  We aim at extending the definition of the Weyl calculus to an infinite
dimensional setting, by replacing the phase space $ \mathbb{R}^{2n}$ by $B^2$,
where $(i,H,B)$ is an abstract Wiener space. A first approach is to generalize
the integral definition using the Wigner function. The symbol is then a
function defined on $B^2$ and belonging to a $L^1$ space for a gaussian
measure, the Weyl operator is defined as a quadratic form on a dense subspace
of $L^2(B)$. For example, the symbol can be the stochastic extension on $B^2$,
in the sense of L. Gross, of a function $F$ which is continuous and bounded on
$H^2$.
  In the second approach, this function $F$ defined on $H^2$ satisfies
differentiability conditions analogous to the finite dimensional ones. One
needs to introduce hybrid operators acting as Weyl operators on the variables
of finite dimensional subset of $H$ and as Anti-Wick operators on the rest of
the variables. The final Weyl operator is then defined as a limit and it is
continuous on a $L^2$ space. Under rather weak conditions, it is an extension
of the operator defined by the first approach.
  We give examples of monomial symbols linking this construction to the
classical pseudodifferential operators theory and other examples related to
other fields or previous works on this subject.
