Constraints, measure and vertex amplitude in LQG

[tom.stoer]

Have a look at

http://arxiv.org/abs/1202.5039
Degenerate Plebanski Sector and its Spin Foam Quantization
Authors: Sergei Alexandrov
(Submitted on 22 Feb 2012)
Abstract: We show that the degenerate sector of Spin(4) Plebanski formulation of four-dimensional gravity is exactly solvable and describes covariantly embedded SU(2) BF theory. This fact provides its spin foam quantization and allows to test various approaches of imposing the simplicity constraints. Our analysis suggests a unique method of imposing the constraints which leads to a consistent and well defined spin foam model.

Alexandrov clarifies the role of the first and second class constraints, especially of the simplicity constraints. He shows that in order to recover the well-known Crane-Yetter model for the degenerate sector one has to impose the constraints classicaly i.e. by inserting delta functions which changes the measure and therefore the vertex aamplitude.

He shows that the method used by EPRL and FK is not sufficient to suppress the quantum fluctuations related to these constraints and that their method does not lead to the (correct) Crane-Yetter model! In addition he shows that the Immirzi parameter drops out in the final theory and that effects regarding its quantization are artificial.

What next? Application of these methods to modify (i.e. to correct) the quantization of the EPRL and FK model. I am convinced that this will result in a new vertex amplitude.

marcus: I nominate this paper as the most important one for your next poll: the picks for first quarter 2012 - most important QG paper

 

[marcus]

Hi Tom,
I was wondering if anyone would pick up on that paper, I logged it on the bibliography thread several days ago. It seems like an obvious one to include in the poll. Personally I am not sure about the precise relation of the Crane-Yetter model (a topological quantum field theory) to quantum gravity.

A lot was written about the Crane-Yetter TQFT back in 1993. Louis Crane and David Yetter organized a "Quantum Topology" conference which John Baez attended. He reports here:
http://math.ucr.edu/home/baez/week12.html

He wrote more about it a few weeks later in September 1993 http://math.ucr.edu/home/baez/week19.html
and gave links to several papers.

Some 1994 links:
http://arxiv.org/abs/hep-th/9409167
http://arxiv.org/abs/hep-th/9412025
http://arxiv.org/abs/hep-th/9405183

A Baez paper from 1995
http://arxiv.org/abs/q-alg/9507006

You can probably dig up more recent stuff about Crane-Yetter. This is just what I found by a quick search.

I see we had a brief thread about C-Y here at this forum, in 2003:
https://www.physicsforums.com/showthread.php?t=3934

I gather that spinfoam methods can be applied in various ways to other things besides quantum gravity. In particular, one can apply spinfoam techniques to the Crane-Yetter model describing the various topologies of 4D manifolds. This may or may not yield useful lessons applicable to spinfoam quantum gravity. The connection isn't clear. Should we pay attention to those lessons or not?

 

[tom.stoer]

Personally I am not sure about the precise relation of the Crane-Yetter model (a topological quantum field theory) to quantum gravity.

Alexandrov explains this in detail.

It's not about the relation between CY and QG; it's about the quantization method.

Alexandrov shows that using the EPRL and FK method in case of the CY model explicitly gives us a wrong result (there are several ways to derive the CY model, so the SF rep. is well-known); then he uses a different method to implement the constraints (Dirac + delta-function in the measure) and he finds the correct CY vertex.

So the conclusion is that the method used by EPRL and FK is wrong.

 

[marcus]

That was what I was wondering about. It is entirely two different things. Can one draw that conclusion? Apparently you have concluded that you can. You feel that the proceedure that works for CY should be used for QG. Have to go, back later this evening.

 

[atyy]

Engle also believed the constraints were imposed wrongly:"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." He too proposed a new vertex amplitude.

 

[marcus]

Engle also believed the constraints were imposed wrongly:"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." He too proposed a new vertex amplitude.

Yes! I thought that was a very constructive paper. He actually deals with 4D gravity
and makes a helpful adjustment. It surprised me that Alexandrov's paper does not mention Engle's improvement. In fact it looked to me as if he was going back to around 2007 and taking THAT as his idea of current spin foam. Why not discuss something recent? Like for example Engle's version.

Here is one of the Engle papers you mentioned:
http://arxiv.org/abs/1201.2187
A spin-foam vertex amplitude with the correct semiclassical limit
Jonathan Engle
(Submitted on 10 Jan 2012)
Spin-foam models are hoped to provide a dynamics for loop quantum gravity. 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.
4 pages

I think we've discussed this here at PF, I know I've brought it up. Engle was Ashtekar PhD then went to Marseille for postdoc with Rovelli and then took a faculty job at Florida Atlantic.

 

[tom.stoer]

I am not sure whether Alexandrov and Engle are talking exactly about the same issue, but they seem to be rather close. What I do not understand (see also marcus' comment) is that they never mention each other in the citations; they seem to ignore each other ;-(

Alexandrov goes one step backward. He first explains in a series of papers why EPRL and FK is wrong (I discussed this in several threads over the last year); his observation is related to general rules to implement second class constraints and he therefore addresses issues not specific to QG

The problem is that the correct method a la Dirac has been studied (I can remember the delta functions in the measure in some Thiemann papers) but that it was not possible to get a final answer in terms of the physical spin network states and the vertex amplitude; b/c QG is terribly complicated Alexandrov decided to study something different, namely the top. sector and the CY model, for several reasons: it's simpler; it's quantization can be derived by different methods; so it serves as a consistency check for his approach.

Alexandrov finds that the EPRL and FK method results in a wrong vertex amplitude for CY, so he can demonstrate in a very specific example (where the final result is known) that his general objections are justified. Then he applies his (Dirac's) method to this specific example and finds the correct CY result, so he can conclude that his method is correct.

The final step - and this is where Alexandrov and Engle shall meet - is the application of the new measure to full QG. Alexandrov explains what he sees as the major obstacles to apply his method to full QG, namely not conceptual problems but technical difficulties due to the different structure:

"The main difference distinguishing it from our model is the form of the constraints ... In particular, the secondary constraints become explicitly dependent on the B-field ... Although the construction of section 3.3 is still well defined in the presence of such dependence, it gives rise to many complications. The most important one is that the
quantity (3.22) starts to depend on the bivectors ... and its interpretation as a vertex amplitude is not viable anymore. It is not clear whether this is a serious problem or just a minor obstacle."

It's hard for me to compare every step by Alexandrov with Engle's paper; but I'll do my best ...