Observables II : Quantum Observables

In this work we discuss the notion of observable - both quantum and classical - from a new point of view. In classical mechanics, an observable is represented as a function (measurable, continuous or smooth), whereas in (von Neumann's approach to) quantum physics, an observable is represented as a bonded selfadjoint operator on Hilbert space. We will show in the present part II and the forthcoming part III of this work that there is a common structure behind these two different concepts. If $\mathcal{R}$ is a von Neumann algebra, a selfadjoint element $A \in \mathcal{R}$ induces a continuous function $f_{A} : \mathcal{Q}(\mathcal{P(R)}) \to \mathbb{R}$ defined on the \emph{Stone spectrum} $\mathcal{Q}(\mathcal{P(R)})$ (\cite{deg3}) of the lattice $\mathcal{P(R)}$ of projections in $\mathcal{R}$. $f_{A}$ is called the observable function corresponding to $A$. The aim of this part is to study observable functions and its various characterizations.