Lorentzian spinfoam model free of IR divergence (Muxin Han)

[marcus]

http://arxiv.org/abs/1012.4216
4-dimensional Spin-foam Model with Quantum Lorentz Group
Muxin Han
22 pages, 3 figures
(Submitted on 19 Dec 2010)
"We study the quantum group deformation of the Lorentzian EPRL spin-foam model. The construction uses the harmonic analysis on the quantum Lorentz group. We show that the quantum group spin-foam model so defined is free of the infrared divergence, thus gives a finite partition function on a fixed triangulation. We expect this quantum group spin-foam model is a spin-foam quantization of discrete gravity with a cosmological constant."

 

[MTd2]

This guy is from the same place of Rovelli, right?

 

[marcus]

One or two people have indicated that they either didn't think the recent reformulation of LQG was Lorentz covariant, or else they didn't think LQG was UV and IR finite. I got some static when I mentioned recently that LQG might now be in or near satisfactory shape for testing.
https://www.physicsforums.com/showthread.php?t=445727

So in case anyone is in doubt. Here is a recent paper about manifest Lorentz covariance.http://arxiv.org/abs/1012.1739
Lorentz covariance of loop quantum gravity
Carlo Rovelli, Simone Speziale
6 pages, 1 figure
(Submitted on 8 Dec 2010)
"The kinematics of loop gravity can be given a manifestly Lorentz-covariant formulation: the conventional SU(2)-spin-network Hilbert space can be mapped to a space K of SL(2,C) functions, where Lorentz covariance is manifest. K can be described in terms of a certain subset of the 'projected' spin networks studied by Livine, Alexandrov and Dupuis. It is formed by SL(2,C) functions completely determined by their restriction on SU(2). These are square-integrable in the SU(2) scalar product, but not in the SL(2,C) one. Thus, SU(2)-spin-network states can be represented by Lorentz-covariant SL(2,C) functions, as two-component photons can be described in the Lorentz-covariant Gupta-Bleuler formalism. As shown by Wolfgang Wieland in a related paper, this manifestly Lorentz-covariant formulation can also be directly obtained from canonical quantization. We show that the spinfoam dynamics of loop quantum gravity is locally SL(2,C)-invariant in the bulk, and yields states that are preciseley in K on the boundary. This clarifies how the SL(2,C) spinfoam formalism yields an SU(2) theory on the boundary. These structures define a tidy Lorentz-covariant formalism for loop gravity."

In any case there are several earlyuniverse phenomenologists that have recognized LQG as in satisfactory shape for testing---like Aurelien Barrau, Wen Zhao, and their assorted co-authors---so as a practical matter it does not seem to be in doubt.

It is the simple-matter version of LQG which now seems to be rapidly being wrapped up into some kind of conclusive formulation. For example the recent layout given in http://arxiv.org/abs/1008.1939

I think that they will continue to tie up loose ends with simple-matter LQG but also it looks like the community is turning collective attention on how to formulate a full-matter LQG version---maybe in incremental steps.

 

[marcus]

This guy is from the same place of Rovelli, right?

Notice that Han has co-authored 4 papers with Thiemann, who is at Erlangen.
http://arxiv.org/find/gr-qc/1/au:+Han_M/0/1/0/all/0/1
Until this year Han was at the Albert Einstein Institute at Golm (outside Berlin) Just this Fall he took a postdoc fellowship at Marseille. So yes he is currently in Rovelli's group as of Fall 2010. It is good to do postdoc at several different centers.

 

[marcus]

I think that they will continue to tie up loose ends with simple-matter LQG but also it looks like the community is turning collective attention on how to formulate a full-matter LQG version---maybe in incremental steps.

This should be clear from the information given here:
https://www.physicsforums.com/showthread.php?p=3043882#post3043882

 

[tom.stoer]

4-dimensional Spin-foam Model with Quantum Lorentz Group
...
We show that the quantum group spin-foam model ... thus gives a finite partition function on a fixed triangulation. We expect this quantum group spin-foam model is a spin-foam quantization of discrete gravity with a cosmological constant

I am confused. This is another model that explicitly uses the cc as input for the quantization. But looking at AS all couplings including the cc are on equal footing and are subject to the renormalization group flow. Why should in SF
- the Newtonian constant be a "standard" coupling?
- the Barbero-Immirzi be a "topological theta angle"?
- the cc ba a deformation parameter of a quantum group
- everything else drop out from the very beginning?

