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. Sz\H{o}oke arXiv:1004.4863(2) . Formally, if $\nabla$ is the connection of a smooth field of Hilbert spaces we show that $\hat\nabla=[\nabla,\cdot]$ 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 $\mathfrak{Op}(u)$ is a smooth field of operators over the open interval $(0,\infty)$, where $\mathfrak{Op}$ denotes the canonical quantization (Weyl calculus). Moreover, in such case we show that we can compute derivatives using the formula $\hat\nabla_{X_0}(\mathfrak{Op}(u))=\mathfrak{Op}(\tilde\nabla_{X_0}(u))$, where $\tilde\nabla$ is a Poisson connection on the Poisson algebra of constants of motion and $X_0=2\lambda\frac{\partial}{\partial \lambda}$. We also introduce a notion of smooth field of $C^*$-algebras and we give an example using Hilbert modules theory.