Hermiticity of the Volume Operators in Loop Quantum Gravity

TL;DR

This paper rigorously proves the hermiticity and other key properties of the volume operators in Loop Quantum Gravity using an angular momentum approach, clarifying subtleties and providing a formal spectrum computation method.

Contribution

It introduces a simplified, rigorous proof of hermiticity for the volume operators and details their spectral properties and matrix representations in Loop Quantum Gravity.

Findings

Volume operators are hermitian, real, symmetric, and positive semi-definite.

Eigenvalues of the operators come in pairs for even dimensions, with zero eigenvalues for odd dimensions.

Explicit computational examples confirm the spectral properties and match previous results.

Abstract

The aim of this article is to provide a rigorous-but-simple steps to prove the hermiticity of the volume operator of Rovelli-Smolin and Ashtekar-Lewandowski using the angular momentum approach, as well as pointing out some subleties which have not been given a lot of attention previously. Besides of being hermitian, we also prove that both volume operators are real, symmetric, and positive semi-definite, with respect to the inner product defined on the Hilbert space over SU(2). Other special properties follows from this fact, such as the possibility to obtain real orthonormal eigenvectors. Moreover, the matrix representation of the volume operators are degenerate, such that the real positive eigenvalues always come in pairs for even dimension, with an additional zero if the dimension is odd. As a consequence, one has a freedom in choosing the orthonormal eigenvectors for each 2-dimensional eigensubspaces. Furthermore, we provide a formal procedure to obtain the spectrum and matrix representation of the volume operators. In order to compare our procedure with the earlier ones existing in the literature, we give explicit computational examples for the case of monochromatic quantum tetrahedron, where the eigenvalues agrees with the standard earlier procedure.