The Cauchy–Schwarz Inequality in a Nutshell (or a "Quarkshell" if you're into that whole Number Theory "Integer" thing)

There are few equations as important in Quantum Mechanics (and therefore Physics in General) as this one from Mathematics. I will present a few bits from its entry in Wikipedia, here, then I'll have a few comments, following:

In mathematics, the Cauchy–Schwarz inequality (also known as the Bunyakovsky inequality, the Schwarz inequality, or the Cauchy–Bunyakovsky–Schwarz inequality), is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in analysis applied to infinite series and integration of products, and in probability theory, applied to variances and covariances. The general formulation of the Heisenberg uncertainty principle is derived using the Cauchy–Schwarz inequality in the Hilbert space of pure quantum states.

The inequality for sums was published by Augustin Cauchy (1821), while the corresponding inequality for integrals was first stated by Viktor Yakovlevich Bunyakovsky (1859) and rediscovered by Hermann Amandus Schwarz (1888) (often misspelled "Schwartz").

The Cauchy–Schwarz inequality states that for all vectors x and y of an inner product space,
where < . , . > is the inner product. Equivalently, by taking the square root of both sides, and referring to the norms of the vectors, the inequality is written as Moreover, the two sides are equal if and only if x and y are linearly dependent (or, in a geometrical sense, they are parallel or one of the vectors is equal to zero).

... OK, I'm back. If that looks like mumbo jumbo to you, then let me be the first to welcome you back to the wonderful world of Mathematics and Physics from the money fields of Finance, Accounting, and Law, and assure you that although you have a lot of catching up to do, this would be a good place to start.

The reason many of us left Maths + Physics in the first place was due the weirdness of Quantum Mechanics, specifically Indeterminacy also known as Heisenberg's Uncertainty Principle. The Cauchy–Schwarz inequality is key to understanding it.

Uncertainty is taught SO badly to undergrads by Physics professors it is mind-boggling. It got me to quit my boyhood love of Math and Physics, and I'm not the only one. It seems unbelievable that such an important and fundamental thing in its field would be taught so badly, I know, but it's very much taught in a most horrible fashion. Compounding the problem is another unbelievability which I feel is best expressed by Lee Smolin's honest comment that half of all PhD's in Physics aren't sure they believe it, and every crackpot scientific theory also slams it and calls it flat out wrong.

But it is true. It's true, experiment backs it up.

Good luck, and may the Schwartz be with you!
A photograph of actor Rick Moranis as Darth Helmet in Mel Brooks' film, "Spaceballs", reacting to his Physics professor's comment "Shut up and calculate!" in response to Darth's probing question re Uncertainty. The Schwartz was not with him that day.

Augustin-Louis Cauchy (1789-1857)

Victor Bunyakovsky (1804-1889)

Hermann Schwarz (1843-1921)

Werner Heisenberg ( 1901-1976)

YIN, YANG, and YONG

To continue the 3 Universes discussion …

Most of us are familiar with the intertwining paisley shapes that describe the Eastern symbol of duality called Yin and Yang (Male and Female), but what if there were three … let’s call them Yin, Yang, and … Yong (I submit).

Could three intertwining Universes describe what we see in the one we inhabit?

Perhaps.

One basic problem (and solution) would be the difference in gravity between all three. Consider if you will that one has a much stronger gravity than ours, and the other much weaker, and further consider this may be a “relative” thing.

More specifically, if we could magically travel to the heavier Gravitic Universe, then our universe, the one we just left, would be the Lighter Universe because in the Heavier Universe, we would be of "normal" gravity there, relatively speaking. Once in the Heavy-verse, the Lighter-verse (in our Normal-verse) would appear heavier.

How could this be! I don’t know, but it would create a tension between all three Universes. The tension would exert itself then as a “pull” in each Universe from its relatively heavier neighbor-verse, around and around.

How this might manifest itself is “matter” in the form of say, a Higgs boson, the particle that Standard Modelers hope will explain mass, since the simpler versions of the Standard Model cannot account for that. The attraction would exert itself along the 5th dimension (4th Spatial dimension) that connects the Universes. Neutrinos would be pulled with the least strength, Black holes the strongest.

