This is what Baez was talking about earlier

[selfAdjoint]

Spin foams and MCP are from the same stable, just moving by different roads. From the Baez introduction paper you cited above;

Since loop quantum gravity is based on canonical quantization, states in this formalism describe the geometry of space at a fixed time. Dynamics
enters the theory only in the form of a constraint called the Hamiltonian constraint. Unfortunately this constraint is still poorly understood. Thus until recently, we had almost no idea what loop quantum gravity might say about the geometry of spacetime.

And recall that Thiemann, in his Phoenix paper, noted that some had gone off to study path integral methods because the Hamiltonian constaint had not been solved. But the networks come from the same source and they both started from holonomies along an edge (I should have said that instead of Wilson loops).

 

[marcus]

Spin foams and MCP are from the same stable, just moving by different roads. From the Baez introduction paper you cited above;

And recall that Thiemann, in his Phoenix paper, noted that some had gone off to study path integral methods because the Hamiltonian constaint had not been solved. But the networks come from the same source and they both started from holonomies along an edge (I should have said that instead of Wilson loops).

I don't think that Spin Foams is about "holonomies along an edge". I don't see a space of connections and spinfoams being a Hilbertspace of fuctionals on connections.
My understanding is that Hamiltonian LQG is not rigorously connected to SpinFoam (that is a problem to work on, that people have tried to resolve)

On the other hand, as I see it, Spin Foam is clearly linked to BF theory. See e.g. some mid 1990s papers of Baez and of Smolin. ( bibliography). It seems clear to me that the move into spin foam (even the invention of that approach) was inspired by BF. A paper of Witten. the Barrett-Crane model.

I would say it this way: the contact between BF and SpinFoam is logical and mathematical and evident in papers from the mid 1990s. And it is historical. Spinfoam approach was developed by people thinking TQFT and BF.

On the other hand the link between SpinFoam and Ashtekar variable and holonomies is somewhat vague and "sociological". It involves some imperfect analogies and things brought over when people moved. To some extent we can say SpinFoam is a part of LQG because at a certain point Loop PEOPLE moved over and did Spin Foam.

that reasoning would apply also to any future surge of LQG papers in constrained BF theory if we didnt already have the connection of LQG and BF going back to the 1990s

what I mean is this, you reason that LQG includes SpinFoam because Loop PEOPLE moved over, as you say: "...noted that some had gone off to study path integral methods because the Hamiltonian constaint had not been solved..." Well, if you allow that kind of argument, then definitions become community based. Loop is what Loop people do. String is what String people do, and so on.

It would be a lot clearer if everybody would just take verbatim what Smolin says in "Invitation". See how he defines terms and try to be consistent with that.

I will get a bit from page 12 that I already posted, but some may have missed.

 

[marcus]

Smolin
An Invitation to Loop Quantum Gravity
http://arxiv.org/hep-th/0408048

this gives an overview of LQG in the context of topological field theories
which begins on page 12, section 3.2 "Dynamics of Constrained Topological Field Theories"

BTW selfAdjoint, by LQG here I mean primarily SpinFoam (very different from hamiltonian or masterconstraint narrowlydefined canonical QG). when he says "invitation to LQG" I read it as largely an invitation to spinfoam and BF and stuff like that.

on page 12

"...Observation IV means in details that all gravitational theories of interest, including general relativity and supergravity in 4 dimensions, can be described by an action, which is generically of the form,

S = S^topological + S^constraints + S^matter

To describe the detailed form, its simplest first to fix the dimension to be four, in which case G = SU(2). The first term describes a topological theory called BF theory. It depends on a 2 form Bⁱ and the field strength Fⁱ of a connection, Aⁱ, all valued in a Lie algebra of SU(2). Thus, i = 1, 2, 3. The action is,..."

this is how Smolin does the exposition of what he calls "LQG"
Is there something here that is unclear, or that you disagree with?

My inclination is since he developed the field (with Ashtekar and Rovelli) that we should let him define the field the way he wants (and not insist on defining it the way we want)

 

[Chronos]

I am not comfortable with the S-constraints. The evidence is not very compelling.

 

[marcus]

I am not comfortable with the S-constraints...

