Three Dimensional Canonical Quantum Gravity

General aspects of vielbein representation, ADM formulation and canonical quantization of gravity are reviewed using pure gravity in three dimensions as a toy model. The classical part focusses on the role of observers in general relativity, which will later be identified with quantum observers. A precise definition of gauge symmetries and a classification of inequivalent solutions of Einstein's equations in dreibein formalism is given as well. In the quantum part the construction of the physical Hilbert space is carried out explicitly for a torus and cylinder type space manifold, which has not been done so far. Some conceptual problems of quantum gravity are discussed from the point of view of an observer sitting inside the universe.

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