Constraint algebra in LQG reloaded : Toy model of a U(1)^{3} Gauge Theory I

TL;DR

This paper constructs an anomaly-free Hamiltonian constraint operator in a 3D U(1)^3 gauge theory within Loop Quantum Gravity, demonstrating a consistent quantum algebra that mirrors classical Poisson brackets.

Contribution

It introduces a new quantization scheme for the Hamiltonian constraint in LQG that achieves anomaly freedom and aligns with classical structure functions.

Findings

Anomaly-free representation of the constraint algebra achieved.

Hamiltonian constraint operator constructed with a geometric interpretation.

Continuum limit defined on a subspace of distributions.

Abstract

We analyze the issue of anomaly-free representations of the constraint algebra in Loop Quantum Gravity (LQG) in the context of a diffeomorphism-invariant gauge theory in three spacetime dimensions. We construct a Hamiltonian constraint operator whose commutator matches with a quantization of the classical Poisson bracket involving structure functions. Our quantization scheme is based on a geometric interpretation of the Hamiltonian constraint as a generator of phase space-dependent diffeomorphisms. The resulting Hamiltonian constraint at finite triangulation has a conceptual similarity with the "mu-bar"-scheme in loop quantum cosmology and highly intricate action on the spin-network states of the theory. We construct a subspace of non-normalizable states (distributions) on which the continuum Hamiltonian constraint is defined which leads to an anomaly-free representation of the Poisson bracket of two Hamiltonian constraints in loop quantized framework.