What are the current developments in spin foam renormalization?

[atyy]

So far the spin foam renormalization programme has been very sparse. There's an old intriguing paper by Markopoulou that no one knows what to do with http://arxiv.org/abs/gr-qc/0203036 , and several recent papers from a group associated with Rivasseau http://arxiv.org/abs/0905.3772 , http://arxiv.org/abs/0906.5477 , http://arxiv.org/abs/1103.1900 . But there is increasing talk about spin foam renormalization. I hope this leads to something that tom.stoer will be happy about. Maybe even finbar.

Dupuis, http://arxiv.org/abs/1104.2765 "For example, one issue not really taken into account yet concerns the issue of the renormalization in the spin foam models. Indeed, an important aspect to keep in mind is that it is fundamental to develop a framework with the appropriate tools to study the coarse-graining of spin foam amplitudes in order to truly define the continuum limit of spinfoam models and their semi-classical limit. And beyond this, we need to identify a family of models parametrized by a finite number of parameters. That is, we need to understand what are the physical relevant coupling constants such as it has been done in standard quantum eld theory and to identify a family of spin foam models stable under coarse-graining."

Magliaro & Perini, http://arxiv.org/abs/1103.4602 "Define the semiclassical limit as the asymptotic regime ... The correctness of this approximation has to be checked against concrete computations in specific examples, and possibly be justified and derived from the full amplitude (defined on the infinite 2-complex) as the result of the iteration of some kind of renormalization group transformation at the level of the spinfoam ‘lattice’."

Bahr, Dittrich & Ryan http://arxiv.org/abs/1103.6264 "This also applies for renormalization and coarse graining techniques which need to be developed to access the large scale limit of spin foams. With finite group models it might be in particular possible to access the many–particle (that is many simplices or building blocks in the triangulation) and small spin (corresponding to small geometrical size of the building blocks) regime. This is in contrast to the few particle and large spin (semi–classical) regime [36] which is accessible so far."

According to AdS/CFT, they should do this for a 3D spin foam like http://arxiv.org/abs/0710.3540, right? :tongue2:

 

[marcus]

... I hope this leads to something that tom.stoer will be happy about. Maybe even finbar.
...
Dupuis, http://arxiv.org/abs/1104.2765 "For example, one issue not really taken into account yet concerns the issue of the renormalization in the spin foam models. Indeed, an important aspect to keep in mind is that it is fundamental to develop a framework with the appropriate tools to study the coarse-graining of spin foam amplitudes in order to truly define the continuum limit of spinfoam models and their semi-classical limit. And beyond this, we need to identify a family of models parametrized by a finite number of parameters. That is, we need to understand what are the physical relevant coupling constants such as it has been done in standard quantum eld theory and to identify a family of spin foam models stable under coarse-graining."
...

I think Maité Dupuis PhD thesis is a nice piece of work. 162 pages well written. Some original parts. Some clear exposition. You have chosen to quote the very last paragraph. At the end of her "conclusions" section.

I have to hand it to the French research establishment and educational system. In QG they have intelligent leadership and policies that are way ahead of USA institutions in some ways. A large portion of the current generation of Loop researchers who are now getting academic jobs in various countries are French or, if not, have done the PhD or postdoc in France.

She is Livine's student, at Lyon. With that quality thesis I reckon she can go where she wants. Let's see where she chooses to go. (Perimeter? Marseille? Potsdam?...)

 

[marcus]

Atyy, I can't think how renormalization could work in Spinfoam gravity. Can you give me a simple rudimentary description of a possible way it might be incorporated into the theory? (As formulated, say, in http://arxiv.org/abs/1102.3660 ?)

 

[tom.stoer]

w/o any reference to ongoing research: for me spinfoam renormalization should work along the lines of Kadanoff's block spin coarse graining.

 

[atyy]

Atyy, I can't think how renormalization could work in Spinfoam gravity. Can you give me a simple rudimentary description of a possible way it might be incorporated into the theory? (As formulated, say, in http://arxiv.org/abs/1102.3660 ?)

