Canonical Commutation Relation Preserving Maps

TL;DR

This paper introduces a one-parameter deformation of the Heisenberg algebra using a non-local coordinate operator and Jackson's finite-difference derivative, leading to isospectral finite-difference operators and potential applications in polynomial deformations.

Contribution

It presents a novel deformation of the Heisenberg algebra involving a non-local coordinate and Jackson derivative, extending to the q-deformed algebra and applications to polynomial deformations.

Findings

Deformation involves a non-local coordinate operator and Jackson derivative.

Substituting into differential operators yields isospectral finite-difference operators.

Extensions to q-deformed algebra and polynomial deformations are discussed.

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.