Uniform Approximation from Symbol Calculus on a Spherical Phase Space

TL;DR

This paper introduces a novel method using symbol calculus and quantum normal form theory to derive uniform asymptotic approximations, exemplified by the $6j$-symbol in terms of rotation matrices, based on spherical phase space mappings.

Contribution

It develops a general approach for uniform asymptotic approximations using symbol correspondence on spherical phase space, extending previous results on the $6j$-symbol.

Findings

Derived uniform approximation of the $6j$-symbol in terms of rotation matrices

Introduced a method based on symbol calculus and quantum normal form theory

Established a canonical map between level sets on spherical phase space

Abstract

We use symbol correspondence and quantum normal form theory to develop a more general method for finding uniform asymptotic approximations. We then apply this method to derive a result we announced in an earlier paper, namely, the uniform approximation of the $6j$-symbol in terms of the rotation matrices. The derivation is based on the Stratonovich-Weyl symbol correspondence between matrix operators and functions on a spherical phase space. The resulting approximation depends on a canonical, or area preserving, map between two pairs of intersecting level sets on the spherical phase space.