Regularization of the Hamiltonian constraint and the closure of the constraint algebra

TL;DR

This paper examines the regularization process of the Hamiltonian constraint in loop quantum gravity, demonstrating how the constraint algebra closes when properly regulated, ensuring consistency in the quantum theory.

Contribution

It provides a detailed calculation of the regulated Hamiltonian's action on Wilson loops and shows the algebra closes with redefined regulating functions.

Findings

Poisson bracket between Hamiltonian and Diffeomorphism constraints closes with regulated constraints

Detailed derivation of the Hamiltonian constraint's action on Wilson loops

Closure of the constraint algebra with redefined regulating functions

Abstract

In the paper we discuss the process of regularization of the Hamiltonian constraint in the Ashtekar approach to quantizing gravity. We show in detail the calculation of the action of the regulated Hamiltonian constraint on Wilson loops. An important issue considered in the paper is the closure of the constraint algebra. The main result we obtain is that the Poisson bracket between the regulated Hamiltonian constraint and the Diffeomorphism constraint is equal to a sum of regulated Hamiltonian constraints with appropriately redefined regulating functions.