Master Constraint Operator in Loop Quantum Gravity

TL;DR

This paper introduces a positive, symmetric master constraint operator in loop quantum gravity, establishing its mathematical properties and supporting the viability of the master constraint programme in the field.

Contribution

The paper constructs and analyzes a new master constraint operator in loop quantum gravity, demonstrating its self-adjointness and compatibility with existing proposals.

Findings

The operator is positive and symmetric.

It admits a Friedrichs self-adjoint extension.

Supports the master constraint programme in loop quantum gravity.

Abstract

We introduce a master constraint operator $\hat{\mathbf{M}}$ densely defined in the diffeomorphism invariant Hilbert space in loop quantum gravity, which corresponds classically to the master constraint in the programme. It is shown that $\hat{\mathbf{M}}$ is positive and symmetric, and hence has its Friedrichs self-adjoint extension. The same conclusion is tenable for an alternative master operator $\hat{\mathbf{M'}}$, whose quadratic form coincides with the one proposed by Thiemann. So the master constraint programme for loop quantum gravity can be carried out in principle by employing either of the two operators.