The constraint algebra of quantum gravity in the loop representation

TL;DR

This paper demonstrates that the quantum constraint algebra in loop quantum gravity reproduces the classical algebra when regulators are removed, confirming the consistency of the loop representation with classical gravity.

Contribution

It provides a direct computation showing the quantum constraint algebra matches the classical Poisson algebra in the loop representation of quantum gravity.

Findings

Quantum commutator algebra reproduces classical Poisson bracket algebra

Calculation illustrates computational techniques for the loop representation

Supports consistency of loop quantum gravity with classical constraints

Abstract

We study the algebra of constraints of quantum gravity in the loop representation based on Ashtekar's new variables. We show by direct computation that the quantum commutator algebra reproduces the classical Poisson bracket one, in the limit in which regulators are removed. The calculation illustrates the use of several computational techniques for the loop representation.