Feynman Path Integral Formulation

6.8 Gravitational Wilson Loop 189where U μν (C) is now an area bivector perpendicular to the loop - which will workonly if the loop is close to planar so that U μν can be taken to be approximatelyconstant along the path C.If that is true, then one can define, again in analogy with the infinitesimal loopcase, an appropriate coordinate scalar by contracting the above rotation matrix R(C)with the bivector of Eq. (6.8), namelyW(C) =ω αβ (C)R αβ (C) , (6.65)where the loop bivector, ω αβ (C) =(d − 2)!V (d−2) U αβ = 2A C U αβ (C) in four dimensions,is now intended as being representative of the overall geometric featuresof the loop. For example, it can be taken as an average of the hinge bivector ω αβ (h)along the loop.In the quantum theory one is of course interested in the average of the aboveloop operator W(C), as in Eq. (3.144). The previous construction is indeed quiteanalogous to the Wilson loop definition in ordinary lattice gauge theories (Wilson,1973), where it is defined via the trace of path ordered products of SU(N) colorrotation matrices. In gravity though the Wilson loop does not give any informationabout the static potential (Modanese, 1993; Hamber, 1994). It seems that the Wilsonloop in gravity provides instead some insight into the large-scale curvature of themanifold, just as the infinitesimal loop contribution entering the lattice action ofEqs. (6.38) and (6.39) provides, through its averages, insight into the very shortdistance, local curvature. Of course for any continuum manifold one can definelocally the parallel transport of a vector around a near-planar loop C. Indeed paralleltransporting a vector around a closed loop represents a suitable operational way ofdetecting curvature locally. If the curvature of the manifold is small, one can treatthe larger loop the same way as the small one; then the expression of Eq. (6.64)for the rotation matrix R(C) associated with a near-planar loop can be re-written interms of a surface integral of the large-scale Riemann tensor, projected along thesurface area element bivector A αβ (C) associated with the loop,∫[ 1R μ 2R··αβν(C) ≈ eAαβ (C)] μS . (6.66)νThus a direct calculation of the Wilson loop provides a way of determining the effectivecurvature at large distance scales, even in the case where short distance fluctuationsin the metric may be significant. Conversely, the rotation matrix appearingin the elementary Wilson loop of Eqs. (6.29) and (6.32) only provides informationabout the parallel transport of vectors around infinitesimal loops, with size comparableto the ultraviolet cutoff.One would expect that for a geometry fluctuating strongly at short distances (correspondingtherefore to the small k limit) the infinitesimal parallel transport matricesR(s,s ′ ) should be distributed close to randomly, with a measure close to the uniformHaar measure, and with little correlation between neighboring hinges. In such instanceone would have for the local quantum averages of the infinitesimal lattice