Wave Functional of Quantum Black Holes in Two Dimensions

The Wheeler-DeWitt method is applied to the quantization of the 1 + 1 dimensional dilaton gravity coupled with the conformal matter fields. Exact solutions to the WD equations are found, which are classically interpreted as right(left)-moving black holes. E-mail address: [email protected] The two dimensional dilaton gravity recently attracts much attention as a sample model of the four dimensional theory. Since the pioneering work by Callan et al. [1] many authors have studied the two dimensional models (see ref.[2] for a review) and attempted to unravel the mysteries concerning the black hole evaporation and the resulting information loss [3], which imply a fundamental conflict between the gravity and the quantum theories. In quantizing the four dimensional Einstein gravity the canonical procedure leads to a beautiful scheme known as the Wheeler-Dewitt (WD) quantization [4] [5]. In the two dimensional dilaton gravity a WD-like equation is derived [6] as a physical state condition. A non-vanishing hamiltonian is derived in a unitary gauge and a WD equation of Schrödinger type is obtained [7]. There are some other approaches to the full quantum theory [9]. As far as we know, however, an exact solution to the WD equation in the dilaton gravity has not yet been found. In this note we report that there exist exact functional solutions to the WD equation of the two dimensional dilaton gravity coupled with the conformal matter fields (we do not consider the cosmological term). The wave functionals are interpreted as representing quantum black holes moving toward right or left directions, where the conformal matter fields are present inside. We make the assumption that the spatial dimension is compact, which seems favourable for the cosmological applications. The absence of the asymptotic Minkowski realm, however, brings forth a difficult conceptual problem in interpreting the wave functional, since then the ADM approach [8] cannot be applied. To make contact with the extrinsic time in the WD formalism one put a time dependent gauge condition which can be reduced to a genuine constraint by a suitable canonical transformation. This results in a non-vanishing hamiltonian [7]. However, the extrinsic time has no gauge independent meaning and, in a purely formal point of view, cannot be connected directly to physical observables.