  The evaluation of relativistic spin networks plays a fundamental role in the
Barrett-Crane state sum model of Lorentzian quantum gravity in 4 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 do an integral; if this
integral converges we say 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 5 vertices). This
proves a conjecture of Barrett and Crane, whose validity is required for the
convergence of their state sum model.