Not technically, but the idea is pretty much what tom.stoer says above - something a block spin coarse graining.

But in what "space" should one coarse grain? GFT seems to explore coarse graining in a different space.

If every spin foam is in some sense a lattice gauge theory, then maybe we can use the lattice gauge theory ideas (something like block spin renormalization).

Livine and Ryan have an example of a supersymmetric spin foam. Can they formulate lattice ABJM?

 

[marcus]

w/o any reference to ongoing research: for me spinfoam renormalization should work along the lines of Kadanoff's block spin coarse graining.

...
If every spin foam is in some sense a lattice gauge theory, then maybe we can use the lattice gauge theory ideas (something like block spin renormalization).
...

Would anyone care to say in a few words what the picture or main idea is with Kadanoff "block spin coarse graining"?

BTW the Lyon QG group has a page here: http://www.ens-lyon.fr/PHYSIQUE/index.php?langue=anglais&page=equipe4&souspage=gravite
I gather that it currently consists of Livine, Tambornino, Dupuis, and that Freidel is also there part of the time---shared with Perimeter/Waterloo.

I think that some others (Aristide Baratin or Valentin Bonzom?) may have also come out of Lyon but are no longer there

 

[marcus]

The Magliaro Perini paper that Atyy quoted in post #1 was one of the choices offered in our first quarter 2011 "most important/interesting paper" poll.
https://www.physicsforums.com/showthread.php?t=486501

The Bahr Dittrich Ryan http://arxiv.org/abs/1103.6264 was unfortunately missed.
I may have overlooked it because so many good QG papers appeared Jan-March of this year. The ideas are provocative and intriguing. They use finite groups instead of the usual SU(2) and SL(2, C).

It's a fascinating paper---I want to quote at length from pages 36 and 37 so as to have something for shared study. Atyy already has a passage from page 3 in post #1, here's more:

==quote Bahr Dittrich Ryan of March 2011, pages 36, 37==
There has been some very interesting work exploring coarse graining in the spin foam context [121, 122]. Also the related concept of tensor networks [123], a generalization of spin nets introduced here, has been developed as a tool for coarse graining (to specify topological order in condensed matter systems). Mostly these however concentrate on frameworks and truncations in which the local form of the spin foams does not change. One might however need to accommodate also for non–local19 couplings, in particular if one attempts to regain diffeomorphism symmetry, which is broken by the discretizations employed by many quantum gravity models [16, 70].
Indeed applying block spin transformations to a non–topological lattice gauge system one can expect to obtain a partition function of the form

...[see the paper for the equation for Z = sum of product of products of holonomies...]

where h_li are holonomies around loops (that is around plaquettes but also around other surfaces made of several plaquettes) and w_l1,...lM is a class function of its arguments, describing possible couplings between (Wilson) loops. A character expansion would not only introduce representation labels on the basic plaquettes but also on all the other surfaces encircled by loops appearing in (8.1). The resulting structure is akin to a two–dimensional generalization of a graph. Graphs would appear as effective descriptions for the coarse graining of the Ising like models discussed in section 2. Such non–local ‘spin graphs’ would be generalizations of the spin nets introduced there. Similarly, graphs have been introduced in [12] to accommodate non–local couplings and to describe phase transitions between geometric (i.e a phase where for instance a space time dimension can be defined) and non–geometric phases. We leave the exploration of the ensuing structures for future work.

Section 9 Outlook

In this work we discussed several concepts and tools which arose in the spin foam, loop quantum gravity and group field theory approach to quantum gravity and applied these to finite groups. We encountered different classes of theories, one is the well known example of Yang Mills like theories another are topological BF theories. The latter are also well known in condensed matter and quantum computing. A third class of theories are the constrained models discussed in section 5.3 which mimic the construction of the gravitational models...

[Related material from page 3 of the paper was already quoted.]

