Introduction

Several speakers at this conference have emphasized the conceptual difficulties of quantum gravity (see particularly [1-3]). As they pointed out, when we bring in gravity, some of the basic premises of quantum field theory have to undergo radical changes: we must learn to do physics in the absence of a background space-time geometry. This immediately leads to a host of technical difficulties as well, for the familiar mathematical methods of quantum field theory are deeply rooted in the availability of a fixed space-time metric, which, furthermore, is generally taken to be flat. The purpose of this contribution is to illustrate how these conceptual and technical difficulties can be overcome.

For concreteness, we will use a specific non-perturbative approach and, furthermore, limit ourselves to just one set of issues: exploration of the nature of quantum geometry. Nonetheless, the final results have a certain degree of robustness and the constructions involved provide concrete examples of ways in which one can analyze genuine field theories, with an infinite number of degrees of freedom, in absence of a background metric. As we will see, the underlying diffeomorphism invariance is both a curse and a blessing. On the one hand, since there is so little background structure, concrete calculations are harder and one is forced to invent new regularization methods. On the other hand, when one does succeed, the final form of results is often remarkably simple since the requirement of diffeomorphism invariance tends to restrict the answers severely.