Asymptotics of the Wigner 9j symbol

Details Asymptotics of the Wigner 9j symbol Asymptotics of the Wigner 9j symbol

2009-12-29

arxiv.org/abs/0912.5384v1

Class.Quant.Grav.27:135010,2010

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

17 pages, 7 figures

Scientific paper

10.1088/0264-9381/27/13/135010

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.

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