Integrability for Relativistic Spin Networks

Details Integrability for Relativistic Spin Networks Integrability for Relativistic Spin Networks

2001-01-27

arxiv.org/abs/gr-qc/0101107v2

Class.Quant.Grav. 18 (2001) 4683-4700

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

22 pages AMSLaTeX, 11 encapsulated Postscript figures

Scientific paper

10.1088/0264-9381/18/21/316

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.

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