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 $S$.

arXiv admin note: text overlap with arXiv:1705.06577, arXiv:1706.01011, arXiv:1705.00396

References in corpus (3)

• Quantum Nature of the Big Bang
• Quantum Nature of the Big Bang: An Analytical and Numerical Investigation
• Area Operator in Loop Quantum Gravity

Cited by in corpus (1)

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