A calculus based on a q-deformed Heisenberg algebra

Cerchiai, B. L.; Hinterding, R.; Madore, J.; Wess, J.

doi:10.1007/s100529901097

Title: A calculus based on a q-deformed Heisenberg algebra

Abstract

We show how one can construct a differential calculus over an algebra where position variables $$x$$ and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra has a subalgebra generated by cursive Greek chi and its inverse which we call the coordinate algebra. A physical field is considered to be an element of the completion of this algebra. We can construct a derivative which leaves invariant the coordinate algebra and so takes physical fields into physical fields. A generalized Leibniz rule for this algebra can be found. Based on this derivative differential forms and an exterior differential calculus can be constructed.

Authors:

Cerchiai, B. L. ^[1]; Hinterding, R. ^[1]; Madore, J. ^[2]; Wess, J. ^[1]

Ludwig Maximilian Univ., Munich (Germany). Sektion Physik; Max Planck Institut fur Physik, Munich (Germany). Werner-Heisenberg-Inst.
Max Planck Institut fur Physik, Munich (Germany). Werner-Heisenberg-Inst.; Univ. Paris-Sud, Orsay (France). Lab. de Physique Theorique et Hautes Energies

Publication Date:: Tue Apr 27 00:00:00 EDT 1999

Research Org.:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC)

OSTI Identifier:: 1410300

Grant/Contract Number:: AC02-05CH11231

Resource Type:: Accepted Manuscript

Journal Name:: European Physical Journal. C, Particles and Fields

Additional Journal Information:: Journal Volume: 8; Journal Issue: 3; Journal ID: ISSN 1434-6044

Publisher:: Springer

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

Citation Formats


                    Cerchiai, B. L., Hinterding, R., Madore, J., and Wess, J. A calculus based on a q-deformed Heisenberg algebra.  United States: N. p., 1999. 
Web.  doi:10.1007/s100529901097.

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                    Cerchiai, B. L., Hinterding, R., Madore, J., & Wess, J. A calculus based on a q-deformed Heisenberg algebra.  United States.  https://doi.org/10.1007/s100529901097

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                    Cerchiai, B. L., Hinterding, R., Madore, J., and Wess, J. Tue .  
"A calculus based on a q-deformed Heisenberg algebra".  United States.  https://doi.org/10.1007/s100529901097.  https://www.osti.gov/servlets/purl/1410300.

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@article{osti_1410300,

  title        = {A calculus based on a q-deformed Heisenberg algebra},

  author       = {Cerchiai, B. L. and Hinterding, R. and Madore, J. and Wess, J.},

  abstractNote = {We show how one can construct a differential calculus over an algebra where position variables $x$ and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra has a subalgebra generated by cursive Greek chi and its inverse which we call the coordinate algebra. A physical field is considered to be an element of the completion of this algebra. We can construct a derivative which leaves invariant the coordinate algebra and so takes physical fields into physical fields. A generalized Leibniz rule for this algebra can be found. Based on this derivative differential forms and an exterior differential calculus can be constructed.},

  doi          = {10.1007/s100529901097},

  journal      = {European Physical Journal. C, Particles and Fields},

  number       = 3,

  volume       = 8,

  place        = {United States},

  year         = {Tue Apr 27 00:00:00 EDT 1999},

  month        = {Tue Apr 27 00:00:00 EDT 1999}

}

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Works referenced in this record:

A q-deformation of the harmonic oscillator
journal, April 1997

Lorek, A.; Ruffing, A.; Wess, J.
Zeitschrift f�r Physik C Particles and Fields, Vol. 74, Issue 2
DOI: 10.1007/s002880050399

A q-deformed quantum mechanical toy model
journal, September 1992

Schwenk, J.; Wess, J.
Physics Letters B, Vol. 291, Issue 3, p. 273-277
DOI: 10.1016/0370-2693(92)91044-a

Integration on quantum Euclidean space and sphere
journal, September 1996

Steinacker, Harold
Journal of Mathematical Physics, Vol. 37, Issue 9, p. 4738-4749
DOI: 10.1063/1.531658

On 𝑞-analogues of the Fourier and Hankel transforms
journal, January 1992

Koornwinder, Tom H.; Swarttouw, René F.
Transactions of the American Mathematical Society, Vol. 333, Issue 1, p. 445-461
DOI: 10.1090/s0002-9947-1992-1069750-0

q-deformed Hermite Polynomials in q-Quantum Mechanics
journal, January 1998

Hinterding, Ralf; Wess, Julius
European Journal of Physics C
DOI: 10.48550/arxiv.math/9803050

Works referencing / citing this record:

Gauge theory on noncommutative spaces
journal, August 2000

Madore, J.; Schraml, S.; Schupp, P.
The European Physical Journal C, Vol. 16, Issue 1
DOI: 10.1007/s100520050012

Conservation laws for a $q$-deformed nonrelativistic particle
report, March 2021

Wachter, Hartmut
DOI: 10.48550/arxiv.2103.03356

Noncommutative Geometry for Pedestrians
report, June 1999

Madore, J.
DOI: 10.48550/arxiv.gr-qc/9906059

q-Translations on quantum spaces
report, January 2004

Wachter, Hartmut
DOI: 10.48550/arxiv.hep-th/0410205

Analysis on q-deformed quantum spaces
report, January 2006

Wachter, Hartmut
DOI: 10.48550/arxiv.math-ph/0604028

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Title: A calculus based on a q-deformed Heisenberg algebra

Abstract

Citation Formats

A q-deformation of the harmonic oscillator journal, April 1997

A q-deformed quantum mechanical toy model journal, September 1992

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On 𝑞-analogues of the Fourier and Hankel transforms journal, January 1992

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Conservation laws for a $q$-deformed nonrelativistic particle report, March 2021

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A q-deformation of the harmonic oscillator
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A q-deformed quantum mechanical toy model
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Integration on quantum Euclidean space and sphere
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journal, January 1992

q-deformed Hermite Polynomials in q-Quantum Mechanics
journal, January 1998

Gauge theory on noncommutative spaces
journal, August 2000

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