Asymptotics of 10j Symbols

John Baez, Dan Christensen and Greg Egan

August 5, 2002

Abstract: The Riemannian 10j symbols are spin networks that assign an amplitude to each 4-simplex in the Barrett-Crane model of Riemannian quantum gravity. This amplitude is a function of the areas of the 10 faces of the 4-simplex, and Barrett and Williams have shown that one contribution to its asymptotics comes from the Regge action for all non-degenerate 4-simplices with the specified face areas. However, we show numerically that the dominant contribution comes from degenerate 4-simplices. As a consequence, one can compute the asymptotics of the Riemannian 10j symbols by evaluating a `degenerate spin network', where the rotation group SO(4) is replaced by the Euclidean group of isometries of R3. We conjecture formulas for the asymptotics of a large class of Riemannian and Lorentzian spin networks, including the Lorentzian 10j symbols, in terms of these degenerate spin networks.

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