There are two established speculative models in Physics that may assist in understanding this, and which I intend to explore in greater Mathematical detail. They are:

- Randall-Sundrum model (5D Gravity Brane theory)
and
- Ekpyrotic Universe

There are no "problems," only ... "challenges."

Remember that for the rest of your life, please.

Lisa Randall of Harvard University

Raman Sundrum of John Hopkins University

 

Paul Steinhardt of  Princeton University

Neil Turok of The Perimeter Institute

SU(4) Spin(6) Identity Analysis

My current mathematical fetish, I'm currently working on this and will present when I'm finished and not before.

But aw, that's no fun! Click here for a serious hint and here for an impish hint.

If it's not obvious, well here's another clue for you all ...

{{familytree/start}}
{{familytree| | | | |TOE| | TOE=Theory of Everything }}
{{familytree| | | | | |!| | }}
{{familytree| |REL|-|^|-|GUT| | REL=[[Gravity]] | GUT=[[Electronuclear force]] ([[Grand Unification Theory|GUT]]) }}
{{familytree| | | | | | | | |!| | }}
{{familytree| | | | |QCD|-|^|-|-|EWT| | QCD=[[Strong interaction|Strong force]]
[[Special unitary group|su(3)]] | EWT=[[Electroweak force]]
[[Special unitary group|su(2)]] x [[Unitary group|u(1)]] }}
{{familytree| | | | | | | | | | | | |!| | }}
{{familytree| | | | | | | | |WNF|-|^|-|EMF||WNF=[[Weak force]]
[[special unitary group|su(2)]]|EMF=[[Electromagnetism]]
[[Unitary group|u(1)]]}}
{{familytree| | | | | | | | | | | | | | | |!| | | | | | }}
{{familytree| | | | | | | | | | | |EF|-|^|-|MF| |EF=[[Electric force]]|MF=[[Magnetic force]] }}
{{familytree/end}}

... the Walrus was Paul.

No!

I'm not trying to solve T.O.E., The "Theory of Everything", partly because such as a thing may not exist, or it does but our species lacks the IQ to understand it, but if it does anything that gets us one step closer in understanding is a good thing, and I intend to give it my best shot. I strongly suggest we work on a G.U.T. before tackling T.O.E., but it never hurts to look a bit ahead, and try hard not thinking about Gravity ... I don't think that's possible. :-)

I told you about strawberry fields
You know the place where nothing is real
Well here's another place you can go
Where everything flows.
Looking through the bent backed tulips
To see how the other half live
Looking through a glass onion.

I told you about the walrus and me-man
You know that we're as close as can be-man
Well here's another clue for you all
The walrus was Paul.
Standing on the cast iron shore-yeah
Lady Madonna trying to make ends meet-yeah
Looking through a glass onion.

I told you about the fool on the hill
I tell you man he living there still
Well here's another place you can be
Listen to me.
Fixing a hole in the ocean
Trying to make a dove-tail joint-yeah
Looking through a glass onion.

... John Lennon (1940-1980), Accidental Physicist

John Lennon, circa early 1970's, exploring the Universe from the ground up

Charles Howard Hinton and the Fourth Spatial Dimension

Sure enough, if you think you have an original idea and search around a bit, you'll find someone thought of it first. (Thanks to Dr. Andrew Thomas of Swansea, Wales for making me aware of this fellow)

In this case, it's Charles Hinton, who from his Wikipedia entry was: a British mathematician who was interested and wrote about higher dimensions, particularly a fourth dimension of space, and is known for coining the word tesseract and for his work on methods of visualizing the geometry of higher dimensions.

Also from that entry:

In an 1880 article entitled "What is the Fourth Dimension?", Hinton suggested that points moving around in three dimensions might be imagined as successive cross-sections of a static four-dimensional arrangement of lines passing through a three-dimensional plane, an idea that anticipated the notion of world lines, and of time as a fourth dimension (although Hinton did not propose this explicitly, and the article was mainly concerned with the possibility of a fourth spatial dimension), in Einstein's theory of relativity. Hinton later introduced a system of colored cubes by the study of which, he claimed, it was possible to learn to visualize four-dimensional space (Casting out the Self, 1904). Rumors subsequently arose that these cubes had driven more than one hopeful person insane.

