Smooth Fields of Operators and Some Examples Coming from Canonical Quantization

TL;DR

This paper develops a framework for smooth fields of operators inspired by smooth fields of Hilbert spaces, providing examples from canonical quantization and introducing a notion of smooth fields of C*-algebras.

Contribution

It introduces a new concept of smooth fields of operators, connects it with canonical quantization, and extends the framework to smooth fields of C*-algebras with examples.

Findings

Defined a connection on fields of operators using Hilbert space connections.

Provided examples where constants of motion lead to smooth operator fields.

Established formulas for derivatives of operator fields via Poisson connections.

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.