Symmetry reduction for quantizeddiffeomorphism-invariant theories of connections

Abstract

Given a symmetry group acting on a principal fibre bundle, symmetricstates of the quantum theory of a diffeomorphism-invariant theory ofconnections on this fibre bundle are defined. These symmetricstates, equipped with a scalar product derived from theAshtekar-Lewandowski measure for loop quantum gravity, form aHilbert space of their own. Restriction to this Hilbert space yieldsa quantum symmetry reduction procedure within the framework ofspin-network states, the structure of which is analysed in detail. Threeillustrating examples are discussed: reduction of (3+1)- to(2+1)-dimensional quantum gravity, spherically symmetric quantumelectromagnetism and spherically symmetric quantum gravity. In thelatter system the eigenvalues of the area operator applied to thespherically symmetric spin-network states have the form An∝(n(n + 2))1/2, n = 0,1,2,..., giving An∝n for large n.This result clarifies (and reconciles) the relationship between themore complicated spectrum of the general (non-symmetric) areaoperator in loop quantum gravity and the old Bekenstein proposal thatAn∝n.