Smooth Fields of Operators and Some Examples Coming from Canonical Quantization

Abstract

We introduce a notion of smooth fields of operators following the notion of smooth fields of Hilbert spaces recently defined by L. Lempert and R. Szőoke [16]. Formally, if ∇ is the connection of a smooth field of Hilbert spaces we show that ∇ = [∇, •] defines a connection on a suitable space of fields of operators. In order to provide examples we prove that, if u is a suitable constant of motion of h(q, p) = q 2 (i.e. {u, h} = 0), then Op(u) is a smooth field of operators over the open interval (0, ∞), where Op denotes the canonical quantization (Weyl calculus). Moreover, in such case we show that we can compute derivatives using the formula ∇X 0 (Op(u)) = Op( ∇X 0 (u)), where ∇ is a Poisson connection on the Poisson algebra of constants of motion and X0 = 2λ ∂ ∂λ . We also introduce a notion of smooth field of C * -algebras and we give an example using Hilbert modules theory.