Feynman Path Integral Formulation

6.9 Lattice Regularized Path Integral 191
dimensions is
δg ij (l 2 )= 1 2 (δl2 0i + δl 2 0 j − δl 2 ij) . (6.68)
For one d-dimensional simplex labeled by s the integration over the metric is thus
equivalent to an integration over the edge lengths, and one has the identity
( ) 1 σ∏
detg ij (s) dg ij (s)=
d!√
( − 1 ) d(d−1)
2
2
i≥ j
[
Vd (l 2 ) ] σ d(d+1)/2
dlk 2 . (6.69)
∏
k=1
There are d(d + 1)/2 edges for each simplex, just as there are d(d + 1)/2 independent
components for the metric tensor in d dimensions (Cheeger, Müller and
Schrader, 1982). Here one is ignoring temporarily the triangle inequality constraints,
which will further require all sub-determinants of g ij to be positive, including the
obvious restriction l 2 k > 0.
Let us discuss here briefly the simplicial inequalities which need to be imposed
on the edge lengths (Wheeler, 1964). These are conditions on the edge lengths l ij
such that the sites i can be considered the vertices of a d-simplex embedded in flat
d-dimensional Euclidean space. In one dimension, d = 1, one requires trivially for
all edge lengths
l 2 ij > 0 . (6.70)
In two dimensions, d = 2, the conditions on the edge lengths are again lij 2 > 0asin
one dimensions, as well as
( ) 1 2
A 2 Δ = detg (2)
ij
(s) > 0 , (6.71)
2!
which is equivalent, by virtue of Heron’s formula for the area of a triangle A 2 Δ =
s(s − l ij )(s − l jk )(s − l ki ) where s is the semi-perimeter s = 1 2 (l ij+ l jk + l ki ),tothe
requirement that the area of the triangle be positive. In turn Eq. (6.71) implies that
the triangle inequalities must be satisfied for all three edges,
l ij + l jk > l ik
l jk + l ki > l ji
l ki + l ij > l kj . (6.72)
In three dimensions, d = 3, the conditions on the edge lengths are again such that
one recovers a physical tetrahedron. One therefore requires for the individual edge
lengths the condition of Eq. (6.70), the reality and positivity of all four triangle areas
as in Eq. (6.71), as well as the requirement that the volume of the tetrahedron be real
and positive,
( ) 1 2
Vtetrahedron 2 = detg (3)
ij
(s) > 0 . (6.73)
3!
The generalization to higher dimensions is such that one requires all triangle inequalities
and their higher dimensional analogs to be satisfied,