Lewandowski et al's generalized spinfoams

[atyy]

Why does it "look like space is discrete in EPRL and FK"?

It has nothing to do with spin networks per se. It has to do with the aspects of the semiclassical limit of spin foams shown by Conrady and Freidel, and Barrett et al.

Barrett et al (http://arxiv.org/abs/0907.2440) "In this paper, a particular spin foam amplitude for Lorentzian signature quantum gravity in constructed and studied in the asymptotic regime. It is shown that, in suitable cases, the asymptotics can be expressed in terms of the action of Regge calculus." and Ambjorn et al (http://xxx.lanl.gov/abs/gr-qc/9704079) say "Classical Regge Calculus is a coordinate independent discretization of General Relativity."

Here is the comment by Kaminski et al - again, nothing to do with spin networks per se: "The first difference can be found already at the conceptual level. Loop Quantum Gravity is a quantum theory of gravitational field with all its local degrees of freedom. This is not a discretized theory. The discreteness of LQG is a quantum effect. Of course a role is played here by the choice of the algebra of the kinematical observables labelled by curves and 2-surfaces, but the observables form a complete set of the continuum theory. The SFMs on the other hand, are quantizations of discretized classical theories."

 

[atyy]

BTW, what is the difference in pronunciation between "ń" and "n" in eg. "Wojciech Kamiński, Marcin Kisielowski, Jerzy Lewandowski"

 

[marcus]

It has nothing to do with spin networks per se. It has to do with the aspects of the semiclassical limit of spin foams shown by Conrady and Freidel, and Barrett et al.

...Ambjorn et al (http://xxx.lanl.gov/abs/gr-qc/9704079) say "Classical Regge Calculus is a coordinate independent discretization of General Relativity."
...

I believe now I understand better how you are thinking. In what sense do you think that Classical Regge Calculus is a "discretization"? What is meant by that? Does that mean that according to Tullio Regge the space/spacetime consists of little lumps? I think for Tullio Regge, space was a smooth continuum. A smooth differential manifold. Maybe I am wrong. Do you happen to know?

 

[atyy]

Now I understand better. In what sense do you think that Classical Regge Calculus is a "discretization"? What is meant by that? Does that mean that according to Tullio Regge the space or the spacetime consists of little lumps? I think for Tullio Regge, space was a smooth continuum. A smooth differential manifold. Maybe I am wrong. Do you happen to know?

Not off the top of my head - I too thought Regge calculus was smooth until I saw Kaminski et al's comments about LQG being continuous but spin foams being discrete, and that the aim of their paper was to take the continuum limit of spin foams. And then came across the Ambjorn paper on the Regge action. I'll have to study Regge more carefully.

But my main question is actually do Kaminski et al discuss if the nice bits of EPRL and FK shown by Conrady and Freidel and Barrett et al survive the continuum limit. Conrady and Freidel were uncertain at the time of http://arxiv.org/abs/0809.2280: "One might want to take a continuum limit, where the number of boundary vertices of the spin network grows. It is not clear if such a limit commutes with the semiclassical limit taken here."

 

[marcus]

Not off the top of my head - I too thought Regge calculus was smooth until I saw Kaminski et al's comments about LQG being continuous but spin foams being discrete, and that the aim of their paper was to take the continuum limit of spin foams. And then came across the Ambjorn paper on the Regge action. I'll have to study Regge more carefully.

Words are tricky. I wouldn't worry. Kaminski are completing the picture, deepening the joint between LQG and EPRL and they had to say something to highlight the usefulness of what they were doing. You may have too simple an idea of what is meant by the word "discrete". It doesn't always mean the same thing. Regge discretizes in order to calculate but he does not pretend that space is made of dots. The dots and triangles are only provisional. He is in accord with Lorentz. His calculation method is discrete but his idea of space is not discrete.

But my main question is actually do Kaminski et al discuss if the nice bits of EPRL and FK shown by Conrady and Freidel and Barrett et al survive the continuum limit. Conrady and Freidel were uncertain at the time of http://arxiv.org/abs/0809.2280: "One might want to take a continuum limit, where the number of boundary vertices of the spin network grows. It is not clear if such a limit commutes with the semiclassical limit taken here."

I see, so we are not worrying so much about discrete space and Lorentz now? It sounds from your quote like work in progress. There is always more to prove. I am not sure what nice bits of EPRL you are talking about. Kaminski et al are only talking about EPRL, not FK. So let's pick out a nice bit of EPRL and try to see!

