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 analogous features 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 conformal gauge by using the technique developed in the two dimensional quantum gravity. We argue the relations to the ADM formalism. The physical state conditions reduce to the usual Wheeler-DeWitt equations when the dilaton φ2 (= e−2φ) is large enough compared with κ = (N − 51/2)/12, where N is the number of matter fields. This corresponds to the large mass limit in the black hole geometry. A singularity appears at φ2 = κ(> 0). The final stage of the black hole evaporation corresponds to the region φ2 ∼ κ, where the Liouville term becomes important, which just comes from the measure of the metrics. If κ < 0, the singularity disappears.