Gravitational Wilson loop and large scale curvature

In a quantum theory of gravity the gravitational Wilson loop, defined as a suitable quantum average of a parallel transport operator around a large near-planar loop, provides important information about the large-scale curvature properties of the geometry. Here we shows that such properties can be systematically computed in the strong coupling limit of lattice regularized quantum gravity, by performing local averages over loop bivectors, and over lattice rotations, using an assumed near-uniform measure in group space. We then relate the resulting quantum averages to an expected semiclassical form valid for macroscopic observers, which leads to an identification of the gravitational correlation length appearing in the Wilson loop with an observed large-scale curvature. Our results suggest that strongly coupled gravity leads to a positively curved (de Sitter-like) quantum ground state, implying a positive effective cosmological constant at large distances.

Figure 1

Elementary polygonal path around a hinge (triangle) in four dimensions. The hinge $ABC$, contained in the simplex $ABCDE$, is encircled by the polygonal path $H$ connecting the surrounding vertices which reside in the dual lattice. One such vertex is contained within the simplex $ABCDE$.

Figure 2

Gravitational analog of the Wilson loop. A vector is parallel transported along the larger outer loop. The enclosed minimal surface is tiled with parallel transport polygons, here chosen to be triangles for illustrative purposes. For each link of the dual lattice, the elementary parallel transport matrices $R(s,s^{\prime })$ are represented by arrows. In spite of the fact that the (Lorentz) matrices $\mathbf{R}$ can fluctuate strongly in accordance with the local geometry, two contiguous, oppositely oriented arrows always give $R{R}^{-1}=1$.

Figure 3

A parallel transport loop with four oriented links on the boundary. The parallel transport matrices $\mathbf{R}$ along the links, represented here by arrows, appear in pairs and are sequentially integrated over using the uniform measure.

Figure 4

A parallel transport loop with six oriented links on the boundary.

Figure 5

A larger parallel transport loop with 12 oriented links on the boundary. As before, the parallel transport matrices along the links appear in pairs and are sequentially integrated over using the uniform measure. The new ingredient in this configuration is an elementary loop at the center not touching the boundary.

Figure 6

Correlations between action contributions on hinge $h$ and hinge $h^{\prime }$ arise to lowest order in the strong coupling expansions from diagrams describing a narrow tube connecting the two hinges. Here vertices represent points in the dual lattice, with the tubelike closed surface tiled with parallel transport polygons. For each link of the dual lattice, the $SO(4)$ parallel transport matrices $\mathbf{R}$ are represented by an arrow.