9-j Symbol - Asymptotics

Asymptotics

Like the 6-j symbol, the 9-j symbol has a rich asymptotic structure valid when all 9 j's are taken large. The analog of the Ponzano-Regge formula for the 9-j symbol is

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} \sim A_1 \cos{\left( \sum_{i=1}^{9} J_i \theta_i \right)}+A_2 \sin{\left( \sum_{i=1}^{9} J_i \theta_i \right)}.$

where J_i = j_i+1/2 (i=1,…,9) and the angles in the two functions are calculated with respect to two different geometries. The amplitudes are both given by,

$A_l = \frac{1}{4 \pi \sqrt{|V_{124} V_{542}-V_{451} V_{215}|}} \quad (l=1,2)$

and are distinguished by the fact that the volume factors are again calculated with respect to two different geometries labelled here by l.