Baratin-Freidel: Hidden Quantum Gravity in 4d Feynman diagrams

[marcus]

http://arxiv.org/abs/hep-th/0611042
Hidden Quantum Gravity in 4d Feynman diagrams: Emergence of spin foams
Aristide Baratin, Laurent Freidel
28 pages, 7 figures

"We show how Feynman amplitudes of standard QFT on flat and homogeneous space can naturally be recast as the evaluation of observables for a specific spin foam model, which provides dynamics for the background geometry. We identify the symmetries of this Feynman graph spin foam model and give the gauge-fixing prescriptions. We also show that the gauge-fixed partition function is invariant under Pachner moves of the triangulation, and thus defines an invariant of four-dimensional manifolds. Finally, we investigate the algebraic structure of the model, and discuss its relation with a quantization of 4d gravity in the limit where the Newton constant goes to zero."

this is the one that John Baez started a thread about earlier this year, for us to discuss and get the basic concepts in hand before the paper appeared

 

[Kea]

Hidden Quantum Gravity in 4d Feynman diagrams: Emergence of spin foams
Aristide Baratin, Laurent Freidel
28 pages, 7 figures

Hah, Marcus! Page 20: 2-categories...

 

[marcus]

there is this odd coincidence that BF theory sounds like it was made up especially for Baratin and Freidel to use.

Here is their table of contents:
===quote BF===

Contents

I. Introduction 2

II. Dynamical geometry in Feynman amplitudes 4

III. Symmetries and gauge fixing 7
    A. Classical solutions and zero modes 8
    B. Edge symmetry 9
    C. Face symmetry 10
    D. Reducibility 13
    E. The gauge-fixed model 15

IV. Feynman diagrams as spin foam amplitudes 15
    A. Topological invariance 15
    B. Observables and partial gauge-fixing 17
    C. Feynman diagrams on homogeneous spaces 19

V. Algebraic structure: discussion 20
    A. A new kind of spin foam model 20
    B. A duality relation 21
    C. [b]Gravity and BF theory[/b] 21

VI. Conclusion 22
    A. Pachner moves identities 22
    B. Computation of determinants 25

References 27

===endquote===

I bet Kea onetime that this paper when it came out would not be all categorickky
let me see where is that smilie with the tongue:tongue:

OK Kea you win. they DO mention categories, but not a whole lot, so there.

BTW here is John Baez earlier thread that he started back on 16 JUNE about this very paper, almost 5 months before it came out!
https://www.physicsforums.com/showthread.php?t=123902
Baratin and Freidel: a spin foam model of ordinary particle physics

 

[Kea]

they DO mention categories, but not a whole lot, so there

But they promise an upcoming paper with Baez on 2-representation theory...

 

[marcus]

Yes! their reference [23] "to appear"

and their reference [22] is to a paper you did with Crane

 

[marcus]

I think in this paper Freidel is predicting an energy dependent speed of light. It is not stated very explicitly and one has to look carefully to find it. There may be qualifications that I am missing.

I'll gather some quotes together and we can see if there is a prediction and if so how firm it is.

At the bottom of page 3 he lists four "checks" or conditions which a spinfoam model should satisfy and says the paper demonstrates that his model satisfies the first three of them.

the second of the four is that the spinfoam model should "agree with the structure predicted by [7,8].

[7] is the Freidel Starodubtsev paper of January 2005 which is pretty familiar here---we discussed it quite a bit.

[8] is a paper in which the DeSitter group played a considerable part.
The paper was by Freidel, Starodubtsev, and Kowalski-Glikman (expert with kappa-Poincaré, DeSitter group, DSR, and the like).

There is no rush, so I will quote a lot of the paper, and see just how it points to reference [8] and what that signifies.

