  Let P and Q be the canonical operators acting on the Hilbert space of all L^2
functions on the real line, defined appropriately on a common dense domain. The
derivations D_P(A) = i(PA - AP) and D_Q(A) = i(QA - AQ) act on the *-algebra of
all integral operators having smooth kernels of compact support, for example,
and one may consider the noncommutative "Laplacian"
  L(A) = D_P^2(A) + D_Q^2(A), as a linear mapping of this *-algebra into
itself.
  L generates a semigroup of normal completely positive maps on B(H), and we
establish some basic properties of this semigroup and its minimal dilation to
an E_0-semigroup. In particular, we show that the minimal dilation is pure, has
no normal invariant states, and we discuss the significance of those facts for
the interaction theory developed in a previous paper (appearing in the current
issue of Comm. Math. Phys.).
  There are similar results for the canonical commutation relations with n
degrees of freedom, n = 2, 3, ....
