We analyze the quantum two-dimensional dilaton gravity model, which is described by the $\frac{\mathrm{SL}\mathrm{}(2,R)\mathrm{}}{\mathrm{U}\mathrm{}\left(1\right)\mathrm{}}$ gauged Wess-Zumino-Witten model deformed by a (1,1) operator. We show that the curvature singularity does not appear when the central charge $c_{\mathrm{matter}}$ of the matter fields is given by $22<\mathrm{}{c}_{\mathrm{matter}}<24\mathrm{}$ When $22<\mathrm{}{c}_{\mathrm{matter}}<24\mathrm{}$ the matter shock waves, whose energy-momentum tensors are given by $T_{\mathrm{matter}}\propto \delta (x^{+}-x_0^{}{}_{}{}^{+})$, create a kind of wormholes, i.e., causally disconnected regions. Most of the quantum information in past null infinity is lost in future null infinity but the lost information would be carried by the wormholes. We also discuss the problem of defining the mass of quantum black holes. On the basis of the argument by Regge and Teitelboim, we show that the ADM mass measured by the observer who lives in one of the asymptotically flat regions is finite and does not vanish in general. On the other hand, the Bondi mass is ill defined in this model. Instead of the Bondi mass, we consider the mass measured by observers who live in an asymptotically flat region at first. A class of observers finds the mass of the black hole created by a shock wave changes as the observers' proper time goes by, i.e., they observe Hawking radiation. The measured mass vanishes after the infinite proper time and the black hole evaporates completely. Therefore the total Hawking radiation is positive even when $N<\mathrm{}24\mathrm{}$.

• Received 25 June 1993

DOI:https://doi.org/10.1103/PhysRevD.49.4066