The heat flow of the CCR algebra

Abstract

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, ....