A numerical study of spectral properties of the area operator in loop quantum gravity

Abstract

The lowest 37000 eigenvalues of the area operator in loop quantum gravity is calculated and studied numerically. We obtain an asymptotical formula for the eigenvalues as a function of their sequential number. The multiplicity of the lowest few hundred eigenvalues is also determined and the smoothed spectral density is calculated. The spectral density is presented for various number of vertices, edges and SU(2) representations. A scaling form of spectral density is found, being a power law for one vertex, while following an exponential for several vertices. The latter case is explained on the basis of the one vertex spectral density.