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Dynamical Chaos and the Volume Gap (Haggard's ILQGS talk) 
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#1
Feb813, 02:53 PM

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PF Gold
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Hal Haggard's ILQGS talk should be quite interesting.
The plan is to have the slides PDF uploaded sometime Monday at http://bohr.physics.berkeley.edu/hal...gard021213.pdf It's potentially helpful to let people look at the slides a day in advance of the Tuesday 12 February talk so they can get used to any unfamiliar ideas. I suppose there could be enhanced questions from people who have had the opportunity to think about the topic ahead of time. It's important to the UV finiteness of LQG that the volume operator has a smallest positive eigenvalue. In other words there is a "gap" in the spectrum of volume between measuring zero volume and the smallest nonzero volume measurement. The question naturally arises whether Nature is actually this way! Do we have some evidencesome indication say from classical physicsthat we are prevented from measuring volume below a certain point? Like in a hydrogen atom you don't have a lower energy orbital, below a certain level. Curiously, there is some indication from classical physics. Consider a pentahedron whose shape is "oscillating" all over the placestretching in this direction, shrinking in that direction, skewing this way and that, but staying the same volume. A bit like a cartoon creature expressing excitementanimated movie artists sometimes draw sequences like that. It turns out, when you construct the phase space and set up a dynamical system for the shapefluctuating pentahedron, that it is chaotic. A small change in initial conditions can result in a drastic change of shapetrajectory. This is somewhat unintuitive, it does not happen with simpler shapes like the tetrahedron. This is classical evidence that volume can be hard to get your hands on. Hard to nail down. It was interesting to see that Berndt Müller, a QFT physicist at Duke, recently got interested in pentahedron chaos and put out a paper. It appears to confirm and elaborate on some earlier results by Haggard. Some references: http://arxiv.org/abs/1211.7311 Pentahedral volume, chaos, and quantum gravity http://arxiv.org/abs/1212.1930 (Müller et al paper) A "Helium Atom" of Space: Dynamical Instability of the Isochoric Pentahedron http://arxiv.org/abs/1208.2228 (Bianchi and Haggard) BohrSommerfeld Quantization of Space 


#2
Feb813, 05:33 PM

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But isn't everything here semiclassical? Haggard's reasoning about the relationship between chaos and quantum levels is based on his reference 17 http://arxiv.org/abs/0906.1960:
"This phenomenological RMT result leaves open the question why chaotic systems behave universally, and that question we want to address in the present paper. Taking a semiclassical approach one can follow Gutzwiller [7] and express the level density as a sum over contributions of classical periodic orbits. The correlation function then turns into a sum over pairs of orbits. Systematic contributions to that double sum are due to pairs of orbits whose actions are suﬃciently close for the associated quantum amplitudes to interfere constructively. The task of ﬁnding correlations in quantum spectra is translated into a classical one, namely to understand the correlations between actions of periodic orbits [8]." Also the Poisson distribution in Fig 4 for a "generic" integrable system does not include the harmonic oscillator. That system is integrable, but does not have a Poisson distribution of energy level spacings. I do agree the harmonic oscillator is not "generic", so his claim is technically correct. However, I wonder how one can tell in quantum gravity whether a system is "generic" or not. I would be surprised if this line of argumentation can overide http://arxiv.org/abs/0706.0469 and http://arxiv.org/abs/0706.0382. 


#3
Feb813, 06:23 PM

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PF Gold
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Pentahedral volume, chaos, and quantum gravity Hal M. Haggard (Submitted on 30 Nov 2012, last revised 17 Jan 2013) We show that chaotic classical dynamics associated to the volume of discrete grains of space leads to quantal spectra that are gapped between zero and nonzero volume. This strengthens the connection between spectral discreteness in the quantum geometry of gravity and tame ultraviolet behavior. We complete a detailed analysis of the geometry of a pentahedron, providing new insights into the volume operator and evidence of classical chaos in the dynamics it generates. These results reveal an unexplored realm of application for chaos in quantum gravity. 8 pages, 5 figures I wouldn't say "everything" is semiclassical, Atyy. I have more to learn about this subject before I can speak confidently, but I'd say that there is a classical dynamics basis herewith phase space and Hamiltonian. And a purely classical chaos. And then of course one wants to use semiclassical arguments to BRIDGE from that and draw plausible conclusions about Quantum Mechanics from what one sees happening classically. I think this stratagem may have proven useful before in other physics situations. But I'm not sure of the specifics. Maybe we will find out more when the ILQGS talk goes online. 


