QUANTIZED CURVATURE IN LOOP QUANTUM GRAVITY

A hyperlink is a finite set of nonintersecting simple closed curves in R x R-3. Let S be an orientable surface in R x R-3. The Einstein-Hilbert action S(e, w) is defined on the vierbein e and an su(2) x su(2)-valued connection w, which are the dynamical variables in general relativity. Define a functional F-S(w), by integrating the curvature dw + w Lambda w over the surface S, which is su(2) x su(2)-valued. We integrate F-S(w) 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 (su(2) x su(2))-valued connections w. Using our earlier work done on Chern-Simons path integrals in 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 quantized curvature can be computed from the linking number between L and S.