[Paper Review] Observables III: Classical Observables

de Groote, F Hans|arXiv (Cornell University)|Jan 6, 2006

Quantum Mechanics and Applications3 citations

TL;DR

This paper extends the spectral representation of observables to classical systems, showing that classical observables—measurable and continuous functions—can be represented as bounded continuous functions on the Stone spectrum $ Q(R) $, mirroring the quantum case. The key result is that this representation is linear if and only if the algebra $ R $ is abelian, recovering the classical Gelfand transformation.

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.

Motivation & Objective

• To extend the spectral representation framework from quantum observables to classical observables.
• To investigate whether classical observables—measurable and continuous functions—can be represented on the Stone spectrum $ Q(R) $, analogous to the quantum case.
• To determine under what conditions the representation of classical observables on $ Q(R) $ is linear.
• To clarify the structural similarities between quantum and classical observables through this unified spectral perspective.

Proposed method

• Represent classical observables as bounded continuous functions on the Stone spectrum $ Q(R) $ of a von Neumann algebra $ R $.
• Use the Stone spectrum $ Q(R) $, a topological space associated with the Boolean algebra of projections in $ R $, to define the state space for classical observables.
• Apply techniques from functional analysis and Stone duality to establish the correspondence between functions on $ Q(R) $ and classical observables.
• Analyze the linearity of the representation map from observables to functions on $ Q(R) $, showing it holds precisely when $ R $ is abelian.
• Demonstrate that in the abelian case, the representation coincides with the classical Gelfand transformation.
• Compare the structure of the representation for classical and quantum observables to highlight unifying features.

Experimental results

Research questions

• RQ1 Can classical observables be represented as continuous functions on the Stone spectrum $ Q(R) $, similar to quantum observables?
• RQ2 Under what conditions is the representation of classical observables on $ Q(R) $ linear?
• RQ3 How does the representation of classical observables on $ Q(R) $ relate to the Gelfand transformation in the abelian case?
• RQ4 What structural parallels exist between the spectral representation of classical and quantum observables?
• RQ5 Does the Stone spectrum framework unify the treatment of classical and quantum observables?

Key findings

• Classical observables—measurable and continuous functions—can be represented as bounded continuous functions on the Stone spectrum $ Q(R) $, analogous to the quantum case.
• The representation of classical observables on $ Q(R) $ is linear if and only if the von Neumann algebra $ R $ is abelian.
• In the abelian case, the representation on $ Q(R) $ coincides exactly with the Gelfand transformation of $ R $.
• The framework reveals deep structural similarities between classical and quantum observables through their spectral representations.
• The Stone spectrum provides a unified topological setting for analyzing both classical and quantum observables.