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

We derive the Barrett-Crane spin foam model for Euclidean 4-dimensional quantum gravity from a discretized $\mathrm{BF}$ 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.