Spin Foams Without Spins

Abstract: We formulate the spin foam representation of discrete SU(2) gauge theory as a product of vertex amplitudes each of which is the spin network generating function of the boundary graph dual to the vertex. In doing so the sums over spins have been carried out. The boundary data of each n-valent node is explicitly reduced with respect to the local gauge invariance and has a manifest geometrical interpretation as a framed polyhedron of fixed total area. Ultimately, sums over spins are traded for contour integrals over simple poles and recoupling theory is avoided using generating functions.