{
  "id": "1103.0034",
  "version": "v3",
  "published": "2011-02-28T21:45:26.000Z",
  "updated": "2013-07-05T15:39:58.000Z",
  "title": "On quantum mechanics with a magnetic field on R^n and on a torus T^n, and their relation",
  "authors": [
    "Gaetano Fiore"
  ],
  "comment": "Latex2e file, 22 pages. Final version appeared in IJTP",
  "journal": "International Journal of Theoretical Physics 52 (2013), 877-896",
  "doi": "10.1007/s10773-012-1396-z",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "We show in elementary terms the equivalence in a general gauge of a U(1)-gauge theory of a scalar charged particle on a torus T^n = R^n/L to the analogous theory on R^n constrained by quasiperiodicity under translations in the lattice L. The latter theory provides a global description of the former: the quasiperiodic wavefunctions defined on R^n play the role of sections of the associated hermitean line bundle E on T^n, since also E admits a global description as a quotient. The components of the covariant derivatives corresponding to a constant (necessarily integral) magnetic field B = dA generate a Lie algebra g_Q and together with the periodic functions the algebra of observables O_Q . The non-abelian part of g_Q is a Heisenberg Lie algebra with the electric charge operator Q as the central generator; the corresponding Lie group G_Q acts on the Hilbert space as the translation group up to phase factors. Also the space of sections of E is mapped into itself by g in G_Q . We identify the socalled magnetic translation group as a subgroup of the observables' group Y_Q . We determine the unitary irreducible representations of O_Q, Y_Q corresponding to integer charges and for each of them an associated orthonormal basis explicitly in con?guration space. We also clarify how in the n = 2m case a holomorphic structure and Theta functions arise on the associated complex torus. These results apply equally well to the physics of charged scalar particles on R^n and on T^n in the presence of periodic magnetic field B and scalar potential. They are also necessary preliminary steps for the application to these theories of the deformation procedure induced by Drinfel'd twists.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-07-05T15:39:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55R10",
      "55R91",
      "81S10",
      "53D50"
    ],
    "keywords": [
      "magnetic field",
      "quantum mechanics",
      "global description",
      "necessary preliminary steps",
      "heisenberg lie algebra"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "International Journal of Theoretical Physics",
      "year": 2013,
      "month": "Mar",
      "volume": 52,
      "number": 3,
      "pages": 877
    },
    "note": {
      "typesetting": "LaTeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013IJTP...52..877F",
      "inspire": 891036
    }
  }
}