Area Operator in Loop Quantum Gravity

A hyperlink is a finite set of non-intersecting simple closed curves in . Let S be an orientable surface in . The dynamical variables in general relativity are the vierbein e and a -valued connection . Together with Minkowski metric, e will define a metric g on the manifold. Denote as the area of S, for a given choice of e. The Einstein-Hilbert action is defined on e and . We will quantize the area of the surface S by integrating 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 -valued connections . Using our earlier work done on Chern-Simons path integrals in , 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 to each component of L, the area operator gives the total net momentum impact on the surface S.