## 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:

- 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:

- Research Org.:
- Lawrence Berkeley National Lab. (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.
```

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

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

```
@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 = {1999},

month = {4}

}

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