Canonical Commutation Relation Preserving Maps

We study maps preserving the Heisenberg commutation relation ab − ba = 1. We find a one-parameter deformation of the standard realization of the above algebra in terms of a coordinate and its dual derivative. It involves a non-local “coordinate” operator while the dual “derivative” is just the Jackson finite-difference operator. Substitution of this realization into any differential operator involving x and d dx , results in an isospectral deformation of a continuous differential operator into a finite-difference one. We extend our results to the deformed Heisenberg algebra ab− qba = 1. As an example of potential applications, various deformations of the Hahn polynomials are briefly discussed. Present address (on sabbatical leave): Laboratoire de Physique Theorique, Université Paris Sud, Orsay 91405, France. On leave of absence from the Institute for Theoretical and Experimental Physics, Moscow 117259, Russia. 2 C. Chryssomalakos and A. Turbiner