Positivity of relativistic spin network evaluations

TL;DR

This paper proves that certain relativistic spin network evaluations are non-negative under specific conditions, but provides a counterexample showing not all such evaluations are non-negative, especially in finite group cases.

Contribution

The paper establishes positivity results for GxG-symmetric spin networks with specific representations and intertwiners, and presents a counterexample for a broader conjecture.

Findings

Spin network evaluations with V⊗V* labels are non-negative for compact Lie groups.

Counterexample with finite group S_3 shows some evaluations can be negative.

Product of five 6j-symbols can be negative for specific representations.

Abstract

Let G be a compact Lie group. Using suitable normalization conventions, we show that the evaluation of GxG-symmetric spin networks is non-negative whenever the edges are labeled by representations of the form V\otimes V^* where V is a representation of G, and the intertwiners are generalizations of the Barrett--Crane intertwiner. This includes in particular the relativistic spin networks with symmetry group Spin(4) or SO(4). We also present a counterexample, using the finite group S_3, 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 6j-symbols which appears in the spin foam model of the S_3-symmetric BF-theory on the two-complex dual to a triangulation of the sphere S^3 using five tetrahedra. We show that this product is negative real for a particular assignment of representations to the edges.