Asymptotics of 10j symbols

TL;DR

This paper investigates the asymptotic behavior of Riemannian 10j symbols in quantum gravity, revealing that degenerate configurations dominate and proposing a new approach using degenerate spin networks for computation.

Contribution

It introduces a novel method to compute 10j symbol asymptotics via degenerate spin networks, challenging previous assumptions about non-degenerate contributions.

Findings

Degenerate 4-simplices dominate the asymptotics.

Asymptotics can be computed using Euclidean group spin networks.

Lorentzian 10j symbols are asymptotic to 1/16 of Riemannian ones.

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.