Most likely some bits will have been already shown to survive in limit. And some other bits will not yet have been shown. The whole thing is gradual and incremental. It is a lot of work and requires a certain amount of patience from the onlookers. But my impression is it's going along fine. In any case pick a nice bit, and we can try to check to see.

 

[atyy]

Classical Regge is a second order approximation to GR, but it does converge to GR in the continuum limit (http://arxiv.org/abs/gr-qc/0408006). So for any finite size triangulation, classical Regge will deviate from GR in some respects - what I don't know is whether Lorentz invariance specifically is broken - but naively I expect it to.

Here is a nice bit of EPRL from Barrett et al (http://arxiv.org/abs/0907.2440): "It is shown that for boundary data corresponding to a Lorentzian simplex, the asymptotic formula has two terms, with phase plus or minus the Lorentzian signature Regge action for the 4-simplex geometry, multiplied by an Immirzi parameter." Will this survive (or in Conrady and Freidel's language "commute with") the continuum limit?

 

[MTd2]

Marcus, does what I said above make sense?

 

[marcus]

MTd2 I understand that you are asking about your post 30 which is replying to Atyy post 26. I am not sure I understand the exchange well enough to comment. What you say sounds reasonable but I can't say anything decisive. I'd have to shift gears and go back and study those posts.

Right now I am trying to understand what Atyy is saying. Atyy I begin dimly to get a sense of what you are saying.

The issue for me is whether Lewandowski's embedded spinfoam reform will be adopted. My original attitude when I came into the discussion was not well thought out.

I assumed that Lewandowski's critique of earlier spinfoam was valid and that this was automatically an improvement and I took it for granted that everything carries over and this is what spinfoams really are now. It was an improvement that would automatically be adopted.

But now I see more work that needs to be done, and inconsistencies that have to be worked out. And I am even wondering if, for example, Rovelli will accept the embedded spinfoam that Lewandowski is offering.

It bothers me that the paper seems to treat the Euclidean case, in section IV, and then it goes immediately to section V conclusions. It seems incomplete in that regard.

Now you have got me worried about all these many issues, and I'm getting tired. I think I need to sleep on this and take a fresh look in the morning.

I see that the way things are scheduled at the Corfu QG School, on Saturday 19 September Rovelli will give and hour talk, and then after lunch and a discussion period Lewandowski will give a 45 minute talk about exactly this paper's topic "Spin foam from a LQG point of view". And I guess there will be quite a lot of questions asked and discussion of this. The management says they are going to put all the talks on line. I hope very much that happens. I want to hear the questions to Lewandowski and his responses. Here is the schedule with the titles of the various talks:
http://www.physics.ntua.gr/corfu2009/Program/3rdWeekSchool.html

 

[marcus]

BTW, what is the difference in pronunciation between "ń" and "n" in eg. "Wojciech Kamiński, Marcin Kisielowski, Jerzy Lewandowski"

I read that the Polish ń is like the spanish tilda n, the "enyay" ñ. The normal n is slightly "padded" by the tongue to the roof of the mouth.

If it's followed by a consonant, the effect is quite subtle. Here are some samples:
http://www.graspr.com/videos/Polish-Lessons-Unit-27-1

 

[atyy]

Classical Regge calc does not reproduce eg. red shifts in the exact Schwarzschild solution. However, there is an interesting comment from Sorkin's 1975 work on classical Regge calc that although the continuum limit for classical Regge calc plus electromagnetic field is wrong in many cases, some sort of average will make it correct. So I suppose the argument that EPRL and FK spin foams do not necessarily violate Lorentz invariance although the semiclassical limit of a vertex is Regge-like, is that somehow the summation when the path integral is performed will give the classical limit as Einstein gravity rather than the classical Regge calc approximation. Of course, I don't know if this will work, but that is good enough heuristic for me at the moment to reserve judgement.

Edit: So this is what Henson (http://arxiv.org/abs/0901.4009) discussed, which I didn't understand when I first read it. It is uncontested that classical Regge calculus is Lorentz violating at some level. "However, in some approaches it is claimed that the problem can be overcome by a quantum sum over many triangulations. The program to which this issue is of most relevance is spinfoams, where fundamental “Planck-scale” discreteness is sometimes claimed."

 

[tom.stoer]

I think ideas like "spin-networks = quantized space" or "spin-foam = quantized spacetime" is totally missleadig. There is a quantum object called spin-network from which (in a certain limit) space can be reconstructed.