I think that equation is a way of writing a generic form that various QG theories (the ones he is interested in) take and to judge the degree of empirical support we need to look at some particular case, like the Plebanski action (which is of this form and reproduces Gen Rel)

In that case i would say the observational evidence is as supportive as it is of Gen Rel, no more no less, because equivalent.

 

[marcus]

http://arxiv.org/abs/hep-th/0501191
Quantum gravity in terms of topological observables
Laurent Freidel, Artem Starodubtsev

"We recast the action principle of four dimensional General Relativity so that it becomes amenable for perturbation theory which doesn't break general covariance. The coupling constant becomes dimensionless (G_{Newton} \Lambda) and extremely small 10^{-120}. We give an expression for the generating functional of perturbation theory. We show that the partition function of quantum General Relativity can be expressed as an expectation value of a certain topologically invariant observable. This sets up a framework in which quantum gravity can be studied perturbatively using the techniques of topological quantum field theory."

Baez gave a report on the October 29-November 1 LQG conference at Perimeter (waterloo Canada) and this was the main development he talked about.

a perturbation series in which the expansion is in powers of a very small number namely the cosmological constant Lambda.

I have been watching and listening to the Feynman (Auckland NZ) lectures on QED
http://www.vega.org.uk/series/lectures/feynman/
thought it full of intuition about perturbative analysis
may someday someone give a talk about quantum gravity with the same assurance and depth as F. explaining electrodynamics.
maybe in the end the quantum theory of spacetime and its shape will resemble that of electron and photon.

what Freidel and Staro say:
"General Relativity so that it becomes amenable for perturbation theory which doesn't break general covariance...framework in which quantum gravity can be studied perturbatively."
suggests maybe this could happen.

watching the Feynman talks gives me hope that it may

 

[marcus]

This is a major advance in the spin foam side of LQG; given Thiemann's apparent major advance in the Canonical side with his Master Constraint Program, we advance both pieces one square. Will one of them capture the other? Stay tuned for the next move!

(Added) Whatever happens with the spin foam path integrals, the deeper understanding of the Barbero-Immirzi parameter contained in this paper is already a major contribution to the field.

Yes, it gives a new role to the B-I parameter and opens up a wide range of possible values. this paper seems to make room for other approaches (undefined as yet) besides spin foam and loop. this idea is vague, for me.
it is as if loop is one wedge driven into the log, and there is room for another, maybe this time one that can split the gnarly thing. (forget this, it is almost too impressionistic to say in public)
Freidel Starodubtsev talk about B-I parameter being anything from zero to infinity

 

[marcus]

the main topic of this thread is this paper

http://arxiv.org/abs/hep-th/0501191
Quantum gravity in terms of topological observables
Laurent Freidel, Artem Starodubtsev

"We recast the action principle of four dimensional General Relativity so that it becomes amenable for perturbation theory which doesn't break general covariance. The coupling constant becomes dimensionless (G_{Newton} \Lambda) and extremely small 10^{-120}. We give an expression for the generating functional of perturbation theory. We show that the partition function of quantum General Relativity can be expressed as an expectation value of a certain topologically invariant observable. This sets up a framework in which quantum gravity can be studied perturbatively using the techniques of topological quantum field theory."

I think I can suggest one other paper to be read with this one (besides the Baez "Introduction"). It is a 1998 article by Freidel and Krasnov

http://arxiv.org/hep-th/9807092
Spin Foam Models and the Classical Action Principle
65 pages, many figures (published version)

"We propose a new systematic approach that allows one to derive the spin foam (state sum) model of a theory starting from the corresponding classical action functional. It can be applied to any theory whose action can be written as that of the BF theory plus a functional of the B field. Examples of such theories include BF theories with or without cosmological term, Yang-Mills theories and gravity in various spacetime dimensions. Our main idea is two-fold. First, we propose to take into account in the path integral certain distributional configurations of the B field in which it is concentrated along lower dimensional hypersurfaces in spacetime. Second, using the notion of generating functional we develop perturbation expansion techniques, with the role of the free theory played by the BF theory. We test our approach on various theories for which the corresponding spin foam (state sum) models are known. We find that it exactly reproduces the known models for BF and 2D Yang-Mills theories. For the BF theory with cosmological term in 3 and 4 dimensions we calculate the terms of the transition amplitude that are of the first order in the cosmological constant, and find an agreement with the corresponding first order terms of the known state sum models. We discuss implications of our results for existing quantum gravity models."