...Finally we hope that these models can be helpful in order to develop techniques for coarse graining and renormalization of spin foam models and in group field theories. Here the connection to standard theories could be exploited and techniques can be taken over from the known examples and with adjustments be applied to quantum gravity models. To this end the finite group models could be an important link and provide a class of toy models on which ideas can be tested more easily. For instance the connection between (real space) coarse graining of spin foams and renormalization in group field theories, which generate spin foams as Feynman diagrams, could be explored more explicitly than in the (divergent) SU(2)–based models.

It will be particularly interesting to access the many–particle regime, for instance using Monte Carlo simulations, which for the full models is yet out of reach. In the ideal case it might be possible to explore the phase structure of the constrained theories and to study the symmetry content of these different phases.
==endquote==

One thing that gets my attention here is the positive reference to [121] at the beginning. This is to the 2002 paper by Markopoulou that I think Atyy already mentioned.
http://arxiv.org/abs/gr-qc/0203036
"an old intriguing paper by Markopoulou that nobody knows what to do with..."

 

[tom.stoer]

http://en.wikipedia.org/wiki/Renormalization_group#Block_spin_renormalization_group

The block spin method simply does the following: suppose you have a Hamiltonian

$$H = J_1 \sum_{<ik>_1} S_i S_k + J_2 \sum_{<ik>_2} S_i S_k + \ldots$$

where the subscripts 1, 2, ... refer to nearest-neighbor, next-to-nearest-neighbor etc.

Now define block spins S' as follows: replace a group of 9 spins S (arranged in a square) via one spin S' representing the whole group; replace the old coupling constants with new ones J' and introduce a new temperature T'; you get a new Hamiltonian

$$H^\prime = J^\prime_1 \sum_{<ik>_1} S^\prime_i S^\prime_k + J^\prime_2 \sum_{<ik>_2} S^\prime_i S^\prime_k + \ldots$$

The relations between the coupling constants {J} and {J'} are described via beta-functions defining the RG flow, the physical observables are subject to renormalization group. The interesting physics of large-sclae systems is given by fixed points of the RG flow. At these fixed points the "fundamental spins" are irrelevant as the system "does not change under renormalization".

The application of the block spin method in condensed matter physics is limited by the fact that there are fundamental spins (located at atoms) whereas in QG it is by no means clear if there are "fundamental atoms" of spacetime. In that sense spinfoams could become "self-similar" or "infinitly nested"; the scaling of observables could perhaps be absorbed via the Immirzi parameter which has to change in the RG flow.

 

[marcus]

Tom, Rovelli talks about the running of Newton G here:

http://arxiv.org/abs/1012.4707 page 6, Section C: Scale

He expects G to run with scale, so there is a difference between "G_Planck"
and "G_usual" which is simply written as a plain G when no confusion can arise.

He says that there is some reason to expect that it does not run a big amount---papers by Frank Wilczek and Finn Larsen, and by Xavier Calmet, and "references therein." The two might be the same order of magnitude.

http://arxiv.org/abs/hep-th/9506066
Renormalization of Black Hole Entropy and of the Gravitational Coupling Constant
Larsen Wilczek

http://arxiv.org/abs/1002.0473
Renormalization of Newton's constant and Particle Physics
Calmet
===========================

Rovelli has a perspective on this that seems to differ from what I get from the papers Atyy referenced.

He doesn't seem to consider that the spinfoam theory needs to be coarse-grained, or that spinfoams need to be manipulated according to some radical renormalization scheme.

He takes what seems like a reasonably simple view that Loop theory is appropriate to its scale, and in recovering the macroscopic picture one should allow for some adjustment of Newton constant. What seems to me a comparativelyk mild UNdrastic kind of renormalization.

