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**Symmetry reduction for quantized diffeomorphism-invariant theories of connections.**
*(English)*
Zbl 0977.83019

The paper contains a description of the method of quantum symmetry reduction for diffeomorphism-invariant theories of connections. Usually the reduction of the \((3+1)\)-dimensional theory is performed on the classical level, to simplify the model to be quantized. The authors propose the method of reduction of the loop quantized \((3+1)\)-dimensional theory. The central role in the construction plays the restrictive definition of the symmetric state introduced by the authors. The quantum reduction is illustrated by several examples: spherically symmetric electromagnetism, \((2+1)\)-dimensional gravity, spherically symmetric quantum gravity. This approach has several motivations: a) the classically reduced Schwarzschild system has not been quantized using loop quantization techniques, b) it gives a possibility to find a way to calculate degeneration of energy levels (which is crucial in the calculation of black hole entropy), c) it possibly gives a new tool for the study of cosmological models within loop quantum gravity. In addition to the strict analysis of the problem the paper contains interesting background information and comments.

Reviewer: A.Frydryszak (Wrocław)