Summary: THE HEAT FLOW OF THE CCR ALGEBRA
Department of Mathematics
University of California
Berkeley CA 94720, USA
29 February, 2000
Abstract. 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 + D2
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, . . . .