Dynamical Chaos and the Volume Gap (Haggard's ILQGS talk)

by marcus
Tags: chaos, dynamical, haggard, ilqgs, talk, volume
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Feb13-13, 05:17 PM
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The discussion has reminded me of and/or clarified several points

The LQG volume operators have discrete spectrum. Already demonstrated. A discrete set of positive numbers does not have to be bounded away from zero. E.g. {1/n | n=1,2,3....}
So the gap is a separate issue from discreteness.

It seems OK for there to be several volume operators. They agree in certain basic cases and this agreement is sufficient--people should use whichever best suits the application.

Ashtekar observed that the LQG area gap has been proven and ensures finiteness in the applications he's interested in. He seemed to be saying at one point that LQG does not need a volume gap---it's interesting but not to get worked up about.

Classical chaos can have regions of phase space which are unstable surrounding small islands of stability. The maps are visually interesting. Having discrete islands of classical stability amidst chaos seem to correspond to having discrete quantum spectra.

If anyone wants to take another look at the color-coded phase-space maps in the Coleman-Smith + Müller paper, here's the link:

Scroll quickly down to Figures 15, 17, 18, 19 at the VERY END. These are a lot of little SLICES of the phase space. Like thin slices of an exotic sausage that would make you wonder what was in it and lose your appetite.

What Hal Haggard did was to take ONE SLICE FROM EACH of those figures, an interesting central slice that you could sort of explain and see what was going on. And blow it up---enlarge that one slice---and discuss that. Whoseever idea that was, Hal, or Coleman-Smith, or Berndt Müller's, it was a good communication idea.

You get more out of Hal Haggard's slides version, focusing on a small manageable amount of information, than you get out of Coleman-Smith Müller's Figures 15-19, or so I think.

COMPARE those figures with Haggard's slide #29! http://relativity.phys.lsu.edu/ilqgs/haggard021213.pdf
which has the enlarged figures.

We should talk a bit about the two-digit adjacency code for classifying pentahedrons, to make sense of the adjacency "orbitals" that the pentahedron can jump around among by various pachneresque moves. The two digits designate the sides that AREN'T in contact with all the others. Three of the sides are quadrilateral and share an edge with all four others, but two are trilateral, and don't. All this geometry is getting a bit overwhelming, I'm going to take a break.
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Feb13-13, 09:04 PM
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Quote Quote by marcus View Post
You've got me curious, Atyy. What do you think are Haggard's claims?
What are the Müller group's claims you say you believe?
Quote Quote by marcus View Post
To take an example, Bianchi&Haggard worked out the Bohr-Sommerfeld quantization of the tetrahedron and found the volume spectrum. This is an entirely different quantization! As I recall, what they found was an UNCANNY RESEMBLANCE between the semiclassical spectrum and the LQG spectrum. What does that prove? I don't know. What can one claim based on the remarkable similarity of volume spectra using two entirely different approaches? I don't know that logically it proves anything, both spectra could be wrong (not how Nature is.) What it does, I think, is whet one's intuition.
I think the Muller group just claims to be at the "intuitive" level like you say - that's why I buy it, especially since the rigrous calculation has been done. In Haggard's calculation, I don't buy "intuition" - or at least he seems to be claiming more than helping intuition. As an example of what might go wrong - he says the same procedure leads to a Poisson distribution for integrable systems (see his slide 21). However, the harmonic oscillator is integrable, but has a uniform energy level spacing. So intuition fails for the harmonic oscillator.

Haggard says BGS has been proved. In fact, there are also proofs of some versions of the Berry-Tabor conjecture that integrable systems have Poisson distributed level spacings. http://www.maths.bris.ac.uk/~majm/bib/3ecm.pdf states that Sarnak proved a version of the Berry-Tabor conjecture. So how can a proved theorem have "counter-examples"? The idea is that the theorem holds only for "generic" systems, and presumably some condition for the proof to hold rules out the harmonic oscillator as "generic". But at the intuitive level, it is very hard to know ahead of time which particular systems are generic. Markolof's article says "This illustrates the subtlety of the problem: “generic” tori — in the measure-theoretic sense — follow the Berry-Tabor conjecture, topologically “generic” tori do not." !!!

http://www.ams.org/notices/200801/tx080100032p.pdf says that counter-examples are also known to naive statements of BGS "Deviations are also seen in the chaotic case in arithmetic examples.". http://arxiv.org/abs/quant-ph/0506082v2 says counter-examples to naive Berry-Tabor and naive BGS are known and points to Zelditch, 1996. The Zelditch reference is available at http://mathnt.mat.jhu.edu/zelditch/P...preprints.html and on p8 says that niave BGS is false for quantized cat maps, and numerical evidence suggests naive BGS is also false for the Laplacian on arithmetic hyperbolic quotients.

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