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Reduced models for quantum gravity

The preceding talks given at this conference have dealt mainly with general ideas for, main problems of and techniques for the task of quantizing gravity canonically. Since one of the major motivations to arrange for this meeting was that it should serve as a beginner's introduction to canonical quantum gravity, we regard it as important to demonstrate the usefulness of the formalism by means of applying it to simplified models of quantum gravity, here formulated in terms of Ashtekar's new variables. From the various, completely solvable, models that have been discussed in the literature we choose those that we consider as most suitable for our pedagogical reasons, namely 2+1 gravity and the spherically symmetric model. The former model arises from a dimensional, the latter from a Killing reduction of full 3+1 gravity. While 2+1 gravity is usually treated in terms of closed topologies without boundary of the initial data hypersurface, the toplogy for the spherically symmetric system is chosen to be asymptotically flat. Finally, 2+1 gravity is more suitably quantized using the loop representation while spherically symmetric gravity is easier to quantize via the self-dual representation. Accordingly, both types of reductions, both types of topologies and both types of representations that are mainly employed in the literature in the context of the new variables come into practice. What makes the discussion especially clear is the fact that for both models the reduced phase space turns out to be finitely dimensional.