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

Physically Consistent Probability Density in Noncommutative Quantum Mechanics

Andrey Gomes Martins, Ronaldo V. Lobato

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

We formulate and solve the problem of boundary values in non-relativistic quantum mechanics in non-commutative boundary spaces-times. The formalism developed can be useful to the formulation of the boundary value problem in in Noncommutative Quantum Mechanics

Related papers

Quantum Mechanics With Noncommutative Amplitudes

Gregory W. Moore

We formulate a Born rule for families of quantum systems parametrized by a noncommutative space of control parameters. The resulting formalism may be viewed as a generalization of quantum mechanics where overlaps take values in a noncommutative algebra, rather than in the complex numbers.

arXiv:1701.07746•Jan 26, 2017•3 citations

Formulation, interpretation and application of non-commutative quantum mechanics

Frederik G. Scholtz, Laure Gouba, A Hafver +1 more

In analogy with conventional quantum mechanics, non-commutative quantum mechanics is formulated as a quantum system on the Hilbert space of Hilbert-Schmidt operators acting on non-commutative configuration space. It is argued that the standard quantum mechanical interpretation based on Positive Operator Valued Measures, provides a sufficient framework for the consistent interpretation of this quantum system. The implications of this formalism for rotational and time reversal symmetry are discussed. The formalism is applied to the free particle and harmonic oscillator in two dimensions and the physical signatures of non commutativity are identified.

arXiv:0812.2803•Journal of Physics A Mathematical and Theoretical•Apr 7, 2009•132 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

Non-relativistic quantum mechanics in a non-commutative space

S. Chaturvedi, R. Jagannathan, Radhakrishnan Sridhar +1 more

Non-relativistic quantum theory of a particle in a non-commutative space is formulated within the framework of the usual quantum mechanics itself. This is accomplished by showing explicitly that the Wess-Zumino formalism of deformed quantum mechanical phase space corresponding to a non-commutative space can be realized, in principle, in terms of the usual quantum mechanical phase space. The case of a slightly non-commutative space is considered in detail. A free particle in such a space is found to behave as if it is a charge particle moving in a self-generated magnetic field.

Journal of Physics A Mathematical and General•Feb 7, 1993•9 citations

Magnetic fields in noncommutative quantum mechanics

F. Delduc, Quentin Duret, François Gieres B +1 more

We discuss various descriptions of a quantum particle on noncommutative space in a (possibly non-constant) magnetic field. We have tried to present the basic facts in a unified and synthetic manner, and to clarify the relationship between various approaches and results that are scattered in the literature.

arXiv:0710.2239•Journal of Physics Conference Series•Feb 1, 2008•57 citations

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
In analogy with conventional quantum mechanics, non-commutative quantum mechanics is formulated as a quantum system on the Hilbert space of Hilbert-Schmidt operators acting on non-commutative configuration space. It is argued that the standard quantum mechanical interpretation based on Positive Operator Valued Measures, provides a sufficient framework for the consistent interpretation of this quantum system. The implications of this formalism for rotational and time reversal symmetry are discussed. The formalism is applied to the free particle and harmonic oscillator in two dimensions and the physical signatures of non commutativity are identified.
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.
Non-relativistic quantum theory of a particle in a non-commutative space is formulated within the framework of the usual quantum mechanics itself. This is accomplished by showing explicitly that the Wess-Zumino formalism of deformed quantum mechanical phase space corresponding to a non-commutative space can be realized, in principle, in terms of the usual quantum mechanical phase space. The case of a slightly non-commutative space is considered in detail. A free particle in such a space is found to behave as if it is a charge particle moving in a self-generated magnetic field.
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.