Hinton created several new words to describe elements in the fourth dimension. According to OED, he first used the word tesseract in 1888 in his book A New Era of Thought. He also invented the words "kata" (from the Greek "down from") and "ana" (from the Greek "up toward") to describe the two opposing fourth-dimensional directions—the 4-D equivalents of left and right, forwards and backwards, and up and down.

Charles H. Hinton  (1853-1907) of Princeton University, University of Minnesota, and US Naval Observatory

Three Universes ?

To continue the previous discussion of 4 Dimensions of Space:

We know that the Universe is constructed with long filaments of galaxies, connected by nodes, and between which lie great voids.

I submit that in the center of those voids, or close to them, the 5th dimension, the fourth dimension of Space, is close to being perfectly symmetrical, a near-perfect in-out dimension, as gravity has little effect far from matter.

As we get closer to the filaments and the massive galaxies, however, we encounter mass and higher levels of gravity. A "tesseract" then gets less symmetrical, it bends.

As we approach the black holes that center most galaxies, any tesseract bends so much that the 4D space becomes parallel to the space in our neighboring Universes, which I speculate are 2 others for a total of 3. Inside black holes' event horizons the 3 Universes merge.

In other words, I speculate that the three Universes, each of which are sandwiched between the other two, merge at black holes which are shared by all three. This may assist in proving 5th dimensionality.

And living where WE live, about 2/3's the way from the Galactic center, we live in a region where yes the Galactic Mass affects us and influences our experimental results, on a cosmological scale.

Why 3 Universes? Why not? The number "3" pops up all the time in Physics.

I am open to the possibility of there being 4 spatial dimensions (and 1 of time), as that would fit nicely with Pyron Theory.

I am equally open to whether or not this idea can be tested or falsified, and how so, if so.

The European Space Agency's Planck and Hershel satellites have achieved the L2 point. Any clue as to when we will have test results?

5D Theory - The Fourth Dimension of Space

We know that we live in, at a minimum, three dimensions of space and one of time, but what about a fifth dimension, to whit, an additional spacial dimension?

Theodor Kaluza in 1919 found that Einstein's General Theory of Relativity worked perfectly in such a 5D Universe. From Wiki:

"In April 1919 Kaluza noticed that when he solved Albert Einstein's equations for general relativity using five dimensions, then James Clark Maxwell's equations for electromagnetism emerged spontaneously. Kaluza wrote to Einstein who, in turn, encouraged him to publish. Kaluza's theory was published in 1921 in a paper, "Zum Unitätsproblem der Physik" with Einstein's support in "Sitzungsberichte Preussische Akademie der Wissenschaften"

The expansion of space, notably Inflation and Dark Energy, would be well-explained if such an extra dimension existed.

Randall & Sundrum's 5D gravity brane theory would also be validated, and a clear explanation of Gravity would be noted. (Read Lisa Randall's wonderful book "Warped Passages" for more on this)

So if such a thing exists, why can't we see this "fifth" dimension?

I submit we do see it, but we're embedded in it as is everything around us, so we don't notice it.

I submit it is a variable as far as the Universe as a whole is concerned, but a constant to each of us personally given the small region we inhabit and our short lifespans.

Consider what a 4th spacial dimension would be, based on what we know of the first three:

1st Spatial Dimension: forward-back

perpendicular to which is:

2nd Spatial Dimension: right-left

perpendicular to which is:

3rd Spatial Dimension: up-down

perpendicular to which is:

4th Spatial Dimension: out-in, but not grow-shrink, as out-in are the directions, and grow-shrink is what happens when we move along them



A tesseract is a hypercube, a shape that consists of a cube within a cube, joined by lines at the corners. This implies expansion and shrinkage.

String theory has gone too far in exploring up to 11 dimensions.

5D is likely more that sufficient to describe our Universe.

Rudy Rucker (1946-    ), author of "The Fourth Dimension" (1984)

Ted Kaluza (1885-1954)