Integrability for relativistic spin networks
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Publication:2769511
DOI10.1088/0264-9381/18/21/316zbMATH Open1063.83568arXivgr-qc/0101107OpenAlexW2058190725WikidataQ62443556 ScholiaQ62443556MaRDI QIDQ2769511
Author name not available (Why is that?)
Publication date: 2001
Published in: (Search for Journal in Brave)
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.
Full work available at URL: https://arxiv.org/abs/gr-qc/0101107
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