Gravitational Collapse in 1+1 Dimensions and Quantum Gravity †

We investigate the quantum theory of 1+1 dimensional dilaton gravity, which is an interesting toy model of the black hole dynamics. The functional measures of gravity part are explicitly evaluated and derive the Wheeler-DeWitt like equations as physical state conditions. In ADM formalism the measures are very ambiguous, but in our formalism they are explicitly defined. Then the new features which are not seen in ADM formalism come out. A singularity appears at ϕ 2 = κ(> 0), where κ = (N − 51/2)/12 and N is the number of matter fields. At the final stage of the black hole evaporation, the Liouville term becomes important, which just comes from the measures of the fields. Behind the singularity the quantum mechanical region κ > ϕ 2 > 0 extends, where the sign of the kinetic term in the Wheeler-DeWitt like equation changes. If κ < 0, the singularity disappears. We briefly discuss the possibility of gravitational tunneling and the issue of the information loss.