Quantum Theories of Dilaton Gravity
  Quantization of two-dimensional dilaton gravity coupled to conformal matter is investigated. Working in conformal gauge about a fixed background metric, the theory may be viewed as a sigma model whose target space is parameterized by the dilaton $\phi$ and conformal factor $\rho$. A precise connection is given between the constraint that the theory be independent of the background metric and conformal invariance of the resulting sigma model. Although the action is renormalizable, new coupling constants must be specified at each order in perturbation theory in order to determine the quantum theory. These constants may be viewed as initial data for the beta function equations. It is argued that not all choices of this data correspond to physically sensible theories of gravity, and physically motivated constraints on the data are discussed. In particular a recently constructed subclass of initial data which reduces the full quantum theory to a soluble Liouville-like theory has energies unbounded from below and thus is unphysical. Possibilities for modifying this construction so as to avoid this difficulty are briefly discussed. 
Steven B. Giddings and Andrew Strominger