Feynman Path Integral Formulation

6.11 Lattice Higher Derivative Terms 199
there is a natural volume associated with each hinge, defined by dividing the volume
of each simplex touching the hinge into a contribution belonging to that hinge,
and other contributions belonging to the other hinges on that simplex (Hamber and
Williams, 1984). The contribution belonging to that simplex will be called dihedral
volume V d (for an illustration, see Fig. 6.13). The volume V h associated with
the hinge h is then naturally the sum of the dihedral volumes V d belonging to each
simplex
V h = ∑ V d . (6.108)
d−simplices
meeting on h
The dihedral volume associated with each hinge in a simplex can be defined using
dual volumes, a barycentric subdivision, or some other natural way of dividing the
volume of a d-simplex into d(d + 1)/2 parts. If the theory has some reasonable
continuum limit, then the final results should not depend on the detailed choice of
volume type.
3
2
4
0
1
Fig. 6.13 Illustration of dual
volumes in two dimensions.
The vertices of the polygons
reside in the dual lattice. The
shaded region describes the
dual area associated with the
vertex 0.
5
6
As mentioned previously, there is a well-established procedure for constructing a
dual lattice for any given lattice. This involves constructing polyhedral cells, known
in the literature as Voronoi polyhedra, around each vertex, in such a way that the cell
around each particular vertex contains all points which are nearer to that vertex than
to any other vertex. Thus the cell is made up from (d − 1)-dimensional subspaces
which are the perpendicular bisectors of the edges in the original lattice, (d − 2)-
dimensional subspaces which are orthogonal to the 2-dimensional subspaces of the
original lattice, and so on. General formulas for dual volumes are given in (Hamber
and Williams, 1986). In the case of the barycentric subdivision, the dihedral volume
is just 2/d(d + 1) times the volume of the simplex. This leads one to conclude that
there is a natural area A Ch associated with each hinge