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

## TL;DR

This paper constructs a path integral formulation for General Relativity in four-dimensional space using holonomy operators and Einstein-Hilbert action, connecting it to Chern-Simons integrals through a limiting process.

## Contribution

It introduces a novel path integral approach for Einstein-Hilbert action in $\

## Key findings

- Path integrals expressed as limits of Chern-Simons integrals.
- Functional types include surface area, volume, and curvature.
- Framework connects gravitational path integrals with topological quantum field theories.

## 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.