{
  "id": "math-ph/0104004",
  "version": "v2",
  "published": "2001-04-02T22:14:31.000Z",
  "updated": "2001-08-28T20:23:27.000Z",
  "title": "Canonical Commutation Relation Preserving Maps",
  "authors": [
    "C. Chryssomalakos",
    "A. Turbiner"
  ],
  "comment": "11 pages. To appear in J. Phys. A., Special Issue on Difference Equations Revised version: an important note, communicated to us by C. Zachos, has been added, giving the similarity transformation between classical and q-deformed coordinates and derivatives",
  "doi": "10.1088/0305-4470/34/48/312",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "math.NA",
    "math.QA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2001-08-28T20:23:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "canonical commutation relation preserving maps",
      "differential operator",
      "heisenberg commutation relation",
      "jackson finite-difference operator",
      "deformation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}