M Theory Lesson 56
Returning to Heisenberg's 1925 paper, we find the basic honeycomb sum rule (on page 3) as the quantum mechanical version of the classical relation
$\omega (n,a) + \omega (n,b) = \omega (n,a+b)$
To a category theorist this classical relation looks like a triangle, where two sides compose to give the third. Moreover, since $n$ is fixed, we may view $\omega(n,-)$ as a functor from a category containing the triangle $(a,b,a+b)$. This source category has addition as composition. There is only one object. In fact, it is an Abelian group in the form of a category on one object. The target category, containing the frequencies as arrows, is also an Abelian group, so the functor $\omega (n,-)$ is just a group homomorphism.
Following Heisenberg, the quantum relation takes the form
$\omega (n, n-a) + \omega (n-a, n-a-b) = \omega (n, n-a-b)$
Setting $a = 0$ we see that $\omega (n,n)$ acts as a (left) identity for frequency composition, replacing the classical $\omega (n,0)$. It is now necessary to consider a variable $n$ in the source category of the classical case, so frequencies are best expressed as 2-arrows between 1-arrow integers. Thus the honeycomb diagrams in the plane are associated with two dimensional categorical structures. Alternatively, and loosely speaking, a set theoretic real number classical physics suggests a categorical complex number quantum physics.
$\omega (n,a) + \omega (n,b) = \omega (n,a+b)$
To a category theorist this classical relation looks like a triangle, where two sides compose to give the third. Moreover, since $n$ is fixed, we may view $\omega(n,-)$ as a functor from a category containing the triangle $(a,b,a+b)$. This source category has addition as composition. There is only one object. In fact, it is an Abelian group in the form of a category on one object. The target category, containing the frequencies as arrows, is also an Abelian group, so the functor $\omega (n,-)$ is just a group homomorphism.
Following Heisenberg, the quantum relation takes the form
$\omega (n, n-a) + \omega (n-a, n-a-b) = \omega (n, n-a-b)$
Setting $a = 0$ we see that $\omega (n,n)$ acts as a (left) identity for frequency composition, replacing the classical $\omega (n,0)$. It is now necessary to consider a variable $n$ in the source category of the classical case, so frequencies are best expressed as 2-arrows between 1-arrow integers. Thus the honeycomb diagrams in the plane are associated with two dimensional categorical structures. Alternatively, and loosely speaking, a set theoretic real number classical physics suggests a categorical complex number quantum physics.
posted by Kea | 8:24 AM
11 Comments:
If translating something from Chinese to Greek means a lot, I would be happy to know why. Honeycombs are very convenient way to think about any kind of quantum mechanics but, they cannot explain why electron on a stable orbit does not emit radiation. This issue is not resolved today as it was not resolved 80 years ago. Last but not the least, the correspondence principle (the saturation conjecture and its solution) is not of much help if we would like to make a comparison between classical and quantum reality since for each quantum reality could be more than one classical and vice versa. Think about Hydrogen atom. Classically it is described by planar Kepler-like configuration. Quantum mechanically this planarity is ignored from the begining. If it would NOT be ignored, we would have the 2d Schrodinger's equation to solve. It does have the same spectrum as 3d BUT it will not allow any chemistry. Chemistry comes from the angular momentum projection absent in 2d problem. Hence, there is no classical analog of the angular momentum projection. Thus, the correspondence principle fails right at the begining of the quantum story. Recall M.Kac problem about hearing of the shape of the drum...
If translating something from Chinese to Greek means a lot, I would be happy to know why.
LOL, anonymous. But I'm afraid we take the idea of a higher categorical formulation for QG (and every component of it, including honeycombs) VERY seriously.
