Semiclassical Limits of Extended Racah Coefficients

TL;DR

This paper investigates the geometry and asymptotics of extended Racah coefficients, revealing their relation to Lorentzian tetrahedra and their potential application in 3D Lorentzian quantum gravity models.

Contribution

It introduces an extension of Racah coefficients linked to Lorentzian tetrahedra and derives asymptotic formulas similar to known quantum gravity models.

Findings

Extension relates to positive discrete series of SU(1,1)

Asymptotics resemble Ponzano-Regge formulae

Potential application in Lorentzian quantum gravity

Abstract

We explore the geometry and asymptotics of extended Racah coeffecients. The extension is shown to have a simple relationship to the Racah coefficients for the positive discrete unitary representation series of SU(1,1) which is explicitly defined. Moreover, it is found that this extension may be geometrically identified with two types of Lorentzian tetrahedra for which all the faces are timelike. The asymptotic formulae derived for the extension are found to have a similar form to the standard Ponzano-Regge asymptotic formulae for the SU(2) 6j symbol and so should be viable for use in a state sum for three dimensional Lorentzian quantum gravity.