Integrability for Relativistic Spin Networks

Barrett, John W.; Baez, John C.

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.

• Journal Article Type: Article
• Publication Date: Jan 1, 2001
• Journal: Classical and Quantum Gravity
• Print ISSN: 0264-9381
• Publisher: IOP Publishing
• Peer Reviewed: Not Peer Reviewed
• Volume: 18
• Issue: 4683-4
• APA6 Citation: Barrett, J. W., & Baez, J. C. (2001). Integrability for Relativistic Spin Networks. Classical and Quantum Gravity, 18(4683-4),
• Copyright Statement: Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf