The volume operator in covariant quantum gravity

TL;DR

This paper derives the volume operator in covariant quantum gravity, demonstrating its equivalence to the loop quantum gravity operator, and clarifies the implementation of classical constraints in the covariant spin-foam formulation.

Contribution

It provides the explicit derivation of the volume operator in covariant quantum gravity and shows its consistency with loop quantum gravity, along with a simplified approach to the constraints.

Findings

The volume operator in covariant quantum gravity matches the loop quantum gravity operator.

The boundary Hilbert space can be derived directly from classical constraints.

All constraints vanish weakly on the boundary Hilbert space.

Abstract

A covariant spin-foam formulation of quantum gravity has been recently developed, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity. In particular, the geometrical observable giving the area of a surface has been shown to be the same as the one in loop quantum gravity. Here we discuss the volume observable. We derive the volume operator in the covariant theory, and show that it matches the one of loop quantum gravity, as does the area. We also reconsider the implementation of the constraints that defines the model: we derive in a simple way the boundary Hilbert space of the theory from a suitable form of the classical constraints, and show directly that all constraints vanish weakly on this space.