In a pedagogical paper he states as an open research problem "how does G scale?" This is #8 in a list of some 17 open problems at the end of the Zako Lectures http://arxiv.org/abs/1102.3660

6. Study the radiative corrections in (55) and their possible (infrared) divergences, following the preliminary investigations in [63]. In particular, the sum can be split into a sum over two-complexes and a sum over labelings (spin and intertwiners) for a given two-complex. The potential divergences of the second are associated to “bubbles” (nontrivial elements of the second homotopy class) in the two-complex. Classify them and study how do deal with these.

7. Use the analysis of the these radiative corrections to study the scaling of the theory.

8. In particular, how does G scale?
​

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

Please let me know if I am missing or failing to consider something. I believe you can see this spelled out in the section Scale on page 6 of the December 2010 review. The Loop theory has a particular scale L_Loop ("the scale at which geometry is quantized" or more impressionistically speaking: "the scale at which spacetime becomes discrete").

He chooses to keep this as a free parameter, but it is probably given approximately by
L_Loop² ~ 8 pi gamma hbar G_Planck/c³

In other words the area scale appropriate to the theory is some constant 8 pi gamma times the Planck area, allowing for some play in the Newton constant and therefore in the Planck area.

 

[Finbar]

Isn't the point just to get from a theory that knows about a minimum length and also some microscopic dynamics to one that doesn't i.e. GR. A nice way to to this is to coarse grain the microscopic theory such that you have a effective theory from which macroscopic observables can be defined.

If spin foams have some other neat way to get the semi-classical limit then fine. But so far I don't see this. Since spin foams are already very similar to lattices it shouldn't be too hard to define a black spinning procedure.

 

[marcus]

Isn't the point just to get from a theory that knows about a minimum length and also some microscopic dynamics to one that doesn't i.e. GR. A nice way to to this is to coarse grain the microscopic theory such that you have a effective theory from which macroscopic observables can be defined.

If spin foams have some other neat way to get the semi-classical limit then fine. But so far I don't see this. Since spin foams are already very similar to lattices it shouldn't be too hard to define a black spinning procedure.

Well you could be right, Finbar, and I can make sense of it by thinking that a graph is about adjacency of measurements and when you give adjacency relations you are already giving suggestions of how to consolidate the data.

I can intuitively understand (or at least dimly picture) coarse graining a spin network.

I see Loop as based on a few principles one of which is:
"Not WHAT it is but how it responds to measurement".
The Loop theory is about geometric measurement, where geometry is to some extent uncertain elusve and evolving. In a modest way the theory also says something about the limitations of geometric measurement.
The general format is of a 4D spinfoam "process" occurring within a spin network representing "boundary conditions".

The 3D spin network forming the boundary of a 4D foam region represents what we can aspire to measure, predetermine and predict----initial conditions, final conditions, side conditions. The spin network is the information "box" within which the foam of process occurs.

So the spin network boundary is ALL about measurement and I guess I can imagine a procedure for coarse-grain consolidating that network.

For me, it's harder to think how to coarse-grain the foam that it surrounds. The foam is not accessible information, but rather hypothetical process---one of millions of possible ways that nature might choose to get from here to there----from this geometry to that geometry---within the box of constraints which we impose.

Let me know if you have any further comment or suggestions. This is where I am at the moment.

 

[atyy]

I take a different perspective from Rovelli. I think renormlization is not the only way to see emergence, eg. in the Levin and Wen models or in AdS/CFT (renormalization plays some role in the latter, but probably other things are required too).

Nonetheless, for Rovelli fans, may I quote:

"An important observation regards radiative corrections. The QED perturbative expansion is viable because the effect of all the radiative corrections due to the higher frequency modes can be absorbed into the renormalization of a few parameters." http://arxiv.org/abs/1004.1780 , p12.

"A hint about the regime where this expansion is effective, namely where the complete sum is well approximated by its lowest terms (possibly renormalized, see below), is given by the fact that in the classical limit the vertex amplitude goes to the Regge action of large simplices." http://arxiv.org/abs/1010.1939 , p6.

