Towards an Anomaly-Free Quantum Dynamics for a Weak Coupling Limit of Euclidean Gravity: Diffeomorphism Covariance

TL;DR

This paper demonstrates how to achieve diffeomorphism covariance in the quantum Hamiltonian constraint for a U(1)^3 gauge theory derived from Euclidean gravity, ensuring anomaly freedom and covariance in the continuum limit.

Contribution

It introduces modifications to previous constructions to ensure diffeomorphism covariance while maintaining anomaly freedom in the quantum dynamics of the theory.

Findings

Achieved diffeomorphism covariance of the Hamiltonian constraint operator.

Maintained anomaly-free continuum limit of the constraint algebra.

Provided a consistent quantum representation respecting spatial covariance.

Abstract

The G-->0 limit of Euclidean gravity introduced by Smolin is described by a generally covariant U(1)xU(1)xU(1) gauge theory. In an earlier paper, Tomlin and Varadarajan constructed the quantum Hamiltonian constraint of density weight 4/3 for this U(1)xU(1)xU(1) theory so as to produce a non-trivial anomaly free LQG-type representation of the Poisson bracket between a pair of Hamiltonian constraints. These constructions involved a choice of regulating coordinate patches. The use of these coordinate patches is in apparent conflict with spatial diffeomorphism covariance. In this work we show how an appropriate choice of coordinate patches together with suitable modifications of these constructions results in the diffeomorphism covariance of the continuum limit action of the Hamiltonian constraint operator, while preserving the anomaly free property of the continuum limit action of its commutator.