A Systematic Derivation of the Riemannian Barrett-Crane Intertwiner

TL;DR

This paper provides a systematic derivation of the Riemannian Barrett-Crane intertwiner, offering an alternative proof of its uniqueness by rigorously imposing the cross-simplicity constraint in quantum gravity.

Contribution

It introduces a systematic derivation method for the Barrett-Crane intertwiner, emphasizing the rigorous imposition of the cross-simplicity constraint, and confirms its uniqueness.

Findings

The Barrett-Crane intertwiner is uniquely derived for Riemannian gravity.

The derivation offers an alternative proof of the intertwiner's uniqueness.

Rigorous imposition of the cross-simplicity constraint is achieved.

Abstract

The Barrett-Crane intertwiner for the Riemannian general relativity is systematically derived by solving the quantum Barrett-Crane constraints corresponding to a tetrahedron (except for the non-degeneracy condition). It was shown by Reisenberger that the Barrett-Crane intertwiner is the unique solution. The systematic derivation can be considered as an alternative proof of the uniqueness. The new element in the derivation is the rigorous imposition of the cross-simplicity constraint.