{
  "id": "1108.5601",
  "version": "v2",
  "published": "2011-08-29T15:23:43.000Z",
  "updated": "2011-11-25T13:58:17.000Z",
  "title": "Quantum theory from the geometry of evolving probabilities",
  "authors": [
    "Marcel Reginatto",
    "Michael J. W. Hall"
  ],
  "comment": "12 pages. Presented at MaxEnt 2011, the 31st International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, July 10-15, 2011, Waterloo, Canada. Updated version: the affiliation of one of the authors was updated, minor changes were made to the text",
  "categories": [
    "quant-ph",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the space of probabilities {P(x)}, where the x are coordinates of a configuration space. Under the action of the translation group there is a natural metric over the space of parameters of the group given by the Fisher-Rao metric. This metric induces a metric over the space of probabilities. Our next step is to set the probabilities in motion. To do this, we introduce a canonically conjugate field S and a symplectic structure; this gives us Hamiltonian equations of motion. We show that it is possible to extend the metric structure to the full space of the {P,S} and this leads in a natural way to a Kaehler structure; i.e., a geometry that includes compatible symplectic, metric and complex structures. The simplest geometry that describes these spaces of evolving probabilities has remarkable properties: the natural, canonical variables are precisely the wave functions of quantum mechanics; the Hamiltonian for the quantum free particle can be derived from a representation of the Galilean group using purely geometrical arguments; and it is straightforward to associate with this geometry a Hilbert space which turns out to be the Hilbert space of quantum mechanics. We are led in this way to a reconstruction of quantum theory based solely on the geometry of probabilities in motion.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-11-25T13:58:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum theory",
      "evolving probabilities",
      "hilbert space",
      "quantum mechanics",
      "quantum free particle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012AIPC.1443...96R"
    }
  }
}