Hi Kea,
1 - I found this blog discussion of MacPherson and Srolovitz from Nature:
Mogadalai Gururajan's blog, imechanica, ‘Going beyond 2D Neumann-Mullins (or, what is popularly known as, solving the beer froth structure)’.
http://imechanica.org/node/1302
RE: L56, L42 and others - honeycomb or hexagon and cubical paths- “von Neumann showed that six sided cells are stable; the grains with sides greater than six sides grow while those with less than six sides shrink.”
2 - RE: Tetractys and Sierpinski's triangle
Look at the MathWorld ‘see also’ [numerous] links for fractal: included is Cantor Set or protofractal, which in turn links to Cantor Function, a “particular case of the Devil’s Staircase”.
http://mathworld.wolfram.com/Fractal.html
NeverEndingBooks, Lieven le Bruyn, ‘devilish symmetries’.
http://www.neverendingbooks.org/?p=314
3 - RE: L55 surreal tree and Witten’s News
Check out some threads of the ‘Good Math, Bad Math’, Mark Chu-Carroll, on “on surreal numbers, surreal arithmetic and the connection with games” linked through NeverEndingBooks..
John H Conway, ’On Numbers and games’ [ONAG], 2ed 2001 is the primary reference.
Conway in chapter 0 briefly reviews number history noting that Cantor began counting with one [clock-face] while von Neumann began counting with zero [classic modulus] where 0={}, 1={0}, 2={0,1}, etc.
4 - To me, this suggests a beginning to the unification of mathematics [Game Theory perhaps a node operator algebra NOA, Borcherds-Kac-Moody as a vertex operator algebra VOA and with other algebras, calculi and geometries] which probably needs to be done before any physics TOE or GUT can occur?
...which probably needs to be done before any physics TOE or GUT can occur?
Hi Doug. Well, physicists really don't care about rigorous mathematics. They only care about matching experimental data, and I'm pretty sure that will start happening long before the details of the maths is sorted out - especially considering how few mathematicians are thinking about these (relatively unpopular) things. Of course, people like Lieven are notable exceptions.
I'm glad you're seeing the Game Theory aspects of this ... but that just isn't my focus, because I'm more interested in, eg., amplitudes for the Large Hadron Collider.
Hi Kea, you know that I am for mathematics as much as for physics. Any mathematical developments related to honeycoms are the most welcome. The issue is what one can do with all this after all. Soon, I am going to put into arxiv yet another physical use of honeycombs. You will see that, when confronting real problems, mathematics can help a lot but, without broader context, can lead totally into wrong direction. (Surely this only concerns physicists willing to spend time studying mathematics. Mathematicians are not confronting reality and, hence, can afford whatever fantasy comes to their minds).
Soon, I am going to put into arxiv yet another physical use of honeycombs.
Fantastic, anonymous! I look forward to it.
...mathematics can help a lot but, without broader context, can lead totally into wrong direction.
Oh, but we (Matti, Carl, Mike, Louise, Me et al) have a broader context. The whole aim of this blog is to investigate it from my personal focus, which is higher operads and topos theory for QG, and it is only in that context that I came across honeycombs.
Theoretical physicist must be able to tolerate schizophrenic splitting into two totally different sub-personalities, which cannot tolerate each other. Mathematician and a person willing to play with simple physical numbers and build simple models. These two modes of thinking about physics are fundamentally different but both necessary.
"... Honeycombs ... cannot explain why electron on a stable orbit does not emit radiation. This issue is not resolved today as it was not resolved 80 years ago. Last but not the least, the correspondence principle (the saturation conjecture and its solution) is not of much help if we would like to make a comparison between classical and quantum reality since for each quantum reality could be more than one classical and vice versa. Think about Hydrogen atom. Classically it is described by planar Kepler-like configuration. Quantum mechanically this planarity is ignored from the begining. If it would NOT be ignored, we would have the 2d Schrodinger's equation to solve. It does have the same spectrum as 3d BUT it will not allow any chemistry. Chemistry comes from the angular momentum projection absent in 2d problem. Hence, there is no classical analog of the angular momentum projection. Thus, the correspondence principle fails right at the begining of the quantum story..." - anonymous
This is totally false: the reason why the electron in orbit around a hydrogen atom "doesn't radiate" is that all the electrons in the universe radiate, so the electron receives as much power from the background Yang-Mills U(1) radiation field as it loses. This prevents it spiralling into the nucleus.
