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By John C. (-car) Christensen, J. Daniel (-won) Egan, Reviewed Etera and Richard Livine

Abstract

(2001g:83048)] is a state sum extending to four dimensions the Ponzano-Regge model for 3d gravity. As a spin foam model, it is believed to provide a path integral for (loop) quantum gravity. It is based on a simplicial decomposition of the space-time manifold and associates an amplitude to each 4-simplex which depends on 10 representations labelling its faces (triangles). One uses representations of SO(4) for Riemannian gravity and of SO(3, 1) for Lorentzian gravity. These representations are required to have a vanishing Casimir operator and are called simple or balanced. These 10j symbols for the 4-simplex replace the 6j symbols associated to tetrahedra (3-simplex) by the Ponzano-Regge model. In 3d, the asymptotics of the 6j symbols was a simple function of the volume of the tetrahedron and the Regge calculus version of the Einstein action. This hinted at a link between the Ponzano-Regge model and 3d quantum gravity. Following this logic, the authors study the asymptotics of the 10j symbols. They show numerically that the asymptotics are surprisingly dominated by degenerate 4-simplex configurations instead of the genuine 4-simplex configuration. Then they analytically compute the asymptotics in term of the evaluation of degenerate spin networks, where the rotation group SO(4) is replaced by the Eclidean group of isometries of R 3. The same procedure can be applied to the Lorentzian case and allow the authors to deal more generally with a large class of Riemannian and Lorentzian spin networks and to conjecture formulae for their asymptotics. This paper was followed by two papers, [J. W. Barrett and C. M. Steele, Classical Quantu

Year: 2013
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