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Scaling of variables and the relation between noncommutative parameters in Noncommutative Quantum Mechanics

Bertolami, O, Rosa, J G, de Aragao, C M L, P. Castorina, Zappalà, D

Published Sep 28, 2005• 56 citations
OpenAlex

Abstract

We consider Noncommutative Quantum Mechanics with phase space noncommutativity. In particular, we show that a scaling of variables leaves the noncommutative algebra invariant, so that only the self-consistent effective parameters of the model are physically relevant. We also discuss the recently proposed relation of direct proportionality between the noncommutative parameters, showing that it has a limited applicability.

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We consider noncommutative quantum mechanics with phase space noncommutativity. In particular, we show that a scaling of variables leaves the noncommutative algebra invariant, so that only the self-consistent effective parameters of the model are physically relevant. We also discuss the recently proposed relation of direct proportionality between the noncommutative parameters, showing that it has a limited applicability.

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News and views

AAPPS

The article gives an introductory summary of our recent results from the perspective of the new conceptual notion of noncommutative values of quantum observables, where basic quantum mechanics is seen a theory of particle dynamics that is intrinsically noncommutative.Real numbers as an algebra is really an abstract mathematical model for the description of values of physical quantities.Quantum mechanics illustrates the limitation of that perspective.The noncommutative values preserve the algebraic relations among the observables in their values.Such a value contains the complete information the theory presents for an observable on a given state.All physical quantities, including coordinates of the physical space for describing the location of a particle take definite noncommutative values.

AAPPS bulletin•Feb 8, 2022•0 citations

Berry's Phase in Noncommutative Spaces

S. A. Alavi

We discuss the perturbative aspects of noncommutative quantum mechanics. Then we study Berry's phase within the framework of noncommutative quantum mechanics. The results show deviations from the usual quantum mechanics, which depend on the parameter of space/space noncommutativity.

Chinese Physics Letters•Apr 24, 2003•2 citations

Quantum Observables and a Model of Noncommutativity

Tung Ten Yong

This paper considers a generalization of the notion of quantum observables in ontological models of quantum mechanics. Within this framework it is possible to construct physical models where quantum noncommutativity can arise dynamically. Unlike quantum systems, the basic entities in this model have definite properties. Relations with no-go theorems and other hidden variable theories are also discussed.

arXiv:0712.1634•Dec 11, 2007•0 citations
We consider Noncommutative Quantum Mechanics with phase space noncommutativity. In particular, we show that a scaling of variables leaves the noncommutative algebra invariant, so that only the self-consistent effective parameters of the model are physically relevant. We also discuss the recently proposed relation of direct proportionality between the noncommutative parameters, showing that it has a limited applicability.
In two-dimensional noncommutive space for the case of both position - position and momentum - momentum noncommuting, a constraint between noncommutative parameters is investigated. The related topic of guaranteeing Bose - Einstein statistics in noncommutive space in the general case are elucidated: Bose - Einstein statistics is guaranteed by the deformed Heisenberg - Weyl algebra itself, independent of dynamics. A special character of a dynamical system is represented by a constraint between noncommutative parameters. The general feature of the constraint for any system is a direct proportionality between noncommutative parameters with a proportional coefficient depending on characteristic parameters of the system under study. The constraint for a harmonic oscillator is illustrated.
The article gives an introductory summary of our recent results from the perspective of the new conceptual notion of noncommutative values of quantum observables, where basic quantum mechanics is seen a theory of particle dynamics that is intrinsically noncommutative.Real numbers as an algebra is really an abstract mathematical model for the description of values of physical quantities.Quantum mechanics illustrates the limitation of that perspective.The noncommutative values preserve the algebraic relations among the observables in their values.Such a value contains the complete information the theory presents for an observable on a given state.All physical quantities, including coordinates of the physical space for describing the location of a particle take definite noncommutative values.