Prospects of the canonical formalism in loop quantum gravity

[tom.stoer]

As I said earlier there is a definite LQG theory. It takes about a page to specify and it has become the main focus of LQG research.

marcus, the fact that no proof of equivalence between canonical LQG as SF does exist shows that there may be more than one theory; the fact that there are quantization and regularization ambiguities in Hamiltonian and the fact that we do not understand how they show up in the SF approach demonstrates that there are some fundamental open issues in the theory (in the theories :-). Think about QM with Heisenberg and Schrödinger picture but w/o proof of their equivalence; would you really call it a single theory?

The fact that the majority of publications is about SFs does not prove that canonical LQG is of no relevance. Ignoring problems is not solving problems.

 

[atyy]

It seems to be no problem as the entropy calculation is reasonable and agrees (except for the PI parameter ambiguity) e.g. with string- / M-theory (which has other limitations).

The problem is that we don't know if it's correct to use a classical horizon as long as one cannot prove that full LQG produces this horizon (in low-energy effective theories of QCD you are allowed to you use mesons - not b/c QCD produces meson states - which is very hard to prove mathematically - but b/c we observe meson states in nature; so meson states are justified phenomenologically; this hint is missing in LQG b/c neither does LQG produce a horizon, nor do we observe it experimentally.) It is interesting that already Hawking calculation produces an entropy w/o any QG; therefore entropy in itself is not such a big success. If we want to understand its microscopic origin it may be dangerous to use a classical horizon as input.

I sometimes wonder whether the undetermined IP and so-far unjustified use of the classical horizon means that actually another entropy that obeys an area law is being calculated? Maybe one of these http://arxiv.org/abs/0808.3773 ? OTOH, some of those might be related to BH entropy via AdS/CFT.

I know there has been some re-examination of the LQG BH entropy recently eg. http://arxiv.org/abs/1103.2723 , but I haven't read those. Do they also use a classical horizon assumption?

 

[marcus]

... that there may be more than one theory;

I have always been allowing for that in what I say. If you look back you will see that I have even been discussing some of the alternatives.

would you really call it a single theory?

What a question! I have never said there was just a single theory.
What I am pointing out is a new development. There is a definite LQG. In the main LQG talks and review articles, Rovelli describes the theory in about one page and he says "This is the theory."
He also makes clear that there are alternative lines of investigation and open research problems etc etc.
He stresses that there are all these interesting problems (some about relation to ham. Loop) and there is plenty of research to be done!
The point is that there is a definite clear concise formulation that one sees repeatedly over the past couple of years. And it is the prevailing one used by Loop researchers.

The fact that the majority of publications is about SFs does not prove that canonical LQG is of no relevance.

No, but you hardly need to tell me or anyone else this! I have never said that ham.Loop was of no relevance!
I don't think anyone has, have they?

Personally I like to see alternative lines being explored and it is extremely interesting what equivalences they find or do not find!

So far I have not seen a clear definite formulation of hamiltonian LQG, that Thiemann or anybody sticks to consistently for a couple of years. I would like to see one. Let it be equivalent or not equivalent. Just let it be definite, and testable.

 

[marcus]

The fact that the majority of publications is about SFs ...

That is indeed a fact. The vast majority of LQG research now uses spin network states and spin foam dynamics.

That was one of the points I wanted to make, and also that this formulation is definite and concise. One can say definitively what it is and how to calculate (which I do not see with some of the interesting alternatives.)

Other things you refer to are not my statements. Thiemann's hamiltonian effort is a small minority, but I would never disparage it. I think it is important. Even though it has attracted very few grad students/post docs in the past couple of years.

So when you say it is important, you are preaching to the converted!

I think it would be splendid if Thiemann would arrive at a clear definite formulation of ham-LQG and even better if it turned out to NOT EQUIVALENT and made definite but different predictions. Then one would have two distinct theories to test by observation and one might then EXCLUDE one. Everybody would I think benefit from this.

 

[tom.stoer]

marcus, I think you still don't understand my point. Of course I expect that there is a LQG hamiltonian which is equivalent to a SF PI. I expect that we will learn more from the proof of equivalence (or from the disproof!) than from numerous 36757j-symbol-sorcery.

I love QCD. I was in a similar situation when we tried to quantize it canonically 20 years ago. I was surrounded by people trying to calculate 3-loop integrals in perturbative QCD. They never understood why we tried to renormalize a Hamiltonian b/c they already had their perturbatively 'defined' QCD for decades. I don't now if they succeeded with their 3-loop integrals and if they managed (using some clever tricks, Mathematica, Cray XYZ or something) to go to 4- or 5-loops. But I now for sure that they will never be able to understand confinement, even if they learned how to calculate 42-loop integrals in the meantime.

The problem is that when you don't have phenomenology or experiments you have to be more careful with the maths.

 

[marcus]

marcus, I think you still don't understand my point. Of course I expect that there is a LQG hamiltonian which is equivalent to a SF PI..
.

This may be the exact place our perspectives differ. I understand you to say abstractly THERE MUST EXIST a hamiltonian version which is equivalent to presentday LQG as Rovelli presents it.