On the level of spin-networks Lorentz invariance is a property of an operator algebra. As long as this operator algebra has no anomalies Lorentz invariance is correctly implemented on the quantum level.

Whether it is broken in a semiclassical limit is something totally different. A certain planetary orbit violates full SO(3) symmetry, but the set of all possible planetary orbits respects this symmetry.

 

[atyy]

Whether it is broken in a semiclassical limit is something totally different. A certain planetary orbit violates full SO(3) symmetry, but the set of all possible planetary orbits respects this symmetry.

That's the point. The semiclassical limit is related to the Regge action (http://arxiv.org/abs/0907.2440). The Regge action is only an approximation to general relativity, so although the theory may have formal Lorentz invariance, in what we are calling observational tests of Lorentz invariance, why wouldn't there be deviations from the predictions of general relativity?

From what I understand, although the semiclassical limit of a vertex is related to the Regge action, maybe the sum in the path integral may have a semiclassical limit which is general relativity, rather than an approximiation to it.

 

[tom.stoer]

... "A priori, a spin foam model of gravity need not be related to canonical loop quantum gravity (LQG). That is, a given model could be a viable quantization of gravity, and nevertheless do not have the kinematical boundary variables of canonical LQG.

Sure, in principle this is right.

Nevertheless a path integral (action + measure) in ordinary q.m. is a derived object. If the fundamental Hamiltonian is not known (as it is the case in LQG due to quantization ambiguities) you cannot derive a path integral.

Unfortunately the other way round does not work, either. You have a class of path integrals / spin-foams but you are neither able to derive the (a) LQG Hamiltonian, nor do you have something like a selection principle for uniqueness.

So there is a missing link between these two approaches. Either it will be established (it should work in both directions), then we should be able to select the correct Hamiltonian or spin foam, or it will be shown that due to some reason this link will not work or willbe replaced by something different, then we will certainly learn a lot from this failure.

 

[atyy]

Sure, in principle this is right.

Nevertheless a path integral (action + measure) in ordinary q.m. is a derived object. If the fundamental Hamiltonian is not known (as it is the case in LQG due to quantization ambiguities) you cannot derive a path integral.

Unfortunately the other way round does not work, either. You have a class of path integrals / spin-foams but you are neither able to derive the (a) LQG Hamiltonian, nor do you have something like a selection principle for uniqueness.

So there is a missing link between these two approaches. Either it will be established (it should work in both directions), then we should be able to select the correct Hamiltonian or spin foam, or it will be shown that due to some reason this link will not work or willbe replaced by something different, then we will certainly learn a lot from this failure.

How about if you start from the path integral, and hope the problem of non-uniqueness will be solved by something like Asymptotic Safety with a finite dimensional critical manifold?

 

[MTd2]

why wouldn't there be deviations from the predictions of general relativity?

There are huge deviations from general relativity, depending in what dark matter ends up being. Dark matter may well be a gravitational feature absent from general relativy, for example, like the emerging scalar fields that cannot get rid of in Horava Gravity.

Horava Gravity in the classical limit does not behave like General relativity mainly because it has
a scalar field that does not go away. It turns out that people notice that this has a behavior very similar, almost exactly to, dark matter. So, in this case, General Relativity is missing the 2nd most important feature of the universe.

 

[marcus]

I'm a bit reluctant to interrupt the Tom-Atyy dialog because you two are tuned to each others' ideas enough to have a real discussion and it's interesting to try to follow. So feel free just to skip over this post and continue (don't get sidetracked!).
Mainly I just wanted to say that Jerzy Lewandowski will be talking exactly on this topic at the ILQGS on October 20
http://relativity.phys.lsu.edu/ilqgs/schedulefa09.html
These telephone seminars linking Perimeter Marseille Warsaw and PennState are often very interesting because you get real experts firing real questions, interrupting the speaker, sometimes commenting at length, and so on. It's likely that when Jerzy L gives his talk there will be probing questions from Rovelli, Ashtekar, and Laurent Freidel. Just judging from past experience with the ILQGS.
The title of the October 20 talk is
Spin foams from loop quantum gravity perspective
The audio and the slides PDF both stay on line so you can catch them later, or listen a second time.

