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 L2(R). The derivations DP (A) = i(PA−AP ) and DQ(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 = D2 P + D 2 Q as a linear mapping of A into itself. L generates a semigroup of normal completely positive linear maps on B(L2(R)), and we establish some basic properties of this semigroup and its minimal dilation to an E0-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, . . . . 1. Discussion, basic results. Consider the canonical operators P,Q acting on an appropriate common dense domain in L(R) P = 1 i · d dx , Q = multiplication by x. These operators can be used to define unbounded derivations (say on the dense ∗algebra A of all integral operators having kernels which are smooth and of compact support) by DP (X) = i(PX −XP ), DQ(X) = i(QX −XQ), X ∈ A. Thinking of these derivations as noncommutative counterparts of ∂/∂x and ∂/∂y we define a “Laplacian” L : A → A by (1.1) L = D P +D 2 Q. On appointment as a Miller Research Professor in the Miller Institute for Basic Research in Science. Support is also acknowledged from NSF grant DMS-9802474 1