Discrete quantum gravity: the Lorentz invariant weight for the Barrett-Crane model

TL;DR

This paper develops a Lorentz invariant weight for the Barrett-Crane model in quantum gravity by constructing complexified Clebsch-Gordan coefficients and spherical functions for the Lorentz group.

Contribution

It introduces a new Lorentz invariant weight for the Barrett-Crane model using complexified Clebsch-Gordan coefficients and spherical functions.

Findings

Derived the Lorentz invariant weight for the model.

Constructed complexified Clebsch-Gordan coefficients for SO(3,1).

Established the spherical function as the invariant weight.

Abstract

In a recent paper [1] we have constructed the spin and tensor representations of SO(4) from which the invariant weight can be derived for the Barrett-Crane model in quantum gravity. By analogy with the SO(4) group, we present the complexified Clebsch-Gordan coefficients in order to construct the Biedenharn-Dolginov function for the SO(3,1) group and the spherical function as the Lorentz invariant weight of the model.