{
  "id": "1003.1156",
  "version": "v2",
  "published": "2010-03-04T22:41:08.000Z",
  "updated": "2010-09-04T22:39:32.000Z",
  "title": "Feynman-diagrammatic description of the asymptotics of the time evolution operator in quantum mechanics",
  "authors": [
    "Theo Johnson-Freyd"
  ],
  "comment": "21 pages. Many diagrams drawn in TikZ. To appear in Letters in Mathematical Physics",
  "journal": "Lett.Math.Phys.94:123-149,2010",
  "doi": "10.1007/s11005-010-0424-2",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "We describe the \"Feynman diagram\" approach to nonrelativistic quantum mechanics on R^n, with magnetic and potential terms. In particular, for each classical path \\gamma connecting points q_0 and q_1 in time t, we define a formal power series V_\\gamma(t,q_0,q_1) in \\hbar, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V_\\gamma) satisfies Schr\\\"odinger's equation, and explain in what sense the t\\to 0 limit approaches the \\delta distribution. As such, our construction gives explicitly the full \\hbar\\to 0 asymptotics of the fundamental solution to Schr\\\"odinger's equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman's path integral in diagrams.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-09-04T22:39:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81T18",
      "81S40",
      "81Q15"
    ],
    "keywords": [
      "time evolution operator",
      "feynman-diagrammatic description",
      "asymptotics",
      "represent finite-dimensional convergent integrals",
      "nonrelativistic quantum mechanics"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Letters in Mathematical Physics",
      "year": 2010,
      "month": "Nov",
      "volume": 94,
      "number": 2,
      "pages": 123
    },
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 867311,
      "adsabs": "2010LMaPh..94..123J"
    }
  }
}