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

Journal of Mathematical Physics

Fuzzy observables in quantum mechanics

S. Twareque Ali, Gérard G. Emch

Published Feb 1, 1974• 111 citations
Publisher

Abstract

The formalism of covariant conditional expectations is described as leading to an operational definition of generalized observables in quantum mechanics, wide enough to account for the fuzziness inherent in actual measurement processes, relative to a multidimensional physical continuum. As an application, a position operator for the photon is defined and its intrinsic fuzziness is discussed.

DOI
10.1063/1.1666616

Related papers

The Intersections Based on Joint Observables In Fuzzy Probability

Murat Ozcek

“Fuzzy probability theory” appeared as a smooth extension of classical probability theory in 1995. It was expected that it will be of great importance in quantum mechanics, but the theory doesn’t keep its development as it was expected. This necessitates revising some of its fundamental basic concepts. We argue that if quantum probability theory should have less constrained than classical probability theory as can be seen in the case of joint random variables, we surely need to weaken the definition of the intersection operation. In this paper, discuss the definition validity in quantum probability theory and to discuss the consistency of the given definitions with the whole theory and the possibility to have a more suitable definition.

•Jan 1, 2023•0 citations

Conditional expectations in Quantum Mechanics and causal interpretations: the Bohm momentum as a best predictor

Raymond Brummelhuis

Given a normalized state-vector $ψ$, we define the conditional expectation $\mathbb{E }_{ψ} (A | B ) $ of a Hermitian operator $A $ with respect to a strongly commuting family of self-adjoint operators $B $ as the best approximation, in the operator mean square norm associated to $ψ$, of $A $ by a real-valued function of $B . $ A fundamental example is the conditional expectation of the momentum operator $P $ given the position operator $X $, which is found to be the Bohm momentum. After developing the Bohm theory from this point of view we treat conditional expectations with respect to general $B $, which we apply to non-relativistic spin 1/2-particles. We derive the dynamics of the conditional expectations of momentum and spin with respect to position and the third spin component. These dynamics can be interpreted in terms of classical continuum mechanics as a two-component fluid whose components carry intrinsic angular momentum. Interpreting the joint spectrum of the conditioning operators as a space of beables, we can introduce a classical-stochastic particle dynamics on this space which is compatible with the time-evolution of the Born probability, by combining the de Broglie-Bohm guidance condition with a Markov jump process, following an idea of J. Bell. This results in a new Bohm-type model for particles with spin. A basic problem is that such auxiliary particle dynamics are far from unique. We finally examine the relation of our conditional expectations with the conditional expectations of the theory of $C^* $-algebras and, as an application, derive a general evolution equation for conditional expectations for operators acting on finite dimensional Hilbert spaces. Two appendices re-interpret the classical Bohm model as an integrable constrained Hamiltonian system, and provide the details of the two-fluid interpretation.

arXiv:2411.08532•Nov 13, 2024•0 citations

Fuzzy position and momentum observables

Teiko Heinonen

The notion of a fuzzy observable is a possible way to describe imprecise measurements. We show that fuzzy versions of the usual position and momentum observables are exactly those observables which satisfy the natural covariance and invariance requirements. We characterize the class of fuzzy position and momentum observables which are informationally equivalent with the canonical position and momentum observables.

AIP conference proceedings•Jan 1, 2005•0 citations

A physical distinction between a covariant and non covariant description of the measurement process in relativistic quantum theories

Rodolfo Gambini, Rafael A. Porto

Causality imposes strong restrictions on the type of operators that may be observables in relativistic quantum theories. In fact, causal violations arise when computing conditional probabilities for certain partial causally connected measurements using the standard non covariant procedure. Here we introduce another way of computing conditional probabilities, based on an intrinsic covariant relational order of the events, which differs from the standard one when these type of measurements are included. This alternative procedure is compatible with a wider and very natural class of operators without breaking causality. If some of these measurements could be implemented in practice, as predicted by our formalism, the non covariant, conventional approach should be abandoned.

arXiv:quant-ph/0209044•Sep 5, 2002•0 citations

Intrinsic and Operational Observables in Quantum Mechanics

Piotr Kochański, Krzysztof Wódkiewicz

The concept of intrinsic and operational observables in quantum mechanics is introduced. It is argued that, in any realistic description of a quantum measurement that includes a detecting device, it is possible to construct from the statistics of the recorded raw data a set of operational quantities that correspond to the intrinsic quantum mechanical observable. Using the concept of the propensity and the associated operational positive operator valued measure (POVM) a general description of the operational algebra of quantum mechanical observables is derived for a wide class of realistic detection schemes. This general approach is illustrated by the example of an operational Malus measurement of the spin phases and by an analysis of the operational homodyne detection of the phase of an optical field with a squeezed vacuum in the unused ports.