The title of the November 17 talk is Asymptotics of the new vertex. The speaker is someone who used to post quite a lot here at Beyond forum, some years back. He has coauthored some with John Barrett now, and the general topic of the new spinfoam vertex is relevant.
If you go to the main page
http://relativity.phys.lsu.edu/ilqgs/
you will see slides and audio for past seminars like
Apr 21st--John Barrett--Asymptotics of spin foam models with an Immirzi parameter
(the problem being that the field does not hold still and may have moved on from where it was on April 21, hopefully the November 17 talk will bring us up to date)

 

[marcus]

My general thoughts on what is being discussed are that people are currently making measurements of what emanates from extreme events like gravitational collapse and the start of expansion. We've been doing this for some time already.

We must want to infer from these measurements what happens to geometry and matter at the bottom of a black hole.

We must want to infer what geometry and matter were doing around the time of the bang.

It is primarily a problem of relating measurements. We have all this data to fit together into an understanding. The purpose of QG is to get a pragmatic grasp of the behavior of geometry and matter under extreme conditions so that we can talk reasonably about the observations already made (and predict future ones).

That means terms like Lorentz and non-Lorentz need to have an operational meaning, to mean anything at all. You need to be able to describe an experiment/observation which is at least in principle do-able. So far at least from Loop/foam nobody succeeded in deriving a prediction of some non-Lorentz outcome of some practical experiment/observation. They might in the future, and it could go either way. But I don't think we can see that well into the future just by arguing from first principles or based on what someone who may be on the periphery happens to have said about it. Ultimately I trust no one's judgment about the future of research, though I'm more apt to pay attention to what core people say and do.

Bee Hossenfelder has an interesting blog entry today 12 September on the minimal length idea--relating to the Lorentz issue:
http://backreaction.blogspot.com/2009/09/minimal-length-in-quantum-gravity.html

The central question for me is not Lorentz invariance but can one of these spinfoam models describe the big bang? Can it describe inflation? Does it have a place in it for dark energy? Can you get numbers out? Does it know about gravitational waves and did g-waves leave an imprint on the microwave background? Or is that too ridiculous to hope? I may be being dense or naive, but that's how I personally approach it. Reuter AsymSafe doesn't have a bounce yet. I'd like a spinfoam that describes bounce AND has running constants to take care of inflation.
Maybe I want this by christmas of next year.

 

[tom.stoer]

How about if you start from the path integral, and hope the problem of non-uniqueness will be solved by something like Asymptotic Safety with a finite dimensional critical manifold?

This is a totally different issue.

LQG is well-behaved in the UV. You do not add extra terms in the action, so you do not have to restrict their behaviour or the coupling constants. The non-uniqueness I am talking about is essentially due to the construction of the Hamiltopnian (starting with the Einstein-Cartan action): epsilon-regularization, operator-ordering. They are related to the interaction vertex in the spin-foam approach.

 

[atyy]

This is a totally different issue.

LQG is well-behaved in the UV. You do not add extra terms in the action, so you do not have to restrict their behaviour or the coupling constants. The non-uniqueness I am talking about is essentially due to the construction of the Hamiltopnian (starting with the Einstein-Cartan action): epsilon-regularization, operator-ordering. They are related to the interaction vertex in the spin-foam approach.

Let me see if I'm getting you right.

-You are concerned with LQG. There is non-uniqueness in the Hamiltonian constraint, but if only one spin foam matches LQG, one can use that to select the Hamiltonian constraint.

-On the other hand, I am not concerned with LQG, just spin foams. A priori there is no reason to think spin foams are connected with Asymptotic Safety because spin foams actions give exactly the Einstein equations, whereas Asymptotic Safety adds to the Einstein-Hilbert action all possible terms consistent with diffeomorphism invariance.

If that's what you're saying, I agree. My intuition for why Asymptotic Safety might come in is by analogy with CDT which uses the Einstein-Hilbert action with no extra terms, but is nonetheless suspected to be linked to Asymptotic Safety.

 

[tom.stoer]

There is non-uniqueness in the Hamiltonian constraint, but if only one spin foam matches LQG, one can use that to select the Hamiltonian constraint.

Yes - hopefully! And vive versa - hopefully!

... because spin foams actions give exactly the Einstein equations, whereas Asymptotic Safety adds to the Einstein-Hilbert action all possible terms consistent with diffeomorphism invariance.

Yes!

So we are essentially talking about two different issues of (quantization) ambiguities.
1) If we start with one fixed action (e.g. Einstein-Cartan due to fermions) we face conceptual difficulties due to quantization ambiguities. We expect these ambiguities to be related to ambiguities in the path integral approach.
2) But there is an additional ambiguity on the level of the classical action; if the classical action is only the lowest order term there is no principle which selects the higher order terms or fixes their constants. This is were asymptotic gravity might help.

