Prospects of the canonical formalism in loop quantum gravity

[marcus]

I thought that the covariant or Spin-Foam approach (which is what I think of Rovelli as being behind these days) as trying to side-step the entire issue. Here, one sets up an obviously well-defined quantum theory, which is motivated but not derived in any sense from GR, and then the difficulty is to show that GR is given in the right limits. Clearly, one of the issues which has only recently become apparent is what that limit actually is --- and it seems to involve the IP. This theoretical structure also has the benefit that one feels free to play around with the basic constructs, and e.g. come up with entirely different intertwiner structures, q-deformations, etc. and just go ahead and compute the outcome and see if it's interesting or relevant.
...

I agree. There is a definite LQG theory. Rovelli lays it out in about one page, defining the hilbertspace and basic operators and the dynamics, and says that's the theory. Period.

If you think of canonical LQG ALSO as having been definitively formulated, then that is another LQG theory. So then Tom is right, there is more than one. More than one is fine! My point was that there is a definite theory. If there are two different ones, so much the better!

Rovelli's is getting quite a lot of research attention currently. There are interesting problems to explore in it. And possibilities for empirical test. So we'll see how it goes.

Genneth, I do want to mention Eugenio Bianchi's new formulation of LQG as the dynamics of topological defects in a manifold. He presents it as a third alternative---to covariant (SF) LQG and canonical LQG. You may have seen his PIRSA talk about it. He gets a lot of questions from Freidel and Smolin and Afshordi and others I can't identify.
I think they may have invited him to Perimeter. (But why would he leave Marseille?)
The formulation is not fully worked out but seems very interesting to me. I don't think it is actually new with Bianchi but he has gone further with it than others have.
You get the video if you search PIRSA with the name Bianchi.

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

Tom, of course there is a lot of good research to be done. There is no reason that canonical Lqg as formulated by Thiemann in say 2012 or 2013 should be logically equivalent to Bianchi's formulation using networks of topological defects in a manifold, or to what Rovelli has formulated in 2011.

These are nice questions to explore! It gives PhD students and postdocs something to work on that is really interesting and could shed light on the subject. Are this and that equivalent?

It is also an advantage. One might make correct empirical predictions and the other might make incorrect ones! and there is the advantage that the tree can grow by sprouting new branches.

so I think it is not a big deal whether this is equivalent to that at some point in history. Nature does not make for us a RULE that all of our human formulations of theories called by the same verbal name should be equivalent at all times

I must say that at the moment I am not so interested in Thiemann's attempt to make a canonical formulation. I am more interested in Bianchi's new formulation. So I would like to hear more about whether it is equivalent or inequivalent to covariant (sf) LQG. AFAICS it would be great if it turned out INequivalent. I suspect that Bianchi's version allows something like knotting. Not sure of this. Is knotting good, or bad? I have no idea! But it would be different.

I am glad that you are interested in the canonical approach. Success may happen there and no in covariant LQG! We need to watch it carefully and I just am not following it so closely at the moment.

 

[atyy]

@atyy: At the same time, I don't think being only an effective theory is a problem. Again, from a condensed matter point of view, one can certain go ahead and quantise non-microscopic degrees of freedom and hope to get something reasonable.

Gravity is already an effective theory. Predictivity is the problem. If LQG is not triangulation independent, then there is a different LQG for each of an infinite number of triangulations, and predictivity is not solved. If LQG cannot solve that, it is not solving anything.

 

[tom.stoer]

@tom: the issue of anomaly cancellation shouldn't be a problem here because anomaly issues are fundamentally a perturbative problem.

The relation with the Atiyah-Singer index theorem shows that anomalies may come up in perturbation theory but are essentially non-perturbative.

In the current Rovellian view he simply defines a manifestly finite theory (on finite graphs) and takes the limit ... as long as the admitted graphs are not too "wild", which probably means that one has to restrict the structure, ...

