Simplification of the Spectral Analysis of the Volume Operator in Loop Quantum Gravity

Abstract

The Volume Operator plays a crucial role in the definition of the quantum dynamics of Loop Quantum Gravity (LQG). Efficient calculations for dynamical problems of LQG can therefore be performed only if one has sufficient control over the volume spectrum. While closed formulas for the matrix elements are currently available in the literature, these are complicated polynomials in 6j symbols which in turn are given in terms of Racah's formula which is too complicated in order to perform even numerical calculations for the semiclassically important regime of large spins. Hence, so far not even numerically the spectrum could be accessed. In this article we demonstrate that by means of the Elliot -- Biedenharn identity one can get rid of all the 6j symbols for any valence of the gauge invariant vertex, thus immensely reducing the computational effort. We use the resulting compact formula to study numerically the spectrum of the gauge invariant 4 -- vertex. The techniques derived in this paper could be of use also for the analysis of spin -- spin interaction Hamiltonians of many -- particle problems in atomic and nuclear physics.