Long-awaited Freidel Baratin (QG related to Feynman diagrams)

[marcus]

http://arxiv.org/abs/gr-qc/0604016
Hidden Quantum Gravity in 3d Feynman diagrams
Aristide Baratin, Laurent Freidel
35 pages, 4 figures
"In this work we show that 3d Feynman amplitudes of standard QFT in flat and homogeneous space can be naturally expressed as expectation values of a specific topological spin foam model. The main interest of the paper is to set up a framework which gives a background independent perspective on usual field theories and can also be applied in higher dimensions. We also show that this Feynman graph spin foam model, which encodes the geometry of flat space-time, can be purely expressed in terms of algebraic data associated with the Poincare group. This spin foam model turns out to be the spin foam quantization of a BF theory based on the Poincare group, and as such is related to a quantization of 3d gravity in the limit where the Newton constant G_N goes to 0. We investigate the 4d case in a companion paper where the strategy proposed here leads to similar results."

 

[marcus]

the extension to 4D, which has not yet been posted, is their reference [1]

"[1] A. Baratin, L. Freidel Hidden Quantum Gravity in 4d Feynman diagrams: Emergence of spin foams, To appear. "

In the 3D paper, introduction on page 3, they say:

"In this paper we focus our study to the case of 3-dimensional Feynman diagrams, but as should be clear from our work and as shown in a companion paper [1], most of the results obtained here can be extended to the more interesting and challenging case of 4-dimensional amplitudes. What makes the analysis work is not that gravity is a topological theory but the fact that, when coupled to matter, quantum gravity in the limit G_N -> 0 behaves as a topological field theory. This is a trivial fact in dimension 3 and has been argued to be true in dimension 4 when one takes this limit while preserving diffeomorphism invariance [12]. "the reference [12] is, as one might have guessed, the Freidel Starodubtsev paper of a year ago
[12] L. Freidel and A. Starodubtsev, Quantum gravity in terms of topological observables, arXiv:hep-th/0501191.

 

[marcus]

From the Freidel Baratin introduction, page 3

===quote===
There has been a recent understanding and new results concerning the consistent coupling of matter fields to 3d quantum gravity amplitudes, in the context of the spin foam approach to 3d quantum gravity [2, 3, 4, 5, 6, 7]. These results have led to the construction of an effective field theory, describing the coupling of quantum fields to quantum gravity [8, 9], which arises after the exact integration of quantum gravitational degrees of freedom. This effective field theory lives on a non-commutative space-time whose geometry is studied in detail in [10]. When the no gravity limit G_N -> 0 is taken, the geometry becomes commutative and usual field theory is recovered.

From the spin foam side, the way usual Feynman diagrams are recovered is highly non trivial, since quantum gravity amplitudes are expressed as purely algebraic objects in terms of spin foam models, a language quite remote from usual field theory language. The fact that Feynman diagrams arise from this picture effectively, amounts to show that usual field theory can be expressed in a background independent manner, where flat space-time emerges dynamically from the choice of spin foam amplitudes. Namely, this amounts to show that Feynman amplitudes can be expressed as the expectation value of certain observables in a topological spin foam model.

Note that this last statement is not a statement regarding quantum gravity but one regarding well known Feynman diagrams in 3d. It is then natural to wonder if one can reach a deeper understanding of this property; whether it is tied up to the topological nature of gravity in dimension 3 or if it can be extended to field theory in higher dimensions. It also begs the question, whether or not Feynman diagrams contain some seeds of information about quantum gravity dynamics.

A natural strategy to address successfully these questions is to start from a careful study of the structure of Feynman graph amplitudes, without any assumption about the nature of quantum gravity dynamics. The goal is to see why and how such familiar objects can be interpreted in terms of spin foam models, which have appeared to be a suitable language to address dynamical question in a background independent approach to quantum gravity [11].

Usual Feynman diagrams are natural observables that allow one in principle to probe the geometry of space-time at arbitrarily small scales. Moreover, they can be naturally promoted to physical observables in any theory of quantum gravity. Therefore, the preferred spin foam model - if any which is related to Feynman diagrams would give us some important seeds of information about the structure of quantum gravity amplitudes. In other words, if one assumes that it is possible to consistently describe quantum gravity amplitudes coupled to matter in terms of a spin foam model, a necessary requirement for the spin foam amplitudes, in order to be physically relevant, is to reduce to the ones appearing in the study of Feynman diagrams when G_N -> 0. This gives strong restrictions on the admissible, physically viable spin foam models.
===endquote===