Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space

TL;DR

This paper derives the Barrett-Crane quantum gravity model from a Boulatov-Ooguri field theory over a homogeneous space, providing a new perspective on quantum spacetime as Feynman graphs and offering an alternative model for quantum gravity.

Contribution

It shows how the Barrett-Crane model naturally emerges from a group field theory restricted to a homogeneous space, linking spin foam models to Feynman graph expansions.

Findings

Barrett-Crane model derived from group field theory

Quantum spacetime represented as Feynman graphs

Introduction of an alternative quantum gravity model

Abstract

Boulatov and Ooguri have generalized the matrix models of 2d quantum gravity to 3d and 4d, in the form of field theories over group manifolds. We show that the Barrett-Crane quantum gravity model arises naturally from a theory of this type, but restricted to the homogeneous space S^3=SO(4)/SO(3), as a term in its Feynman expansion. From such a perspective, 4d quantum spacetime emerges as a Feynman graph, in the manner of the 2d matrix models. This formalism provides a precise meaning to the ``sum over triangulations'', which is presumably necessary for a physical interpretation of a spin foam model as a theory of gravity. In addition, this formalism leads us to introduce a natural alternative model, which might have relevance for quantum gravity.