On the constraint algebra of quantum gravity in the loop representation

TL;DR

This paper investigates the algebra of constraints in loop quantum gravity, showing that expressing constraints as shift operators simplifies their algebraic relations, which is crucial for consistent quantization.

Contribution

It introduces a new approach to representing the constraint algebra using shift operators, simplifying calculations in the loop representation of quantum gravity.

Findings

Constraint algebra simplifies with shift operators

Explicit computation aligns with previous formal results

Enhances understanding of quantum gravity constraints

Abstract

Although an important issue in canonical quantization, the problem of representing the constraint algebra in the loop representation of quantum gravity has received little attention. The only explicit computation was performed by Gambini, Garat and Pullin for a formal point-splitting regularization of the diffeomorphism and Hamiltonian constraints. It is shown that the calculation of the algebra simplifies considerably when the constraints are expressed not in terms of generic area derivatives but rather as the specific shift operators that reflect the geometric meaning of the constraints.