A New Recursion Relation for the 6j-Symbol

TL;DR

This paper introduces a novel recursion relation for the square of the 6j-symbol, linking it to tetrahedral geometry and potentially advancing quantum gravity models.

Contribution

It presents a new recursion formula for the 6j-symbol's square and explores its implications for quantum gravity and tetrahedral geometry.

Findings

Recursion relation characterizes tetrahedral closure in asymptotic limit

Method can be generalized to full amplitudes in quantum gravity models

Enhances understanding of 6j-symbols in quantum geometry

Abstract

The 6j-symbol is a fundamental object from the re-coupling theory of SU(2) representations. In the limit of large angular momenta, its asymptotics is known to be described by the geometry of a tetrahedron with quantized lengths. This article presents a new recursion formula for the square of the 6j-symbol. In the asymptotic regime, the new recursion is shown to characterize the closure of the relevant tetrahedron. Since the 6j-symbol is the basic building block of the Ponzano-Regge model for pure three-dimensional quantum gravity, we also discuss how to generalize the method to derive more general recursion relations on the full amplitudes.