#4
Feb813, 06:54 PM

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Dynamical Chaos and the Volume Gap (Haggard's ILQGS talk)
http://arxiv.org/abs/0706.0382 p32
"We ignore zero eigenvalues in all histograms. (In general there are many more zero eigenvalues than the number of eigenvalues in any histogram bin that we display.)" Haggard's conclusions contradict Brunnemann and Rideout's. 


#6
Feb813, 07:09 PM

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http://arxiv.org/abs/0706.0469 Here we find that the geometry of the underlying vertex characterizes the spectral properties of the volume operator, in particular the presence of a `volume gap' (a smallest nonzero eigenvalue in the spectrum) is found to depend on the vertex embedding. 


#7
Feb813, 07:32 PM

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BTW, marcus, since you're a mathematician, do you know the speculative connection between random matrix eigenvalues that Haggard's using and the Riemann zeros? :)



#8
Feb813, 08:22 PM

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PF Gold
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There are several (classical or quantum) volume operators to choose from. This may come up on Tuesday in discussion at the seminar. We'll see what Hal says, if the subject does come up. We'll see if there is an actual contradiction, or whether it simply comes down to the fact that they are using different operators. Riemann Zeta function is a whole other topic, Atyy 


#9
Feb813, 08:39 PM

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PF Gold
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As background to the upcoming ILQGS talk on Tuesday, I'll quote a bit from this paper:
http://arxiv.org/abs/1208.2228 BohrSommerfeld Quantization of Space ==quote page 2== Both the RovelliSmolin and the AshtekarLewandowski proposals, defined here on the node Hilbert space, admit classical versions: we dequantize the operators E⃗_{l} to obtain vectors A⃗_{l} ∈ R^{3}. This results in two distinct functions on phase space V_{RS} (A⃗_{l}) and V_{AL} (A⃗_{l}). (4) Recently, a third proposal for the volume operator at a node has emerged [17]. Motivated by the geometry of the Minkowski theorem, Bianchi, Dona and Speziale suggest the promotion of the classical volume of the polyhedron associated to {A⃗_{l}} to an operator V_{poly} (A⃗_{l}) → Vˆ_{poly} (E⃗_{l}). (5) The number N of links at the node determines the number of faces of the polyhedron. One advantage of this proposal is that it is closer in structure to the spin foam formulation of the dynamics of loop gravity [18, 16]. In the case of a node with four links, N = 4, all three of these proposals for the volume operator coincide and match the operator introduced by Barbieri [19] for the volume of a quantum tetrahedron. The heart of this paper is a study of the semiclassics of this operator. ==endquote== It's interesting that details of the volume operator are still being worked out and that classical (and semiclassical) analysis can help validate the proposed operators. After dealing with the tetrahedron case, it's natural to proceed to more complicated polyhedra, where differences may show up. If you looked at the Bianchi Haggard paper 1208.2228 you may have noticed the remarkable agreement of the semiclassical Bohr Sommerfeld volume spectrum with that of the three proposed LQG operators. What happens in the next highest case may give a clue as to which proposed operator is to be preferred. 


#10
Feb1113, 05:04 PM

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PF Gold
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I don't see the main ILQGS link anywhere in this thread, so I'll post it:
http://relativity.phys.lsu.edu/ilqgs/ The upcoming talk will, I think, be Haggard's first on ILQGS but as I recall he gave a talk last year at Perimeter and one at the Atlanta meeting of the APS (American Physical Society). There is an interesting direction in his researchit tends to dig up independent evidence that Nature really has a volume gap. Of course this is not easy to do because if there is a minimal measurable volume (of the size indicated by LQG) then it is Planck scale. So direct experimentation might not be able to detect a volume gap. But there are other ingenious ways to get a grip on this issue. I'll go fetch the Perimeter talk to have for reference. WHOA! I just noticed that Hal's slides are posted (for tomorrow's talk): http://bohr.physics.berkeley.edu/hal...gard021213.pdf They look like the basis for a very interesting talk! 


