Reduced models for quantum gravity

As outlined in various other lectures given at this meeting, it seems that a quantum theory of gravity can only be constructed in a non perturbative manner (compare, in particular, Ashtekar’s lectures). Because of that, no calculations, as for example cross sections, decay rates and so on, can be done for the full theory of 3+1 gravity unless one has the full solution space of the quantum constraints and, derived from that, the physical Hilbert space. For a non-gravitational quantum field theory that can be attacked via a perturbative approach one can make quantitative predictions and even make estimates of the error due to higher order corrections while for quantum gravity ’one would have to consider all orders’. So for the former theories one does have a very good idea of how the exact quantum theory should look like and this is important because intuition gives rise to new lines of attack. Hence unfortunately, for the quantum theory of gravity, we lack this general picture of how the exact theory should look like completely. The only way that might help to uncover some of the secrets of how to solve the technical and/or conceptual problems of quantum gravity seems to be to study model systems, ideally those that can be solved exactly. Of course, the lessons that models teach us might be totally misleading and extreme care is due when transferring results from the model to the physical theory of full quantum gravity. This is the point of view that we adopt in the sequel : The models that we are going to discuss capture some of the technical and conceptual problems of gravity and we will pin these down. We will attempt at drawing some conclusions from the solutions we found but we stress the limitations that arise from the various special features of the models we choose and which are not shared by the full theory of quantum gravity.