6-j Symbol - Asymptotics

Asymptotics

A remarkable formula for the asymptotic behavior of the 6-j symbol was first discovered by Ponzano and Regge. The asymptotic formula applies when all six quantum numbers j₁, ..., j₆ are taken to be large and associates to the 6-j symbol the geometry of a tetrahedron. If the 6-j symbol is determined by the quantum numbers j₁, ..., j₆ the associated tetrahedron has edge lengths J_i = j_i+1/2 (i=1,...,6) and the asymptotic formula is given by,

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} \sim \frac{1}{\sqrt{12 \pi |V|}} \cos{\left( \sum_{i=1}^{6} J_i \theta_i +\frac{\pi}{4}\right)}.$

The notation is as follows: Each θ_i is the external dihedral angle about the edge J_i of the associated tetrahedron and the amplitude factor is expressed in terms of the volume, V, of this tetrahedron.