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Title: Separation of noncommutative differential calculus on quantum Minkowski space

Abstract

Noncommutative differential calculus on quantum Minkowski space is not separated with respect to the standard generators, in the sense that partial derivatives of functions of a single generator can depend on all other generators. It is shown that this problem can be overcome by a separation of variables. We study the action of the universal L-matrix, appearing in the coproduct of partial derivatives, on generators. Powers of the resulting quantum Minkowski algebra valued matrices are calculated. This leads to a nonlinear coordinate transformation which essentially separates the calculus. A compact formula for general derivatives is obtained in form of a chain rule with partial Jackson derivatives. It is applied to the massive quantum Klein-Gordon equation by reducing it to an ordinary q-difference equation. The rest state solution can be expressed in terms of a product of q-exponential functions in the separated variables.

Authors:
;  [1];  [2]
  1. Arnold Sommerfeld Center for Theoretical Physics, Fakultaet fuer Physik, Universitaet Muenchen, Theresienstrasse 37, 80333 Munich (Germany)
  2. (Germany)
Publication Date:
OSTI Identifier:
20768728
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 2; Other Information: DOI: 10.1063/1.2165793; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; COMMUTATION RELATIONS; COORDINATES; DIFFERENTIAL CALCULUS; FUNCTIONS; KLEIN-GORDON EQUATION; MATHEMATICAL SOLUTIONS; MATRICES; MINKOWSKI SPACE; NONLINEAR PROBLEMS; SOLUTIONS; TRANSFORMATIONS

Citation Formats

Bachmaier, Fabian, Blohmann, Christian, and Department of Mathematics, University of California, Berkeley, California 94720-3840 and International University Bremen, School of Engineering and Science, Campus Ring 1, 28759 Bremen. Separation of noncommutative differential calculus on quantum Minkowski space. United States: N. p., 2006. Web. doi:10.1063/1.2165793.
Bachmaier, Fabian, Blohmann, Christian, & Department of Mathematics, University of California, Berkeley, California 94720-3840 and International University Bremen, School of Engineering and Science, Campus Ring 1, 28759 Bremen. Separation of noncommutative differential calculus on quantum Minkowski space. United States. doi:10.1063/1.2165793.
Bachmaier, Fabian, Blohmann, Christian, and Department of Mathematics, University of California, Berkeley, California 94720-3840 and International University Bremen, School of Engineering and Science, Campus Ring 1, 28759 Bremen. Wed . "Separation of noncommutative differential calculus on quantum Minkowski space". United States. doi:10.1063/1.2165793.
@article{osti_20768728,
title = {Separation of noncommutative differential calculus on quantum Minkowski space},
author = {Bachmaier, Fabian and Blohmann, Christian and Department of Mathematics, University of California, Berkeley, California 94720-3840 and International University Bremen, School of Engineering and Science, Campus Ring 1, 28759 Bremen},
abstractNote = {Noncommutative differential calculus on quantum Minkowski space is not separated with respect to the standard generators, in the sense that partial derivatives of functions of a single generator can depend on all other generators. It is shown that this problem can be overcome by a separation of variables. We study the action of the universal L-matrix, appearing in the coproduct of partial derivatives, on generators. Powers of the resulting quantum Minkowski algebra valued matrices are calculated. This leads to a nonlinear coordinate transformation which essentially separates the calculus. A compact formula for general derivatives is obtained in form of a chain rule with partial Jackson derivatives. It is applied to the massive quantum Klein-Gordon equation by reducing it to an ordinary q-difference equation. The rest state solution can be expressed in terms of a product of q-exponential functions in the separated variables.},
doi = {10.1063/1.2165793},
journal = {Journal of Mathematical Physics},
number = 2,
volume = 47,
place = {United States},
year = {Wed Feb 15 00:00:00 EST 2006},
month = {Wed Feb 15 00:00:00 EST 2006}
}