Asymptotic Phase of the 6j Symbol from Racah's Sum — E8 Intelligence Research

Abstract

FINDING: The Ponzano-Regge asymptotic formula for the 6j symbol — proven directly from Racah's single-sum formula — yields a phase factor \((-1)^{j_1+j_2+j_3+j_4+j_5+j_6} e^{i(\pi/4 + S/\hbar)}\) for large spins, where \(S\) is the Regge action of a Euclidean tetrahedron. | MATH: The asymptotic form is \( \{6j\} \sim \frac{1}{\sqrt{12\pi V}} \cos\left(S + \frac{\pi}{4}\right) \) (up to sign conventions), with \(V\) the tetrahedron volume. The phase \(\pi/4\) arises from the stationary-phase evaluation of Racah's single sum, where the critical point corresponds to the dihedral angles \(\theta_i\) satisfying \(\cos\theta_i = \frac{\partial S}{\partial j_i}\). The proof treats half-integer and integer spins uniformly, and extends directly to Minkowskian signature (where \(\cos\) becomes \(\cosh\) and the phase shifts to \(i\pi/4\) or similar). | CONNECTION: The \(\pi/4\) phase is the same universal Maslov index appearing in WKB quantisation of a single turning point — but here it emerges Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com