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:1408.5697 · Lecture notes in computer science

Quantum Mechanics: Harbinger of a Non-commutative Probability Theory?

Basil J. Hiley

Published Jan 1, 2014• 3 citations
arXivHTMLPDF
DOI
10.1007/978-3-642-54943-4_2

Related papers

On non-commutativity in quantum theory (I): from classical to quantum probability

Luca Curcuraci

A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. In this article, we present a critical analysis on the role of non-commutativity in quantum theory, focusing on its consequences in the probabilistic description. Typically, a random phenomenon is described using the measure-theoretic formulation of probability theory. Such a description can also be done using algebraic methods, which are capable to deal with non-commutative random variables (like in quantum mechanics). Here we propose a method to construct a non-commutative probability theory starting from an ordinary measure-theoretic description of probability. This will be done using the entropic uncertainty relations between random variables, in order to evaluate the presence of non-commutativity in their algebraic description.

arXiv:1803.04913•Mar 13, 2018•0 citations

Quantum Mechanics as Quantum Probability

Bassano Vacchini

Graduate texts in physics•Jan 1, 2024•0 citations

Beyond Kolmogorov: Towards a Non-Commutative Measure Theory of Quantum Probability Spaces

Revista, Zen, MATH, 10

This paper explores the limitations of Kolmogorov's measure-theoretic foundation of probability in the context of quantum mechanics and proposes an extension towards a non-commutative framework. We delve into the mathematical structures underlying quantum probability spaces, focusing on operator algebras and their associated states. We analyze the challenges in defining joint probabilities for non-commuting observables and present an overview of alternative approaches based on operator algebras, quantum logic, and non-commutative integration. Furthermore, we investigate the applications of non-commutative measure theory in quantum information theory, quantum field theory, and the foundations of quantum mechanics, highlighting the potential for a more comprehensive understanding of probabilistic phenomena in quantum systems.

•Dec 5, 2025•0 citations

Quantum Mechanics as an Exotic Probability Theory

Saul Youssef

arXiv:quant-ph/9509004•Springer eBooks•Jan 1, 1996•10 citations

A possible origin of the quantum mechanical behavior

Luca Curcuraci

The thesis is about the peculiar probabilistic structure of quantum mechanics. Typically, a random phenomenon is described using the measure-theoretic formulation of probability theory. Such a description can also be done using algebraic methods, which are capable to deal with non-commutative random variables. A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. A method to construct a non-commutative probability theory starting from an ordinary measure-theoretic description of probability is proposed and applied to a series of model to reobtain the quantum behaviour od a particle. In particular, using ordinary probability theory, the kinematics of a point-like particle jumping at random over a discrete random space is studied. After the removal of the random space from the model, the position and velocity of the particle do not commute, when represented as operators on the same Hilbert space. Using a suitable random process for the space one is able to recover non-relativistic quantum mechanics in a suitable limit.

•Feb 15, 2019•0 citations

Quantum Mechanics as a Theory of Probability

Itamar Pitowsky

arXiv:quant-ph/0510095•Springer eBooks•Nov 2, 2006•129 citations
A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. In this article, we present a critical analysis on the role of non-commutativity in quantum theory, focusing on its consequences in the probabilistic description. Typically, a random phenomenon is described using the measure-theoretic formulation of probability theory. Such a description can also be done using algebraic methods, which are capable to deal with non-commutative random variables (like in quantum mechanics). Here we propose a method to construct a non-commutative probability theory starting from an ordinary measure-theoretic description of probability. This will be done using the entropic uncertainty relations between random variables, in order to evaluate the presence of non-commutativity in their algebraic description.
This paper explores the limitations of Kolmogorov's measure-theoretic foundation of probability in the context of quantum mechanics and proposes an extension towards a non-commutative framework. We delve into the mathematical structures underlying quantum probability spaces, focusing on operator algebras and their associated states. We analyze the challenges in defining joint probabilities for non-commuting observables and present an overview of alternative approaches based on operator algebras, quantum logic, and non-commutative integration. Furthermore, we investigate the applications of non-commutative measure theory in quantum information theory, quantum field theory, and the foundations of quantum mechanics, highlighting the potential for a more comprehensive understanding of probabilistic phenomena in quantum systems.
The thesis is about the peculiar probabilistic structure of quantum mechanics. Typically, a random phenomenon is described using the measure-theoretic formulation of probability theory. Such a description can also be done using algebraic methods, which are capable to deal with non-commutative random variables. A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. A method to construct a non-commutative probability theory starting from an ordinary measure-theoretic description of probability is proposed and applied to a series of model to reobtain the quantum behaviour od a particle. In particular, using ordinary probability theory, the kinematics of a point-like particle jumping at random over a discrete random space is studied. After the removal of the random space from the model, the position and velocity of the particle do not commute, when represented as operators on the same Hilbert space. Using a suitable random process for the space one is able to recover non-relativistic quantum mechanics in a suitable limit.