  An operator formalism for bosonic $\beta-\gamma$ systems on arbitrary
algebraic curves is introduced. The classical degrees of freedom are identified
and their commutation relations are postulated. The explicit realization of the
algebra formed by the fields is given in a Hilbert space equipped with a
bilinear form. The construction is based on the "gaussian" representation for
$\beta-\gamma$ systems on the complex sphere [Alvarez-Gaum\' e et al, Nucl.
Phys. B 311 (1988) 333]. Detailed computations are provided for the two and
four points correlation functions.
