A Note on the Asymptotic Limit of the Four Simplex

TL;DR

This paper demonstrates that the asymptotic limit of the Barrett-Crane models inherently satisfies the essential constraints, including the Schlaffi identity, without imposing them explicitly.

Contribution

It provides a direct analysis showing the asymptotic limit naturally yields bivectors meeting Barrett-Crane constraints, simplifying the understanding of the model's geometric properties.

Findings

Bivectors satisfying Barrett-Crane constraints can be extracted from the asymptotic limit.

The Schlaffi identity is implied by the asymptotic limit, not imposed.

The approach offers a more direct understanding of the model's geometric structure.

Abstract

Recently the asymptotic limit of the Barrett-Crane models has been studied by Barrett and Steele. Here by a direct study, I show that we can extract the bivectors which satisfy the essential Barrett-Crane constraints from the asymptotic limit. Because of this the Schlaffi identity is implied by the asymptotic limit, rather than to be imposed as a constraint.