Spacetime as a Feynman diagram: the connection formulation

TL;DR

The paper advances a program to render spacetime topology dynamical by summing spin foam models over all admissible 2-complexes, recasting the sum as a perturbative expansion of a field theory on $G^4$ whose Feynman diagrams enumerate the 2-complex spacetimes. It provides an explicit action $I[psi] = I_0[psi] - \lambda \mathcal{V}[psi]$ with a quadratic kinetic term and a nonlocal 5-vertex interaction, showing that the boundary observable amplitudes coincide with a sum over spin foams on all admissible 2-complexes, weighted by representation data and intertwiners. The formalism unifies and extends prior field-theoretic formulations of BF theory and gravity-spin foam models, explains the translation between connection and spin foam pictures via Peter–Weyl expansions, and discusses regularization through quantum groups and potential finiteness of regulated theories. It also outlines generalizations to other atom types and links to matrix model structures, suggesting a broad and flexible framework for dynamical topology in quantum gravity with potential computational handles via adapted spin-network bases.

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.

Figures (2)

• Figure 1: This figure illustrates simplicial complexes and their dual skeletons in two and three dimensions, as an aid to the reader contemplating the the four dimensional simplicial complexes and their dual 2-skeletons that appear in our discussions. On the left a complex of seven triangles (2-simplices) along with its dual 1-skeleton is shown. On the right a complex of six 3-simplices (sharing a common 1-simplex) and its dual 2-skeleton is shown, with the dual 2-cells shaded in. In both cases the dual skeleton is cut off at the boundary of the simplicial complex.
• Figure 2: Atoms and their relation to simplices are illustrated in three dimensions. Panel a) shows an atom and one complete dual 2-cell of which the atom has a wedge. Panel b) shows the 3-simplex inside which the atom would live were it part of a dual 2-complex of a three dimensional simplicial complex. The "four dimensional" atoms that we deal with in the text differs from that shown in that it is identical to the portion inside a 4-simplex of the 2-skeleton dual to a four dimensional simplicial complex. This means that the atom has five one dimensional cells and ten wedges, as opposed to the four one dimensional cells and six wedges shown here.