Hadamard states and two-dimensional gravity

We have used a two-dimensional analog of the Hadamard state-condition to st udy the local constraints on the two-point function of a linear quantum fie ld conformally coupled to a two-dimensional gravitational background. We de velop a dynamical model in which the determination of the state of the quan tum field is essentially related to the determination of a conformal frame. A particular conformal frame is then introduced in which a two-dimensional gravitational equation is established.