I think you can't restrict the class of graphs.

I thought that the covariant or Spin-Foam approach ... Here, one sets up an obviously well-defined quantum theory, which is motivated but not derived in any sense from GR, and then the difficulty is to show that GR is given in the right limits.

I agree that the SFs defined on graphs should not suffer from these problems. Anyway - the relation between SF and H should be clarified.

Finally, Fra's persistent issue of observer/system dichotomy is, as always, a good issue and it's unclear how any of the existing ideas deals with or side-steps the issue. ... but a more physical one ... is simply the observation that in usual field theories ...

Rovelli states clearly that these issues are not addressed in LQG.

 

[Haelfix]

"@tom: the issue of anomaly cancellation shouldn't be a problem here because anomaly issues are fundamentally a perturbative problem. "

Anomalies are fundamentally nonperturbative problems!

 

[marcus]

...
@tom: the issue of anomaly cancellation shouldn't be a problem here because anomaly issues are fundamentally a perturbative problem. In the current Rovellian view he simply defines a manifestly finite theory (on finite graphs) and takes the limit. I agree that the existence of this limit is problematic, but as a condensed matter theorist I've seen and done much worse without too much problem, so :shrug:? My intuition says that the limit is probably fine as long as the admitted graphs are not too "wild", which probably means that one has to restrict the structure, which I believe is indeed an open question currently anyway, and actively investigated.
...

I agree with the spirit of your remark. Technically though, isn't the limit people are working on a limit over two-complexes.
Technically there is no need to take a limit over graphs in (covariant) LQG. The graph supplies the boundary state conditions of the experiment. Then one sums over two-complexes (foams) to find the amplitude.

And as you point out, there are various ways to control spin foam dynamics. Indeed one can e.g. restrict the class of foams to be dual to a triangulation---people often do. One can limit how complicated the vertices can be. Also there is the business of a cosmological constant. This seems to control 'bubble divergences'. You may know more about this than I do: q-deformation, quantum group labeling. Basically it bounds spin so one gets rid of the divergence caused by spins running to infinity.

The basic thing is what you suggested---from a condensed matter perspective Atyy's worry (What if the spin foams amplitudes don't converge!) sounds a bit alarmist. It is something that they are on their way to solving---where there's been progress over the past year with inclusion of a cosmological constant. I just wait further developments on the convergence issue.

 

[tom.stoer]

marcus et al., I agree with most of your statements regarding SFs, but the thread is about the canonical formalism in LQG (and possibly its relation with SFs).

 

[atyy]

So does LQC indicate that LQG may actually work? Or does LQC indicate that pure quantum gravity cannot be "background indepedent" (since LQC is not background independent)?

I think it's interesting that Ashtekar thinks, based on LQC, that many apparently plausible LQG Hamiltonian constraints may be ruled out on purely theoretical grounds.

 

[tom.stoer]

LQC is based on a spherical symmetric truncation of classical GR, not of LQG, so it's "first truncate - then quantize" instead of "first quantize - then truncate". Therefore I would say that nearly all problems of canonical LQG goaway simply by truncation.

 

[atyy]

LQC is based on a spherical symmetric truncation of classical GR, not of LQG, so it's "first truncate - then quantize" instead of "first quantize - then truncate". Therefore I would say that nearly all problems of canonical LQG goaway simply by truncation.

Is what you mean by truncation the same as what I mean by background (degrees of freedom are frozen to enforce some symmetry)? BTW, doesn't something like that happen in the black hole entropy calculation too (which makes me wonder whether it is right)?

 

[suprised]

And there is also a definite LQG theory.

Actually from my experience it seems that depending whom you ask from the LQG camp, you get different answers about what approach is right. Check for example, p. 233 of Oriti's review. A whole list of different models is listed there. I really fail to see any definite theory here; maybe we have different notions of what a "theory" is.

