Regge symmetry of 6j-symbols of the Lorentz group

TL;DR

This work extends Regge symmetry from $su(2)$ to the Lorentz group $SL(2, C)$ by analyzing $6j$-symbols in the unitary principal series labeled by $(\sigma,N)$. It derives a limiting identity from hyperbolic gamma-function structures to produce a Lorentzian analogue of Regge symmetry, yielding a new closed form in terms of a complex hypergeometric function $J_{cr}$ and a Regge transformation with ${\\mathcal S} = \\frac{1}{2}(\\\ extsigma_1+\\\\textsigma_2+\\\\textsigma_3-\\\\textsigma_4)$ and ${\\mathcal N}=\\frac{1}{2}(N_1+N_2+N_3-N_4)$, up to a sign $(-1)^{(N_1+N_3-N_2+N_4)/2}$. The result makes the Regge symmetry explicit for Lorentzian $6j$-symbols and lays the groundwork for large-spin asymptotics and potential non-compact Ponzano–Regge-type formulas.

Abstract

In this paper we derive new symmetry and new expression for $6j$-symbols of the unitary principal series representations of the $SL(2,\mathbb{C})$ group. This allowed us to derive for them the analogue of the Regge symmetry.