# 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:

- Arnold Sommerfeld Center for Theoretical Physics, Fakultaet fuer Physik, Universitaet Muenchen, Theresienstrasse 37, 80333 Munich (Germany)
- (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}

}