{
  "id": "1307.5303",
  "version": "v2",
  "published": "2013-07-19T18:42:16.000Z",
  "updated": "2016-03-22T18:58:29.000Z",
  "title": "Invariant Connections in Loop Quantum Gravity",
  "authors": [
    "Maximilian Hanusch"
  ],
  "comment": "33 pages. Revised version: Proof of Theorem 4.8 simplified; comments added to Sect. 1 and Sect. 5",
  "journal": "Commun. Math. Phys. (2016) 343(1):1-38",
  "doi": "10.1007/s00220-016-2592-0",
  "categories": [
    "math-ph",
    "gr-qc",
    "math.MP"
  ],
  "abstract": "Given a group $G$ and an abelian $C^*$-algebra $\\mathfrak{A}$, the antihomomorphisms $\\Theta\\colon G\\rightarrow \\mathrm{Aut}(\\mathfrak{A})$ are in one-to-one with those left actions $\\Phi\\colon G\\times \\mathrm{Spec}(\\mathfrak{A})\\rightarrow \\mathrm{Spec}(\\mathfrak{A})$ whose translation maps $\\Phi_g$ are continuous; whereby continuities of $\\Theta$ and $\\Phi$ turn out to be equivalent if $\\mathfrak{A}$ is unital. In particular, a left action $\\phi\\colon G \\times X\\rightarrow X$ can be uniquely extended to the spectrum of a $C^*$-subalgebra $\\mathfrak{A}$ of the bounded functions on $X$ if $\\phi_g^*(\\mathfrak{A})\\subseteq \\mathfrak{A}$ holds for each $g\\in G$. In the present paper, we apply this to the framework of loop quantum gravity. We show that, on the level of the configuration spaces, quantization and reduction in general do not commute, i.e., that the symmetry-reduced quantum configuration space is (strictly) larger than the quantized configuration space of the reduced classical theory. Here, the quantum-reduced space has the advantage to be completely characterized by a simple algebraic relation, whereby the quantized reduced classical space is usually hard to compute.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-19T18:42:16.000Z",
      "abstract": "Given a group $G$ and an abelian C*-algebra $A$ the antihomomorphisms $\\Theta: G --> Aut(A)$ are in one-to-one with left actions $\\Phi: G x Spec(A) --> Spec(A)$ for which the translations $\\Phi_g$ are continuous. Under the assumption that $A$ is unital continuities of $\\Theta$ and $\\Phi$ turn out to be equivalent. Then a left action $\\phi: G x X --> X$ can be uniquely extended to the spectrum of a C*-subalgebra $B$ of the bounded functions on $X$ if ${\\phi*f | f in B}=B$ for each map $\\phi_g$. In the present paper we apply this fact to the framework of loop quantum gravity. Here we show that quantization and reduction in general do not commute. More precisely, we prove that the symmetry-reduced quantum configuration space is (strictly) larger than the quantized space of the reduced classical theory. The former one has the advantage to be completely characterized by a simple algebraic relation that can encode measure theoretical information. We also show that there cannot exist any Haar measure on the cosmological quantum configuration space $\\mathbb{R} \\sqcup \\mathbb{R}_Bohr$.",
      "comment": "38 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-03-22T18:58:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L60",
      "46L65",
      "53C05",
      "57S25",
      "81T05",
      "83F05"
    ],
    "keywords": [
      "loop quantum gravity",
      "invariant connections",
      "left action",
      "encode measure theoretical information",
      "simple algebraic relation"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer",
      "journal": "Communications in Mathematical Physics",
      "year": 2016,
      "month": "Apr",
      "volume": 343,
      "number": 1,
      "pages": 1
    },
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1243640,
      "adsabs": "2016CMaPh.343....1H"
    }
  }
}