Asymptotic Limits of the Wigner $15J$-Symbol with Small Quantum Numbers

TL;DR

This paper derives new asymptotic formulas for the Wigner 15j-symbol with small quantum numbers, linking them to geometric figures and paving the way for semiclassical analysis in quantum gravity models.

Contribution

It introduces novel WKB-type asymptotic formulas for the 15j-symbol with multiple small quantum numbers, extending geometric interpretations similar to the Ponzano-Regge formula.

Findings

Formulas are validated numerically.

Formulas relate to geometric figures of angular momentum.

Facilitates analysis of semiclassical limits in quantum gravity.

Abstract

We present new asymptotic formulas for the Wigner $15j$-symbol with two, three, or four small quantum numbers, and provide numerical evidence of their validity. These formulas are of the WKB form and are of a similar nature as the Ponzano-Regge formula for the Wigner $6j$-symbol. They are expressed in terms of edge lengths and angles of geometrical figures associated with angular momentum vectors. In particular, the formulas for the $15j$-symbol with two, three, and four small quantum numbers are based on the geometric figures of the $9j$-, $6j$-, and $3j$-symbols, respectively, The geometric nature of these new asymptotic formulas pave the way for further analysis of the semiclassical limits of vertex amplitudes in loop quantum gravity models.