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Canonical Commutation Relation Preserving Maps

C. Chryssomalakos·A. Turbiner
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Abstract

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 $\frac{d}{dx}$, results in an {\em 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.

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BibTeXRIS

C. Chryssomalakos, A. Turbiner. 2001-04-02. Canonical Commutation Relation Preserving Maps. https://doi.org/10.1088/0305-4470%2F34%2F48%2F312

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