KNOWN to be true. Even Wiki cannot disagree with them.

Jérôme CHAUVET said...

You're right saying that many articles lack relevant information. Anyway, I am used to reading articles in three languages (French, German and English), which most of the time are significantly different one from another, and I get a more complete overview of the subject :)

I however think controversial themes should always be in Wikipedia provided the controversial nature of the subject is mentioned in the article.

Best,

Kurtis Hallman said...

This conversation is true to the necessity of skepticism in contrast to other strategies at arriving at conclusions. The conclusions in the simplest form should be approachable and so they inherently are. Conversations do not need full verifiability beyond what is useful. Supposing use of any real argument is translated into a relative idea of happiness, the proximity effect is far more beneficial than the negativity effect or the positivity when considering information to be true or not. The over whelming amount of information naturally creates problems that for all intents and purposes should be put off for a later time to solve. The right amount of information about the right topic at the right time does require responsibility to understanding the original purpose of that information. This should lead any inquiring mind to avoid over whelming information and stick to the information that is simple and produces practical results that are consistent and proven both quality wise and quantity wise as the inquiring mind is likely to need to distinguish their own intentions from the intentions of information and the informations creator or original poster. The open source platform is inherent to the philosophy that the pursuit of truth should be free but is prone to hyperbole because free has a social inflection (communication being rational or irrational i.e. extreme or radical), a physical inflection (being an object or subject that is more or less contained), and a numeric inflection (no resource transaction fee implied) There are other ways to categorize free but to perfectly define the compound word 'open-source' the rational pursuit of happiness through the exercise communicative and physical freedom must be consistently observed to be as positive as possible. Where the lack of resource or the lack in the ability to obtain resource is through content, a tendency to be responsible for being closed off from the ability to make these distinctions about the information we approach incidentally or with purpose; however dry it may seem to be.

Kurtis Hallman said...

Particularly Mr. Chauvet and Mr. Colyer your statements about the 'free' encyclopedia uncommonly thoughtful;)