#11
Feb1113, 05:27 PM

Astronomy
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PF Gold
P: 23,090

Here is the Pirsa video of Hal Haggards May 2012 talk. People who want to understand tomorrow's ILQGS online talk should probably watch this one in preparation:
http://pirsa.org/12050084/ Pentahedral Volume, Chaos, and Quantum Gravity Speaker(s): Hal Haggard Abstract: The space of convex polyhedra can be given a dynamical structure. Exploiting this dynamics we have performed a BohrSommerfeld quantization of the volume of a tetrahedral grain of space, which is in excellent agreement with loop gravity. Here we present investigations of the volume of a 5faced convex polyhedron. We give for the first time a constructive method for finding these polyhedra given their face areas and normals to the faces and find an explicit formula for the volume. This results in new information about cylindrical consistency in loop gravity and a couple of surprises about polyhedra. In particular, we are interested in discovering whether the evolution generated by this volume is chaotic or integrable as this will impact the interpretation of the spin network basis in loop gravity. BohrSommerfeld method gives a way to quantize a tetrahedral bit of volume which is completely different from Loop. So it's like getting a "second opinion" on the question of Nature's volume discreteness. They found a remarkable agreement of the volume spectra, not just a gap but across a wide range. This tended to confirm that Loop quantization of geometry was on the right track. So then Haggard went on to study the pentahedron volume. It's sort of like first you study the energy spectrum of the hydrogen atom, and then you go on to study the electron orbitals and energy levels of the helium atom. Or maybe that is the wrong analogy. Maybe the tetrahedron and the pentahedron are like two different "orbitals" of a fuzzy chunk of volume (which can't decide what shape it is, or even how many faces it has.) Anyway that is the May 2012 Pirsa talk. Now let's have a look at the slides PDF for tomorrow's talk. http://bohr.physics.berkeley.edu/hal...gard021213.pdf 


#12
Feb1213, 02:19 PM

Astronomy
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PF Gold
P: 23,090

The audio for Hal Haggard's talk is now online:
http://relativity.phys.lsu.edu/ilqgs/haggard021213.wav If you listen to the talk be sure you have the slides PDF up as well so you can scroll through the slides along with the talk. Links to both are now available here: http://relativity.phys.lsu.edu/ilqgs/ 


#13
Feb1213, 07:41 PM

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Who are the jokers at 10 minutes? Great Snaky last slide! Happy New Year! 


#14
Feb1213, 08:08 PM

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PF Gold
P: 23,090

The person who comes in at 9:30 minutes and says "we are just joining you, can you give us a summary?" is Laurent Freidel. I recognize his voice. I guess that would mean that the Perimeter Institute connection was delayed. Freidel is at PI. I suppose Hal Haggard is speaking from Marseille, and the seminar is being chaired either from Louisiana State or from Penn State. 


#15
Feb1213, 08:28 PM

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Is the questioner at 54 also Laurent Freidel?



#16
Feb1213, 10:08 PM

Astronomy
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PF Gold
P: 23,090

You probably recognize some of the others who are contributing to the discussion: Ashtekar, Rovelli, Lee Smolin, and there is one person whose name I did not catch who may be Thomas Thiemann. I'll listen to the talk again in the next few days and will be able to mention others. Hal did a great job communicating, I thought. The audience were more than usual excited and involved. He organized the material in a way that made it exceptionally understandable and wellmotivated. The quote from Hermann Nicolai at the beginning was effective. The brief sketches of historical context (Einstein, Wigner...) and the reference to string theoretical work (Susskind...) were as well, I thought. The graphics were generally very intuitive. The coverage of the work at Duke University by Müller's group was fascinating. In fact maybe I'll watch it a second time sooner rather than later. 


#17
Feb1213, 11:03 PM

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#18
Feb1313, 09:54 AM

Astronomy
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PF Gold
P: 23,090

What are the Müller group's claims you say you believe? BTW have you looked at the ILQGS blog? After an interesting talk often some other expert in the field will be invited to write a commentary and post it on the blog. As I recall, after Derek Wise's seminar talk there was quite a long blog commentary by Jeff Morton (another former PhD student of John Baez). It can be like a second seminar on the same topic but from a different angle. I'm interested to see who they get to write the ILQGS blog post commenting on Hal's talk. If they do. About "claims", I don't know how one would translate what Haggard and the Duke authors (ColemanSmith and Müller) into "claims" about quantum gravity. Maybe you can translate. To take an example, Bianchi&Haggard worked out the BohrSommerfeld 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 just checked the ILQGS blog: http://ilqgs.blogspot.com The last 3 commentary posts have been by Jeff Morton, Astrid Eichhorn (!) and Edward WilsonEwing (!!). AE was the author of that paper that just came up on Asym Safe Unimodular QG. EWE we've seen a lot of and in particular just recently a paper on matterbounce effect that changes the LQC bounce substantially 


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