A calculus based on a q-deformed Heisenberg algebra
B.L. Cerchiai1,2 - R. Hinterding 1,2 - J. Madore2,3 - 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
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.