But I do not see one on paper.

I take a practical attitude. For me a theory does not exist until it is clearly written down. Sometimes in mathematics one can convince onesself that such and such exists, but cannot find a concrete example. This is not enough. I want to see a ham-Loop theory written down as clearly as ordinary Rovelli-type Loop.

I do not know that presentday majority LQG is right, only has a clear simple concise formulation and seems testable.
My first wish is that another different formulation be clear and definite, able to make predictions whether or not it is equivalent. Simple existence must come first.
I cannot assume the existence of a theory that does not exist yet.

For me, the question of equivalence is secondary to the concrete existence of a theory.
I would like to see Thiemann present a ham-Loop theory and say "This is the theory. I will stand by this!" That would make me happy.

After that one can consider the important and interesting question are they equivalent or not. I would be happy if they were NOT equivalent, of course. We would learn something, and we would have two definite theories. I would also be happy if they WERE equivalent. And then, they could, for example both be wrong and be falsified by observation! Or they might (both) not be falsified. All that is nice.

But it is still just a fantasy until we actually see a definite hamiltonian-LQG.

Maybe we already have one and I just have not heard! Do you have one you can show me?
Has Thiemann arrived at a definite formulation? Something widely acknowledged to be the best hamiltonian? If so please give me the link. I would be delighted to hear about it.
But if such a thing exists, why was it not featured at the Zakopane March 2011 school?
And why was it not showcased at the May 2011 Loops conference? Or was it, and I somehow missed it?

The problem is that when you don't have phenomenology or experiments you have to be more careful with the maths.

I agree with your emphasis on the importance of phenom'y and of empirical observation.
Testing is all-important. Any LQG theory should be testable, or I would hardly call it a theory.
This is why Ashtekar's papers are so important in the overall picture. And those of Julien Grain and Aurelien Barrau and others. They explicitly compare the past CMB data with what LQG tells us to expect, and they look forward to more CMB in future.

And even if you do have phenom'y it makes sense to be careful with the maths.

You probably recall that a substantial fraction of the talks at Loops 2011 were about phenomenology. Loop cosmology and its potential for testing is one of the strengths of the program.

 

[tom.stoer]

You will never see a Hamiltonian of LQG if you don't construct it :-)

Thiemann and others are working on canonical LQG - and regardless whether they succeed or fail, it will be a major step forward! Either b/c we learn how the famous Hamiltonian will look like (and I bet we can derive many useful results from it) or we learn why the construction of H fails (even if SFs are still sound) - or we learn that SFs also fail b/c no well-defined H does exist!

Being careful with the maths just means you have to prove the existence of what you are writing down! Now you write down a Z which is usually constructed via exp(iH). If it turns out that H (or exp iH) does not exist I doubt that you will succeed with your Z.

Currently the difference between constructing H and applying SFs ist just "shut up and calculate". Proving the existence of H (and of a anomaly-free, consistent operator algebra + observables) or proving its non-existence is core for the success of the whole program.

 

[marcus]

Currently the difference between constructing H and applying SFs ist just "shut up and calculate".

A lot of what you just said makes sense to me and is not too different from my original view, but I don't share this attitude towards spinfoam dynamics.
I do not see what you call "j-symbol sorcery" (post #55) or something like that. For that matter, the spinfoam amplitudes can be defined without "j-symbols". One has a choice between Feynman rules or a method attributed to Bianchi that, to me, seems intuitive and appropriate---the way you would like the dynamical evolution of quantum geometry to work.
So I do not see, in the path integral approach, any "shut up and calculate".

 

[tom.stoer]

perhaps 'shut up and calculate' and 'j-symbol sorcery' was a bit crude.

Let's cite Rovelli: http://arxiv.org/PS_cache/arxiv/pdf/1012/1012.4707v4.pdf

"The proper definition of C [Hamiltonian or Wheeler-deWitt constraint] requires a regularization. Several regularizations were studied."

"A second potential difficulty with the hamiltonian approach is the fact that the detailed construction of the Wheeler-deWitt operator is intricate and a bit 'baroque' ..."

"The perception of it as more in the rigorous mathematical style of constructive field theory than in the direct computationally friendly language of theoretical physics may have contributed to growing involvement of a substantial part of the loop community with an alternative method of constructing the theory’s dynamics."

"The kinematics of the canonical theory and the covariant theory ... the dynamics defined in the two versions of the theory ... ... but it is has not yet been possible to clearly derive the relation in the 4d theory ... This is another form of incompleteness of the theory."

 

[marcus]

Recalling a September 2011 PIRSA talk by Bianchi:

There certainly are a lot of open questions to be worked on in QG! The field is in active ferment and going through a creative period of growth.

I want to note that Eugenio Bianchi has promoted a third perspective to stand beside the two main others (abstract SF and canonical).
http://pirsa.org/11090125/

For what seems a long time we have been hearing suggestions about this---but I have the impression always as a side remark or footnote or lowerdimension toy illustration. I never saw it so clearly developed as in Eugenio's talk. So I think of it as his project.

