The volume operator in covariant quantum gravity

TL;DR

The paper constructs a covariant volume operator within the Euclidean spin-foam framework and proves its equivalence to the LQG volume operator, complementing the area-volume correspondence. It derives the physical boundary Hilbert space by enforcing linear simplicity and closure constraints, showing that the resulting space is isomorphic to the SU(2) spin-network space of canonical LQG. The boundary space is built from gamma-simple Spin(4) representations, and, after weakly imposing the constraints, yields K_ph = Inv_SU(2) of the extended boundary data, aligning covariant and canonical descriptions. The volume observable is then defined in terms of B (or equivalently J) and shown to reduce to the LQG volume operator on K_ph, with the expected gamma-dependence. This work strengthens the bridge between covariant spin-foam models and canonical LQG, demonstrating consistent geometric operator spectra across formulations.

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.