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

TL;DR

This work derives the Barrett-Crane spin foam model for Euclidean quantum gravity in four dimensions by discretizing $SO(4)$ BF theory and imposing Barrett-Crane constraints at the quantum level on representations. It provides a precise, gluing-driven prescription for edge amplitudes, showing how interior tetrahedra arise from the assembly of 4-simplices and yielding a finite state sum consistent with previous matrix-model approaches. The method extends to arbitrary dimensions with $SO(D)$ and discusses boundary conditions, potential Lorentzian generalizations, and the semiclassical limit, offering a concrete route to a background-independent quantum gravity path integral. These results illuminate how gravity can emerge from constrained topological field theory and furnish a concrete, finite spin foam framework with control over amplitudes and boundaries.

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.