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Dynamical Chaos and the Volume Gap (Haggard's ILQGS talk) 
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#19
Feb1313, 05:17 PM

Astronomy
Sci Advisor
PF Gold
P: 23,124

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 sufficientpeople 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 gapit'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 colorcoded phasespace maps in the ColemanSmith + Müller paper, here's the link: http://arxiv.org/abs/1212.1930 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 upenlarge that one sliceand discuss that. Whoseever idea that was, Hal, or ColemanSmith, 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 ColemanSmith Müller's Figures 1519, 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 twodigit 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. 


#20
Feb1313, 09:04 PM

Sci Advisor
P: 8,395

Haggard says BGS has been proved. In fact, there are also proofs of some versions of the BerryTabor 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 BerryTabor conjecture. So how can a proved theorem have "counterexamples"? 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 measuretheoretic sense — follow the BerryTabor conjecture, topologically “generic” tori do not." !!! http://www.ams.org/notices/200801/tx080100032p.pdf says that counterexamples are also known to naive statements of BGS "Deviations are also seen in the chaotic case in arithmetic examples.". http://arxiv.org/abs/quantph/0506082v2 says counterexamples to naive BerryTabor 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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