{
  "id": "1501.02506",
  "version": "v1",
  "published": "2015-01-11T22:57:52.000Z",
  "updated": "2015-01-11T22:57:52.000Z",
  "title": "Special-case closed form of the Baker-Campbell-Hausdorff formula",
  "authors": [
    "Alexander Van-Brunt",
    "Matt Visser"
  ],
  "comment": "5 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "The Baker-Campbell-Hausdorff formula is a general result for the quantity $Z(X,Y)=\\ln( e^X e^Y )$, where $X$ and $Y$ are not necessarily commuting. For completely general commutation relations between $X$ and $Y$, (the free Lie algebra), the general result is somewhat unwieldy. However in specific physics applications the commutator $[X,Y]$, while non-zero, might often be relatively simple, which sometimes leads to explicit closed form results. We consider the special case $[X,Y] = u X + vY + cI$, and show that in this case the general result reduces to \\[ Z(X,Y)=\\ln( e^X e^Y ) = X+Y+ f(u,v) \\; [X,Y]. \\] Furthermore we explicitly evaluate the symmetric function $f(u,v)=f(v,u)$, demonstrating that \\[ f(u,v) = {(u-v)e^{u+v}-(ue^u-ve^v)\\over u v (e^u - e^v)}, \\] and relate this to previously known results. For instance this result includes, but is considerably more general than, results obtained from either the Heisenberg commutator $[P,Q]=-i\\hbar I$ or the creation-destruction commutator $[a,a^\\dagger]=I$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-11T22:57:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "special-case closed form",
      "baker-campbell-hausdorff formula",
      "general result reduces",
      "explicit closed form results",
      "general commutation relations"
    ],
    "publication": {
      "doi": "10.1088/1751-8113/48/22/225207",
      "journal": "Journal of Physics A Mathematical General",
      "year": 2015,
      "month": "Jun",
      "volume": 48,
      "number": 22,
      "pages": 225207
    },
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1340118,
      "adsabs": "2015JPhA...48v5207V"
    }
  }
}