Asymptotic Analysis of the Wigner $3j$-Symbol in the Bargmann Representation

TL;DR

This paper derives the leading asymptotic behavior of the Wigner 3j-symbol using stationary phase approximation in the Bargmann representation, revealing geometric and phase properties related to the Hopf fibration.

Contribution

It provides a novel asymptotic analysis of the Wigner 3j-symbol through a stationary phase approach in the Bargmann framework, connecting quantum numbers with geometric structures.

Findings

Asymptotic limit of the 3j-symbol derived from stationary phase approximation.

Geometric interpretation of stationary phase conditions via Hopf fibration.

Relation between the imaginary part of Bargmann wavefunctions and the 3j-symbol phase.

Abstract

We derive the leading asymptotic limit of the Wigner $3j$-symbol from a stationary phase approximation of a twelve dimensional integral, obtained from an inner product between two exact Bargmann wavefunctions. We show that, by the construction of the Bargmann inner product, the stationary phase conditions have a geometric description in terms of the Hopf fibration of ${\mathbb C}^6$ into ${\mathbb R}^3 \times {\mathbb R}^3 \times {\mathbb R}^3$. In addition, we find that, except for the usual modification of the quantum numbers by 1/2, the imaginary part of the logarithm of a Bargmann wavefunction, evaluated at the stationary points, is equal to the asymptotic phase of the $3j$-symbol.