Spacetime as a Feynman diagram: the connection formulation

TL;DR

This paper demonstrates that spin foam models of quantum gravity can be represented as sums over Feynman diagrams of a quantum field theory on a group manifold, generalizing previous models and connecting to matrix models.

Contribution

It provides an explicit formula for the field theory action corresponding to spin foam models, extending previous work and unifying various approaches to quantum gravity.

Findings

Spin foam sums can be realized as Feynman diagram sums of a quantum field theory.

The work generalizes previous models including Boulatov's and Ooguri's.

Provides a detailed derivation based on the connection formulation of spin foam models.

Abstract

Spin foam models are the path integral counterparts to loop quantized canonical theories. In the last few years several spin foam models of gravity have been proposed, most of which live on finite simplicial lattice spacetime. The lattice truncates the presumably infinite set of gravitational degrees of freedom down to a finite set. Models that can accomodate an infinite set of degrees of freedom and that are independent of any background simplicial structure, or indeed any a priori spacetime topology, can be obtained from the lattice models by summing them over all lattice spacetimes. Here we show that this sum can be realized as the sum over Feynman diagrams of a quantum field theory living on a suitable group manifold, with each Feynman diagram defining a particular lattice spacetime. We give an explicit formula for the action of the field theory corresponding to any given spin foam model in a wide class which includes several gravity models. Such a field theory was recently found for a particular gravity model [De Pietri et al, hep-th/9907154]. Our work generalizes this result as well as Boulatov's and Ooguri's models of three and four dimensional topological field theories, and ultimately the old matrix models of two dimensional systems with dynamical topology. A first version of our result has appeared in a companion paper [gr-qc\0002083]: here we present a new and more detailed derivation based on the connection formulation of the spin foam models.