2001/01/31 by John C. Baez, John W. Barrett · 1 citation
Physics and Astronomy · #gr-qc
paper · pdf · doi:10.1088/0264-9381/18/21/316
published as Class.Quant.Grav. 18 (2001) 4683-4700 · 22 pages AMSLaTeX, 11 encapsulated Postscript figures
arxiv created 2001/08/10 · arxiv updated 2009/11/30
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 L2 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.