Feynman Path Integral Formulation

6.4 Rotations, Parallel Transports and Voronoi Loops 173
Next, in order to introduce curvature, one needs to define the dihedral angle between
faces in an n-simplex. In an n-simplex s two n − 1-simplices f and f ′ will intersect
on a common n − 2-simplex h, and the dihedral angle at the specified hinge h is
defined as
cosθ( f , f ′ )= ω( f ) n−1 · ω( f ′ ) n−1
V 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 squared
edge lengths using
ω 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 arccosine
of the expression on the r.h.s. if the orientation for the e’s had been chosen in such
a way that the ω’s would all point from the face f inward into the simplex s). As an
example, in two dimensions and within a given triangle, two edges will intersect at
a vertex, giving θ as the angle between the two edges. In three dimensions within
a given simplex two triangles will intersect at a given edge, while in four dimension
two tetrahedra will meet at a triangle. For the special case of an equilateral
n-simplex, one has simply θ = arccos 1 n
. A related and often used formula for the
sine of the dihedral angle θ is
sinθ( 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 unambiguously
determine the angle itself, which is needed in order to compute the local
curvature.
In a piecewise linear space curvature is detected by going around elementary
loops which are dual to a (d − 2)-dimensional subspace. From the dihedral angles
associated with the faces of the simplices meeting at a given hinge h one can compute
the deficit angle δ(h), defined as
δ(h) =2π − ∑ θ(s,h) , (6.13)
s⊃h
where the sum extends over all simplices s meeting on h. It then follows that the
deficit angle δ is a measure of the curvature at h. The two-dimensional case is illustrated
in Fig. 6.4, while the three- and four-dimensional cases are shown in Fig. 6.5.
6.4 Rotations, Parallel Transports and Voronoi Loops
Since the interior of each simplex s is assumed to be flat, one can assign to it a
Lorentz frame Σ(s). Furthermore inside s one can define a d-component vector