I think there was even a paragraph or two about it in the Zako lectures 1102.3660. But as a side comment: the main line of development there was abstract SF (with abstract SN boundary).

http://pirsa.org/11090125/
Loop Gravity as the Dynamics of Topological Defects
Eugenio Bianchi
A charged particle can detect the presence of a magnetic field confined into a solenoid. The strength of the effect depends only on the phase shift experienced by the particle's wave function, as dictated by the Wilson loop of the Maxwell connection around the solenoid. In this seminar I'll show that Loop Gravity has a structure analogous to the one relevant in the Aharonov-Bohm effect described above: it is a quantum theory of connections with curvature vanishing everywhere, except on a 1d network of topological defects. Loop states measure the flux of the gravitational magnetic field through a defect line. A feature of this reformulation is that the space of states of Loop Gravity can be derived from an ordinary QFT quantization of a classical diffeomorphism-invariant theory defined on a manifold. I'll discuss the role quantum geometry operators play in this picture, and the prospect of formulating the Spin Foam dynamics as the local interaction of topological defects.
21 September 2011

Who knows if this will succeed? Progress is made by branching out and trying new ways.

Now a paper by Freidel et al, http://arxiv.org/abs/1110.4833
==Freidel et al, page 2==
Let us stress that the classical picture of the loop gravity phase space that we develop here is, when quantized, related to the picture first proposed by Bianchi in [8]. In this precursor work, it is argued that the spin network Hilbert space can be identified with the state space of a topological theory on a flat manifold with defects. Our analysis makes the same type of identification at the classical level...


[8] E. Bianchi, Loop quantum gravity à la Aharonov-Bohm, (2009), arXiv:0907.4388 [gr-qc].
==endquote==

I think this paper by Freidel et al is important and it is interesting that what it cites is the paper which Bianchi essentially presented in that PIRSA seminar video I mentioned. Bianchi has only one paper and one seminar talk on this and yet it is the formulation of LQG which the authors choose to work out their equivalence from.

==quote pages 26==
Our approach gives a precise understanding of which set or equivalence class of continuous geometries is represented by the discrete geometrical data (h_e,X_e) on a graph. It provides a classical understanding of the work by Bianchi [8], who showed that the spin network states can be understood as states of a topological field theory living on the complement of the dual graph. It also allows us to reconcile the tension...

==quote page 24==
This means that at the quantum level we can represent the quantization of holonomies and fluxes in terms of operators acting on holonomies of flat connections. This interpretation has already proposed by Bianchi in [8]. It is interesting to note that this is reminiscent of the geometry considered by Hitchin in [25].
==endquote==

Here's the abstract of the Freidel paper:
http://arxiv.org/abs/1110.4833
Continuous formulation of the Loop Quantum Gravity phase space
Laurent Freidel, Marc Geiller, Jonathan Ziprick
(Submitted on 21 Oct 2011)
In this paper, we study the discrete classical phase space of loop gravity, which is expressed in terms of the holonomy-flux variables, and show how it is related to the continuous phase space of general relativity. In particular, we prove an isomorphism between the loop gravity discrete phase space and the symplectic reduction of the continuous phase space with respect to a flatness constraint. This gives for the first time a precise relationship between the continuum and holonomy-flux variables. Our construction shows that the fluxes depend on the three-geometry, but also explicitly on the connection, explaining their non commutativity. It also clearly shows that the flux variables do not label a unique geometry, but rather a class of gauge-equivalent geometries. This allows us to resolve the tension between the loop gravity geometrical interpretation in terms of singular geometry, and the spin foam interpretation in terms of piecewise flat geometry, since we establish that both geometries belong to the same equivalence class. This finally gives us a clear understanding of the relationship between the piecewise flat spin foam geometries and Regge geometries, which are only piecewise-linear flat: While Regge geometry corresponds to metrics whose curvature is concentrated around straight edges, the loop gravity geometry correspond to metrics whose curvature is concentrated around not necessarily straight edges.
27 pages

 

[marcus]

There is a lot in this paper of Freidel Geiller Ziprick. It will probably turn out to be in the handfull of most-cited Loop papers of 2011.
It seems to me that it makes the "prospects of the canonical formalism" look very good. But I am still struggling to understand and cannot be sure. Perhaps you will disagree.