This is not a convincing picture!

 

[marcus]

But looking at AS all couplings including the cc are on equal footing and are subject to the renormalization group flow...

I too find the Wilsonian approach to unification appealing. I won't use the word "convincing" because I suppose that would have something to do with realworld observations.

AS is certainly appealing---maybe convincing in an aesthetic/philosophical sense. It is the way one expects and the way one would like nature to be. One grand Lagrangian with all couplings equally subject to the RG flow.

On the other hand, what I find attractive about LQG (so far just simple-matter LQG) is that it offers---right or wrong---a quantum geometry of the early universe.

It offers---whether right or wrong we will see---something that the e.u. phenomenologists can get their teeth into.

I do not think of LQG as a "quantization" of General Relativity. It is a new theory in its own right---like GR it is a theory of geometry. Inspired by GR and hopefully/apparently reproducing GR in the appropriate limit. But not a ritualistic quantization of it.

 

[tom.stoer]

marcus,

I am so so far away from your thinking; I think the LQG program is appealing as quantum geometry.

But the mathematical method of LQG could be applied to a lot of different terms (operators) and therefore something like the AS approach should be possible within the LQG framework as well. What I don't like is that they treat different terms differently. Perhaps I miss something, but I have the feeling that this is a little bit ad hoc. Currently there is no coherent picture what LQG really IS when going beyond the vacuum Einstein Hilbert sector.

 

[marcus]

...LQG could be applied to a lot of different terms (operators) and therefore something like the AS approach should be possible within the LQG framework...

That would be great! Combine the best of both and have running couplings in Wilsonian LQG. Unfortunately my imagination is not good enough and I can't picture it.

What I don't like is that they treat different terms differently. Perhaps I miss something, but I have the feeling that this is a little bit ad hoc.

It seems very much ad hoc to have the cosmological constant Lambda serve as the deformation parameter of a quantum group. Maybe someone here can point to a justification of it in the literature.

I can't say that I'm happy---to me quantum groups come across as a headache for which there is no aspirin. At least so far I know of no aspirin.

For me, what is of paramount importance is that any QG should offer a geometric model of the early universe. So that even if one cannot prove it right, one can at least prove it wrong. I mean by observation of the primordial light and whatever else we can contrive to detect.
Besides observation of the e.u. I can't think of any other immediately available way to test a QG.

Weinberg has apparently been trying to get an e.u. model with AS but he sounds frustrated.
If he or someone else would succeed, I'd certainly be happy. Then we would have two different geometric models of the e.u. and they could fight it out. AS cosmology versus LQG cosmology.

Though also, as you say, one might have a convergence of AS and LQG---a Wilsonian LQG where all the couplings run: equally subject to the RG.

 

[marcus]

If anyone is not familiar with e.u. phenomenologists' proposed tests of LQG cosmology they could glance at this list of papers by Barrau
http://arxiv.org/find/grp_physics/1/au:+barrau_A/0/1/0/all/0/1
or at this one by Wen Zhao
http://arxiv.org/abs/1007.2396
By now I guess most people at Beyond forum have some awareness of this trend in the literature.

 

[MTd2]

I can't say that I'm happy---to me quantum groups come across as a headache for which there is no aspirin. .

That is a wonderful thing. You`ve got the possibility of implementing fractional statistics. Quantum computation? Possiblity of transforming bosons into fermions?

