\abstract
Let $Pf(x)=-if^\prime(x)$ and
$Qf(x) = xf(x)$ be the canonical
operators acting on an appropriate common dense
domain in $L^2(\Bbb R)$. The derivations
$D_P(A)=i(PA - AP)$ and $D_Q(A)=i(QA-AQ)$ act on the $*$-algebra
$\Cal A$ of all integral operators having smooth kernels of compact
support, for example, and one may consider the
noncommutative ``Laplacian" $L=D_P^2 + D_Q^2$ as a
linear mapping of $\Cal A$ into itself.
$L$ generates a semigroup of normal completely positive
linear maps on $\Cal B(L^2(\Bbb R))$, and we establish some basic
properties of this semigroup and its minimal dilation to
an $E_0$-semigroup. In particular, we show that its minimal
dilation is pure, has no normal invariant states, and in section 3
we discuss
the significance of those facts for the interaction theory
introduced in a previous paper.
There are similar results for the canonical commutation relations with
$n$ degrees of freedom, $n=2,3,\dots$.
\endabstract