...
Here's the abstract of the Freidel paper:
http://arxiv.org/abs/1110.4833
Continuous formulation of the Loop Quantum Gravity phase space
Laurent Freidel, Marc Geiller, Jonathan Ziprick
(Submitted on 21 Oct 2011)
In this paper, we study the discrete classical phase space of loop gravity, which is expressed in terms of the holonomy-flux variables, and show how it is related to the continuous phase space of general relativity. In particular, we prove an isomorphism between the loop gravity discrete phase space and the symplectic reduction of the continuous phase space with respect to a flatness constraint. This gives for the first time a precise relationship between the continuum and holonomy-flux variables. Our construction shows that the fluxes depend on the three-geometry, but also explicitly on the connection, explaining their non commutativity. It also clearly shows that the flux variables do not label a unique geometry, but rather a class of gauge-equivalent geometries. This allows us to resolve the tension between the loop gravity geometrical interpretation in terms of singular geometry, and the spin foam interpretation in terms of piecewise flat geometry, since we establish that both geometries belong to the same equivalence class. This finally gives us a clear understanding of the relationship between the piecewise flat spin foam geometries and Regge geometries, which are only piecewise-linear flat: While Regge geometry corresponds to metrics whose curvature is concentrated around straight edges, the loop gravity geometry correspond to metrics whose curvature is concentrated around not necessarily straight edges.
27 pages

I looked up the March 2011 Loop workshop in Paris that Geiller and Oriti organized. It was a strong program. This site gives the participants and the 3-day schedule of talks:
http://indico.cern.ch/conferenceDisplay.py?ovw=True&confId=124857
Geiller is at the APC Lab (Laboratoire - AstroParticule & Cosmologie) of University of Paris-7, where the workshop was held.

Geiller gave a talk at Madrid:
http://loops11.iem.csic.es/loops11/index.php?option=com_content&view=article&id=146
A new look at Lorentz-covariant canonical loop quantum gravity.
Marc Geiller
We construct a Lorentz-covariant connection starting from the canonical analysis of the Holst action in which the second class constraints have been solved explictely. We show in a very simple way that this connection is unique, and commutative in the sense of the Poisson bracket. Furthermore, it has the nice property of being gauge-equivalent to a pure su(2)-valued connection, which can be interpreted as a non-time gauge generalization of the Ashtekar-Barbero connection. As a consequence, the Lorentz-covariant formulation of canonical gravity leads to SU(2) loop quantum gravity without imposing the time gauge. Furthermore, we show that the action of the Lorentz-invariant area operator on the connection is diagonal, and therefore leads to the discrete SU(2) spectrum.
[this page links to the SLIDES]

 

[suprised]

And there is also a definite LQG theory. ..
So LQG is both a program and a definite theory...
.

As I said earlier there is a definite LQG theory.

Your information seems to be out of date. LQG is mathematically welldefined. ...
..There is no reason to say that LQG is not well-defined but one can certainly say that it is "not unique". There are several versions!

... now what?

 

[tom.stoer]

... now what?

let's continue here: https://www.physicsforums.com/showthread.php?t=544728

I checked Alexandrov's paper from 2010 especially for the canonical quantization; I think his issues are still 100% relevant, nothing has been fixed since (perhaps I overlooked something in Thiemann's papers; I admit I have to check them more carefully; perhaps there is a new construction where he does not mention Alexandrov and which I do not fully understand) ...

Regarding SFs which suffer from the same problems (secondary second class constraints) I will continue asap.

 

[marcus]

In post #62 Suprised asked a good question "Now what?" I have already given part of my answer in post #61.
==quote==
There is a lot in this paper of Freidel Geiller Ziprick. It will probably turn out to be in the handfull of most-cited Loop papers of 2011.
It seems to me that it makes the "prospects of the canonical formalism" look very good ...

http://arxiv.org/abs/1110.4833
Continuous formulation of the Loop Quantum Gravity phase space
Laurent Freidel, Marc Geiller, Jonathan Ziprick
(Submitted on 21 Oct 2011)
In this paper, we study the discrete classical phase space of loop gravity, which is expressed in terms of the holonomy-flux variables, and show how it is related to the continuous phase space of general relativity. In particular, we prove an isomorphism between the loop gravity discrete phase space and the symplectic reduction of the continuous phase space with respect to a flatness constraint. This gives for the first time a precise relationship between the continuum and holonomy-flux variables. Our construction shows that the fluxes depend on the three-geometry, but also explicitly on the connection, explaining their non commutativity. It also clearly shows that the flux variables do not label a unique geometry, but rather a class of gauge-equivalent geometries. This allows us to resolve the tension between the loop gravity geometrical interpretation in terms of singular geometry, and the spin foam interpretation in terms of piecewise flat geometry, since we establish that both geometries belong to the same equivalence class. This finally gives us a clear understanding of the relationship between the piecewise flat spin foam geometries and Regge geometries, which are only piecewise-linear flat: While Regge geometry corresponds to metrics whose curvature is concentrated around straight edges, the loop gravity geometry correspond to metrics whose curvature is concentrated around not necessarily straight edges.
27 pages​

==endquote==
I think you want to understand what comes next this paper is a good place to start.
I am not sure the stuff at the CERN workshop was representative or relevant to actual LQG except possibly for some general remarks in Nicolai's talk. Nor am I sure that the work of Alexandrov or Thiemann is relevant to where the field is going.
We will, I expect, now see some significant progress in the canonical approach. It will, I expect, proceed by way of this FGZ paper.

It looks to me as if the canonical line of development has been basically stagnant for 5-10 years, while the spinfoam line has made significant advances, especially since 2007. Now it is time for a major advance in the canonical sector.

