Classical and Quantum Aspects of 1 + 1 Gravity

We present a classification of all global solutions (with Lorentzian signature) for any general 2D dilaton gravity model. For generic choices of potential-like terms in the Lagrangian one obtains maximally extended solutions on arbitrary non-compact two-manifolds, including various black-hole and kink configurations. We determine all physical quantum states in a Dirac approach. In some cases the spectrum of the (black-hole) mass operator is found to be sensitive to the signature of the theory, which may be relevant in view of current attempts to implement a generalized Wickrotation in 4D quantum gravity. There is some good news and some bad news about the subject of this talk. To start with the bad news: I am going to speak about pure gravity-Yang-Mills systems. This means that matter fields are not included (with the exception of dilaton fields, which are not regarded as matter fields in our context). Furthermore, I am going to speak about (1+1) dimensional theories. As we seemingly live in a (3 + 1) dimensional universe, this might not be precisely what one eventually is interested in. And here is the good news: We can give a comprehensive treatment of the subject, encompassing a large variety of different gravity models. We deduce a method to calculate all classical solutions of all of these models on the local as well as on the global level. And in some setting, made more precise later in this talk, we are able to calculate all physical quantum states and to analyze the quantum spectra of observables. We arrive at a family supported by Fonds zur Förderung der wissenschaftlichen Forschung (FWF), project P10221-PHY. talk delivered at Journees Relativistes ‘96 (Ascona, Switzerland).