Discrete Dynamics in Nature and Society
Volume 5 (2000), Issue 2, Pages 97-106
doi:10.1155/S1026022600000455
Differential representations of dynamical oscillator symmetries in discrete Hilbert space
Zentrum Mathematik, Technische Universität München, Arcisstrasse, 21/H4, München D-80333, Germany
Received 10 January 2000
Copyright © 2000 Andreas Ruffing. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
As a very important example for dynamical symmetries in the context of q-generalized quantum mechanics the algebra aa†−q−2a†a=1 is investigated. It represents the oscillator symmetry SUq(1,1) and is regarded as a commutation phenomenon of the q-Heisenberg algebra which provides a discrete spectrum of momentum and space, i.e., a discrete Hilbert space structure. Generalized q-Hermite functions and systems of creation and annihilation operators are derived. The classical limit q→1 is investigated. Finally the SUq(1,1) algebra is represented by the dynamical variables of the q-Heisenberg algebra.