{
  "id": "1705.06577",
  "version": "v1",
  "published": "2017-05-10T01:31:23.000Z",
  "updated": "2017-05-10T01:31:23.000Z",
  "title": "Area Operator in Loop Quantum Gravity",
  "authors": [
    "Adrian P. C. Lim"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1701.04397, arXiv:1705.00396",
  "categories": [
    "math-ph",
    "gr-qc",
    "math.DG",
    "math.MP"
  ],
  "abstract": "A hyperlink is a finite set of non-intersecting simple closed curves in $\\mathbb{R} \\times \\mathbb{R}^3$. Let $S$ be an orientable surface in $\\mathbb{R}^3$. The dynamical variables in General Relativity are the vierbein $e$ and a $\\mathfrak{su}(2)\\times\\mathfrak{su}(2)$-valued connection $\\omega$. Together with Minkowski metric, $e$ will define a metric $g$ on the manifold. Denote $A_S(e)$ as the area of $S$, for a given choice of $e$. The Einstein-Hilbert action $S(e,\\omega)$ is defined on $e$ and $\\omega$. We will quantize the area of the surface $S$ by integrating $A_S(e)$ against a holonomy operator of a hyperlink $L$, disjoint from $S$, and the exponential of the Einstein-Hilbert action, over the space of vierbeins $e$ and $\\mathfrak{su}(2)\\times\\mathfrak{su}(2)$-valued connections $\\omega$. Using our earlier work done on Chern-Simons path integrals in $\\mathbb{R}^3$, we will write this infinite dimensional path integral as the limit of a sequence of Chern-Simons integrals. Our main result shows that the area operator can be computed from a link-surface diagram between $L$ and $S$. By assigning an irreducible representation of $\\mathfrak{su}(2)\\times\\mathfrak{su}(2)$ to each component of $L$, the area operator gives the total net momentum impact on the surface $S$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-10T01:31:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "83C45",
      "81S40",
      "81T45",
      "57R56"
    ],
    "keywords": [
      "loop quantum gravity",
      "area operator",
      "infinite dimensional path integral",
      "einstein-hilbert action",
      "total net momentum impact"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}