Properties of the Volume Operator in Loop Quantum Gravity I: Results

TL;DR

This paper investigates the spectral properties of the volume operator in Loop Quantum Gravity, focusing on generic graph vertices of valence greater than four, revealing how vertex geometry influences the spectrum and the existence of a volume gap.

Contribution

It extends the analysis of the volume operator to higher valence vertices and characterizes how vertex geometry affects spectral properties and the volume gap in Loop Quantum Gravity.

Findings

Presence of a volume gap depends on vertex embedding.

Spectral properties are influenced by vertex geometry.

Analytical proof of a volume gap for 4-valent vertices.

Abstract

We analyze the spectral properties of the volume operator of Ashtekar and Lewandowski in Loop Quantum Gravity, which is the quantum analogue of the classical volume expression for regions in three dimensional Riemannian space. Our analysis considers for the first time generic graph vertices of valence greater than four. Here we find that the geometry of the underlying vertex characterizes the spectral properties of the volume operator, in particular the presence of a `volume gap' (a smallest non-zero eigenvalue in the spectrum) is found to depend on the vertex embedding. We compute the set of all non-spatially diffeomorphic non-coplanar vertex embeddings for vertices of valence 5--7, and argue that these sets can be used to label spatial diffeomorphism invariant states. We observe how gauge invariance connects vertex geometry and representation properties of the underlying gauge group in a natural way. Analytical results on the spectrum on 4-valent vertices are included, for which the presence of a volume gap is proved. This paper presents our main results; details are provided by a companion paper arXiv:0706.0382v1.