Wilson loops in four-dimensional quantum gravity

TL;DR

This paper defines and computes Wilson loops in four-dimensional quantum gravity, revealing their invariance and non-local behavior, which contrasts with gauge fields and explains the absence of localized curvature correlations.

Contribution

It introduces a Wilson loop formulation in 4D quantum gravity and analyzes its properties, highlighting the non-local nature of quantum gravitational fields.

Findings

Wilson loops are invariant under diffeomorphisms and local Lorentz transformations.

The gravitational Wilson loop does not develop localized curvature configurations at low temperature.

The results explain the absence of invariant curvature correlations in quantum gravity.

Abstract

A Wilson loop is defined, in 4-D pure Einstein gravity, as the trace of the holonomy of the Christoffel connection or of the spin connection, and its invariance under the symmetry transformations of the action is showed (diffeomorphisms and local Lorentz transformations). We then compute the loop perturbatively, both on a flat background and in the presence of an external source; we also allow some modifications in the form of the action, and test the action of ``stabilized'' gravity. A geometrical analysis of the results in terms of the gauge group of the euclidean theory, $SO(4)$, leads us to the conclusion that the correspondent statistical system does not develope any configuration with localized curvature at low temperature. This ``non-local'' behavior of the quantized gravitational field strongly contrasts with that of usual gauge fields. Our results also provide an explanation for the absence of any invariant correlation of the curvature in the same approximation.