Gluing 4-simplices: a derivation of the Barrett-Crane spin foam model for Euclidean quantum gravity

TL;DR

This paper derives the Barrett-Crane spin foam model for Euclidean quantum gravity by discretizing BF theory and imposing quantum constraints, providing a clear prescription for the state sum and edge amplitudes.

Contribution

It offers a novel derivation of the Barrett-Crane model from first principles, clarifying the gluing of 4-simplices and extending to higher dimensions.

Findings

Derived the Barrett-Crane model from discretized BF theory.

Provided a natural procedure for gluing 4-simplices.

Extended the derivation to higher-dimensional cases.

Abstract

We derive the the Barrett-Crane spin foam model for Euclidean 4 dimensional quantum gravity from a discretized BF theory, imposing the constraints that reduce it to gravity at the quantum level. We obtain in this way a precise prescription of the form of the Barrett-Crane state sum, in the general case of an arbitrary manifold with boundary. In particular we derive the amplitude for the edges of the spin foam from a natural procedure of gluing different 4-simplices along a common tetrahedron. The generalization of our results to higher dimensions is also shown.