Finbar: is our universe supposed to be near an IR fixed point in the Asymptotic Safety picture (ie. which trajectory do we pick that flows out and down from the putative UV fixed point)?

Is Rovelli suggesting an IR fixed point here? I assume that unlike AS, he is not assuming a UV fixed point, since spin foams are fundamentally discrete due to lp (but then, how does that tie in with KKLI?)

Edit: I believe in AS, the flow down doesn't go to an IR fixed point, since the cc should be positive. So maybe we shouldn't expect one here either.

 

[Finbar]

Correct me if I'm wrong but in

http://arxiv.org/abs/1012.4707

doesn't Rovelli construct the theory and get to equation (13) without introducing any dimensionful quantities? It seems that way.

"As defined so far, there is no scale in the theory. Everything has been defined in terms of dimensionless quantities."

This implies to me that he has a theory which has no scale and is therefore scale invariant by construction. Essentially the theory sits directly at a fixed point. The job for the LQG guys is then to embedded this theory into a spacetime such that one can go away from the fixed point and recover GR as an effective theory at low energies. In a sense this step should probably be a finite RG step as supposed to an infitessimal perturbation away from the fixed point. Something like embedding the spinfoam in a triangulated manifold and inducing the coarse grain geometrical information
on to the manifold at a set scale. As long as this scale (the initial size of the triangles) is suitably close to where the the effective theory becomes scale invariant not much information will be lost in this step. Once this is done the theory might hopefully look like CDT and one can coarse grain the theory further by taking the triangles larger and larger; coarse graining the geometry. With a bit of luck then they can recover something like de-sitter space.

 

[Finbar]

Correct me if I'm wrong but in

http://arxiv.org/abs/1012.4707

Doesn't Rovelli construct the theory and get to equation (13) without introducing any dimensionful quantities? It seems that way.

"As defined so far, there is no scale in the theory. Everything has been defined in terms of dimensionless quantities."

This implies to me that he has a theory which has no scale and is therefore scale invariant by construction. Essentially the theory sits directly at a fixed point. The job for the LQG guys is then to embedded this theory into a spacetime such that one can go away from the fixed point and recover GR as an effective theory at low energies. In a sense this step should probably be a finite RG step as supposed to an infitessimal perturbation away from the fixed point. Something like embedding the spinfoam in a triangulated manifold and inducing the coarse grain geometrical information
on to the manifold at a set scale. As long as this scale (the initial size of the triangles) is suitably close to where the the effective theory becomes scale invariant not much information will be lost in this step. Once this is done the theory might hopefully look like CDT and one can coarse grain the theory further by taking the triangles larger and larger; coarse graining the geometry. With a bit of luck then they can recover something like de-sitter space.

 

[atyy]

There is a free parameter which I think of as "scale", even though that's not quite right. Rovelli comments on its physical interpretation and units of this parameter in footnote 3 on p4 of http://arxiv.org/abs/1010.1939 .

 

[marcus]

Correct me if I'm wrong but in

http://arxiv.org/abs/1012.4707

doesn't Rovelli construct the theory and get to equation (13) without introducing any dimensionful quantities? It seems that way.

"As defined so far, there is no scale in the theory. Everything has been defined in terms of dimensionless quantities."

This implies to me that he has a theory which has no scale and is therefore scale invariant by construction. Essentially the theory sits directly at a fixed point. The job for the LQG guys is then to embedded this theory into a spacetime such that one can go away from the fixed point and recover GR as an effective theory at low energies. In a sense this step should probably be a finite RG step as supposed to an infitessimal perturbation away from the fixed point. Something like embedding the spinfoam in a triangulated manifold and inducing the coarse grain geometrical information
on to the manifold at a set scale. As long as this scale (the initial size of the triangles) is suitably close to where the the effective theory becomes scale invariant not much information will be lost in this step. Once this is done the theory might hopefully look like CDT and one can coarse grain the theory further by taking the triangles larger and larger; coarse graining the geometry. With a bit of luck then they can recover something like de-sitter space.

