Discrete Quantum Gravity: II. Simplicial complexes, irreps of SL(2,C), and a Lorentz invariant weight in a state sum model

TL;DR

This paper extends the Lorentz invariant state sum model in quantum gravity by classifying simplicial edges, analyzing SL(2,C) irreducible representations, and introducing a new Lorentz invariant weight using zonal spherical functions.

Contribution

It introduces a novel Lorentz invariant weight in the state sum model based on SL(2,C) irreps and zonal spherical functions, advancing the quantum gravity framework.

Findings

Classification of edges and subspaces in simplicial decomposition

Application of SL(2,C) irreps to the model

Introduction of a Lorentz invariant weight using zonal spherical functions

Abstract

In part I of [1] we have developed the tensor and spin representation of SO(4) in order to apply it to the simplicial decomposition of the Barrett-Crane model. We attach to each face of a triangle the spherical function constructed from the Dolginov-Biedenharn function. In part II we apply the same technique to the Lorentz invariant state sum model. We need three new ingredients: the classification of the edges and the corresponding subspaces that arises in the simplicial decomposition, the irreps of SL(2,C) and its isomorphism to the bivectors appearing in the 4-simplices, the need of a zonal spherical function from the intertwining condition of the tensor product for the simple representations attached to the faces of the simplicial decomposition.