{
  "id": "0807.2837",
  "version": "v1",
  "published": "2008-07-17T17:23:23.000Z",
  "updated": "2008-07-17T17:23:23.000Z",
  "title": "Variations on a theme of Heisenberg, Pauli and Weyl",
  "authors": [
    "M. R. Kibler"
  ],
  "comment": "Dedicated to the memory of Mosh\\'e Flato on the occasion of the tenth anniversary of his death",
  "journal": "Journal of Physics A Mathematical and Theoretical 41 (2008) 375302",
  "doi": "10.1088/1751-8113/41/37/375302",
  "categories": [
    "quant-ph"
  ],
  "abstract": "The parentage between Weyl pairs, generalized Pauli group and unitary group is investigated in detail. We start from an abstract definition of the Heisenberg-Weyl group on the field R and then switch to the discrete Heisenberg-Weyl group or generalized Pauli group on a finite ring Z_d. The main characteristics of the latter group, an abstract group of order d**3 noted P_d, are given (conjugacy classes and irreducible representation classes or equivalently Lie algebra of dimension d**3 associated with P_d). Leaving the abstract sector, a set of Weyl pairs in dimension d is derived from a polar decomposition of SU(2) closely connected to angular momentum theory. Then, a realization of the generalized Pauli group P_d and the construction of generalized Pauli matrices in dimension d are revisited in terms of Weyl pairs. Finally, the Lie algebra of the unitary group U(d) is obtained as a subalgebra of the Lie algebra associated with P_d. This leads to a development of the Lie algebra of U(d) in a basis consisting of d**2 generalized Pauli matrices. In the case where d is a power of a prime integer, the Lie algebra of SU(d) can be decomposed into d-1 Cartan subalgebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-07-17T17:23:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lie algebra",
      "generalized pauli group",
      "weyl pairs",
      "generalized pauli matrices",
      "variations"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Physics A Mathematical General",
      "year": 2008,
      "month": "Sep",
      "volume": 41,
      "number": 37,
      "pages": 375302
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008JPhA...41K5302K"
    }
  }
}