Properties of the Volume Operator in Loop Quantum Gravity II: Detailed Presentation
arXiv:0706.0382 · doi:10.1088/0264-9381/25/6/065002
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.
Companion to arXiv:0706.0469. Version as published in CQG in 2008. More compact presentation. Sign factor combinatorics now much better understood in context of oriented matroids, see arXiv:1003.2348, where also important remarks given regarding sigma configurations. Subsequent computations revealed some minor errors, which do not change qualitative results but modify some numbers presented here
References in corpus (5)
- Algebraic Quantum Gravity (AQG) I. Conceptual Setup
- Spectral Analysis of the Volume Operator in Loop Quantum Gravity
- Algebraic Quantum Gravity (AQG) II. Semiclassical Analysis
- Algebraic Quantum Gravity (AQG) III. Semiclassical Perturbation Theory
- Properties of the Volume Operator in Loop Quantum Gravity I: Results
Cited by in corpus (26)
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- From Classical To Quantum Gravity: Introduction to Loop Quantum Gravity
- Loop Quantum Gravity Phenomenology: Linking Loops to Observational Physics
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- Spectra of geometric operators in three-dimensional LQG: From discrete to continuous
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- Hermiticity of the Volume Operators in Loop Quantum Gravity
- Categorical Quantum Volume Operator