Is Holonomy Spinfoam even more likely to succeed than Twistor Networks?

[marcus]

Holonomy Spinfoam models address several of the issues left open by EPRL and I suspect HS could become the new "EPRL" phenomenon: the new Loop pace-setter, instead of another strong candidate (Twistor Networks) we were discussing in another thread.

I'm pretty sure that anyone who wants to follow Loop gravity research would be well advised to read Hellmann Kaminski 1210.5276 and get prepared to understand Dittrich's ILQGS talk on 27 November.

So I'll list some titles and links in this thread. But also it would be very interesting if someone disagrees and thinks that some other reformulation of LQG that is currently being actively pursued has a better chance and could make a stronger showing at the upcoming Loops 2013 conference.

I should emphasize, if what I just said doesn't make it clear enough, that the topic here is nearterm Loop gravity development---in particular the 9 months from now until the 2013 Loops conference. It's a fast moving field and I'm skeptical of anyone (including me) pretending to see its longterm future.

First of all here's the main paper.
http://arxiv.org/abs/1210.5276
Geometric asymptotics for spin foam lattice gauge gravity on arbitrary triangulations
Frank Hellmann, Wojciech Kaminski
(Submitted on 18 Oct 2012)
We study the behavior of holonomy spin foam partition functions, a form of lattice gauge gravity, on generic 4d-triangulations using micro local analysis. To do so we adapt tools from the renormalization theory of quantum field theory on curved space times. This allows us, for the first time, to study the partition function without taking any limits on the interior of the triangulation.
We establish that for many of the most widely used models the geometricity constraints, which reduce the gauge theory to a geometric one, introduce strong accidental curvature constraints. These limit the curvature around each triangle of the triangulation to a finite set of values. We demonstrate how to modify the partition function to avoid this problem. Finally the new methods introduced provide a starting point for studying the regularization ambiguities and renormalization of the partition function.
4+6 pages, 1 figure
edit: I just realized, thanks to Atyy pointing it out, that I mistakenly typed "tensor" in the title when I intended "twistor".

 

[atyy]

The thread title should be "Twistor Networks", not "Tensor Networks"?

 

[marcus]

The thread title should be "Twistor Networks", not "Tensor Networks"?

Shoot! Typing too fast. I was able to edit within the thread and make that correction, but wish it could be changed to "Twistor" in the title as well. Thanks for pointing out my mistake!

Holonomy Spinfoams just means you have a foam (i.e. a 2 complex of vertices, edges, faces) which, instead of being labeled by half-integer spins and the like, is labeled by GROUP ELEMENTS. So the group elements living on, say, the edges of the foam describe what happens to you as you travel through the foam on some path along its edges.

HS looks a lot like Lattice Gauge Theory generalized to include the dynamic geometry needed for gravity.

It's a fairly recent development (mostly this year) and besides these two authors (Hellmann Kaminski) one of the main people working on it has been Bianca Dittrich. So this fall, since it is fairly new, TWO related seminar talks were scheduled at ILQGS. One by Hellmann (4 September) and one by Dittrich (27 November).
http://relativity.phys.lsu.edu/ilqgs/ (has links to Hellmann's talk)
http://relativity.phys.lsu.edu/ilqgs/schedulefa12.html

Hellmann's talk was really clear and organized, anybody at all interested in nearterm future of LQG should hear it while following thru the slides. Audio and slides PDF are online.
I don't know the title of Dittrich's talk but it's bound to be related to this in some way.

I'll get links to the recent HS papers, which will give a more precise concrete idea of the Holonomy Spinfoams approach.

 

[marcus]

Here is a good introduction to Holonomy Spinfoams:

http://arxiv.org/abs/1208.3388
Holonomy Spin Foam Models: Definition and Coarse Graining
Bahr, Dittrich, Hellmann, Kaminski
We propose a new holonomy formulation for spin foams, which naturally extends the theory space of lattice gauge theories. This allows current spin foam models to be defined on arbitrary two-complexes as well as to generalize current spin foam models to arbitrary, in particular finite groups. The similarity with standard lattice gauge theories allows to apply standard coarse graining methods, which for finite groups can now be easily considered numerically. We will summarize other holonomy and spin network formulations of spin foams and group field theories and explain how the different representations arise through variable transformations in the partition function. A companion paper will provide a description of boundary Hilbert spaces as well as a canonical dynamic encoded in transfer operators.