You can say that what FGZ do in this paper is what should have been done some years ago, to avoid the blockage that we have all seen in the canonical LQG program.

 

[tom.stoer]

I think you want to understand what comes next this paper is a good place to start.

I'll check that.

Nor am I sure that the work of Alexandrov or Thiemann is relevant to where the field is going.

Let's discuss the issues in the canonical formalism in the other thread

 

[marcus]

I think you want to understand what comes next this paper is a good place to start.

I'll check that.

Nor am I sure that the work of Alexandrov or Thiemann is relevant to where the field is going.

Let's discuss the issues in the canonical formalism in the other thread

I'm glad to know you will check out the FGZ paper (Freidel Geiller Ziprick)! It is a deep paper. I was excited to see they cite a result of Alan Weinstein, whom I remember as a graduate student at Berkeley.

I think PROSPECTS is a key word here. If one is going to be forward-looking and think about the reformulation of the canonical version LQG, which has begun and which I expect will be compatible with the Zakopane spinfoam version, then I think one should start with the FGZ paper and try to imagine where it is going.

 

[marcus]

I suppose that the way forward (towards canonical formulation of LQG) does not lie, to take an example, in studying the heroic, if largely frustrated, effort of Thomas Thiemann. This largely solo effort has continued for something like a decade, so one can get in the mental habit of associating the canonical approach with TT. Nor does it lie in studying the persistent criticism by Sergei Alexandrov which has also continued for many years.

I think we should break those habits---we should not get the prospects of canonical formalism confused with a fixed cast of people. The situation is fluid, so which ideas and people are the main players can shift rapidly.

I propose to look at the prospects of canonical formalism in a fresh light, not tying it to a particular agenda, scenario, or cast of characters.

I was really surprised this week by the paper 1110.4833 by Freidel Geiller Ziprick. This was one I did not expect. It seems to open up a way to REDO the Hamiltonian formulation in a way that is both more elegant and more likely to be compatible with the boundary amplitude spinfoam history formulation (e.g. 1102.3660 or more specifically what was presented in 1005.2927, which FGZ cite as their key reference [9].)

 

[atyy]

I suppose that the way forward (towards canonical formulation of LQG) does not lie, to take an example, in studying the heroic, if largely frustrated, effort of Thomas Thiemann.

But maybe it can go somewhere - like Barrett-Crane being the forerunner of EPRL-FK: http://arxiv.org/abs/1005.0817, http://arxiv.org/abs/1110.6150.

Or maybe even the initial regularization is ok http://arxiv.org/abs/1109.1290.

 

[marcus]

But maybe it can go somewhere - like Barrett-Crane being the forerunner of EPRL-FK: http://arxiv.org/abs/1005.0817, http://arxiv.org/abs/1110.6150...

I was just reading 1005.0817, the Alesci Rovelli paper you mentioned, last night! I am puzzling over how a proper canonical formulation might come about. It looks like an important step in the right direction---to at least get the valence right, include the 1-4 Pachner move etc.

I easily get discouraged about canonical prospects, it seems like such a hard problem. Anything new, that changes the game a little bit, can be a source of hope.

Thanks for pointing out the brief summary presentation by Alesci solo: 1110.6150. I did not see anything new in it that was not already in the earlier longer Alesci Rovelli paper. But it is always good to have continued signs of life from an idea.

 

[atyy]

I was just reading 1005.0817, the Alesci Rovelli paper you mentioned, last night! I am puzzling over how a proper canonical formulation might come about. It looks like an important step in the right direction---to at least get the valence right, include the 1-4 Pachner move etc.

I easily get discouraged about canonical prospects, it seems like such a hard problem. Anything new, that changes the game a little bit, can be a source of hope.

Thanks for pointing out the brief summary presentation by Alesci solo: 1110.6150. I did not see anything new in it that was not already in the earlier longer Alesci Rovelli paper. But it is always good to have continued signs of life from an idea.

Me too - and perhaps Alesci himself doesn't know. His other line of investigation http://arxiv.org/abs/1109.1290, using Thiemann's regularization seems to be getting good results with KKL and the canonical formalism - very much what KKL intended.

 

[marcus]

That last one does not interest me so much, at least at the moment. It is very much at the toy model stage.

I hope you will glance at the reference to a Wen problem here, and at the next slide where Bianchi proposes a dual formulation of LQG which does not use spin networks and SF.
I think you are already familiar with this, but let's refresh.
http://pirsa.org/11090125

I am talking about slides 23/24 and 24/24. The penult and last slides of PIRSA 11090125.
In the PDF, so you can go directly to them without watching the video, they are on pages
46/48 and 48/48 of the PDF.

