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

[marcus]

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/pubs/Talks/ILQGS2013/haggard021213.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 evidence---some indication say from classical physics---that 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 place---stretching in this direction, shrinking in that direction, skewing this way and that, but staying the same volume. A bit like a cartoon creature expressing excitement---animated movie artists sometimes draw sequences like that. It turns out, when you construct the phase space and set up a dynamical system for the shape-fluctuating pentahedron, that it is chaotic.
A small change in initial conditions can result in a drastic change of shape-trajectory. 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)
Bohr-Sommerfeld Quantization of Space

 

[atyy]

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.

 

[marcus]

But isn't everything here semiclassical? ...

http://arxiv.org/abs/1211.7311
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 here---with 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.

 

[atyy]

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.

 

[marcus]

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.

How so?

 

[atyy]

How so?

Sorry, wrong quote. Here's the right one.

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 non-zero eigenvalue in the spectrum) is found to depend on the vertex embedding.

 

[atyy]

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? :)

 

[marcus]

http://arxiv.org/abs/0706.0382...
Haggard's conclusions contradict Brunnemann and Rideout's.

...
http://arxiv.org/abs/0706.0469

I still would not say that Haggard contradicts B&R. After all they are studying different volume operators, and Haggard discusses this at length, describing the difference between the operator Brunnemann and Rideout use and the one he uses (which is classical, and specific to the pentahedron). He suggests an explanation for the difference.

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

 

[marcus]

As background to the upcoming ILQGS talk on Tuesday, I'll quote a bit from this paper:
http://arxiv.org/abs/1208.2228
Bohr-Sommerfeld Quantization of Space
==quote page 2==
Both the Rovelli-Smolin and the Ashtekar-Lewandowski proposals, defined here on the node Hilbert space, admit classical versions: we dequantize the operators E⃗_l to obtain vectors A⃗_l ∈ R³. 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.

 

[marcus]

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 research---it 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/pubs/Talks/ILQGS2013/haggard021213.pdf

They look like the basis for a very interesting talk!

 

[marcus]

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 Bohr-Sommerfeld 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 5-faced 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.

Bohr-Sommerfeld 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/pubs/Talks/ILQGS2013/haggard021213.pdf

 

[marcus]

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/

 

[atyy]

Riemann Zeta function is a whole other topic, Atyy

He still couldn't resist putting it on his 20th slide ...

Who are the jokers at 10 minutes?

Great Snaky last slide! Happy New Year!

 

[marcus]

...
Who are the jokers at 10 minutes?

Sometimes people don't get connected to a conference call right away, and join the call late. It could be due to some technical difficulty.

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.

 

[atyy]

Is the questioner at 54 also Laurent Freidel?

 

[marcus]

Is the questioner at 54 also Laurent Freidel?

If it sounds the same, Atyy. Laurent's voice (and French accent) are distinctive. He comments and asks questions several times in this ILQGS session.

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 well-motivated. 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.

 

[atyy]

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.

Haggard mentions Norbert Bodendorfer, but I'm not sure which he is.

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.

I have to admit I'm skeptical that Haggard's semiclassical calculation can support his claims. I do believe the Muller group's claims - but they seem more modest than Haggard's. I'll agree that Haggard's work does usefully highlight the discussion about the two different volume operators.

 

[marcus]

...I have to admit I'm skeptical that Haggard's semiclassical calculation can support his claims. I do believe the Muller group's claims...

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?

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 (Coleman-Smith and Müller) into "claims" about quantum gravity. Maybe you can translate.

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 just checked the ILQGS blog:
http://ilqgs.blogspot.com
The last 3 commentary posts have been by Jeff Morton, Astrid Eichhorn (!) and Edward Wilson-Ewing (!).
AE was the author of that paper that just came up on Asym Safe Unimodular QG.
EW-E we've seen a lot of and in particular just recently a paper on matter-bounce effect that changes the LQC bounce substantially

 

[marcus]

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:
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 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.

 

[atyy]

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?

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/Preprints/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.