Symmetry reduction for quantized diffeomorphism-invariant theories of connections

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Abstract
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 An∝(n(n + 2))1/2, n = 0,1,2,..., giving An∝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 An∝n.

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