A length operator for canonical quantum gravity

TL;DR

This paper introduces a well-defined length operator in four-dimensional Lorentzian quantum gravity that is discrete, self-adjoint, and provides detailed metric information, advancing the understanding of quantum geometric measurements.

Contribution

The paper constructs a new length operator in quantum gravity that is densely defined, free of factor ordering issues, and has a discrete spectrum, enhancing quantum geometric tools.

Findings

The length operator is densely defined and self-adjoint.

Its spectrum is discrete and quantized in units of the Planck length.

The operator encodes all metric components, enabling new weave states.

Abstract

We construct an operator that measures the length of a curve in four-dimensional Lorentzian vacuum quantum gravity. We work in a representation in which a $SU(2)$ connection is diagonal and it is therefore surprising that the operator obtained after regularization is densely defined, does not suffer from factor ordering singularities and does not require any renormalization. We show that the length operator admits self-adjoint extensions and compute part of its spectrum which like its companions, the volume and area operators already constructed in the literature, is purely discrete and roughly is quantized in units of the Planck length. The length operator contains full and direct information about all the components of the metric tensor which faciliates the construction of a new type of weave states which approximate a given classical 3-geometry.