Integrability for Relativistic Spin Networks

TL;DR

This paper proves that a broad class of relativistic spin networks, including the 4-simplex, are integrable, confirming a key conjecture necessary for the convergence of the Barrett-Crane quantum gravity model.

Contribution

It demonstrates the integrability of a large class of relativistic spin networks, including the 4-simplex, validating a conjecture crucial for quantum gravity models.

Findings

Many relativistic spin networks are integrable.

The 4-simplex spin network is integrable.

Confirmed the Barrett-Crane conjecture.

Abstract

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.