Feynman Path Integral Formulation

238 7 Analytical Lattice Expansion MethodsIf the number of zero modes for each triangulation of the sphere is denoted by N z.m. ,then the results can be re-expressed asN z.m. = 2N 0 − 6 , (7.48)which agrees with the expectation that in the continuum limit, N 0 → ∞, N z.m. /N 0should approach the constant value d in d space-time dimensions, which is the numberof local parameters for a diffeomorphism. On the lattice the diffeomorphismscorrespond to local deformations of the edge lengths about a vertex, which leavethe local geometry physically unchanged, the latter being described by the values oflocal lattice operators corresponding to local volumes, and curvatures. The lessonis that the correct count of zero modes will in general only be recovered asymptoticallyfor sufficiently large triangulations, where N 0 is roughly much larger thanthe number of neighbors to a point in d dimensions. A similar pattern is expectedin higher dimensions, although in general one would expect such results to holdonly for deformations of flat space which are not too large. In particular one shouldalways keep in mind the presence of the triangle inequalities, which do not allowdeformations of the edges past a certain configuration space boundary.3q 022Fig. 7.5 Notation for an arbitrarysimplicial lattice, wherethe edge lengths meeting atthe vertex 0 have been deformedaway from a regularlattice by a small amount q i(minimally deformed equilaterallattice).q 0364q 01q 04015q 05q 06The previous discussion dealt with the expansion of the gravitational action abouta 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 particularsymmetry, but this can be shown not to be the case. To complete our discussion, weturn therefore to the slightly more complex task of exhibiting explicitly the local latticeinvariance for an arbitrary background simplicial complex. The idea here is tolook at lattices that are deformations of a regular lattice, and small edge fluctuationsaround them. To this end we write for the edge length deformations