Integrability for Relativistic Spin NetworksTools Barrett, John W. and Baez, John C. (2001) Integrability for Relativistic Spin Networks. Classical and Quantum Gravity, 18 (46834).
AbstractThe evaluation of relativistic spin networks plays a fundamental role in the BarrettCrane 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 threedimensional 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 4simplex (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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