This is keen observation and thoughtful reflection. I like this post a lot. I recall that he defines a length scale L _Loop for the theory around that point, and leaves it unspecified. This may be what Atyy is saying in his post, can't read it now.
Turns out I have to go out right now and can't comment more. Back later.

 

[marcus]

New spinfoam paper by Dupuis Livine has a bearing on renormalization:

==quote Dupuis Livine 1104.3683 ==
This new spinfoam model is formulated without reference to spin labels but directly through a discrete action principle and integrals over spinor variables. The diagonal simplicity constraints are not strongly enforced and we are no more restricted to simple irreducible representations of Spin(4). A possible side-effect is that this might allow to discuss in more details the possible renormalization and running of the Immirzi parameter in this spinfoam model.
This new model naturally opens the door to various questions:...
==endquote==

In case anyone wants to check out the details, here is the abstract:
http://arxiv.org/abs/1104.3683
Holomorphic Simplicity Constraints for 4d Spinfoam Models
Maité Dupuis, Etera R. Livine
27 pages
(Submitted on 19 Apr 2011)
"Within the framework of spinfoam models, we revisit the simplicity constraints reducing topological BF theory to 4d Riemannian gravity. We use the reformulation of SU(2) intertwiners and spin networks in term of spinors, which has come out from both the recently developed U(N) framework for SU(2) intertwiners and the twisted geometry approach to spin networks and spinfoam boundary states. Using these tools, we are able to perform a holomorphic/anti-holomorphic splitting of the simplicity constraints and define a new set of holomorphic simplicity constraints, which are equivalent to the standard ones at the classical level and which can be imposed strongly on intertwiners at the quantum level. We then show how to solve these new holomorphic simplicity constraints using coherent intertwiner states. We further define the corresponding coherent spin network functionals and introduce a new spinfoam model for 4d Riemannian gravity based on these holomorphic simplicity constraints and whose amplitudes are defined from the evaluation of the new coherent spin networks."

This work is preliminary in the sense that it deals with the Riemannian 4D case. It must be extended to the Lorentzian case, and they cite work in preparation on that, with two additional authors:
[33] M. Dupuis, L. Freidel, E.R. Livine and S. Speziale, Intertwiners for SO(3, 1) and Holomorphic Simplicity Constraints, in preparation

 

[atyy]

Here is some work from Robert Oeckl.

http://cgpg.gravity.psu.edu/events/.../Proceedings/QG/Thursday/Oeckl/oeckl_talk.pdf
http://arxiv.org/abs/gr-qc/0212047
http://arxiv.org/abs/gr-qc/0401087

Rather amazingly, he says he "renormalizes" a Barrett-Crane-like model and gets BF theory as a "UV" fixed point, and the BC model as an IR fixed point.

There is a definite tension between "Rovellian" and "Rivasseauesque" (what is the adjectival form of "Rivasseau"?) approaches. Rovelli's theory, as it stands, is probably divergent. It is related to the Ponzano-Regge model, which is a spin foam based on normal groups related to 3D Euclidean gravity, which is also divergent. The Turaev-Viro model replaced Ponzano-Regge normal groups with quantum groups and became nice. So Rovelli is pushing this as the means to regularize his theory (http://arxiv.org/abs/1012.4216, http://arxiv.org/abs/1012.4784). Rivasseau, however, says "Regularization by going to a quantum group at a root of unity leads to well-defined topological invariants of the triangulation, namely the Turaev-Viro invariants. However the theory seems unsuited for a true RG analysis, as the propagator has spectrum limited to 0 and 1." (http://arxiv.org/abs/1103.1900).

The Levin-Wen models (which are related to the Turaev-Viro model) were also constructed as fixed points of some sort of renormalization flow. I wonder if there's any relationship between that sense of renormalization, and any of the various spin foam renormalization ideas.
http://arxiv.org/abs/0806.4583
http://arxiv.org/abs/0809.2821