The Area Operator in the Spherically Symmetric Sector of Loop Quantum Gravity

Abstract

Utilizing the previously established general formalism for quantum symmetry reduction in the framework of loop quantum gravity the spectrum of the area operator acting on spherically symmetric states in 4 dimensional pure gravity is investigated. The analysis requires a careful treatment of partial gauge fixing in the classical symmetry reduction and of the reinforcement of SU(2)-gauge invariance for the quantization of the area operator. The eigenvalues of that operator applied to spherically symmetric spin network states have the form A_n propor. sqrt{n(n+2)}, n=0,1,2..., giving A_n propor. n for large n. The result clarifies (and reconciles!) the relationship between the more complicated spectrum of the general (non-symmetric) area operator in loop quantum gravity and the old Bekenstein proposal that A_n propor. n.