{
  "id": "0905.0944",
  "version": "v1",
  "published": "2009-05-07T02:20:58.000Z",
  "updated": "2009-05-07T02:20:58.000Z",
  "title": "The mathematical structure of quantum real numbers",
  "authors": [
    "John V. Corbett"
  ],
  "comment": "24 pages, 0 figures",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "The mathematical structure of the sheaf of Dedekind real numbers $\\RsubD(X)$ for a quantum system is discussed. The algebra of physical qualities is represented by an $O^{*}$ algebra $\\mathcal M$ that acts on a Hilbert space that carries an irreducible representation of the symmetry group of the system. $X =\\EsubS(\\mathcal M)$, the state space for $\\mathcal M$, has the weak topology generated by the functions $ a_{Q}(\\cdot)$, defined for $\\hat A \\in \\mathcal M_{sa} $ and $\\forall \\hat \\rho \\in \\EsubS(\\mathcal M) $, by $ a_{Q}(\\hat \\rho) = Tr \\hat A \\hat \\rho $. For any open subset $W$ of $\\EsubS(\\mathcal M)$, the function $ a_{Q}|_{W}$ is the numerical value of the quality $\\hat A$ defined to the extent $W$. The example of the quantum real numbers for a single Galilean relativistic particle is given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-05-07T02:20:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47L90",
      "32L81"
    ],
    "keywords": [
      "quantum real numbers",
      "mathematical structure",
      "single galilean relativistic particle",
      "dedekind real numbers",
      "hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.0944C"
    }
  }
}