 

[marcus]

Here is a 38-page paper about the same general topic. This by Fairbairn and Meusberger. It is high quality work---they put a lot into this:

http://arxiv.org/abs/1012.4784
Quantum deformation of two four-dimensional spin foam models
Winston J. Fairbairn, Catherine Meusburger
38 pages, 3 figures
(Submitted on 21 Dec 2010)
"We construct the q-deformed version of two four-dimensional spin foam models, the Euclidean and Lorentzian EPRL model. The q-deformed models are based on the representation theory of two copies of U_q(su(2)) at a root of unity and on the quantum Lorentz group with a real deformation parameter. For both models we give a definition of the quantum EPRL intertwiners, study their convergence and braiding properties and construct an amplitude for the four-simplexes. We find that both of the resulting models are convergent."MTd2 you may be right that quantum groups are good stuff.

 

[tom.stoer]

As said elsewhere this is just a landscape of SF models.

 

[marcus]

You mean the landscape you get by varying the parameter q?

 

[tom.stoer]

You mean the landscape you get by varying the parameter q?

First of all it's not a real landscape as there is no altitude or "potential".

I mean the landscape of G's and possible deformations G_q

It's strange. 3+1 dim spacetime fixes SL(2,C) or SU(2), SO(3,1) as variants which are our G. But once G is fixed and the SF is constructed, the dimension itself disappeares; it can be recovered by something like an emergent dimension defined on the graph. But this dimension (e.g. a spectral dimension in a diffusion process) does not depend on the intertwiner structure but only on the topology of the graph. Therefore at the level of the SF G is no longer fixed via the dimension; G could be everything! In addition one can use q-deformations or whatever.

In addition one can add matter on top of it which (see your other thread) looks similar to an extension of the concept what a vertex is (which rep.s it carries). So one goes from G to G*H where H comes with the matter fields. I do not see how to constrain H.

One go even one step further and add SUGRA in this way ...

 

[marcus]

First of all it's not a real landscape as there is no altitude or "potential".

I mean the landscape of G's and possible deformations G_q

It's strange. 3+1 dim spacetime fixes SL(2,C) or SU(2), SO(3,1) as variants which are our G. But once G is fixed and the SF is constructed, the dimension itself disappeares; it can be recovered by something like an emergent dimension defined on the graph. But this dimension (e.g. a spectral dimension in a diffusion process) does not depend on the intertwiner structure but only on the topology of the graph. Therefore at the level of the SF G is no longer fixed via the dimension; G could be everything! In addition one can use q-deformations or whatever.

In addition one can add matter on top of it which (see your other thread) looks similar to an extension of the concept what a vertex is (which rep.s it carries). So one goes from G to G*H where H comes with the matter fields. I do not see how to constrain H.

One go even one step further and add SUGRA in this way ...

This is unquestionably a fair and insightful criticism of the theory. As you point out, LQG does not, at the level of specifics, seem to be deduced from earlier theory, or from the fact that we experience 3 spatial dimensions.

I would say that it carries over some general principles from Quantum Mechanics and GR. And then the job is to find at least one covariant or general relativistic quantum field theory that recovers GR in the appropriate limit.

If one could find two such theories, one would truly have a landscape problem.

So far it seems to have been a struggle to get even just one, and they are still checking details. The emphasis is on general principles and recovering, rather than on deducing--by a chain of logical syllogisms and conventional mathematical proceedures.

I suppose if one could get such a theory, and it had stabilized in a conclusive form, one could test it observationally. I may be wrong in thinking that we are nearing that stage with LQG (I want to see what Rovelli says concerning that---does he say the theory has reached a conclusive testable stage?)

Today, time permitting, I'm going to read some in a new review article on LQG he just posted on arxiv. It gives valuable perspective----what the (limited) aims of the program are, what has so far been achieved, what surprises have turned up, what is unsatisfactory/missing, what questions remain open. It also gives a compact definition of the theory in 3 equations.

This paper is "December 4707" I use that way to remember the URL so I can refer to it without looking it up. http://arxiv.org/abs/1012.4707. It supersedes April 1780 and October 1939 as the year's main LQG paper. As a review article it has many (156 in fact) references which will make it a convenient doorway to the literature.

My own perspective may change, in reading the review article.

 

[tom.stoer]

I would say that it carries over some general principles from Quantum Mechanics and GR. And then the job is to find at least one covariant or general relativistic quantum field theory that recovers GR in the appropriate limit.

