  In this letter we develope an operator formalism for the $b-c$ systems with
conformal weight $\lambda=1$ defined on a general closed and orientable Riemann
surface. The advantage of our approach is that the Riemann surface is
represented as an affine algebraic curve. In this way it is possible to perform
explicit calculations in string theory at any perturbative order. Besides the
obvious applications in string theories and conformal field theories, (the
$b-c$ systems at $\lambda=1$ are intimately related to the free scalar field
theory), the operator formalism presented here sheds some light also on the
quantization of field theories on Riemann surfaces. In fact, we are able to
construct explicitly the vacuum state of the $b-c$ systems and to define
creation and annihilation operators. All the amplitudes are rigorously computed
using simple normal ordering prescriptions as in the flat case.
