Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model

TL;DR

This paper develops a modified group field theory for quantum gravity that incorporates bivector simplicity constraints, leading to a spin foam model akin to Barrett-Crane, and revisits previous criticisms of this model.

Contribution

It introduces a new GFT extension with projected fields and constraints, producing a Barrett-Crane-like spin foam model from a dual BF theory formulation.

Findings

Feynman amplitudes are simplicial path integrals for constrained BF theory.

The spin foam formulation aligns with a variant of the Barrett-Crane model.

Re-examination of criticisms against Barrett-Crane model in light of new construction.

Abstract

A dual formulation of group field theories, obtained by a Fourier transform mapping functions on a group to functions on its Lie algebra, has been proposed recently. In the case of the Ooguri model for SO(4) BF theory, the variables of the dual field variables are thus so(4) bivectors, which have a direct interpretation as the discrete B variables. Here we study a modification of the model by means of a constraint operator implementing the simplicity of the bivectors, in such a way that projected fields describe metric tetrahedra. This involves a extension of the usual GFT framework, where boundary operators are labelled by projected spin network states. By construction, the Feynman amplitudes are simplicial path integrals for constrained BF theory. We show that the spin foam formulation of these amplitudes corresponds to a variant of the Barrett-Crane model for quantum gravity. We then re-examin the arguments against the Barrett-Crane model(s), in light of our construction.