Casimir constraints in the EPRL spin foam model: the Y_γ map at small and large spin, and a proposed recolored embedding of SU(2) boundary spins into SL(2,ℂ) irreps

The EPRL spin foam model embeds an SU(2) boundary spin j into the lowest SU(2) subspace of a unitary principal-series representation of SL(2,ℂ) with (p,k) = (γj, j). This paper quantifies, for all j, how well this map and the alternative map p = γ(j+1) satisfy the Casimir form of the simplicity constraints. At small j the sign of the first Casimir is wrong (for γ = 0.274 and 0.2375 at j = 1/2 and 1), and at large j the absolute deviations grow linearly. Using Schur’s lemma, we show that no superposition of representations containing the SU(2) spin j can satisfy both weak conditions even on average. We give the exact continuous solution, and propose a heuristic “recolored” embedding at small j that removes the sign problem. We specify a numerical test and invite practitioners with spin foam codes to carry it out.