Closed formula for the matrix elements of the volume operator in canonical quantum gravity

TL;DR

This paper derives a closed-form expression for the matrix elements of the volume operator in canonical Lorentzian quantum gravity, introducing a new state-dependent regularization technique that preserves diffeomorphism covariance.

Contribution

It provides a novel closed formula for the volume operator's matrix elements and presents a state-dependent regularization method maintaining covariance.

Findings

Derived a closed formula for volume operator matrix elements

Introduced a state-dependent regularization technique

Connected results to Ashtekar and Lewandowski's volume operator

Abstract

We derive a closed formula for the matrix elements of the volume operator for canonical Lorentzian quantum gravity in four spacetime dimensions in the continuum in a spin-network basis. We also display a new technique of regularization which is state dependent but we are forced to it in order to maintain diffeomorphism covariance and in that sense it is natural. We arrive naturally at the expression for the volume operator as defined by Ashtekar and Lewandowski up to a state independent factor.