On the volume simplicity constraint in the EPRL spin foam model

TL;DR

This paper introduces a quantum formulation of the volume simplicity constraint in the EPRL spin foam model, which becomes non-trivial for complex boundary graphs and aligns with geometricity conditions in the semi-classical limit.

Contribution

It presents a novel quantum version of the volume simplicity constraint based on a volume formula involving bivectors and boundary knotting, applicable to complex boundary graphs.

Findings

Constraint is trivial for 4-simplex but non-trivial for complex graphs.

In semi-classical limit, constraint matches geometricity conditions.

Provides a new perspective on quantum geometric constraints in spin foam models.

Abstract

We propose a quantum version of the quadratic volume simplicity constraint for the EPRL spin foam model. It relies on a formula for the volume of 4-dimensional polyhedra, depending on its bivectors and the knotting class of its boundary graph. While this leads to no further condition for the 4-simplex, the constraint becomes non-trivial for more complicated boundary graphs. We show that, in the semi-classical limit of the hypercuboidal graph, the constraint turns into the geometricity condition observed recently by several authors.