Asymptotics of 10j symbols
John C. Baez, J. Daniel Christensen, Greg Egan
TL;DR
The paper shows that the large-spin asymptotics of the Riemannian $10j$ symbol in the Barrett–Crane model are dominated by degenerate 4-simplices, not the nondegenerate stationary points. It introduces degenerate spin networks based on the Euclidean group $\mathcal{E}(3)$ to compute these asymptotics and formulates conjectures that extend to broad classes of Riemannian and Lorentzian spin networks, with the Lorentzian case predicted to be a constant multiple of the Riemannian one. Numerical evidence using VEGAS supports the $\lambda^{-2}$ scaling and the degenerate-network coefficients, while border-admissibility cases show different (slower) decays. The work connects the asymptotics to Euclidean-group representations and suggests that the Barrett–Crane model may be governed by degenerate geometries in the large-spin limit, with implications for the underlying quantum gravity dynamics.
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 R^3. We conjecture formulas for the asymptotics of a large class of Riemannian and Lorentzian spin networks in terms of these degenerate spin networks, and check these formulas in some special cases. Among other things, this conjecture implies that the Lorentzian 10j symbols are asymptotic to 1/16 times the Riemannian ones.