==quote Bianchi's last slide [slightly elucidated :)]==
Summary: Loop Gravity [as the Dynamics of] Topological Defects

* Dual formulation of Loop Gravity:
not in terms of Spin Networks and Spin Foams
[but instead as] local Quantum Field Theory with topological defects

* Derivation of the Loop Gravity functional measure via QFT methods

* New light on the main technical assumptions of Loop gravity
the microscopic d.o.f. of classical and quantum Loop Gravity are
gravitational connections A with distributional magnetic field on defects
==endquote==

See also the earlier slide 6/24 or PDF page 12/48
where he says "Canonical Quantization as above + require also:
[a flatness constraint on the connection in the bulk of the 3-manifold]"

In other words he says that the Canonical Q of HIS version of LQG can be just like the Canonical Q of the OLD version of LQG, if you please, except that his 3-manifold is shot thru with a web of hairline fractures and the connection is required to be trivial except (distributionally) on the defects.

Laurent Schwartz distributions. Takes me back to 1960s grad school days. Happy. Bianchi is a talented mathematician as well as a smart creative physicist. I guess he is postdoc at Perimeter now and might team up with Freidel on some work.

 

[atyy]

==quote Bianchi's last slide==
Summary: Loop Gravity [as the Dynamics of] Topological Defects

* Dual formulation of Loop Gravity:
not in terms of Spin Networks and Spin Foams
[but instead as] local Quantum Field Theory with topological defects

* Derivation of the Loop Gravity functional measure via QFT methods

* New light on the main technical assumptions of Loop gravity
the microscopic d.o.f. of classical and quantum Loop Gravity are
gravitational connections A with distributional magnetic field on defects
==endquote==

What does dual mean here? Does it mean unitarily equivalent? Or is it one of the potentially many inequivalent theories that tom.stoer (and others) bring up?

 

[marcus]

What does dual mean here? Does it mean unitarily equivalent? Or is it one of the potentially many inequivalent theories that tom.stoer (and others) bring up?

I don't know how Eugenio would paraphrase. To me it just means another approach.

Hopefully as different as possible!

Particle theory has become highly ritualized, or so it seems to me.

 

[atyy]

I don't know how Eugenio would paraphrase. To me it just means another approach.

Hopefully as different as possible!

Particle theory has become highly ritualized, or so it seems to me.

Seems to be dual in the sense of unitarily equivalent - at least from Rovellli's commentary on it in the Zakopane lectures http://arxiv.org/abs/1102.3660, p7?

I remember being quite excited about Rovelli's attempt to frame the new spin foams as TQFTs, but remember later thinking it unnatural. Of course I may be wrong. Witten would pick up any successful push in this direction quickly - it's his childhood dream

Edit 1: Link corrected - thanks, marcus.

Edit 2: Oh yes, an indication I'm wrong is that FGZ actually took the time to write a paper about it AND the link between spin foams and canonical LQG. I have no understanding how FGZ fits into the big picture at the moment. I didn't even know the problem the were trying to solve existed!

 

[marcus]

Seems to be dual in the sense of unitarily equivalent - at least from Rovellli's commentary on it in the Zakopane lectures http://arxiv.org/abs/1102.3660, p7?

I remember being quite excited about Rovelli's attempt to frame the new spin foams as TQFTs, but remember later thinking it unnatural. Of course I may be wrong. Witten would pick up any successful push in this direction quickly - it's his childhood dream

I remember some discussion of the "defects" formulation of LQG in the Zakopane lectures
I will go review page 7 of 1102.3660.

===============================

Yes, what you point to on page 7 is such a concise account of the Bianchi "topological defects" presentation of LQG that I'm inclined just to quote the whole thing:

==Zako 1102.3660 page 7==
There is another very interesting way of interpreting the Hilbert space H_Γ, pointed out by Eugenio Bianchi [40]. Consider a Regge geometry in three (euclidean) dimensions. That is, consider a triangulation (or, more in general, a cellular decomposition) of a 3d manifold M, where every cell is flat and curvature, determined by the deficit angles, is concentrated on the bones. Let ∆₁ be one-skeleton of the cellular decomposition, namely the union of all the bones.
Notice that the spin connection of the Regge metric is flat everywhere except on ∆₁. Consider the space M^∗ = M − ∆₁ obtained removing all the bones from M. Let A be the moduli space of the flat connections on M^∗ modulo gauge trasformations.
A moment of reflection will convince the reader that this is precisely the configuration space [SU(2)^L/SU(2)^N] considered above, determined by the graph Γ which is dual to the cellular decomposition. This is the graph obtained by representing each cell by a node and connecting any two nodes by a link if the corresponding cells are adjacent. It is the graph capturing the fundamental group of M^∗.
Therefore the Hilbert space H_Γ is naturally a quantization of a 3d Regge geometry. Since Regge geometries can approximate Riemanian geometries arbitrarily well, this can be seen as a way to capture quantum states of 3d geometries.
The precise relation between these variables and geometry becomes more clear in light of the Ashtekar formulation of GR. Ashtekar has shown that GR can be formulated using the kinematics of an SU(2) YM theory. The canonical variable is an SU(2) connection and the corresponding conjugate momentum is the triad field. Accordingly, we might expect that the quantum derivative operators on the wave functions on H_Γ represent the triad, namely metric information. We’ll see below that this in indeed the case.
A word of caveat: in the Ashtekar formalism, the SU(2) connection is not the spin connection Γ of the triad: it is a linear combination of Γ and the extrinsic curvature. Therefore the momentum conjugate the connection will code information about the metric, while the information about the conjugate variable, namely the extrinsic curvature, is included in the connection itself, or, in the discretization, in the group elements h_l.
==endquote==

 

[marcus]

Edit 2: Oh yes, an indication I'm wrong is that FGZ actually took the time to write a paper about it AND the link between spin foams and canonical LQG. I have no understanding how FGZ fits into the big picture at the moment. I didn't even know the problem the were trying to solve existed!