This was proposed by T.H. Boyer in 1975 (Physical Review D, v11, p790) and proved rigorously by H.E. Puthoff in 1987 (Physical Review D v35, p3266, "Ground state of hydrogen as a zero-point-fluctuation-determined state").
See my comment which Kea kindly didn't delete at the post http://kea-monad.blogspot.com/2007/05/wittens-news.html (the comment was for a different post about Mach's principle, which she later separated from the comments form this one).
The reason why orbits aren't planar is partly that spacetime is lumpy with pair-production occurring inside a shell in the strong electric field close to electrons, which causes zig-zagging deflections on small scales. This pair-production is why the orbit of an electron in a hydrogen atom isn't classical.
If you have 2+ electrons in addition to the nucleus, it's a 3+ body problem, resulting in chaotic orbits anyway:
‘... the Heisenberg formulae can be most naturally interpreted as statistical scatter relations [between virtual particles in the quantum foam vacuum and real electrons, etc.], as I proposed [in the 1934 book The Logic of Scientific Discovery]. ... There is, therefore, no reason whatever to accept either Heisenberg’s or Bohr’s subjectivist interpretation ...’
– Sir Karl R. Popper, Objective Knowledge, Oxford University Press, 1979, p. 303.
‘... the ‘inexorable laws of physics’ ... were never really there ... Newton could not predict the behaviour of three balls ... In retrospect we can see that the determinism of pre-quantum physics kept itself from ideological bankruptcy only by keeping the three balls of the pawnbroker apart.’
– Dr Tim Poston and Dr Ian Stewart, ‘Rubber Sheet Physics’ (science article, not science fiction!) in Analog: Science Fiction/Science Fact, Vol. C1, No. 129, Davis Publications, New York, November 1981.
Classical theory is the two-body solution, so it's generally wrong, except for the solar system which is classical because the sun has 99.8% of the mass of the solar system. If you had two planets with equal mass to the sun, orbiting a star with a mass equal to the sum of the planetary masses (the gravitational force equivalent of the Coulomb situation of two electrons each of electric charge e orbiting a nucleus of +2e charge), the result would be chaos and some kind of Schroedinger distribution to describe where you'd be likely to find the planets at any time. The reason why planets don't massively The statistical/probabilistic quantum theory is just a mathematical model for the multibody effects which become extremely important on atomic and subatomic size scales, where stable particles have similar electric charges. I can't understand why Bohr and Einstein didn't grasp this back in 1927!
To make the discussion on honeycombs a bit more lively, I suggest to use honeycombs for explanation of the structure of the Saturn rings. Incidentally, Jupiter, Uranus and Neptune also have rings. Try to use honeycombs in order to figure out what is going on.
Kea
I recently came across a fascinating arxiv paper, in which the authors extend the Freudenthal-Tits construction to fields of characteristic three. Since the construction of the corresponding Lie superalgebras involve simple degree three Jordan algebras, it's likely this leads to a new class of magical supergravities.
I suggest to use honeycombs for explanation of the structure of the Saturn rings.
Great idea, anonymous. This is right up Louise's alley, actually. My first guess would be that the reason Saturn has such prominent rings is linked to the alignment of the magnetic and rotational poles. We've already seen a hexagon at the hot pole of Saturn.
Louise says that little black holes ('dark matter') created and stabilised the rings. For other planets these small holes must mostly have fallen into the central hole, because the misaligned magnetic field sets up a kind of chaotic attractor. But planarity works for Saturn.
Matti might already have calculated DM orbits for Saturn, but not using honeycombs. Cool problem.
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