We evaluate the influence functional for two dimensional models of dilaton
gravity. This functional is exactly computed when the conformal invariance is
preserved, and it can be written as the difference between the Liouville
actions on each closed-time-path branch plus a boundary term. From the
influence action we derive the covariant form of the semiclassical field
equations. We also study the quantum to classical transition in cosmological
backgrounds. In the conformal case we show that the semiclassical approximation
is not valid because there is no imaginary part in the influence action.
Finally we show that the inclusion of the dilaton loop in the influence
functional breaks conformal invariance and ensures the validity of the
semiclassical approximation.Comment: 25 pages, RevTex, no figures. Minor changes has been added. To appear
in Physical Review
