The length operator in Loop Quantum Gravity

TL;DR

This paper introduces a new length operator in Loop Quantum Gravity, explores its properties, spectrum, and semiclassical behavior, enhancing the understanding of quantum geometry in the theory.

Contribution

It presents the first detailed construction and analysis of a length operator within Loop Quantum Gravity, including its spectrum and semiclassical properties.

Findings

The length operator has a discrete spectrum.

It is diagonalized by specific superpositions of spin network states.

The operator's semiclassical properties are explicitly verified.

Abstract

The dual picture of quantum geometry provided by a spin network state is discussed. From this perspective, we introduce a new operator in Loop Quantum Gravity - the length operator. We describe its quantum geometrical meaning and derive some of its properties. In particular we show that the operator has a discrete spectrum and is diagonalized by appropriate superpositions of spin network states. A series of eigenstates and eigenvalues is presented and an explicit check of its semiclassical properties is discussed.