The Heat Flow of the CCR Algebra

Let Pf ( x ) = − if ′( x ) and Qf ( x ) = xf ( x ) be the canonical operators acting on an appropriate common dense domain in L 2 (R). The derivations D P ( A ) = i ( PA − AP ) and D Q ( A ) = i ( QA − AQ ) act on the *‐algebra 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 A into itself. L generates a semigroup of normal completely positive linear maps on B( L 2 (R)), and this paper establishes some basic properties of this semigroup and its minimal dilation to an E 0 ‐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 ⩽ n < ∞. 2000 Mathematics Subject Classification 46L57 (primary), 46L53, 46L65 (secondary).