Correct; besides polymere quantization (and taking diff. inv. into account) it does nothing else but quantizing GR in a rather old-fashioned way.

If one could find two such theories, one would truly have a landscape problem.

No; the landscape already exists if there are two consistent but inequivalent theories; then you cannot answer (w/o a further principle) why to chose theory X and not Y. For example you cannot answer the question why SU(2) but not SU(3) SFs.

So far it seems to have been a struggle to get even just one, and they are still checking details. The emphasis is on general principles and recovering, rather than on deducing--by a chain of logical syllogisms and conventional mathematical proceedures.

You can't deduce a quantum theory from a classical one. There may always be quantum structures which are hidden / washed awy / smoothed out / ... in the classical limit and which cannot be constructed rigorously via quantization. A very simple example is the Barbero-Immirzi parameter.

I suppose if one could get such a theory, and it had stabilized in a conclusive form, one could test it observationally. I may be wrong in thinking that we are nearing that stage with LQG (I want to see what Rovelli says concerning that---does he say the theory has reached a conclusive testable stage?)

Not yet; but I think this becomes clear from his "25 years LQG" paper.

 

[atyy]

Does this address the divergence of Eq 26 of Rovelli's simple model?

Or is it limited to addressing the IR divergence? In the simple model paper he explicitly says q-deformation would address the IR divergence, which is what Han's and Fairbairn and Meusberger are addressing. He also says he doesn't know if that would address the Eq 26 divergence.

Also, what does the q-deformation do to Rovelli's summing=refining?

Furthermore, why are they studying the KKL construction also? May I guess the top paper of the year is Rovelli's simple model, as it marks the death of Rovellian LQG. They are clearly going in random directions now. My interest remains in what Rivasseau's school does, as well as with Livine and Oriti (despite the coherent state twistor distraction).

 

[tom.stoer]

@atyy; did I miss something? what do you mean by Eq. 26 divergence? and what is the "death of Rovellian LQG" and his "simple model"?

 

[atyy]

I meant Eq 26 in http://arxiv.org/abs/1010.1939 .

A few paragraphs after Eq 26 he says there are IR divergences and the Eq 26 divergence (which I think of as a "UV" divergence, since summing=refining).

What I dislike about Rovellian LQG is that it is so tied to the EH action. At the very least, if pure gravity is to be quantized, then why not the AS approach of all possible terms consistent with the hypothesized symmetry? In fact, why not some version of AS? My take is that the true worth of LQG is to point to GFTs, and that if GFT is fundamental, then there is no need for a pure gravity GFT, and some form of unification shuold be hoped for. Obviously, I don't have any technical idea how that could be achieved. But my understanding is that GFTs should be different from AS, which is a hope that pure gravity can be quantized.

 

[marcus]

I think what Atyy is talking about is eq (26) in October 1939 which is the same as eq (16) in the recent review paper December 4707.

I don't think it matter that they have not yet shown convergence of the "continuum limit"---also could be described as the "limit of infinite refinement" where you take larger and larger spinfoams with more and more vertices.

Correct me if I'm wrong but I think there are two very different kinds of convergence/divergence issues here. Of primary concern is where you have a fixed graph or fixed 2-complex and the labels can vary. You want to make sure the formulas make sense---the transition amplitudes for example---that they give finite numbers as written.

When for example Barrett et al present a paper on spinfoam assymptotics I think that is what they are addressing. A fixed 2-complex.
The first requirement for a finite theory is that the formulas for that make sense.

Equation (26) or in the recent review paper eq. (16) is where you consider all possible spinfoams with a given spin network as boundary. You start with a simple spinfoam enclosed within a given boundary, and you keep adding vertices and making the spinfoam more and more complicated!

My intuitive sense is that this is likely to converge and it might be an interesting math project to prove it.

But it does not seem like the first order of business---it looks rather secondary. What is more important practically speaking is to take some one spinfoam, possibly very complicated but finite, enclosed by the boundary, and make sure IT's transition amplitude is well defined.