It looks to me like these three papers (the two here plus the Baez) tell a sufficiently complete story so one could learn from them. It looks to me as if it is not impossible that QG is going to develop in this direction.

for completeness here is the Baez (1999) link again
http://arxiv.org/gr-qc/9905087
John Baez
An Introduction to Spin Foam Models of Quantum Gravity and BF Theory
55 pages, 31 figures

"In loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks. The concept of 'spin foam' is intended to serve as a similar picture for the quantum geometry of spacetime. In general, a spin network is a graph with edges labelled by representations and vertices labelled by intertwining operators. Similarly, a spin foam is a 2-dimensional complex with faces labelled by representations and edges labelled by intertwining operators. In a 'spin foam model' we describe states as linear combinations of spin networks and compute transition amplitudes as sums over spin foams. This paper aims to provide a self-contained introduction to spin foam models of quantum gravity and a simpler field theory called BF theory."

 

[marcus]

the main topic of this thread is this paper

http://arxiv.org/abs/hep-th/0501191
Quantum gravity in terms of topological observables
Laurent Freidel, Artem Starodubtsev

"We recast the action principle of four dimensional General Relativity so that it becomes amenable for perturbation theory which doesn't break general covariance. ... This sets up a framework in which quantum gravity can be studied perturbatively using the techniques of topological quantum field theory."

Today Freidel posted another paper. It is number III in a series he started last year with David Louapre (who has sometimes come to PF)
the series is called "Ponzano-Regge Revisited"
http://arxiv.org/hep-th/0502106
However this 3rd in the series is co-authored with Etera Livine

Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory
Laurent Freidel, Etera R. Livine
46 pages

"We study the no gravity limit G_{N}-> 0 of the Ponzano-Regge amplitudes with massive particles and show that we recover in this limit Feynman graph amplitudes (with Hadamard propagator) expressed as an abelian spin foam model. We show how the G_{N} expansion of the Ponzano-Regge amplitudes can be resummed. This leads to the conclusion that the dynamics of quantum particles coupled to quantum 3d gravity can be expressed in terms of an effective new non commutative field theory which respects the principles of doubly special relativity. We discuss the construction of Lorentzian spin foam models including Feynman propagators"

am currently experiencing difficulty getting the PDF for this paper due to some mixup at Arxiv.

 

[marcus]

the mixup was fixed at arxiv so the paper of Freidel and Livine
is available
http://arxiv.org/hep-th/0502106
Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory

"... This leads to the conclusion that the dynamics of quantum particles coupled to quantum 3d gravity can be expressed in terms of an effective new non commutative field theory which respects the principles of doubly special relativity. We discuss the construction of Lorentzian spin foam models including Feynman propagators"

this one detail, about respecting the principles of DSR, ups the ante.
DSR is exposed to observational testing and refutation by experiments like GLAST. remember this Freidel/Livine paper studies 3D gravity not the eventual 4D, but results about DSR in one lower dimension are strongly suggestive that Loop-and-allied QG as it is currently being developed depends on GLAST or something like it detecting some variation in the speed of gammaray with energy. Here is a relevant quote from Freidel/Livine introduction near top of page 4

---quote from Freidel and Livine---
Then, at G=0, the spin foam amplitudes are to be interpreted as providing the Feynman graph evaluation of particles coupled to quantum gravity. We study the perturbative G expansion of the spin foam amplitudes. Remarkably, this expansion can be re-summed and expressed as the Feynman graphs of a non-commutative braided quantum field theory with deformation parameter G, which thus describes the effective theory for matter in quantum gravity.

Any deformed Poincaré theory usually suffers from a huge ambiguity [5] coming from what should be identify as the physical energy and momenta since the introduction of the Planck scale allows non-linear redefinitions. This ambiguity can also be understood as an ambiguity in the identification of the non-commutative space-time. Our work shows that the Ponzano-Regge model naturally defines a star product and a duality between space and momenta, therefore no ambiguity remains once we identify quantum gravity as being responsible for the effective deformation of the Poincaré symmetry.

