Review on hermiticity of the volume operators in Loop Quantum Gravity

The aim of this article is to provide rigorous-but-simple steps to prove the hermiticity of the volume operator of Rovelli–Smolin and Ashtekar–Lewandowski from the angular momentum calculation, as well as pointing out some subtleties 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 of a fixed graph. Other special properties follow from this fact, such as the possibility to obtain real orthonormal eigenvectors. Moreover, the matrix representations of the volume operators on a vertex could have degeneracy, such that the real positive eigenvalues always come in pairs for even dimension, with an additional zero if the dimension is odd. This is general for the Ashtekar–Lewandowski volume operator, but special for the Rovelli–Smolin case. 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 agree with the standard earlier procedure, but differs on the eigenvectors.