Asymptotics of the Wigner 9j symbol

TL;DR

This paper derives the asymptotic behavior of the Wigner 9j-symbol for large quantum numbers, linking it to geometric figures and analyzing singularities like caustics and catastrophes.

Contribution

It provides the first detailed asymptotic formula for the 9j-symbol, extending geometric and analytical methods used for simpler symbols.

Findings

Asymptotic formula expressed via geometric lengths and angles

Identification of caustics and catastrophes in the asymptotics

Symmetries of the 9j-symbol are preserved in the formula

Abstract

We present the asymptotic formula for the Wigner 9j-symbol, valid when all quantum numbers are large, in the classically allowed region. As in the Ponzano-Regge formula for the 6j-symbol, the action is expressed in terms of lengths of edges and dihedral angles of a geometrical figure, but the angles require care in definition. Rules are presented for converting spin networks into the associated geometrical figures. The amplitude is expressed as the determinant of a 2x2 matrix of Poisson brackets. The 9j-symbol possesses caustics associated with the fold and elliptic and hyperbolic umbilic catastrophes. The asymptotic formula obeys the exact symmetries of the 9j-symbol.