Quantized Curvature in Loop Quantum Gravity

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 $\mathbb{R} \times \mathbb{R}^3$. The Einstein-Hilbert action $S(e,\omega)$ is defined on the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $\omega$, which are the dynamical variables in General Relativity. Define a functional $F_S(\omega)$, by integrating the curvature $d\omega + \omega \wedge \omega$ over the surface $S$, which is $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued. We integrate $F_S(\omega)$ 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 quantized curvature can be computed from the linking number between $L$ and $S$.