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

TL;DR

The paper links the Barrett-Crane spin foam model to a Boulatov-Ooguri-type group field theory defined on the homogeneous space $S^3=SO(4)/SO(3)$ and shows that the Barrett-Crane amplitudes arise as Feynman amplitudes of a colored 2-complex, i.e., $Z_{BC}[\Delta] = Z[\Gamma(\Delta)]$. By restricting to simple $SO(4)$ representations and organizing amplitudes through either Case A (15-$j$ type) or Case B (Barrett-Crane intertwiners), the authors obtain a sum over spin foams that confers covariance beyond a fixed triangulation. The framework generalizes the matrix-model idea to 4D, providing a concrete implementation of a “sum over triangulations” via the field theory’s Feynman expansion and offering a natural regularization path via quantum groups. It also points to a second, alternative model that implements the BF-to-Einstein constraints differently, highlighting the flexibility and potential richness of group field theories for quantum gravity.

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.

Figures (3)

• Figure 1: The structure of the kinetic and potential terms in the action.
• Figure 2: The propagator can be represented by a collection of four strands, each carrying a simple representation.
• Figure 3: The structure of the vertex of the fat 5-valent graphs generated by the Feynman expansion.