{
  "id": "2008.12344",
  "version": "v1",
  "published": "2020-08-27T19:26:16.000Z",
  "updated": "2020-08-27T19:26:16.000Z",
  "title": "Heat Semigroups on Weyl Algebra",
  "authors": [
    "Ivan G. Avramidi"
  ],
  "comment": "39 pages",
  "categories": [
    "math-ph",
    "hep-th",
    "math.MP"
  ],
  "abstract": "We study the algebra of semigroups of Laplacians on the Weyl algebra. We consider two sets of operators $\\nabla^\\pm_i$ forming the Lie algebra $[\\nabla^\\pm_j,\\nabla^\\pm_k]= i\\mathcal{R}^\\pm_{jk}$ and $[\\nabla^+_j,\\nabla^-_k] =i\\frac{1}{2}(\\mathcal{R}^+_{jk}+\\mathcal{R}^-_{jk})$ with some anti-symmetric matrices $\\mathcal{R}^\\pm_{ij}$ and define the corresponding Laplacians $\\Delta_\\pm=g_\\pm^{ij}\\nabla^\\pm_i\\nabla^\\pm_j$ with some positive matrices $g_\\pm^{ij}$. We show that the heat semigroups $\\exp(t\\Delta_\\pm)$ can be represented as a Gaussian average of the operators $\\exp\\left<\\xi,\\nabla^\\pm\\right>$ and use these representations to compute the product of the semigroups, $\\exp(t\\Delta_+)\\exp(s\\Delta_-)$ and the corresponding heat kernel.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-27T19:26:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weyl algebra",
      "heat semigroups",
      "corresponding heat kernel",
      "gaussian average",
      "lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}