{
  "id": "0807.0289",
  "version": "v2",
  "published": "2008-07-02T16:03:01.000Z",
  "updated": "2008-09-07T10:31:02.000Z",
  "title": "The Heisenberg-Pauli canonical formalism of quantum field theory in the rigorous setting of nonlinear generalized functions (Part I)",
  "authors": [
    "Jean-Francois Colombeau",
    "Andre Gsponer"
  ],
  "comment": "108 pages. Section on zero-point energy revised and expanded",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "The unmodified Heisenberg-Pauli canonical formalism of quantum field theory applied to a self-interacting scalar boson field is shown to make sense mathematically in a framework of generalized functions adapted to nonlinear operations. The free-field operators, their commutation relations, and the free-field Hamiltonian operator are calculated by a straightforward transcription of the usual formalism expressed in configuration space. This leads to the usual results, which are essentially independent of the regularization, with the exception of the zero-point energy which may be set to zero if a particular regularization is chosen. The calculations for the self-interacting field are more difficult, especially because of the well-known problems due to the unboundness of the operators and their time-dependent domains. Nevertheless, a proper methodology is developed and a differentiation on time-dependent domains is defined. The Heisenberg equations and the interacting-field equation are shown to be mathematically meaningful as operator-valued nonlinear generalized functions, which therefore provide an alternative to the usual Bogoliubov-Wightman interpretation of quantized fields as operator-valued distributions. The equation for the time-evolution operator is proved using two different methods, but no attempt is made to calculate the scattering operator, and the applications to perturbation theory are left to a subsequent report.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-09-07T10:31:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q99",
      "35D99",
      "35Q40",
      "46F30",
      "81T99"
    ],
    "keywords": [
      "quantum field theory",
      "nonlinear generalized functions",
      "heisenberg-pauli canonical formalism",
      "rigorous setting",
      "time-dependent domains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 108,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 789797,
      "adsabs": "2008arXiv0807.0289C"
    }
  }
}