Feynman Path Integral Formulation

6.4 Rotations, Parallel Transports and Voronoi Loops 173Next, in order to introduce curvature, one needs to define the dihedral angle betweenfaces in an n-simplex. In an n-simplex s two n − 1-simplices f and f ′ will intersecton a common n − 2-simplex h, and the dihedral angle at the specified hinge h isdefined ascosθ( f , f ′ )= ω( f ) n−1 · ω( f ′ ) n−1V n−1 ( f )V n−1 ( f ′ , (6.10))where the scalar product appearing on the r.h.s. can be re-written in terms of squarededge lengths usingω n · ω ′ n = 1(n!) 2 det(e i · e ′ j) , (6.11)and e i · e ′ j in turn expressed in terms of squared edge lengths by the use of Eq. (6.3).(Note that the dihedral angle θ would have to be defined as π minus the arccosineof the expression on the r.h.s. if the orientation for the e’s had been chosen in sucha way that the ω’s would all point from the face f inward into the simplex s). As anexample, in two dimensions and within a given triangle, two edges will intersect ata vertex, giving θ as the angle between the two edges. In three dimensions withina given simplex two triangles will intersect at a given edge, while in four dimensiontwo tetrahedra will meet at a triangle. For the special case of an equilateraln-simplex, one has simply θ = arccos 1 n. A related and often used formula for thesine of the dihedral angle θ issinθ( f , f ′ )=n V n (s)V n−2 (h)n − 1 V n−1 ( f )V n−1 ( f ′ ) , (6.12)but is less useful for practical calculations, as the sine of the angle does not unambiguouslydetermine the angle itself, which is needed in order to compute the localcurvature.In a piecewise linear space curvature is detected by going around elementaryloops which are dual to a (d − 2)-dimensional subspace. From the dihedral anglesassociated with the faces of the simplices meeting at a given hinge h one can computethe deficit angle δ(h), defined asδ(h) =2π − ∑ θ(s,h) , (6.13)s⊃hwhere the sum extends over all simplices s meeting on h. It then follows that thedeficit angle δ is a measure of the curvature at h. The two-dimensional case is illustratedin Fig. 6.4, while the three- and four-dimensional cases are shown in Fig. 6.5.6.4 Rotations, Parallel Transports and Voronoi LoopsSince the interior of each simplex s is assumed to be flat, one can assign to it aLorentz frame Σ(s). Furthermore inside s one can define a d-component vector