Positivity of relativistic spin network evaluations

Let G be a compact Lie group. Using suitable normalization conventions, we show that the evaluation of G x G-symmetric spin networks is non-negative whenever the edges are labeled by representations of the form V V* where V is a representation of G, and the inter twiners are generalizations of the Barrett-Crane intertwiner. This includes in particular the relativistic spin networks with symmetry group Spm(4) or 50(4) on a large class of graphs, not restricted to the graph underlying the lOj-symbol. We also present a counterexample, using the finite group 53, to the stronger conjecture that all spin network evaluations are non-negative as long as they can be written using only group integrations and index contractions. This counterexample applies in particular to the product of five Gj-symbols which appears in the spin foam model of the Ss-symmetric .BF-theory on the two-complex dual to a triangulation of the sphere 5 3 using five tetrahedra. We show that this product is negative real for a particular assignment of representations to the edges.