Quantum mechanics on noncommutative plane and sphere from constrained systems

A. A. Deriglazov

Research output: Contribution to journalArticle

37 Citations (Scopus)

Abstract

It is shown that quantum mechanics on noncommutative (NC) spaces can be obtained by canonical quantization of some underlying constrained systems. Noncommutative geometry arises after taking into account the second class constraints presented in the models. It leads, in particular, to a possibility of quantization in terms of the initial NC variables. For a two-dimensional plane we present two Lagrangian actions, one of which admits addition of an arbitrary potential. Quantization leads to quantum mechanics with ordinary product replaced by the Moyal product. For a three-dimensional case we present Lagrangian formulations for a particle on NC sphere as well as for a particle on commutative sphere with a magnetic monopole at the center, the latter is shown to be equivalent to the model of usual rotor. There are several natural possibilities to choose physical variables, which lead either to commutative or to NC brackets for space variables. In the NC representation all information on the space variable dynamics is encoded in the NC geometry. Potential of special form can be added, which leads to an example of quantum mechanics on the NC sphere.

Original languageEnglish
Pages (from-to)235-243
Number of pages9
JournalPhysics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics
Volume530
Issue number1-4
DOIs
Publication statusPublished - 28 Mar 2002

Fingerprint

quantum mechanics
magnetic monopoles
brackets
products
geometry
rotors
formulations

Keywords

  • Noncommutative geometry
  • Quantum mechanics
  • Star product

ASJC Scopus subject areas

  • Nuclear and High Energy Physics

Cite this

Quantum mechanics on noncommutative plane and sphere from constrained systems. / Deriglazov, A. A.

In: Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, Vol. 530, No. 1-4, 28.03.2002, p. 235-243.

Research output: Contribution to journalArticle

@article{ff81c6d7e2544647bd7c3a5406b84e8c,
title = "Quantum mechanics on noncommutative plane and sphere from constrained systems",
abstract = "It is shown that quantum mechanics on noncommutative (NC) spaces can be obtained by canonical quantization of some underlying constrained systems. Noncommutative geometry arises after taking into account the second class constraints presented in the models. It leads, in particular, to a possibility of quantization in terms of the initial NC variables. For a two-dimensional plane we present two Lagrangian actions, one of which admits addition of an arbitrary potential. Quantization leads to quantum mechanics with ordinary product replaced by the Moyal product. For a three-dimensional case we present Lagrangian formulations for a particle on NC sphere as well as for a particle on commutative sphere with a magnetic monopole at the center, the latter is shown to be equivalent to the model of usual rotor. There are several natural possibilities to choose physical variables, which lead either to commutative or to NC brackets for space variables. In the NC representation all information on the space variable dynamics is encoded in the NC geometry. Potential of special form can be added, which leads to an example of quantum mechanics on the NC sphere.",
keywords = "Noncommutative geometry, Quantum mechanics, Star product",
author = "Deriglazov, {A. A.}",
year = "2002",
month = "3",
day = "28",
doi = "10.1016/S0370-2693(02)01262-5",
language = "English",
volume = "530",
pages = "235--243",
journal = "Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics",
issn = "0370-2693",
publisher = "Elsevier",
number = "1-4",

}

TY - JOUR

T1 - Quantum mechanics on noncommutative plane and sphere from constrained systems

AU - Deriglazov, A. A.

PY - 2002/3/28

Y1 - 2002/3/28

N2 - It is shown that quantum mechanics on noncommutative (NC) spaces can be obtained by canonical quantization of some underlying constrained systems. Noncommutative geometry arises after taking into account the second class constraints presented in the models. It leads, in particular, to a possibility of quantization in terms of the initial NC variables. For a two-dimensional plane we present two Lagrangian actions, one of which admits addition of an arbitrary potential. Quantization leads to quantum mechanics with ordinary product replaced by the Moyal product. For a three-dimensional case we present Lagrangian formulations for a particle on NC sphere as well as for a particle on commutative sphere with a magnetic monopole at the center, the latter is shown to be equivalent to the model of usual rotor. There are several natural possibilities to choose physical variables, which lead either to commutative or to NC brackets for space variables. In the NC representation all information on the space variable dynamics is encoded in the NC geometry. Potential of special form can be added, which leads to an example of quantum mechanics on the NC sphere.

AB - It is shown that quantum mechanics on noncommutative (NC) spaces can be obtained by canonical quantization of some underlying constrained systems. Noncommutative geometry arises after taking into account the second class constraints presented in the models. It leads, in particular, to a possibility of quantization in terms of the initial NC variables. For a two-dimensional plane we present two Lagrangian actions, one of which admits addition of an arbitrary potential. Quantization leads to quantum mechanics with ordinary product replaced by the Moyal product. For a three-dimensional case we present Lagrangian formulations for a particle on NC sphere as well as for a particle on commutative sphere with a magnetic monopole at the center, the latter is shown to be equivalent to the model of usual rotor. There are several natural possibilities to choose physical variables, which lead either to commutative or to NC brackets for space variables. In the NC representation all information on the space variable dynamics is encoded in the NC geometry. Potential of special form can be added, which leads to an example of quantum mechanics on the NC sphere.

KW - Noncommutative geometry

KW - Quantum mechanics

KW - Star product

UR - http://www.scopus.com/inward/record.url?scp=0037187756&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037187756&partnerID=8YFLogxK

U2 - 10.1016/S0370-2693(02)01262-5

DO - 10.1016/S0370-2693(02)01262-5

M3 - Article

AN - SCOPUS:0037187756

VL - 530

SP - 235

EP - 243

JO - Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics

JF - Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics

SN - 0370-2693

IS - 1-4

ER -