Quantized Curvature in Loop Quantum Gravity

TL;DR

This paper develops a method to quantize curvature in loop quantum gravity by relating it to linking numbers between hyperlinks and surfaces, using path integrals and Chern-Simons theory.

Contribution

It introduces a novel approach to compute quantized curvature via linking numbers, connecting loop quantum gravity with topological invariants and path integral techniques.

Findings

Quantized curvature is expressed through linking numbers.

Path integrals are approximated by Chern-Simons integrals.

The method bridges loop quantum gravity and topological invariants.

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 $S$.