Feynman Path Integral Formulation

6.12 Scalar Matter Fields 205
A ij is the dual (Voronoi) area associated with the edge ij, and the symbol < ij>
denotes a sum over nearest neighbor lattice vertices. It is immediate to generalize
the action of Eq. (6.133) to higher dimensions, with the two-dimensional Voronoi
volumes replaced by their higher dimensional analogs, leading to
(
I(l 2 ,φ) = 1 2 ∑ V (d) φi − φ
) j 2
ij
. (6.134)
l ij
Here V (d)
ij
is the dual (Voronoi) volume associated with the edge ij, and the sum is
over all links on the lattice.
φ 3
l 2
3
h 1
l 5
4
φ 4
φ 1
1
l 1
l 4
2
l 3
φ 2
Fig. 6.15 Dual area associated with the edge l 1 (shaded area), and the corresponding dual link h 1 .
In two dimensions, in terms of the edge length l ij and the dual edge length
h ij , connecting neighboring vertices in the dual lattice, one has A ij = 1 2 h ijl ij (see
Fig. 6.15). Other choices for the lattice subdivision will lead to a similar formula
for the lattice action, with the Voronoi dual volumes replaced by their appropriate
counterparts for the new lattice. Explicitly, for an edge of length l 1 the dihedral dual
volume contribution is given by
A l1 = l2 1 (l2 2 + l2 3 − l2 1 )
16A 123
+ l2 1 (l2 4 + l2 5 − l2 1 )
16A 234
= 1 2 l 1h 1 , (6.135)
with h 1 is the length of the edge dual to l 1 .
On the other hand the barycentric dihedral area for the same edge would be
simply (see Fig 6.16)
A l1 =(A 123 + A 234 )/3 . (6.136)
It is well known that one of the disadvantages of the Voronoi construction is the lack
of positivity of the dual volumes, as pointed out in (Hamber and Williams, 1984).
Thus some of the weights appearing in Eq. (6.133) can be negative for such an
action. For the barycentric subdivision this problem does not arise, as the areas A ij