$\hat{Q}$ operator for canonical quantum gravity

TL;DR

This paper analyzes the spectral properties of the $ abla$ operator in canonical quantum gravity, demonstrating it is a well-defined, self-adjoint, diagonal operator in the spin network basis, applicable to both kinematical and gauge-invariant Hilbert spaces.

Contribution

It provides a complete spectral analysis of the $ abla$ operator, establishing its self-adjointness and diagonalization in the spin network basis within quantum gravity.

Findings

$ abla$ operator is diagonal in the spin network basis.

It is a self-adjoint operator on the Hilbert space.

Results hold for both kinematical and gauge-invariant spaces.

Abstract

We study the properties of $\hat{Q}[\omega]$ operator on the kinematical Hilbert space ${\cal H}$ for canonical quantum gravity. Its complete spectrum with respect to the spin network basis is obtained. It turns out that $\hat{Q}[\omega]$ is diagonalized in this basis, and it is a well defined self-adjoint operator on ${\cal H}$. The same conclusions are also tenable on the SU(2) gauge invariant Hilbert space with the gauge invariant spin network basis.