Feynman Path Integral Formulation

6.4 Rotations, Parallel Transports and Voronoi Loops 177Fig. 6.6 Elementary polygonalpath around a hinge(triangle) in four dimensions.The hinge ABC, contained inthe simplex ABCDE, isencircledby the polygonal pathH connecting the surroundingvertices, which reside in thedual lattice. One such vertexis contained within the simplexABCDE.ADHBECIn three dimensions the path will encircle an edge, while in two dimensions it willencircle a site. Thus for each hinge h there is a unique elementary closed path C h forwhich one again can define the ordered productR(C h )=R(s 1 ,s m )···R(s 2 ,s 1 ) . (6.29)The hinge h, being geometrically an object of dimension (d − 2), is naturally representedby a tensor of rank (d − 2), referred to a coordinate system in h: an edgevector l μ hin d = 3, and an area bi-vector 1 2 (lμ h l′ νh − lh ν μl′ h) in d = 4 etc. FollowingEq. (6.8) it will therefore be convenient to define a hinge bi-vector U in any dimensionasU μν (h) =N ε μνα1 α d−2l α 1(1) ...lα d−2(d−2) , (6.30)normalized, by the choice of the constant N , in such a way that U μν U μν = 2. Infour dimensionsU μν (h) = 1 ε2A μναβ l1 α lβ 2 , (6.31)hwhere l 1 (h) and l 2 (h) two independent edge vectors associated with the hinge h, andA h the area of the hinge.An important aspect related to the rotation of an arbitrary vector, when paralleltransported around a hinge h, is the fact that, due to the hinge’s intrinsic orientation,only components of the vector in the plane perpendicular to the hinge areaffected. Since the direction of the hinge h is specified locally by the bivector U μνof Eq. (6.31), one can write for the loop rotation matrix RR μ ν(C) =(e δ U) μ, (6.32)νwhere C is now the small polygonal loop entangling the hinge h, and δ the deficitangle at h, previously defined in Eq. (6.13). One particularly noteworthy aspect of