 

[tom.stoer]

Is what you mean by truncation the same as what I mean by background (degrees of freedom are frozen to enforce some symmetry)?

I would not call this a background. Let me explain where the difference is: Suppose you restrict the geometry to spherical symmetry + some spherical harmonics. Now you use a FRW universum as a background and quantize the sperical harmonics. This is "background dependent" calculation. This is not what is done in LQC. The simplest approach is to use spherical symmetry w/o any spherical harmonics. This is what I call a truncation of d.o.f. But then the spherical symmetric d.o.f. is subject to the full quantum evolution.

BTW, doesn't something like that happen in the black hole entropy calculation too (which makes me wonder whether it is right)?

No. The LQC BHs are subject to the same truncation (and show a resolution of the singularity), but the LQC BHs are not the objects for which the entropy is calculated. The entropy calculation is done for a so-called "isolated horizon" which is a geometric approach in classical GR. This is a background-dependent approach as one fixes a piece of classical geometry, but there is no such truncation; the rest of the geometry is again subject to full quantum evolution. Fixing the horizon induces surface degrees of freedom on the horizon which are described by a quantum Chern-Simons theory.

The difference is that in LQC you cannot have any entropy at all (entropy is roughly speaking the number of microstates forming one macroscopic state). In the "isolated horizon" approach the macroscopic state is the isolated horizon, the microscopic states are the full Chern-Simons states induced by punctures of spin networks on the horizon. There are indications that the horizon d.o.f. are nothing else but a "holographic map" of full LQG states within the horizon. That means that the BH is nothing else but a huge single intertwiner on which all the edges puncturing the horizon are attached.

 

[marcus]

Actually from my experience it seems that depending whom you ask from the LQG camp, you get different answers about what approach is right. Check for example, p. 233 of Oriti's review. A whole list of different models is listed there. I really fail to see any definite theory here; maybe we have different notions of what a "theory" is.

There was a major shakeup in Loop starting in 2007 which did not really settle into the new formulation until 2009 after Oriti's book was published. Oriti's book is an interesting snapshot out of the past. Much of the revolution involved what people call EPRL-FK. I don't like the alphabet soup flavor of the name, but that is the most recognizable tag. (Engle-Pereira-Rovelli-Livine-Freidel-Krasnov). It concerns 4D spinfoam.

Here is the TOC of Oriti's book about the various QG approaches (roughy as of 2006 judging from the LQG chapters that start on p233.)
http://assets.cambridge.org/97805218/60451/toc/9780521860451_toc.pdf
You can see about 60 pages devoted to String/M and then starting p233 about 100 pages devoted to LQG.

Livine's chapter was submitted 2006. It is about stuff he doesn't work on any more.
Perez' chapter (on spin foam) was January 2006. He was not a central player in EPRL-FK--his research focus seems to have shifted after 2006 and he hasn't published on spin foam since then. Freidel's chapter is about work he did in 2005 on 3D spinfoam (before the big change.)

The 2009 publication date of Oriti's book is something of an anomaly. It just happened that a substantial fraction of the chapters were posted on archive in 2006 just as Loop was about to undergo a revolution. It is curious that anyone would cite the LQG section of the book (that starts on page 233) as indicative of current status.

It's harmless enough AFAICS, just bizarre. Like wearing blinders on purpose, to make yourself see not very well.

 

[atyy]

I would not call this a background. Let me explain where the difference is: Suppose you restrict the geometry to spherical symmetry + some spherical harmonics. Now you use a FRW universum as a background and quantize the sperical harmonics. This is "background dependent" calculation. This is not what is done in LQC. The simplest approach is to use spherical symmetry w/o any spherical harmonics. This is what I call a truncation of d.o.f. But then the spherical symmetric d.o.f. is subject to the full quantum evolution.