I could have misunderstood---the page to look at in the December review article is page 5. That is where eq. 16 is. Let me know if I'm wrong. Have to go do something else for the moment.

 

[atyy]

I think what Atyy is talking about is eq (26) in October 1939 which is the same as eq (16) in the recent review paper December 4707.

I don't think it matter that they have not yet shown convergence of the "continuum limit"---also could be described as the "limit of infinite refinement" where you take larger and larger spinfoams with more and more vertices.

My take is that the divergence there is very important. It points to the need for GFT renormalization. If GFT renormalization works, I doubt the theory can stay purely gravitational.

Edit: I think some EPRL/FK GFT could be constructed, not entirely sure, but they've gotten things that are close. That may be a pure gravity GFT. The reason I don't think the pure gravity+matter framework can remain in GFT, is that I don't think you can just add "matter fields" to a GFT, since the GFT fields don't have a direct interpretation in spacetime, and so additional fields will show up as spacetime and matter.

 

[marcus]

... I think some EPRL/FK GFT could be constructed, not entirely sure, but they've gotten things that are close.

... I don't think you can just add "matter fields" to a GFT, since the GFT fields don't have a direct interpretation in spacetime, ...

speaking of EPRL-GFT, Rivasseau is expected to be lecturing on that at the 3rd QG school (first two weeks of March) at Zakopane.

You likely saw the program:
http://www.fuw.edu.pl/~kostecki/school3/

It seems to me that including matter in Spinfoam LQG could be a straight shot.
You probably looked at the recent paper "Spinfoam Fermions".
And I suspect anything implemented in Spinfoam can be converted to GFT.

The Loop people seem to have gotten very intererest of late in including matter. And I expect however they eventually turn out to include matter could serve as a guide to including in in GFT.

Does that agree with your take, Atyy?

 

[marcus]

My take is that the divergence there is very important. It points to the need for GFT renormalization...

What divergence? You mentioned eq (26). There is no divergence shown. They just have not shown CONvergence, so far. Maybe they can show it (say in the quantum group case) maybe they can't. You remember what was said about "tempting to suppose" that q-groups would regulate (26).

But your point is interesting and makes me wonder. How do you imagine "GFT renormalization"?

Do you picture it having some analogy with AsymSafe renormalization? The running of couplings with scale?

The limit in eq (26) is not a refinement in the sense of indefinitely shrinking down the scale. It is not a UV limit, as I see it. I don't think spinfoam can get indefinitely smaller and smaller or finer and finer as you add more vertices. There is some smallest non-zero spinlabel below which you can't go.

What you do in that limit is jack up to new 2-complexes which INCLUDE the old 2-complex as a subset. I shouldn't be talking without taking another look at the two papers, though.

 

[tom.stoer]

I don't think that GFT will help since it turns IR divergences into UV ones, but I have never seen that it works the other way round. Rovelli is not consistent here as in other papers he says that the theory is by definition free of UV divergences

I wouldn't care about IR divergence at at all as it disappears as soon as you calculate Dirac observables (which should be defined via finite matter distribution). You calculate an observable X via summing over all nodes where matter sits (which is the physical definition of a volume). If the matter distribution is finite so is X.

My guess is that one has to implement a renormalization group approach a la Kadanoff's block spin transformation http://en.wikipedia.org/wiki/Renormalization_group#Block_spin_renormalization_group i.e. via representing a finite set of nodes (defining a physical volume ones the set is defined via a matter distribution) via one huge single intertwiner plus redefinition of the couplings. I agree that again this will not work w/o matter b/c it sets the scale.

I don't think that Rovelli's approach is restricted to EH action, but that it's more a calculational tool that applies to more complex terms beyond EH as well. My expectattion is that higher order terms will eventually fit into an RG approach as the available couplings are restricted algebraically via the intertwiners. You can have nothing else but a finite number of edges coupled via intertwiners. I don't think that higher order terms can change this at all, therefore they are implicitly there as soon as you allow the most generic SF model.

What I don't like is that the cc is so special in the sense that it's introduced algebraically via q-deformation whereas AS sais that it's an ordinary coupling subject to RG flow.

