{
  "id": "1004.4305",
  "version": "v2",
  "published": "2010-04-24T19:18:29.000Z",
  "updated": "2010-09-05T18:29:08.000Z",
  "title": "The formal path integral and quantum mechanics",
  "authors": [
    "Theo Johnson-Freyd"
  ],
  "comment": "33 pages, many TikZ diagrams, submitted to _Journal of Mathematical Physics_",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "Given an arbitrary Lagrangian function on \\RR^d and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by \"Feynman diagrams,\" although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volume-preserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a \"Fubini theorem\" expressing the standard composition law for the time evolution operator in quantum mechanics. Moreover, we show that when the Lagrangian is inhomogeneous-quadratic in velocity such that its homogeneous-quadratic part is given by a matrix with constant determinant, then the divergences cancel at each order. Thus, by \"cutting and pasting\" and choosing volume-compatible local coordinates, our construction defines a Feynman-diagrammatic \"formal path integral\" for the nonrelativistic quantum mechanics of a charged particle moving in a Riemannian manifold with an external electromagnetic field.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-09-05T18:29:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81T18",
      "81S40",
      "81Q15",
      "04.20.-q",
      "03.65.Ge",
      "03.65.-w"
    ],
    "keywords": [
      "ultraviolet divergences",
      "formal path integral satisfies",
      "divergences cancel",
      "nonrelativistic quantum mechanics",
      "standard composition law"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1063/1.3503472",
      "journal": "Journal of Mathematical Physics",
      "year": 2010,
      "month": "Nov",
      "volume": 51,
      "number": 11,
      "pages": 2105
    },
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 853352,
      "adsabs": "2010JMP....51k2105J"
    }
  }
}