No. The LQC BHs are subject to the same truncation (and show a resolution of the singularity), but the LQC BHs are not the objects for which the entropy is calculated. The entropy calculation is done for a so-called "isolated horizon" which is a geometric approach in classical GR. This is a background-dependent approach as one fixes a piece of classical geometry, but there is no such truncation; the rest of the geometry is again subject to full quantum evolution. Fixing the horizon induces surface degrees of freedom on the horizon which are described by a quantum Chern-Simons theory.

The difference is that in LQC you cannot have any entropy at all (entropy is roughly speaking the number of microstates forming one macroscopic state). In the "isolated horizon" approach the macroscopic state is the isolated horizon, the microscopic states are the full Chern-Simons states induced by punctures of spin networks on the horizon. There are indications that the horizon d.o.f. are nothing else but a "holographic map" of full LQG states within the horizon. That means that the BH is nothing else but a huge single intertwiner on which all the edges puncturing the horizon are attached.

OK, I agree with both of your descriptions although it's different language from what I used. Do you think it's a problem for the black hole entropy calculation that one fixes a piece of classical geometry, ie. is it justified from quantum LQG alone or is there an additional assumption?

 

[marcus]

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.
I think everybody realizes that to a large extent LQG = SF. So one should not make statements which contrast the two as if there were a distinction. Most of the LQG research is about an approach that combines SN and SF (graphs bounding a 2-complex, networks bounding a foam.)
The prevailing LQG Hilbert space is built on spin networks (not foams). The dynamics uses spin foams. In the main prevailing LQG approach these things are inseparable.

There certainly are alternatives that people are investigating, as there always should be! One of the most interesting recent exploratory offshoots is the one Eugenio Bianchi is working on (LQG as dynamics of topological defects).
As always, Thomas Thiemann's continued attempts to develop a Hamiltonian approach are extremely interesting, and I think important. I wouldn't say they involve a substantial number of researchers---but they are significant nevertheless.
Nobody is saying that the main line of Loop gravity development is the only one, just that there IS a definite LQG theory. It's conspicuous--you can't miss it. .

marcus et al., I agree with most of your statements regarding SFs, but the thread is about the canonical formalism in LQG (and possibly its relation with SFs).

That's a good point. The thread should be mainly about the continued effort (esp. by Thiemann) on hamiltonian LQG, and how that might possibly relate to the rest of LQG.

If anyone wants to regularize the relation of hamiltonian LQG to the rest of LQG, they should try to avoid distortng the terminology by suggesting that LQG means hamiltonian LQG. That starts the discussion off in confusion.

So let's talk about hamiltonian LQG. Who is working on it? How many papers in the past two years---say 2010 and 2011? How many grad students/post docs? What have they been looking into? Maybe it will help us assess the prospects of ham. Loop to list some recent research.

I see that Livine has posted 20 papers in 2010-2011 and that TWO bear on ham.Loop
This could be significant, he is one of the younger leaders in the field.
A new Hamiltonian for the Topological BF phase with spinor networks
Valentin Bonzom, Etera R. Livine
40 pages
Effective Hamiltonian Constraint from Group Field Theory
Etera R. Livine, Daniele Oriti, James P. Ryan
14 pages
Hopefully someone will take a closer look at how these bear on hamiltonian Loop prospects.

I also think that TWO of Bianca Dittrich's papers (2010-2011) might have a bearing on ham. Loop prospects.
Canonical simplicial gravity
Bianca Dittrich, Philipp A Hoehn
52 pages, 14 figures, 3 tables
Non-commutative flux representation for loop quantum gravity
Aristide Baratin, Bianca Dittrich, Daniele Oriti, Johannes Tambornino
21 pages, 1 figure

It might be interesting to make this inspection for several of the other younger researchers prominent in LQG. What are they doing that is specifically about the hamilitonian version?

 

[tom.stoer]

Do you think it's a problem for the black hole entropy calculation that one fixes a piece of classical geometry, ie. is it justified from quantum LQG alone or is there an additional assumption?

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.

 

[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.