Let Pf(x) =−if′(x) and Qf(x) = xf(x) be the canonical operators acting on an appropriate common dense domain in L2(ℝ). The derivations DP(A) = i(PA−AP) and DQ(A) = i(QA−AQ) act on the *-algebra [Ascr ] of all integral operators having smooth kernels of compact support, for example, and one may consider the noncommutative ‘Laplacian’, L = D2P+D2Q, as a linear mapping of [Ascr ] into itself.

L generates a semigroup of normal completely positive linear maps on [Bscr ](L2(ℝ)), and this paper establishes some basic properties of this semigroup and its minimal dilation to an E0-semigroup. In particular, the author shows that its minimal dilation is pure and has no normal invariant states, and he discusses 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, where 1 [les ] n < 1.