 

[atyy]

But your point is interesting and makes me wonder. How do you imagine "GFT renormalization"?

Do you picture it having some analogy with AsymSafe renormalization? The running of couplings with scale?

I don't imagine it to be like AS of pure gravity. Pure gravity says that unification does not occur at the quantum gravity scale. This is the hypothesis of Rovellian LQG and AS of pure gravity (AS as stated eg. by Niedemeier and Reuter or Percacci is somewhat more nuanced than that). I dislike LQG because I believe that if pure gravity is the hypothesis, then AS is the research programme.

Since LQG doesn't seem to be AS, if it makes sense, then it must be accidentally a unification programme, like string, ie. the fundamental action is not the EH-like action, but maybe a CFT as in AdS/CFT or a GFT.

GFT is heuristically like QFT, and so must be AS to be fundamental - just like QCD is AS, and so can be fundamental. However, GFT is sufficiently different from QFT that we don't understand how it can be AS. NCFT is another such QFT-like construct for which the existence or not of AS was not understood 10 years ago. AS NCFT is now possible, and Rivasseau hopes to construct similar GFTs. So I am thinking of AS of GFT, so yes AS but not of pure gravity. This is very much the point of view that Rivasseau put forth at the Perimeter AS conference. If I read his latest EPRL/FK GFT papers correctly, a further interesting twist is that maybe classical gravity will emerge as an IR fixed point of an (AS?) GFT.

I don't think that GFT will help since it turns IR divergences into UV ones, but I have never seen that it works the other way round. Rovelli is not consistent here as in other papers he says that the theory is by definition free of UV divergences

I wouldn't care about IR divergence at at all as it disappears as soon as you calculate Dirac observables (which should be defined via finite matter distribution). You calculate an observable X via summing over all nodes where matter sits (which is the physical definition of a volume). If the matter distribution is finite so is X.

My guess is that one has to implement a renormalization group approach a la Kadanoff's block spin transformation http://en.wikipedia.org/wiki/Renormalization_group#Block_spin_renormalization_group i.e. via representing a finite set of nodes (defining a physical volume ones the set is defined via a matter distribution) via one huge single intertwiner plus redefinition of the couplings. I agree that again this will not work w/o matter b/c it sets the scale.

I don't think that Rovelli's approach is restricted to EH action, but that it's more a calculational tool that applies to more complex terms beyond EH as well. My expectattion is that higher order terms will eventually fit into an RG approach as the available couplings are restricted algebraically via the intertwiners. You can have nothing else but a finite number of edges coupled via intertwiners. I don't think that higher order terms can change this at all, therefore they are implicitly there as soon as you allow the most generic SF model.

What I don't like is that the cc is so special in the sense that it's introduced algebraically via q-deformation whereas AS sais that it's an ordinary coupling subject to RG flow.

That's certainly a very coherent outlook. Just to clarify that I don't mean that simply casting it as a GFT will solve it, rather that GFT is divergent and needs renormalization, which is still being worked out. Rovelli does appear to be inconsistent in not calling this a UV divergence, but I think he is technically OK, since the variables are not on a fixed metric background where UV can be defined - I've been saying "UV-like" to fit in with his claim that it is not a UV divergence. I also agree that the technical apparatus of LQG can be used on anything, so when I criticize Rovellian LQG I am criticizing the heuristics in his QG book which emphasizes pure gravity in 4D very strongly, and which doesn't seem to take AS into account at all.

I find your comments on RG of spin foams very interesting - how can this be done - I know only of Freidel et al's and Rivasseau et al's attempts to do that on GFTs and Markopoulou's vrey early and very interesting http://arxiv.org/abs/gr-qc/0203036 , which was followed up by http://arxiv.org/abs/0909.5631 .

 

[MTd2]

ASEQG and LQG will never have UV limit with or without matter. LQG is not the same as GR at the UV, just at low energies. ASEQG is GR to all scales, so if one wants to picture what will happen, LQG behaves like AS up to near Planck scale but not all the way to it.