If we would try to construct LQG from a more general action with higher order terms we expect the ambiguities of class (1) to arise on top of the ambiguities of class (2).

I studied some papers regarding non-commutative geometry and found that there seems to be a relation between NCG, BPHZ-renormalization / dim.-reg. and Riemann-Hilbert correspondence. Connes (http://www.alainconnes.org/docs/einsymp.pdf) states that the BPHZ-procedure for Feynman graphs is nothing else but the Birkhoff decomposition for a Hopf algebra.

The main lesson one learns from the above developments is that one should not consider the divergences of QFT as unwanted nuisances but rather as the signature of subtle symmetries of Galois type which prevent one from making simple predictions unless they are carefully taken into account. It also shows that it is worthwhile to give a precise geometric support to the dimensional regularization and to understand in a
more geometric manner the universal behavior of counterterms as dictated by (2). As
we shall briefly explain below this can be done within the new framework provided by
noncommutative geometry.

I would like to speculate that these results could (must ?) be generalized to include dim.-reg. for gravity as well (I mean as far as I understand the NCG framework, gravity cannot be treated seperately).

 

[tom.stoer]

There are several incompatibilities between Loop Quantum Gravity (LQG) and spin-foam models (SFMs).
The first difference ... LQG is aquantum theory of gravitational field with all its local degrees of freedom. This is not a discretized theory. ... The SFMs on the other hand, are quantizations of discretized classical theories.
Secondly, LQG is a diffeomorphism invariant theory of fields on manifolds. ... The SFMs, on the other hand, are theories of piecewise flat geometries defined on piecewise linear manifolds. ... In order to match the two theories, either LQG should be restricted to the piecewise linear manifolds and piecewise linear spin-networks, or SFMs should be suitably generalized.
The third difference ... In the LQG canonical framework, ... there is no justified way to restrict the graphs to those dual to triangulations of the underlying 3-manifold. The SFMs on the other hand, use only simplicial complexes ... and the spin-networks defined on their boundaries. Therefore they do not define a spin-foam history of a generic spin-network state of LQG. In particular, LQG admits knotted and linked graphs. The simplicial SFMs do not allow such states as well as they do not allow graphs with vertexes more then 4-valent.

I think (3) is the main issues to be addressed in order to establish a link between LQG and SFM. Anyway - even after having studied the paper which claims to resolve (3) and (2) I am not sure if this helps to overcome all LQG ambiguities.

 

[marcus]

Jerzy Lewandowski gave an introductory presentation of embedded spinfoams
at the June 2009 Abhayfest (where also Rovelli, Freidel, and Smolin gave talks.)
The slides are hand-drawn and resemble colored felt-tip sketches on a whiteboard or transparency sheets.
They including his rough pictures (such as any mathematician might draw when explaining.)
http://gravity.psu.edu/events/abhayfest/talks/Lewandowski.pdf

For me it was a congenial presentation. Here is the sound-track:

http://gravity.psu.edu//~media/abhayfest/Lewandowski.rm

The whole menu of talks is here:
http://gravity.psu.edu/events/abhayfest/proceedings.shtml

 

[marcus]

Today Lewandowski and his collaborators posted the follow-up paper to the September 2009 one that this thread is about. I quote from a post by MTd2, who highlighted a part of the abstract which I don't yet fully understand.

...

http://arxiv.org/abs/0912.0540

The EPRL intertwiners and corrected partition function

Wojciech Kamiński, Marcin Kisielowski, Jerzy Lewandowski
(Submitted on 3 Dec 2009)
Do the SU(2) intertwiners parametrize the space of the EPRL solutions to the simplicity constraint? What is a complete form of the partition function written in terms of this parametrization? We prove that the EPRL map is injective in the general n-valent vertex case for the Barbero-Immirzi parameter less then 1. We find, however, that the EPRL map is not isometric. In the consequence, in order to be written in a SU(2) amplitude form, the formula for the partition function has to be rederived. We do it and obtain a new, complete formula for the partition function. The result goes beyond the SU(2) spin-foam models framework.

 

[MTd2]

Ive been trying to think about the highlighted part, and I guess it is just like that. For immirzi smaller than modulus 1, EPRL is merely a necessary description, but is not sufficient at all. One needs spin foams described by much more complicated foams than that.