Semiclassical Analysis of the Wigner $12j$ Symbol with One Small Angular Momentum

TL;DR

This paper derives the first asymptotic formula for the Wigner 12j symbol with one small and eleven large angular momenta, using semiclassical analysis techniques related to the 9j symbol, revealing geometric connections among angular momentum symbols.

Contribution

It presents the first known asymptotic formula for the 12j symbol with one small angular momentum, extending semiclassical analysis methods from the 9j symbol case.

Findings

Derived the first asymptotic formula for the 12j symbol with one small angular momentum

Identified the geometric structure of the 12j asymptotic formula in terms of a 9j vector diagram

Demonstrated the technique's potential for analyzing other 3nj symbols

Abstract

We derive an asymptotic formula for the Wigner $12j$ symbol, in the limit of one small and 11 large angular momenta. There are two kinds of asymptotic formulas for the $12j$ symbol with one small angular momentum. We present the first kind of formula in this paper. Our derivation relies on the techniques developed in the semiclassical analysis of the Wigner $9j$ symbol [L. Yu and R. G. Littlejohn, Phys. Rev. A 83, 052114 (2011)], where we used a gauge-invariant form of the multicomponent WKB wave-functions to derive asymptotic formulas for the $9j$ symbol with small and large angular momenta. When applying the same technique to the $12j$ symbol in this paper, we find that the spinor is diagonalized in the direction of an intermediate angular momentum. In addition, we find that the geometry of the derived asymptotic formula for the $12j$ symbol is expressed in terms of the vector diagram for a $9j$ symbol. This illustrates a general geometric connection between asymptotic limits of the various $3nj$ symbols. This work contributes the first known asymptotic formula for the $12j$ symbol to the quantum theory of angular momentum, and serves as a basis for finding asymptotic formulas for the Wigner $15j$ symbol with two small angular momenta.