  A derivation operator and a divergence operator are defined on the algebra of
bounded operators on the symmetric Fock space over the complexification of a
real Hilbert space $\eufrak{h}$ and it is shown that they satisfy similar
properties as the derivation and divergence operator on the Wiener space over
$\eufrak{h}$. The derivation operator is then used to give sufficient
conditions for the existence of smooth Wigner densities for pairs of operators
satisfying the canonical commutation relations. For
$\eufrak{h}=L^2(\mathbb{R}_+)$, the divergence operator is shown to coincide
with the Hudson-Parthasarathy quantum stochastic integral for adapted
integrable processes and with the non-causal quantum stochastic integrals
defined by Lindsay and Belavkin for integrable processes.