See also by the same authors http://arxiv.org/abs/1209.4539 (Holonomy Spin Foam Models: Boundary Hilbert spaces and canonical dynamics)

 

[tom.stoer]

marcus, thanks for the hint. Here's a list of objections regarding the SF models known so far

http://arxiv.org/abs/1107.0709
We show that the linearized simplicity constraints used in the EPRL and FK models are not sufficient to impose a restriction to a single Plebanski sector, but rather, three Plebanski sectors are mixed. We propose this as the reason for certain extra `undesired' terms in the asymptotics of the EPRL vertex ...

http://arxiv.org/abs/1201.2187
All 4-d spin-foam models of gravity start from the Plebanski formulation, in which gravity is recovered from a topological field theory, BF theory, by the imposition of constraints, which, however, select not only the gravitational sector, but also unphysical sectors. We show that this is the root cause for terms beyond the required Feynman-prescribed exponential of i times the action in the semiclassical limit of the EPRL spin-foam vertex. By quantizing a condition isolating the gravitational sector, we modify the EPRL vertex, yielding what we call the proper EPRL vertex amplitude. This provides at last a vertex amplitude for loop quantum gravity with the correct semiclassical limit.

Please note that the aim here is the correct semiclassical limit - whcih means that the underlying quantum theoiry can still be ambiguous!

http://arxiv.org/abs/1210.5276
Spin foam models for quantum gravity are a form of lattice gauge gravity. They are constructed by a modification of topological lattice theories [1–4]. These are heuristic lattice quantizations of so called BF theory ... A fully satisfactory restriction to the geometric sector is not known
...
Up until now no method for analyzing the full dynamics of the theory that did not rely on the large quantum number approximation throughout the partition function was available. Furthermore the partition functions obtained in this way all require regularization, and the ambiguities of these regularizations remained ill understood. This is a crucial question as, if the ambiguities proliferate as the lattice gets finer, they would render any continuum limit completely unpredictive
...
As a first application we clearly demonstrate that the most studied models, all of which include the so called Immirzi parameter γ suffer from accidental curvature constraints, strongly disfavoring them. ... We illuminate its geometric origin and propose a modified partition function that is likely to encode the correct geometricity constraints.

This is related to the other thread where the twistor approach is been discussed. The lesson is always the same: the way to restrict BF down to LQG is still not sufficiently understood. The main problem is the definition and implementation of the simplicity constraints and the consistent definition of the SF measure. This is directly related to the missing off-shell closure of the constraint algebra in the canonical approach and the complicated Dirac algebra w/o time gauge.

http://arxiv.org/abs/1105.3708
The quantum operators corresponding to the quadratic simplicity constraints have been found to be anomalous both in the covariant [18] as well as in the canonical picture [19, 3].
...
On the covariant side, this lead to one of the major points of critique about the Barrett-Crane model ... The anomalous constraints are imposed strongly, which may imply erroneous elimination of physical degrees of freedom. This triggered the development of the new Spin Foam models, in which the quadratic simplicity constraints are replaced by linear simplicity constraints. The linear version of the constraints is slightly stronger than the quadratic constraints, since in 3 + 1 dimensions the topological solution is absent. ... corresponding quantum operators are still anomalous ... therefore, in the new models (parts of) the simplicity constraints are implemented weakly to account for the anomaly
...
In this paper we have reported on several new ideas of how to treat the simplicity constraints ... None of them is entirely satisfactory at this point and must be further de-
veloped. We hope that the discussion presented in this paper will be useful for an eventually consistent formulation.

