Integrability for relativistic spin networks

The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in four dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lorentz group appearing in the direct integral decomposition of the space of L-2 functions on three-dimensional hyperbolic space. To 'evaluate' such a spin network we must perform an integral; if this integral converges we say that the spin network is 'integrable'. Here we show that a large class of relativistic spin networks are integrable, including any whose underlying graph is the 4-simplex (the complete graph on five vertices). This proves a conjecture of Barrett and Crane, whose validity is required for the convergence of their state sum model.