Feynman Path Integral Formulation

288 8 Numerical Studies
Fig. 8.5 A typical edge length distribution in the smooth phase for which k < k c ,orG > G c .Note
that the lattice gravitational measure of Eq. (6.77) cuts off the distribution at small edge lengths,
while the cosmological constant term prevents large edge lengths from appearing.
Furthermore the bare cosmological constant λ 0 appearing in the gravitational
action of Eq. (6.91) can be fixed at 1 in units of the cutoff, since it just sets the overall
length scale in the problem. The higher derivative coupling a can be set to a value
very close to 0 since one ultimately is interested in the limit a → 0, corresponding
to the pure Einstein theory.
One finds that for the measure in Eq. (6.77) this choice of parameters leads to
a well behaved ground state for k < k c for higher derivative coupling a → 0. The
system then resides in the “smooth” phase, with an effective dimensionality close to
four. On the other hand for k > k c the curvature becomes very large and the lattice
collapses into degenerate configurations with very long, elongated simplices (see
Fig. 8.4). Fig. 8.5 shows an example of a typical edge length distribution in the well
behaved strong coupling phase close to but below k c .
Fig. 8.6 shows the corresponding curvature (δA or √ gR) distribution.
One finds that as k is varied, the average curvature R is negative for sufficiently
small k (“smooth” phase), and appears to go to zero continuously at some finite
value k c .Fork > k c the curvature becomes very large, and the simplices tend to
collapse into degenerate configurations with very small volumes (/ 2 ∼
0). This “rough” or “collapsed” phase is the region of the usual weak field expansion
(G → 0). In this phase the lattice collapses into degenerate configurations with very
long, elongated simplices (Hamber, 1984; Hamber and Williams, 1985; Berg, 1985).
This phenomenon is usually intepreted as a lattice remnant of the conformal mode
instability of Euclidean gravity discussed in Sect. (2.5).
An elementary argument can be given to explain the fact that the collapsed phase
for k > k c has an effective dimension of two. The instability is driven by the Euclidean
Einstein term in the action, and in particular its unbounded conformal mode