 
Summary: THE HEAT FLOW OF THE CCR ALGEBRA
William Arveson
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(PAAP) and DQ(A) = i(QAAQ) 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 E0semigroup. 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, . . . .
