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

TL;DR

The paper revisits Barrett-Crane-type spin foam models through a non-commutative, metric-representation GFT that extends fields with tetrahedron normals and uses a commuting geometricity projector ${\widehat{G}}={\widehat{S}}{\widehat{C}}$ to covariantly impose simplicity and closure constraints without an Immirzi parameter. This leads to a uniquely defined constrained GFT whose Feynman amplitudes are simplicial path integrals for constrained BF theory and admit a spin foam form with a Barrett-Crane–like ${\{10J\}}_{\sigma}$ weight and boundary states in the form of projected spin networks. The framework also provides a model for the topological sector of Plebanski gravity and clarifies how covariance and parallel transport of bivectors and normals are encoded, offering a reinterpretation of the Barrett-Crane criticisms in light of quantum, non-commutative geometry. Overall, the work shows that BC-type quantum geometry can be realized within a covariant, uniquely defined GFT formalism and highlights directions for incorporating the Immirzi parameter and assessing degeneracies and semiclassical limits.

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.