**8**, 547-558

DOI 10.1007/s100529901097

## A calculus based on a *q*-deformed Heisenberg algebra

**B.L. Cerchiai ^{1,2} - R. Hinterding ^{1,2} - J.
Madore^{2,3} - J. Wess ^{1,2}**

^{1} Sektion Physik, Ludwig-Maximilian Universität, Theresienstraße 37,
D-80333 München, Germany

^{2} Max-Planck-Institut für Physik (Werner-Heisenberg-Institut),
Föhringer Ring 6, D-80805 München, Germany

^{3} Laboratoire de Physique Théorique et Hautes Energies,
Université de Paris-Sud, Bâtiment 211, F-91405 Orsay, France

Received: 26 November 1998 / Published online: 27 April 1999

**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 *x* 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.

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