Feynman Path Integral Formulation

238 7 Analytical Lattice Expansion Methods
If the number of zero modes for each triangulation of the sphere is denoted by N z.m. ,
then the results can be re-expressed as
N z.m. = 2N 0 − 6 , (7.48)
which agrees with the expectation that in the continuum limit, N 0 → ∞, N z.m. /N 0
should approach the constant value d in d space-time dimensions, which is the number
of local parameters for a diffeomorphism. On the lattice the diffeomorphisms
correspond to local deformations of the edge lengths about a vertex, which leave
the local geometry physically unchanged, the latter being described by the values of
local lattice operators corresponding to local volumes, and curvatures. The lesson
is that the correct count of zero modes will in general only be recovered asymptotically
for sufficiently large triangulations, where N 0 is roughly much larger than
the number of neighbors to a point in d dimensions. A similar pattern is expected
in higher dimensions, although in general one would expect such results to hold
only for deformations of flat space which are not too large. In particular one should
always keep in mind the presence of the triangle inequalities, which do not allow
deformations of the edges past a certain configuration space boundary.
3
q 02
2
Fig. 7.5 Notation for an arbitrary
simplicial lattice, where
the edge lengths meeting at
the vertex 0 have been deformed
away from a regular
lattice by a small amount q i
(minimally deformed equilateral
lattice).
q 03
6
4
q 01
q 04
0
1
5
q 05
q 06
The previous discussion dealt with the expansion of the gravitational action about
a regular lattice: a regular tessellation of the sphere, a manifold of constant curvature.
One might wonder whether the results depend on the lattice having a particular
symmetry, but this can be shown not to be the case. To complete our discussion, we
turn therefore to the slightly more complex task of exhibiting explicitly the local lattice
invariance for an arbitrary background simplicial complex. The idea here is to
look at lattices that are deformations of a regular lattice, and small edge fluctuations
around them. To this end we write for the edge length deformations