Quantization of the reduced phase space of two-dimensional dilaton gravity

We study some two-dimensional dilaton gravity models using the formal theory of partial differential equations. This allows us to prove that the reduced phase space is two dimensional without an explicit construction. By using a convenient (static) gauge we reduce the theory to coupled ordinary differential equations and we are able to derive for some potentials of interest closed-form solutions. We use an effective (particle) Lagrangian for the reduced field equations in order to quantize the system in a finite-dimensional setting leading to an exact partial differential Wheeler-DeWitt equation instead of a functional one. A WKB approximation for some quantum states is computed and compared with the classical Hamilton-Jacobi theory. The effect of minimally coupled matter is examined. © 1996 The American Physical Society.