Integrability for Relativistic Spin Networks

TL;DR

This work establishes the convergence (integrability) of a broad class of relativistic spin-network evaluations arising in the Lorentzian Barrett–Crane model for 4D quantum gravity. By deriving a uniform bound on the kernel $K_p$ and employing hyperbolic-space barycentres, the authors prove that graphs built from the tetrahedron via adding a vertex with at least three new edges (and related constructions) yield finite, Lorentz-invariant evaluations, with the $4$-simplex being integrable. The results resolve Barrett–Crane's conjecture on convergence for these amplitudes and lay the groundwork for robust Lorentzian state-sum models, while also suggesting conjectures and directions for higher dimensions. The techniques combine geometric analysis on hyperbolic space with representation-theoretic kernels, potentially enabling broader applications in quantum gravity and non-compact group integrals.

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.

Figures (1)

• Figure 1: A relativistic spin network

Theorems & Definitions (17)

• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3
• proof : Proof of Theorem \ref{['bound']}
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3