Observables III: Classical Observables

TL;DR

This paper explores the structural similarities between classical and quantum observables by representing them as functions on certain spectra, revealing common features in their mathematical frameworks.

Contribution

It extends previous work on quantum observables to classical observables, demonstrating analogous representation results and structural similarities.

Findings

Classical observables can be represented as measurable and continuous functions.

Similar structural features are found in classical and quantum observables.

Representation results mirror those established for quantum observables.

Abstract

In the second part of our work on observables we have shown that quantum observables in the sense of von Neumann, i.e.bounded selfadjoint operators in some von Neumann subalgebra $R$ of $L(H)$, can be represented as bounded continuous functions on the Stone spectrum $Q(R)$ of $R$. Moreover, we have shown that this representation is linear if and only if $R$ is abelian, and that in this case it coincides with the Gelfand transformation of $R$. In this part we discuss classical observables, i.e. measurable and continuous functions, under the same point of view. We obtain results that are quite similar to the quantum case, thus showing up the common structural features of quantum and classical observables.