General Covariance and Free Fields in Two Dimensions

We investigate the canonical equivalence of a matter-coupled 2D dilaton gravity theory defined by the action functional $S = \int d^2x \sqrt{-g} (R\phi + V(\phi) - 1/2 H(\phi ) (\nablaf)^2)$, and a free field theory. When the scalar field $f$ is minimally coupled to the metric field$(H(\phi)=1)$ the theory is equivalent, up to a boundary contribution,to a theory of three free scalar fields with indefinite kinetic terms, irrespective of the particular form of the potential $V(\phi)$. If the potential is an exponential function of the dilaton one recovers a generalized form of the classical canonical transformation of Liouville theory. When $f$ is a dilaton coupled scalar $(H(\phi)=\phi)$ and the potential is an arbitrary power of the dilaton the theory is also canonically equivalent to a theory of three free fields with a Minkowskian target space. In the simplest case $(V(\phi)=0)$ we provide an explicit free field realization of the Einstein-Rosen midisuperspace. The Virasoro anomaly and the consistence of the Dirac operator quantization play a central role in our approach.