Geodesic round trips by parallel transport in quantum gravity

We study the vacuum correlations of a gravitational field in any dimension $N>\mathrm{}2\mathrm{}$ using the perturbative expansion to order $G$. It is known that the usual Green's functions are meaningless in quantum gravity; moreover, we show that the traces of the holonomies of the Christoffel connection vanish, revealing a striking difference between gravity and Yang-Mills theories. In order to define meaningful holonomies we make two physical requirements: (1) the contours of the loop integrals must have "geodesic form and size;" (2) the full rotation matrices, not just their traces, must be averaged. Following these ideas we compute a "dumbbell correlation function" which turns out to behave like $\frac{G}{D^{N+2\mathrm{}}}$, where $D$ is the geodesic distance. Finally, we write a gauge-invariant quantity which could replace the Wilson loop in reproducing the static potential energy between two sources.