 

[marcus]

Here's what it says on page 3:
===quote===

The second consequence of the spin foam hypothesis follows from the work [7] where a new background independent approach to quantum gravity perturbation theory was proposed in the language of spin foam model. In this approach, the starting point is to write 4d gravity as a perturbation of a topological BF theory based on the de-Sitter group for positive cosmological constant. The perturbation parameter G_NLambda is dimensionless and the perturbation theory transmutes gauge degree of freedom into physical degrees of freedom in a controlled way, order by order. In particular this means that the theory becomes topological in the limit G_N -> 0 . It has also been shown in this context that the coupling to matter particles can be explicitly performed by computing expectation value of Wilson lines observables [8], which are the most natural gauge invariant observables in this formulation.

The main consequence of interest to us from these works is the fact that, not only Feynman diagram amplitudes should be written as expectation values of certain natural observables in a spin foam model, but, moreover, the corresponding model should be a topological spin foam model based on a Poincaré BF theory.

So in summary, the spin foam hypothesis implies that usual Feynman graph can be expressed as the expectation value of certain observables in a topological spin foam model based on the Poincaré group. The validity of such a statement is for us a non-trivial check in support of the spin foam hypothesis.

The check is fourfold:
first, spin foam should arise naturally in Feynman integrals;

second, the spin model should agree with the structure predicted by [7, 8];

third, it should confirm the idea that the limit G_N -> 0 is a limit where gravity becomes topological;

and fourth the Feynman diagram observables should be understood as a Wilson lines (or more generally spin networks) expectation value in this spin foam model.

In this paper, we show that the first three conditions are indeed satisfied...

...The idea of our derivation is to consistently erase the information about flat space geometry from the Feynman integral and encode this information in terms of a choice of quantum amplitudes that should be summed over, and which dynamically determine flat space geometry. In doing so, a triangulation, and a specific spin foam model living on it, are naturally found; this allows us to express usual field theory amplitude in
a background independent manner. The idea that spin foam models code, in a background independent manner, the integration measure viewed by Feynman diagrams was formulated for the first time in [9] and [10] in the context of 3d-gravity.

An analysis similar to the one done here has already been performed in 3d [12], where it has been shown that the corresponding spin foam model is constructed in terms of 6j symbols of the 3d Euclidean group for flat space. The deformation of this spin foam model using quantum group naturally leads to a formulation of Feynman diagram coupled to 3d quantum gravity amplitudes [10, 11]. This corresponds to a deformation of field theory carrying a deformed action of the Poincaré group.

===endquote===

[8] http://arxiv.org/abs/gr-qc/0607014
Particles as Wilson lines of gravitational field
L. Freidel, J. Kowalski--Glikman, A. Starodubtsev
19 pages, some number of comments and clarifications added, to be published in Phys. Rev. D

"Since the work of Mac-Dowell-Mansouri it is well known that gravity can be written as a gauge theory for the de Sitter group. In this paper we consider the coupling of this theory to the simplest gauge invariant observables that is, Wilson lines. The dynamics of these Wilson lines is shown to reproduce exactly the dynamics of relativistic particles coupled to gravity, the gauge charges carried by Wilson lines being the mass and spin of the particles. Insertion of Wilson lines breaks in a controlled manner the diffeomorphism symmetry of the theory and the gauge degree of freedom are transmuted to particles degree of freedom."

Maybe I am wrong. In any case this is what I have to check to see whether there is some implication that one should expect a QG dispersion effect.

 

[marcus]

Rats! this one eluded me. Reference [8] has only a *suggestion* of DSR, e.g. on page 17 where it talks about the analogous work in 3d. Nothing conclusive.

Myself and any others who care about seeing a prediciton for GLAST are left with the hope that there will be something about this is the paper which Freidel says is "To appear." This is the B&F reference [23]

[23] J. Baez, A. Baratin, L. Freidel, On the representation theory of the Poincaré 2-group, To appear.

 

[Sauron]

Sic, I was updating my knowledge of string theory, i wan´t be a total crackpot you know ;). I am reading: Tomas Ortín "gravity and strings", great book, i would recommend it algso for LQG people, and and Cliffr V. Jonshon "D-Branes", as wels as some shorter mixed review papers. But it seems that it is time for a break in string theory ;).

That coment about a direct result about avring speed of light seems even more than what I expected form these paper. Just hope to understand something :uhh: