Asymptotic Limits of the Wigner $12J$-Symbol in Terms of the Ponzano-Regge Phases

TL;DR

This paper derives a new asymptotic formula for the Wigner 12j-symbol involving two Ponzano-Regge phases, expanding the understanding of angular momentum recoupling in quantum mechanics.

Contribution

It presents the second type of asymptotic formula for the 12j-symbol, expressed via vector diagrams of two tetrahedra sharing a face, involving two Ponzano-Regge phases.

Findings

Derived the second asymptotic formula for the 12j-symbol.

Expressed the formula in terms of vector diagrams of two tetrahedra.

Identified the involvement of two Ponzano-Regge phases.

Abstract

There are two types of asymptotic formulas for the $12j$ symbol with one small and 11 large angular momenta. We have derived the first type of formula previously in [L. Yu, Phys. Rev. A84 022101 (2011)]. We will derive the second type in this paper. We find that this second asymptotic formula for the $12j$ symbol is expressed in terms of the vector diagram associated with two $6j$ symbols, namely, the vector diagram of two adjacent tetrahedra sharing a common face. As a result, two sets of Ponzano-Regge phases appear in the asymptotic formula. This work contributes another asymptotic formula of the Wigner $12j$ symbol to the re-coupling theory of angular momenta.