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A calculus based on a q-deformed Heisenberg algebra

Journal Article · · European Physical Journal. C, Particles and Fields
 [1];  [1];  [2];  [1]
  1. Ludwig Maximilian Univ., Munich (Germany). Sektion Physik; Max Planck Institut fur Physik, Munich (Germany). Werner-Heisenberg-Inst.
  2. Max Planck Institut fur Physik, Munich (Germany). Werner-Heisenberg-Inst.; Univ. Paris-Sud, Orsay (France). Lab. de Physique Theorique et Hautes Energies
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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.

Research Organization:
Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
Sponsoring Organization:
USDOE Office of Science (SC)
Grant/Contract Number:
AC02-05CH11231
OSTI ID:
1410300
Journal Information:
European Physical Journal. C, Particles and Fields, Journal Name: European Physical Journal. C, Particles and Fields Journal Issue: 3 Vol. 8; ISSN 1434-6044
Publisher:
SpringerCopyright Statement
Country of Publication:
United States
Language:
English

References (5)

A q-deformation of the harmonic oscillator journal April 1997
A q-deformed quantum mechanical toy model journal September 1992
Integration on quantum Euclidean space and sphere journal September 1996
On 𝑞-analogues of the Fourier and Hankel transforms journal January 1992
q-deformed Hermite Polynomials in q-Quantum Mechanics journal January 1998

Cited By (5)

Gauge theory on noncommutative spaces journal August 2000
Conservation laws for a $q$-deformed nonrelativistic particle report March 2021
Noncommutative Geometry for Pedestrians report June 1999
q-Translations on quantum spaces report January 2004
Analysis on q-deformed quantum spaces report January 2006

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Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
97 MATHEMATICS AND COMPUTING