Discrete Quantum Gravity: I. Zonal spherical functions of the representationsof the SO(4,R) group with respect to the SU(2) subgroup and their application to the Barrett-Crane model

TL;DR

This paper develops the mathematical foundation of zonal spherical functions for SO(4,R) and applies them to define amplitudes in the Barrett-Crane quantum gravity model, enhancing the understanding of quantum 4-simplices.

Contribution

It constructs the zonal spherical functions of SO(4,R) with respect to the SU(2) subgroup and applies these functions to the Barrett-Crane model in quantum gravity.

Findings

Derived the fundamental, tensor, and spinor representations of SO(4,R).

Constructed the unitary representations using generalized Euler angles.

Applied zonal spherical functions to quantum 4-simplex amplitudes.

Abstract

Starting from the defining transformations of complex matrices for the $SO(4,R)$ group, we construct the fundamental representation and the tensor and spinor representations of the group $SO(4,R)$. Given the commutation relations for the corresponding algebra, the unitary representations of the group in terms of the generalized Euler angles are constructed. The crucial step for the Barrett-Crane model in Quantum Gravity is the description of the amplitude for the quantum 4-simplex that is used in the state sum partition function. We obtain the zonal spherical functions for the construction of the SO(4,R) invariant weight and associate them to the triangular faces of the 4-simplices.