I am just as in the dark as you or, I suspect, more so. I am very happy to be in a state of suspenseful incomprehension in this case. Something to look forward to: a future ahah.

I also suspect that the FGZ paper has in (perhaps small) part an unspoken cultural or tribal purpose. In my eyes it legitimizes the Aharonov-Bohm version of LQG. It makes it official that this has entered into the worthy brotherhood of versions of LQG.

An advisor can now, if he or she so desires, suggest thesis problems to grad students that they may investigate various aspects of the Aharo-Bo LQG, it is on the "interesting problems" board. If I did not realize it before, I am now awake to the respectability of this version of Loop.

I do not know that it is "unitarily equivalent" or even that it should be. From a genepool evolutionary standpoint it might be better for everybody if theories were slightly different, to increase the chances of success.

 

[atyy]

I am just as in the dark as you or, I suspect, more so. I am very happy to be in a state of suspenseful incomprehension in this case. Something to look forward to: a future ahah.

I also suspect that the FGZ paper has in (perhaps small) part an unspoken cultural or tribal purpose. In my eyes it legitimizes the Aharonov-Bohm version of LQG. It makes it official that this has entered into the worthy brotherhood of versions of LQG.

An advisor can now, if he or she so desires, suggest thesis problems to grad students that they may investigate various aspects of the Aharo-Bo LQG, it is on the "interesting problems" board. If I did not realize it before, I am now awake to the respectability of this version of Loop.

I do not know that it is "unitarily equivalent" or even that it should be. From a genepool evolutionary standpoint it might be better for everybody if theories were slightly different, to increase the chances of success.

I hope the future ahah is as good as the past one :)

My understanding of unitarily inequivalent is clearest in the case of Asymptotic Safety. In that approach, we assume "classical gravity" is ok as a quantum theory, just that one needs to look for a non-trivial fixed point. But which classical variables does one use? If the fixed point exists, it seems that the metric variables and the Holst variables give different fixed points http://arxiv.org/abs/1012.4280, so the quantum versions of the theories are not the same, even though their classical limits are.

 

[tom.stoer]

The simplest model of non-equivalent quantizations is the free particle on a circle. The basis is

\psi_{n,\beta}(x) = e^{i(n+\beta)x}

The boundary condition reads

\psi_{n,\beta} (x+2\pi)= e^{2 \pi i \beta} \psi_{n,\beta}(x)

which shows that beta is some kind of winding (for irrational beta this interpretation becomes difficult). You can move beta from the wave functions to the momentum operator; beta affects the spectrum of certain operators; beta cannot be removed by any unitary transformation.

The operator 'x' does not exist on this space as it violates the boundary condition already for the simplest wave function

\psi_{0,0} = 1

x \psi_{0,0} = x

x \psi_{0,0}|_{x=2\pi} = 2\pi \neq 0

Therefore one has to use

U = e^{ix}

and has to quantize this U, using the Poisson brackets for {U,p}.

So I think there are some simple examples where these inequivalent quantizations show up; there is no preferred choice for beta . There is not even a reasonable explanation why beta should 'scale'.

I think we have to live with the fact that quantization classical theories, e.g. GR may result in inequivalent quantum theories. Then there are two possibilities:
a) different quantum theories reproduce GR in the IR
b) different quantum theories lead to diferent classical theories in the IR
In order to understand that and in order to rule out b) we have to take care about the quantization.

\text{GR} \quad \stackrel{\text{quantization}}{\Longrightarrow} \quad \text{QG 1, 2, ...} \quad \quad \stackrel{\text{classical limit}}{\Longrightarrow} \quad \text{GR, ...}

As far as I understand Rovelli he picks one 'QG x' - but I am sure that in the current state of (L)QG we still have to pay more attention to the first arrow instead of starting with one specific model. I do do say that his model 'QG x' is not reasonable, that it cannot be motivated, that it is not correct physically. But limiting the focus to exactly this model would be the wrong way to proceed.

 

[marcus]

I suppose that the 3 of us who recently posted would have different views of the situation. Maybe it would help if I just indicate my perspective.

I think that we do not know if NOW is the right time to invent the canonical formalism for LQG. It could be! It could also be the right moment to invent the quantization problem that LQG solves, the continuous phase space from which one quantizes.

We have not agreed on a standard classical config or phase space for LQG, so the discussion of equiv or inequivalent quantizations is a bit abstract and academic.

What the FGZ paper is doing is primarily to make precise what is the right picture of the classical phase BEFORE one quantizes.

