{
  "id": "2006.09832",
  "version": "v1",
  "published": "2020-06-16T14:59:35.000Z",
  "updated": "2020-06-16T14:59:35.000Z",
  "title": "Nets of standard subspaces on Lie groups",
  "authors": [
    "Karl-Hermann Neeb",
    "Gestur Olafsson"
  ],
  "comment": "50 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "math.OA",
    "math.RT"
  ],
  "abstract": "Let G be a Lie group with Lie algebra $\\mathfrak{g}$, $h \\in \\frak{g}$ an element for which the derivation ad(h) defines a 3-grading of $\\mathfrak{g}$ and $\\tau_G$ an involutive automorphism of G inducing on $\\mathfrak{g}$ the involution $e^{\\pi i ad(h)}$. We consider antiunitary representations $U$ of the Lie group $G_\\tau = G \\rtimes \\{e,\\tau_G\\}$ for which the positive cone $C_U = \\{ x \\in \\mathfrak{g} : -i \\partial U(x) \\geq 0\\}$ and $h$ span $\\mathfrak{g}$. To a real subspace E of distribution vectors invariant under $exp(\\mathbb{R} h)$ and an open subset $O \\subseteq G$, we associate the real subspace $H_E(O) \\subseteq H$, generated by the subspaces $U(\\varphi)E$, where $\\varphi \\in C^\\infty_c(O,\\mathbb{R})$ is a real-valued test function on $O$. Then $H_E(O)$ is dense in $H_E(G)$ for every non-empty open subset $O \\subseteq G$ (Reeh--Schlider property). For the real standard subspace $V \\subseteq H$, for which $J_V = U(\\tau_G)$ is the modular conjugation and $\\Delta_V^{-it/2\\pi} = U(\\exp th)$ is the modular group, we obtain sufficient conditions to be of the form $H_E(S)$ for an open subsemigroup $S \\subseteq G$. If $\\mathfrak{g}$ is semisimple with simple hermitian ideals of tube type, we verify these criteria and obtain nets of standard subspacs $H_E(O)$, $O \\subseteq G$, satisfying the Bisognano--Wichman property. Our construction also yields such nets on simple Jordan space-times and compactly causal symmetric spaces of Cayley type. By second quantization, these nets lead to free quantum fields in the sense of Haag--Kastler on causal homogeneous spaces whose groups are generated by modular groups and conjugations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-16T14:59:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E45",
      "81R05",
      "81T05"
    ],
    "keywords": [
      "lie group",
      "modular group",
      "real subspace",
      "free quantum fields",
      "distribution vectors invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}