## 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 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.

Original language | English (US) |
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Pages (from-to) | 3009-3043 |

Number of pages | 35 |

Journal | Classical and Quantum Gravity |

Volume | 17 |

Issue number | 15 |

DOIs | |

State | Published - Aug 7 2000 |

## All Science Journal Classification (ASJC) codes

- Physics and Astronomy (miscellaneous)