  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.
