Properties of the Volume Operator in Loop Quantum Gravity II: Detailed Presentation

TL;DR

This paper analyzes the spectral properties of the Volume operator in Loop Quantum Gravity at vertices of valence greater than four, using numerical methods and analytical results, to deepen understanding of its eigenvalues and structure.

Contribution

It provides a detailed numerical and analytical analysis of the Volume operator's spectrum at complex vertices, expanding previous work limited to simpler cases.

Findings

Eigenvalues characterized for vertices of valence >4

Analytical spectrum results for 4-valent vertices

Numerical database of volume operator eigenvalues

Abstract

The properties of the Volume operator in Loop Quantum Gravity, as constructed by Ashtekar and Lewandowski, are analyzed for the first time at generic vertices of valence greater than four. The present analysis benefits from the general simplified formula for matrix elements of the Volume operator derived in gr-qc/0405060, making it feasible to implement it on a computer as a matrix which is then diagonalized numerically. The resulting eigenvalues serve as a database to investigate the spectral properties of the volume operator. Analytical results on the spectrum at 4-valent vertices are included. This is a companion paper to arXiv:0706.0469, providing details of the analysis presented there.