arXiv:quant-ph/9607008•Jul 10, 1996•0 citations

On the operational and algebraic quantum correlations

Shun Umekawa, Jaeha Lee

We investigate the intrinsic ambiguity in the definition of correlation functions arising from the inevitable invasiveness of quantum measurements. While algebraic correlations defined as expectation values of products of observables are widely used, their relationship to operational ones defined through actual measurement procedures remain unclear. We demonstrate that the differences among various definitions of correlation functions and those among their underlying (quasi-)joint probability distributions are bounded above by a quantitative measure of measurement invasiveness. We further obtain a lower bound on the discrepancy among operational and algebraic (quasi-)joint probability distributions, providing a new form of the uncertainty relation. In addition, we identify an equivalence condition under which operational and algebraic correlations coincide. As an application, we analyze the quantum violation of the Leggett-Garg inequality and clarify the structural origin of the equivalence among different approaches to observing the violation, including sequential projective measurements and weak-measurement. Our results provide an operational foundation for the commonly used algebraic concepts of quantum theory.

quant-pharXiv:2603.04332•Mar 5, 2026•0 citations
The formalism of covariant conditional expectations is described as leading to an operational definition of generalized observables in quantum mechanics, wide enough to account for the fuzziness inherent in actual measurement processes, relative to a multidimensional physical continuum. As an application, a position operator for the photon is defined and its intrinsic fuzziness is discussed.
“Fuzzy probability theory” appeared as a smooth extension of classical probability theory in 1995. It was expected that it will be of great importance in quantum mechanics, but the theory doesn’t keep its development as it was expected. This necessitates revising some of its fundamental basic concepts. We argue that if quantum probability theory should have less constrained than classical probability theory as can be seen in the case of joint random variables, we surely need to weaken the definition of the intersection operation. In this paper, discuss the definition validity in quantum probability theory and to discuss the consistency of the given definitions with the whole theory and the possibility to have a more suitable definition.
Given a normalized state-vector ψψψ, we define the conditional expectation $\mathbb{E }_{ψ} (A | B ) $ of a Hermitian operator $A $ with respect to a strongly commuting family of self-adjoint operators $B $ as the best approximation, in the operator mean square norm associated to ψψψ, of $A $ by a real-valued function of $B . $ A fundamental example is the conditional expectation of the momentum operator $P $ given the position operator $X $, which is found to be the Bohm momentum. After developing the Bohm theory from this point of view we treat conditional expectations with respect to general $B $, which we apply to non-relativistic spin 1/2-particles. We derive the dynamics of the conditional expectations of momentum and spin with respect to position and the third spin component. These dynamics can be interpreted in terms of classical continuum mechanics as a two-component fluid whose components carry intrinsic angular momentum. Interpreting the joint spectrum of the conditioning operators as a space of beables, we can introduce a classical-stochastic particle dynamics on this space which is compatible with the time-evolution of the Born probability, by combining the de Broglie-Bohm guidance condition with a Markov jump process, following an idea of J. Bell. This results in a new Bohm-type model for particles with spin. A basic problem is that such auxiliary particle dynamics are far from unique. We finally examine the relation of our conditional expectations with the conditional expectations of the theory of $C^* $-algebras and, as an application, derive a general evolution equation for conditional expectations for operators acting on finite dimensional Hilbert spaces. Two appendices re-interpret the classical Bohm model as an integrable constrained Hamiltonian system, and provide the details of the two-fluid interpretation.
Causality imposes strong restrictions on the type of operators that may be observables in relativistic quantum theories. In fact, causal violations arise when computing conditional probabilities for certain partial causally connected measurements using the standard non covariant procedure. Here we introduce another way of computing conditional probabilities, based on an intrinsic covariant relational order of the events, which differs from the standard one when these type of measurements are included. This alternative procedure is compatible with a wider and very natural class of operators without breaking causality. If some of these measurements could be implemented in practice, as predicted by our formalism, the non covariant, conventional approach should be abandoned.
The concept of intrinsic and operational observables in quantum mechanics is introduced. It is argued that, in any realistic description of a quantum measurement that includes a detecting device, it is possible to construct from the statistics of the recorded raw data a set of operational quantities that correspond to the intrinsic quantum mechanical observable. Using the concept of the propensity and the associated operational positive operator valued measure (POVM) a general description of the operational algebra of quantum mechanical observables is derived for a wide class of realistic detection schemes. This general approach is illustrated by the example of an operational Malus measurement of the spin phases and by an analysis of the operational homodyne detection of the phase of an optical field with a squeezed vacuum in the unused ports.
We investigate the intrinsic ambiguity in the definition of correlation functions arising from the inevitable invasiveness of quantum measurements. While algebraic correlations defined as expectation values of products of observables are widely used, their relationship to operational ones defined through actual measurement procedures remain unclear. We demonstrate that the differences among various definitions of correlation functions and those among their underlying (quasi-)joint probability distributions are bounded above by a quantitative measure of measurement invasiveness. We further obtain a lower bound on the discrepancy among operational and algebraic (quasi-)joint probability distributions, providing a new form of the uncertainty relation. In addition, we identify an equivalence condition under which operational and algebraic correlations coincide. As an application, we analyze the quantum violation of the Leggett-Garg inequality and clarify the structural origin of the equivalence among different approaches to observing the violation, including sequential projective measurements and weak-measurement. Our results provide an operational foundation for the commonly used algebraic concepts of quantum theory.