It is interesting that in the meantime the "Erlangen group" Thiemann, Giesel, Sahlmann, ... seems to abandon the constraint quantization approach and tries to succeed via deparametrization in terms of material (dust) frames:

http://arxiv.org/abs/1206.3807
We believe that switching from the earlier operator constraint reduction (Dirac) approach that dominated the research in LQG over the past 20 years to the reduced phase space approach, ... has many merits. ...

 

[f-h]

Hi marcus, I'm glad you thought my talk was clear, I'm afraid I was quite dissatisfied with it myself. For one, I didn't properly anticipate how difficult it would be to describe the underlying geometry without pointing at the slides, for two I was quite sick during it. Only propped up through aspirin.

If you have any questions on this approach I'll be able to briefly answer them. I'm afraid I'm quite busy at the moment though.

 

[tom.stoer]

forgiven my slow-wittedness - now I know who f-h is ;-)

 

[atyy]

Is there a relationship between the tensory Dittrich stuff and the twistory Speziale stuff? In string theory it looks like the tensory stuff is more general, and the twistory stuff is especially useful when the system is integrable. (Along the lines of mitchell porter's point about MHV in the twistory thread.)

It is interesting that in the meantime the "Erlangen group" Thiemann, Giesel, Sahlmann, ... seems to abandon the constraint quantization approach and tries to succeed via deparametrization in terms of material (dust) frames:

Yes, I think that is the only new approach in LQG. The others are variations on an old theme. I would like to see Giesel calculate the black hole entropy in her approach! The Giesel approach is conceptually nice because there is no "gauge", and everything is "physical". I hear it is calculationally even more gnarly than standard constraint based LQG. I would also like to see Dittrich link up to gauge/gravity - that would evade all the constraint concerns;)

 

[tom.stoer]

Yes, I think that is the only new approach in LQG. The others are variations on an old theme.

Exactly!

The Giesel approach is conceptually nice because there is no "gauge", and everything is "physical".

I agree

I hear it is calculationally even more gnarly than standard constraint based LQG.

Isn't it similar to LQC with a scalar clock field?

http://arxiv.org/abs/1206.0658
Linking Covariant and Canonical General Relativity via Local Observers

 

[marcus]

Hi marcus, I'm glad you thought my talk was clear, I'm afraid I was quite dissatisfied with it myself. For one, I didn't properly anticipate how difficult it would be to describe the underlying geometry without pointing at the slides, for two I was quite sick during it. Only propped up through aspirin.

If you have any questions on this approach I'll be able to briefly answer them. I'm afraid I'm quite busy at the moment though.

It's good news for us that you are quite busy, please keep the interesting papers coming!

I do not expect you to necessarily answer questions raised here (don't feel obliged to, given the limited time)---but I'll ask anyway.

In your October paper 1210.5276 on page 4 halfway thru "Discussion" section you say

As a first application we clearly demonstrate that the most studied models, all of which include the so called Immirzi parameter γ suffer from accidental curvature constraints, strongly disfavoring them. This is a significant refinement of the flatness issue raised by Bonzom in [54, 55]. We illuminate its geometric origin and propose a modified partition function that is likely to encode the correct geometricity constraints.

Further development of this method into a full symbolic calculus of distributions on homogeneous spaces, which would be of independent mathematical interest, will allow us finer control of the distributions, establishing whether the subset of geometric configurations on which the partition function is peaked is that satisfying the Regge equations of discrete gravity.​

I'm not sure I've found the place where you propose the modified partition function but I think it is at equations (18) and (19) where you define the distribution D(g, g-tilda) which is to replace ω(g) in the partition function.

However the definition of D given there seems to depend on the Immirzi γ!

So my question is: does the proposed partition function depend on γ? Would that dependence extend to the subset of geometric configurations on which it's peaked? (I'm hoping the answer is no!) And does this affect the prospects of agreement with the subset of configurations satisfying Regge equations?

The larger question is: do you see a possible way Spinfoams could develop that would either dispense with gamma entirely or significantly reduce its role?

