{
  "id": "1807.07172",
  "version": "v1",
  "published": "2018-07-18T22:17:59.000Z",
  "updated": "2018-07-18T22:17:59.000Z",
  "title": "Geometry of the Madelung transform",
  "authors": [
    "Boris Khesin",
    "Gerard Misiolek",
    "Klas Modin"
  ],
  "comment": "27 pages, 2 figures",
  "categories": [
    "math.DG",
    "math-ph",
    "math.MP",
    "math.SG"
  ],
  "abstract": "The Madelung transform is known to relate Schr\\\"odinger-type equations in quantum mechanics and the Euler equations for barotropic-type fluids. We prove that, more generally, the Madelung transform is a K\\\"ahler map (i.e. a symplectomorphism and an isometry) between the space of wave functions and the cotangent bundle to the density space equipped with the Fubini-Study metric and the Fisher-Rao information metric, respectively. We also show that Fusca's momentum map property of the Madelung transform is a manifestation of the general approach via reduction for semi-direct product groups. Furthermore, the Hasimoto transform for the binormal equation turns out to be the 1D case of the Madelung transform, while its higher-dimensional version is related to the problem of conservation of the Willmore energy in binormal flows.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-18T22:17:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "madelung transform",
      "fuscas momentum map property",
      "fisher-rao information metric",
      "semi-direct product groups",
      "binormal equation turns"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}