Constraints and Solutions of Quantum Gravity in Metric Representation

TL;DR

This paper develops a regularised Wheeler-De Witt operator ensuring anomaly-free constraint algebra in quantum gravity, and finds exact solutions based on volume and curvature functionals of three-manifolds.

Contribution

It introduces a regularisation approach for the Wheeler-De Witt operator and identifies exact solutions involving volume and curvature functionals.

Findings

Anomaly-free constraint algebra achieved for specific wavefunctions.

Exact solutions expressed as functionals of volume and average curvature.

Regularisation method applicable to quantum gravity models.

Abstract

We construct the regularised Wheeler-De Witt operator demanding that the algebra of constraints of quantum gravity is anomaly free. We find that for a subset of all wavefunctions being integrals of scalar densities this condition can be satisfied. We proceed to finding exact solutions of quantum gravity being of the form of functionals of volume and average curvature of compact three-manifold.