This realizes explicitly, for the first time from first principles, the now popular idea that quantum gravity will eventually lead to an effective non-commutative field theory incorporating the principle of doubly special relativity [6].
---end quote---

 

[marcus]

to help get an idea of where this line of research is going I will list the papers these two recent ones refer to as in preparation or to appear.

http://arxiv.org/abs/hep-th/0501191
Quantum gravity in terms of topological observables

http://arxiv.org/hep-th/0502106
Ponzano-Regge model revisited III: Feynman diagrams and effective field theory

the following 3 titles are included in the references thereof but have not yet been posted.

Freidel, Starodubtsev
Perturbative gravity via spin foam

Freidel, Kowalski-Glikman, Starodubtsev
Background independent perturbation theory for gravity coupled to particles: classical analysis

Freidel, Noui, Roche
Duality formulas: geometry and asymptotics of 6j symbols

 

[john baez]

E is conjugate to A

this is the right introduction (for me and I hope others)
Page 3 of Baez "introduction" gr-qc/9905087

Well, that's nice to hear!

A is a connection on P
ad(P) is the vector bundle associated to P via the adjoint action of G on its Lie algebra
E is a 2-form on M with values in ad(P)

The curvature of A, called F, is also an ad(P)-valued 2-form

(notice that for reasons best known to himself Baez is using E rather than B as his notation)

The reason is that this field I'm calling E (which everyone else calls B
for some insane reason) is canonically conjugate to the A field - just as the electric field E in electromagnetism is canonically conjugate to the vector potential A in electromagnetism!

In other words, when we quantize A and E, they should satisfy the canonical commutation relations. Or in other words, A is mathematically analogous to the POSITION of a point particle, while E is analogous to the momentum.

This analogy makes a lot of things easy to understand. For example, there's an equation in EF theory saying that the covariant divergence of E is zero. This is analogous to the equation in electromagnetism saying that the divergence of E is zero - at least in the absence of charged matter.

Furthermore, this quantity - the covariant divergence of E - is also the generator of gauge transformations in EF theory, just as it is in electromagnetism.

Furthermore, flux lines of the E field play an important role as "spin network edges", which carry area.

All of these analogies get hopelessly obscured when people call this E field the "B field". It is NOT an analogue of the magnetic field in electromagnetism!

The B field in electromagnetism is just the curl of A. It consists of the space-space components of the electromagnetic field tensor F. It is NOT canonically conjugate to the A field.

Of course, I'll never succeed in convincing the world to call this theory "EF theory" and call the basic fields A and E instead of A and B. People get attached to familiar notation, and it's almost impossible to change.

BUT, if you want to understand BF theory, the first thing to do is think of it as "EF theory", with the basic fields being a connection A, its curvature F, and a canonically conjugate field E. Then you can start using some of your
intuition from electromagnetism!

Only *some* of it, mind you. But every little bit helps.

It's even better if you know Yang-Mills theory, where there are Lie-algebra
valued versions of A, F, and E, and the Gauss law says the covariant divergence of E vanishes.


jb

 

[marcus]

Freidel: Group Field Theory Overview

just out:

http://arxiv.org/abs/hep-th/0505016
Group Field Theory: An overview
Laurent Freidel (PI, ENS-Lyon)
10 pages
"We give a brief overview of the properties of a higher dimensional generalization of matrix model which arises naturally in the context of a background independent approach to quantum gravity, the so called group field theory. We show that this theory leads to a natural proposal for the physical scalar product of quantum gravity. We also show in which sense this theory provides a third quantization point of view on quantum gravity."

Freidel and his co-workers do some of the most interesting research in Quantum Gravity. Baez clued us about this by flagging the Freidel-Starodubtsev paper, which is what this thread was started about
and then there was the Freidel-Livine paper

This paper covers some history and gives a survey of GFT. It is a talk given at a conference June 2004. So it is not as new actually, as some of the work discussed in this thread, which appeared this year. But maybe it can be helpful as a concise map showing Freidel's perspective.