Feynman Path Integral Formulation

7.5 Gravitational Wilson Loop 251
Fig. 7.9 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 ′ ) are represented by arrows. In spite of the fact that the (Lorentz) matrices
R can fluctuate strongly in accordance with the local geometry, two contiguous, oppositely oriented
arrows always give RR −1 = 1.
through each of the simplices that meet at that hinge one has for the total rotation
matrix R ≡ ∏ s R s,s+1 associated with the given hinge
[ ] μ [
∏ R s,s+1 = e δ(h)U(h)]
s
ν
μν , (7.94)
as in Eq. (6.32). This matrix describes the parallel transport of a vector round the
loop.
More generally one might want to consider a near-planar, but non-infinitesimal,
closed loop C, as shown in Fig. (7.9). Along this closed loop the overall rotation
matrix will still be given by
R μ ν(C) =
[ ] μ
∏ R s,s+1 . (7.95)
s⊂C
ν
In analogy with the infinitesimal loop case, one would like to state that for the overall
rotation matrix one has
R μ ν(C) ≈
[e δ(C)U(C))] μ
, (7.96)
ν
where U μν (C) is now an area bivector perpendicular to the loop, which will work
only if the loop is close to planar so that U μν can be taken to be approximately
constant along the path C. By a near-planar loop around the point P, we mean one
that is constructed by drawing outgoing geodesics, on a plane through P.
If that is true, then one can define an appropriate coordinate scalar by contracting
the above rotation matrix R(C) with the some appropriate bivector, namely
W(C) =ω αβ (C)R αβ (C) , (7.97)