Constraint algebra in LQG reloaded : Toy model of an Abelian gauge theory - II Spatial Diffeomorphisms

TL;DR

This paper advances the quantization of the constraint algebra in a toy model of Abelian gauge theory within Loop Quantum Gravity, demonstrating a diffeomorphism covariant Hamiltonian constraint and addressing longstanding issues in canonical LQG.

Contribution

It provides a full representation of the constraint algebra in a loop quantized framework, including spatial diffeomorphisms, for a toy Abelian gauge model.

Findings

Representation of full constraint algebra achieved

Hamiltonian constraint shown to be diffeomorphism covariant

Progress towards resolving Hamiltonian constraint issues in LQG

Abstract

In [1] we initiated an approach towards quantizing the Hamiltonian constraint in Loop Quantum Gravity (LQG) by requiring that it generates an anomaly-free representation of constraint algebra off-shell. We investigated this issue in the case of a toy model of a 2+1-dimensional $U(1)^{3}$ gauge theory, which can be thought of as a weak coupling limit of Euclidean three dimensional gravity. However in [1] we only focused on the most non-trivial part of the constraint algebra that involves commutator of two Hamiltonian constraints. In this paper we continue with our analysis and obtain a representation of full constraint algebra in loop quantized framework. We show that there is a representation of the Diffeomorphism group with respect to which the Hamiltonian constraint quantized in [1] is diffeomorphism covariant. Our work can be thought of as a potential first step towards resolving some long standing issues with the Hamiltonian constraint in canonical LQG.