Symmetry reduction for quantized diffeomorphism-invariant theories of connections

Given a symmetry group acting on a principal fibre bundle, symmetric states of the quantum theory of a diffeomorphism-invariant theory of connections on this fibre bundle are defined. These symmetric states, equipped with a scalar product derived from the Ashtekar-Lewandowski measure for loop quantum gravity, form a Hilbert space of their own. Restriction to this Hilbert space yields a quantum symmetry reduction procedure within the framework of spin-network states, the structure of which is analysed in detail. Three illustrating examples are discussed: reduction of (3 + 1)- to (2 + 1)-dimensional quantum gravity, spherically symmetric quantum electromagnetism and spherically symmetric quantum gravity. In the latter system the eigenvalues of the area operator applied to the spherically symmetric spin-network states have the form A_n ∝ √n(n + 2), n = 0, 1, 2,..., giving A_n ∝ n for large n. This 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 ∝ n.