What I like very much about Freidel's paper is that it disentangles two operations which are truly different and should not be confused! It separates "discretize" from "quantize".
It discretizes classical GR first, before anything quantic happens. This gives a proposed LQG phase space. I think it is right. Better than Regge. Regge limitations show up.

And yet the Freidel et al title is "Continuous formulation of the LQG phase space". That is how close it is to continuous GR. It is quasi continuous blending into discrete. Right on the "cusp". And they keep the map, so that they can go back from classic discrete back to classic continuous.

I am speaking impressionistically and carelessly about very careful math work. So be it.

 

[marcus]

Anyway, the basic situation in Loop now is that one has a definite quantum theory: Zako LQG. It may not be so forever but it is prevailing for the time being.

And one of the things we would like to do is "reverse-quantize" it. Like "reverse-engineering" one might do with a new consumer product out of the box. Zako LQG did not appear as the result of quantizing anything, it just appeared.

Now one would like to go back and figure out what it could be the quantization of.

That will undoubtably be very instructive and will lead to new discoveries!

It will almost surely lead to a Hamiltonian because everybody wants one very badly .
So far the Alesci-Rovelli paper is a first step because it incorporates 4-valent and the 1-4 Pachner. It is a recent hopeful sign. But I would guess that the researchers will first listen to FGZ and other papers like that decide what is the phase space, what is the thing to be quantized, and then they will work out a Hamiltonian from that which involves the Alesci-Rovelli idea. And the eventual canonical theory will be compatible with Zako LQG or whatever path integral formalism it has changed into.

It is a fantasy that the researchers have agreed on something to quantize. So talking about "inequivalent quantizations" is irrelevant.

Zako LQG is very tight. What other alternative has been constructed that is sufficiently like it so one could make a meaningful comparison.

When you tell me by what quantization procedure it could have arisen, from what phase space, then we can see what ambiguities and inequivalent variants we can find. That will be fun! and instructive! But one is not yet at that point. First one must reverse-quantize. As I said, that is where the paper of Freidel Geiller Ziprick comes in.

 

[tom.stoer]

I suppose that the 3 of us who recently posted would have different views of the situation.

At least you (marcus) and me :-)

I think that we do not know if NOW is the right time to invent the canonical formalism for LQG.

As early as possible b/c it shows most clearly if something goes wrong; it does not allow for
any sleight of hand.

We have not agreed on a standard classical config or phase space for LQG, ...

Yes, there are two ambiguities; first the different classical phase spaces (as Alexandrov shows there is a two-parameter family of connections); second the quantization ambiguites itself which again fall into different categories, namely ordering ambiguities and genuine inequivalent quantizations as my S^toy model demonstrates.

This gives a proposed LQG phase space.

One of many, unfortunately ... and each individual choice can lead to different inequivalent quatizations.

 

[marcus]

I've written some stuff responding to some of these issues right before your post, one the previous page. Also Finbar made a related observation in the "What's happening with Loop?" thread, and I just amplified on what he said.

One way to say the essential point is that before we talk about different quantizations we need to establish what is Loop Classical Gravity.

That is a classical form of General Relativity with finite degrees of freedom. (finite d.o.f. so it can be quantized.)

I'm not sure you have a well-defined LCG in mind, as a starting point for quantization. Would you like to describe what LCG is, as you see it? And since it is a matter of consensus what we call LCG, what assurance have we that the Loop community (Marseille, Perimeter, Penn State...) would accept it? They might or might not. I don't know what classical d.o.f. you have in mind.

In case anyone else is interested I will get the links to my previous posts, and Finbar's.

Here's what I said (#80) about "reverse-quantizing" by analogy with "reverse engineering"
https://www.physicsforums.com/showthread.php?p=3588751#post3588751

Here's Finbar's remark, which makes an important point about disentangling the two logically separate processes of discretizing and quantizing
https://www.physicsforums.com/showthread.php?p=3588133#post3588133

Another post of mine (#79) a fragment of which you quoted:
https://www.physicsforums.com/showthread.php?p=3588716#post3588716

 

[tom.stoer]

One way to say the essential point is that before we talk about different quantizations we need to establish what is Loop Classical Gravity.

Of course you are right, but look at this

 

[marcus]

Of course you are right,

Thanks. We both see the need for the Loop community to agree on what is going to be its Loop Classical Gravity

but look at this

However I notice your picture leaves professional consensus out. It is an important element. After all, the picture is not about Nature or about some God-given mathematical absolute. GR is at bottom a human artifact and serves here as heuristic. We do not know that it is right, or how it will be changed as it metamorphoses into a quantum theory.

There are no formal rules to discovering a theory of nature. It is a community function---the self-selecting professional community guides the process by argument and consensus.

So although GR is extremely important, ultimately the community which we call LQG will decide what is the agreed-on classical GR formulation with finite d.o.f. the LCG.

Freidel has made his bid to define it. As I read and reread the FGZ paper, I become persuaded that this LCG will play in Penn State, Perimeter, and Marseille. I confess to being very excited by this and it may be affecting my judgment. I think that a tipping point has been reached.