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 ten 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 Bbb R3. We conjecture formulae for the asymptotics of a large class of Riemannian and Lorentzian spin networks in terms of these degenerate spin networks, and check these formulae in some special cases. Among other things, this conjecture implies that the Lorentzian 10j symbols are asymptotic to 1/16 times the Riemannian ones.
Classical and Quantum Gravity
- Pub Date:
- December 2002
- General Relativity and Quantum Cosmology
- 25 pages LaTeX with 8 encapsulated Postscript figures. v2 has various clarifications and better page breaks. v3 is the final version, to appear in Classical and Quantum Gravity, and has a few minor corrections and additional references