Geodesic round trips by parallel transport in quantum gravity
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We study the vacuum correlations of a gravitational field in any dimension N > 2 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 G/D(N+2), 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.