Wilson loops in four-dimensional 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.

Gravitational Wilson loop and large scale curvature

“…the Wilson loop does not give any information about the static potential [15,16]. It seems that the Wilson loop in gravity provides instead some insight into the large-scale curvature of the manifold, just as the infinitesimal loop contribution entering the lattice action of Eqs.…”

Quantum gravity on the lattice

“…This last expression can be compared directly to the 2 + result of (46), as well as to the σ -model result of (22). The physical dimensions of G can be restored by multiplying the above expression on both sides by the ultraviolet cutoff , if one so desires.…”

Wilson loops in nonperturbative quantum gravity

“…More complete descriptions can be found elsewhere; see [5] and references therein. In standard CDT, each spacetime T has a proper-time slicing with integer label t, and is assembled from four-simplices in a layered fashion, 12 where one layer of thickness Δt ¼ 1 is a piecewise flat piece of spacetime of topology S 3 × I, all of whose vertices are contained in either of its spatial boundary submanifolds at times t or t þ 1. These submanifolds are arbitrary triangulations in terms of equilateral tetrahedra, and all have the topology of a three-sphere.…”

“…An entire fourgeometry of proper-time extension T is obtained by gluing together T subsequent layers along matching three-geometries, and finally identifying the final boundary of the last layer with the initial boundary of the first layer. 12 See [21] for a generalization of CDT geometries, without strict time slicing, but maintaining causality.…”

“…Gravitational Wilson loops on spacetime have been little studied, with the exception of work in perturbative quantum gravity [12] and in the context of the search for a nonAbelian Stokes' theorem [13]. The story is different in canonical quantum gravity, where holonomies along spatial curves play a prominent role in loop quantum gravity [14].…”