{
  "id": "1412.0052",
  "version": "v1",
  "published": "2014-11-29T00:18:09.000Z",
  "updated": "2014-11-29T00:18:09.000Z",
  "title": "On the construction of quantum fields via measures on path spaces",
  "authors": [
    "Rodrigo Vargas Le-Bert"
  ],
  "comment": "40 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We engage the construction problem of quantum fields via Euclidean measures on path spaces. We start by recasting stochastic process theory in algebro-analytic language, revisiting Nelson's reconstruction theorem after Klein and Landau, and developing a version of the theory of cylinder measures and their radonification which is amenable to explicit calculations. Then, we provide a new construction of the $\\phi^4$ field in dimensions $1+1$ and $2+1$, based on the \"re-radonification\" on a space of continuous, distribution-valued paths, of a measure obtained with the aid of Rajput's characterization of Gaussian measures on $L^p$ spaces. Finally, we develop an understanding of renormalization, based on our notion of cylinder measure, which we exploit in two ways. First, we introduce the notion of cylinder perturbation and sketch a proof that the $\\phi^4$ field can exist in dimension $3+1$ only in a weak coupling limit. Second, we devise a perturbation series on the cutoff scale and propose an explicit solution of the renormalization problem, by treating the interaction part alone. As an application, we provide a formula for the cylinder measure of a local field which is effectively $\\phi^4$ at any given fixed scale, in arbitrary dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-29T00:18:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81T08",
      "81T16",
      "60G07",
      "60G60",
      "58D20"
    ],
    "keywords": [
      "quantum fields",
      "path spaces",
      "cylinder measure",
      "recasting stochastic process theory",
      "revisiting nelsons reconstruction theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.0052V"
    }
  }
}