Tensor and spin representations of SO(4) and discrete quantum gravity

TL;DR

This paper develops mathematical representations of the SO(4) group and applies them to quantum gravity, providing new insights into the Barrett-Crane model and the asymptotic behavior of the Regge action.

Contribution

It constructs the fundamental, tensor, and spinor representations of SO(4) and applies these to enhance the understanding of the Barrett-Crane quantum gravity model.

Findings

Complete realization of the weight function for the partition function

New geometrical interpretation of the asymptotic limit of the Regge action

Mathematical framework for SO(4) representations in quantum gravity

Abstract

Starting from the defining transformations of complex matrices for the SO(4) group, we construct the fundamental representation and the tensor and spinor representations of the group SO(4). Given the commutation relations for the corresponding algebra, the unitary representations of the group in terms of the generalized Euler angles are constructed. These mathematical results help us to a more complete description of the Barrett-Crane model in Quantum Gravity. In particular a complete realization of the weight function for the partition function is given and a new geometrical itnerpretation of the asymptotic limit for the Regge action is presented.