Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:math-ph/0507019

Observables

Hans F. de Groote

Published Jul 7, 2005• 0 citations
arXivHTMLPDF

Abstract

This is the extended version of a talk presented at the J.W.Goethe Universitaet Frankfurt a. M. and at the same time a preview at a forthcoming extensive publication on the same subject. It is shown that there is a common background structure for quantum and classical observables. Moreover, a contextual generalization of the notion of quantum observable is proposed.

DOI
10.48550/arxiv.math-ph/0507019

Related papers

Less (precision) is more (information): quantum information in fuzzy probability theory

Paul Busch

A comparison of structural features of quantum and classical physical theories, such as the information capacity of systems subject to these theories, requires a common formal framework for the presentation of corresponding concepts (such as states, observables, probability, entropy). Such a framework is provided by the notion of statistical model developed in the convexity approach to statistical physical theories. Here we use statistical models to classify and survey all possible types of embedding and extension of quantum probabilistic theories subject to certain reasonable constraints. It will be shown that the so-called canonical classical extension of quantum mechanics is essentially the only `good' representation of the quantum statistical model in a classical framework. All quantum observables are thus identified as fuzzy classical random variables.

arXiv:quant-ph/0401027•Jan 7, 2004•3 citations

Observables and Their Values

K. Kong Wan

Q26(1) Discuss the fundamental differences between classical observables and quantum observables.

•Aug 27, 2020•0 citations

On the relation between classical and quantum observables

Abhay Ashtekar

Communications in Mathematical Physics•Feb 1, 1980•33 citations

Observables II : Quantum Observables

Hans F. de Groote

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.

arXiv:math-ph/0509075•Sep 30, 2005•11 citations

Structural Ordering from Projection to Time

Pasquale Camelia

This work clarifies the structural ordering that connects non-invertible finite-rank projection, operational distinguishability, and the emergence of time within the Quantum Gauge Theory of Time (QGT). It is shown that a finite-rank saturated observable domain cannot internally support a Noetherian formulation of conservation. When conservation is imposed as a structural requirement, the introduction of a temporal ordering parameter becomes unavoidable. Time thus emerges as the minimal extension required for structural coherence, without restoring suppressed information or introducing dynamics. The work further clarifies the role of bulk irrationality and of the invariant √5 in stabilizing finite-rank saturation, and positions the QGT as a necessary extension of this foundational framework. Preliminary version made publicly available on Zenodo. This manuscript has notyet undergone peer review and may be subject to revision

•Dec 31, 2025•0 citations

Quantum observables in classical frameworks

Enrico G. Beltrametti, Slawomir Bugajski

International Journal of Theoretical Physics•Aug 1, 1995•33 citations
This is the extended version of a talk presented at the J.W.Goethe Universitaet Frankfurt a. M. and at the same time a preview at a forthcoming extensive publication on the same subject. It is shown that there is a common background structure for quantum and classical observables. Moreover, a contextual generalization of the notion of quantum observable is proposed.
A comparison of structural features of quantum and classical physical theories, such as the information capacity of systems subject to these theories, requires a common formal framework for the presentation of corresponding concepts (such as states, observables, probability, entropy). Such a framework is provided by the notion of statistical model developed in the convexity approach to statistical physical theories. Here we use statistical models to classify and survey all possible types of embedding and extension of quantum probabilistic theories subject to certain reasonable constraints. It will be shown that the so-called canonical classical extension of quantum mechanics is essentially the only `good' representation of the quantum statistical model in a classical framework. All quantum observables are thus identified as fuzzy classical random variables.
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 R\mathcal{R}R is a von Neumann algebra, a selfadjoint element A∈RA \in \mathcal{R}A∈R induces a continuous function fA:Q(P(R))→Rf_{A} : \mathcal{Q}(\mathcal{P(R)}) \to \mathbb{R}fA​:Q(P(R))→R defined on the \emph{Stone spectrum} Q(P(R))\mathcal{Q}(\mathcal{P(R)})Q(P(R)) (\cite{deg3}) of the lattice P(R)\mathcal{P(R)}P(R) of projections in R\mathcal{R}R. fAf_{A}fA​ is called the observable function corresponding to AAA. The aim of this part is to study observable functions and its various characterizations.
This work clarifies the structural ordering that connects non-invertible finite-rank projection, operational distinguishability, and the emergence of time within the Quantum Gauge Theory of Time (QGT). It is shown that a finite-rank saturated observable domain cannot internally support a Noetherian formulation of conservation. When conservation is imposed as a structural requirement, the introduction of a temporal ordering parameter becomes unavoidable. Time thus emerges as the minimal extension required for structural coherence, without restoring suppressed information or introducing dynamics. The work further clarifies the role of bulk irrationality and of the invariant √5 in stabilizing finite-rank saturation, and positions the QGT as a necessary extension of this foundational framework. Preliminary version made publicly available on Zenodo. This manuscript has notyet undergone peer review and may be subject to revision