QUANTUM-THEORY OF DILATON GRAVITY IN 1+1 DIMENSIONS

We discuss the quantum theory of 1 + 1 dimensional dilaton gravity, which is an interesting model with features analogous to the spherically symmetric gravitational systems in 3 + 1 dimensions. The functional measures over the metrics and the dilaton field are explicitly evaluated and the diffeomorphism invariance is completely fixed in the conformal gauge by using the technique developed in two dimensional quantum gravity. We derive the Wheeler-DeWitt like equations as physical state conditions. In the ADM formalism the measures of fields are very ambiguous, but in our formalism they are explicitly defined. A singularity appears at psi2 = kappa(> 0), where kappa = 1/12 (N - 51/2) and N is the number of matter fields. The final stage of the black hole evaporation corresponds to the region psi2 approximately kappa, where the Liouville term becomes important, which just comes from the measure of the metrics. If kappa < 0, the singularity disappears.