Uniform Semiclassical Approximation for the Wigner $6j$ Symbol in Terms of Rotation Matrices

TL;DR

This paper introduces a new uniform asymptotic approximation for the Wigner 6j symbol using Wigner rotation matrices, applicable across all quantum number values including near caustics, supported by geometric insights and numerical validation.

Contribution

It presents a novel uniform approximation for the Wigner 6j symbol in terms of rotation matrices, extending applicability near caustics.

Findings

The approximation is valid for all quantum numbers, including near caustics.

Numerical tests show good agreement with exact 6j-symbols.

Comparison with Ponzano-Regge approximation demonstrates improved accuracy.

Abstract

A new uniform asymptotic approximation for the Wigner $6j$ symbol is given in terms of Wigner rotation matrices ($d$-matrices). The approximation is uniform in the sense that it applies for all values of the quantum numbers, even those near caustics. The derivation of the new approximation is not given, but the geometrical ideas supporting it are discussed and numerical tests are presented, including comparisons with the exact $6j$-symbol and with the Ponzano-Regge approximation.