{
  "id": "1010.0449",
  "version": "v2",
  "published": "2010-10-03T23:18:58.000Z",
  "updated": "2014-09-23T22:48:58.000Z",
  "title": "Loop Quantization and Symmetry: Configuration Spaces",
  "authors": [
    "Christian Fleischhack"
  ],
  "comment": "35 pages, LaTeX. Changes v1 to v2: algebra and spectrum for homogeneous isotropic case corrected (now Thm. 4.21; formerly 0 was missing in the spectrum); unitality assumption added in some parts of Sect. 2; other results basically not affected; presentation improved, including some reshuffling of subsections; former Sect. 3 extracted (enlarged version now as 1409.5273); Sect. 7, refs. added",
  "categories": [
    "math-ph",
    "gr-qc",
    "math.MP"
  ],
  "abstract": "Given two sets $S_1, S_2$ and unital C*-algebras $A_1$, $A_2$ of functions thereon, we show that a map $\\sigma : S_1 \\nach S_2$ can be lifted to a continuous map $\\bar\\sigma : \\spec A_1 \\to \\spec A_2$ iff $\\sigma^\\ast A_2 := \\{\\sigma^\\ast f | f \\in A_2\\} \\subset A_1$. Moreover, $\\overline\\sigma$ is unique if existing, and injective iff $\\sigma^\\ast A_2$ is dense. Then, we apply these results to loop quantum gravity and loop quantum cosmology. Here, the quantum configuration spaces are indeed spectra of certain C*-algebras $A_\\cosm$ and $A_\\grav$, respectively, whereas the choices for the algebras diverge in the literature. We decide now for all usual choices whether the respective cosmological quantum configuration space is embedded into the gravitational one. Typically, there is no embedding, but one can always get an embedding by defining $A_\\cosm := C^\\ast(\\sigma^\\ast A_\\grav)$, where $\\sigma$ denotes the embedding between the classical configuration spaces. Finally, we explicitly determine $C^\\ast(\\sigma^\\ast A_\\grav)$ in the homogeneous isotropic case for $A_\\grav$ generated by the matrix functions of parallel transports along analytic paths. The cosmological quantum configuration space obtained this way, equals the disjoint union of $\\R$ and the Bohr compactification of $\\R$, appropriately glued together.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-10-03T23:18:58.000Z",
      "abstract": "Given two sets $S_1, S_2$ and unital C*-algebras $A_1$, $A_2$ of functions thereon, we show that a map $\\sigma : S_1 --> S_2$ can be lifted to a continuous map $\\bar\\sigma : spec A_1 --> spec A_2$ iff $\\sigma* A_2 := \\{\\sigma* f | f \\in A_1\\} \\subset A_1$. Moreover, $\\bar\\sigma$ is unique if existing and injective iff $\\sigma* A_2$ is dense. Next, we investigate the spectrum of the sum of two C*-algebras of functions on a locally compact $S_1$ having trivial intersection. In particular, for $A_0$ being the algebra of continuous functions vanishing at infinity and outside some open $Y \\subset S_1$ with $A_0 A_1 \\subset A_0$, the spectrum of $A_0 \\dirvsum A_1$ equals the disjoint union of $Y$ and $spec A_1$, whereas the topologies of both sets are nontrivially interwoven. -- Then, we apply these results to loop quantum gravity and loop quantum cosmology. For all usual technical conventions, we decide whether the cosmological quantum configuration space is embedded into the gravitational one where both are given as spectra of certain C*-algebras $A_\\cosm$ and $A_\\grav$. Typically, there is no embedding, but one can always get an embedding by the defining $A_\\cosm := C*(\\sigma* A_\\grav)$, where $\\sigma$ denotes the embedding between the classical configuration spaces. Finally, we explicitly determine $C*(\\sigma* A_\\grav)$ in the homogeneous isotropic case for $A_\\grav$ generated by the matrix functions of parallel transports along analytic paths. The cosmological quantum configuration space so equals the disjoint union of $\\R \\setminus \\{0\\}$ and the Bohr compactification of $\\R$, appropriately glued together.",
      "comment": "33 pages, LaTeX",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-23T22:48:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L60",
      "46L65",
      "34C27",
      "53C05",
      "81T05",
      "83F05"
    ],
    "keywords": [
      "loop quantization",
      "cosmological quantum configuration space",
      "disjoint union",
      "loop quantum gravity",
      "loop quantum cosmology"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 871945,
      "adsabs": "2010arXiv1010.0449F"
    }
  }
}