Einstein-Hilbert Path Integrals in ${\mathbb{R}}^4$

Abstract

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R}\times {\mathbb{R}}^3$. 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. The Einstein-Hilbert action $S(e,\omega )$ is defined using $e$ and $\omega$. We will define a path integral $I$ by integrating a functional $H(e,\omega )$ against a holonomy operator of a hyperlink $L$, and the exponential of the Einstein-Hilbert action, over the space of vierbeins $e$ and $\mathfrak{su}(2)\times \mathfrak{su}(2)$-valued connections $\omega$. Three different types of functional will be considered for $H$, namely area of a surface, volume of a region and the curvature of a surface $S$. Using our earlier work done on Chern-Simons path integrals in ${\mathbb{R}}^3$, we will derive and write these infinite dimensional path integrals $I$ as the limit of a sequence of Chern-Simons integrals.

References in corpus (2)

• Area Operator in Loop Quantum Gravity
• Quantized Curvature in Loop Quantum Gravity