Spacetime as a Feynman diagram:the connection formulation

Abstract

Spin-foam models are the path-integral counterparts toloop-quantized canonical theories. Over the last few years several spin-foammodels of gravity have been proposed, most of which live on finitesimplicial lattice spacetime. The lattice truncates the presumablyinfinite set of gravitational degrees of freedom down to a finite set.Models that can accommodate an infinite set of degrees of freedom and thatare independent of any background simplicial structure, or indeed any a priori spacetime topology, can be obtained from the lattice models bysumming them over all lattice spacetimes. Here we show that this sumcan be realized as the sum over Feynman diagrams of a quantum fieldtheory living on a suitable group manifold, with each Feynman diagramdefining a particular lattice spacetime. We give an explicit formula forthe action of the field theory corresponding to any given spin-foam modelin a wide class which includes several gravity models. Such a field theorywas recently found for a particular gravity model. Our workgeneralizes this result as well as Boulatov's and Ooguri's models of three-and four-dimensional topological field theories, and ultimately the oldmatrix models of two-dimensional systems with dynamical topology. Afirst version of our result has appeared in a companion paper:here we present a new and more detailed derivation based on theconnection formulation of the spin-foam models.