I'm asking this also in the light of the September paper of Dittrich and Ryan 1209:4892 "On the role of the Barbero-Immirzi parameter..." Hoping Dittrich will discuss that paper when she gives her 27 November talk at ILQGS.

 

[f-h]

Quick answers: Yes D depends on \gamma. The solution spaces should differ, but be isomorphic/code the same geometry, and yes, we already have a perfectly good spin foam model: FK without \gamma is what is favoured by our analysis.

The caveat is that not all technical lemmas are available for FK, so this is a (strong) expectation, rather than a theorem.

 

[tom.stoer]

what are your answers w.r.t. the objections of Alexandrov et al. regarding implementation of second-class / simplicity constraints and PI measure in the "new models"?

 

[marcus]

Tom, hopefully f-h will get back to us and reply to your question in post #12, in the meantime let's look again at what he says about the Immirzi parameter.

Quick answers: Yes D depends on \gamma. The solution spaces should differ, but be isomorphic/code the same geometry, and yes, we already have a perfectly good spin foam model: FK without \gamma is what is favoured by our analysis.

The caveat is that not all technical lemmas are available for FK, so this is a (strong) expectation, rather than a theorem.

Among other things this could suggest to us what Dittrich's TBA seminar talk might be about. (A reformulation of LQG that does not depend on gamma, or depends less heavily on it?) As a reminder here is the ILQGS fall schedule [with my marginal comments on some]:

Sept. 4 Holonomy Spin Foam Models: Asymptotic Dynamics Frank Hellmann Albert Einstein Institute [HSF]
Sept. 18 Flux coherent states Lorenzo Sindoni Albert Einstein Institute
Oct. 2 Lifting General Relativity to Observer Space Derek Wise FAU, Erlangen ["dust"]
Oct. 16 Horizon entropy from loop gravity Eugenio Bianchi Perimeter Institute [GR thermo]
Oct. 30 Renormalization of Tensorial Group Field Theories Sylvain Carrozza Albert Einstein Institute [TGFT]
Nov. 6 Twistorial structure of loop quantum gravity transition amplitudes Simone Speziale CPT Marseille [twistor LQG]
Nov. 27 TBA Bianca Dittrich Perimeter Institute [?]

http://relativity.phys.lsu.edu/ilqgs/

Dittrich has taken the lead in developing the Holonomy Spin Foam (HSF) reformulation and she also recently posted a paper about the role of the Immirzi. When f-h says "our analysis favors FK without gamma" I assume he means primarily his work with Kaminski, but it might also include others, such as Dittrich, working on HSF. So she might have something to say about it in the talk.

BTW as time goes on I increasingly appreciate how well Jorge Pullin constructed the Fall 2012 ILQGS lineup. It reflects the various bids to reformulate LQG. (namely TGFT, HSF, dust, GR thermodynamics, twistor LQG)
You mentioned "dust" earlier in this thread and Atyy seemed enthused about that one:

... to abandon the constraint quantization approach and tries to succeed via deparametrization in terms of material (dust) frames...

The talk by Derek Wise is an example that can represent that approach.
You also have (e.g. by your recent thread) indicated the potentially important effect relativistic thermodynamics can have on the development of quantum gravity. Eugenio Bianchi's talk is a concrete example of that.
The twistor approach (your choice on the MIP poll) will be the subject of Speziale's talk in less than a week from today--6 November.
And I agree in viewing these things as the most interesting/impactful initiatives (all five: TGFT, HSF, dust, GR thermodynamics, twistor LQG.) So it seems Jorge Pullin called the shots pretty well back in July, when he set up the Fall ILQGS speaker schedule.

 

[f-h]

Hi tom.stoer,

I personally don't think of spin foams as arising from a quantization construction via simplicity constraints, see the first half of my paper with Bianchi for example. So I never paid too much attention to those results.

And indeed I would interpret them to show that they are not the result of a sensible quantisation procedure.

One formulation that I've grown to like are the holomorphic constructions of Dupuis and Livine. These can be motivated from asymptotic geometry, but also from a commuting set of constraints. As far as I can see it does not suffer from the problem pointed out by Alexandrov.