Feynman Path Integral Formulation

280 8 Numerical Studies
Fig. 8.1 Geodesic distance
and correlations. On each
metric configuration correlation
functions are computed
for lattice vertices within
the physical distance range
[d,d + Δd].
o
with n some exponent characterizing the power law decay, or at very large distances
an exponential decay, characterized by a correlation length ξ ,
< √ gR(x) √ gR(y) δ(|x − y|−d) > c ∼
d ≫ ξ
e −d/ξ . (8.32)
In fact the invariant correlation length ξ is generally defined (in analogy with what
one does for other theories) through the long-distance decay of the connected, invariant
correlations at fixed geodesic distance d. In the pure power law decay case
of Eq. (8.31) the correlation length ξ is of course infinite. One can show from scaling
considerations (see below) that the power n in Eq. (8.31) is related to the critical
exponent ν by n = 4 − 1/ν.
In the presence of a finite correlation length ξ one needs therefore to carefully
distinguish between the “short distance” regime
where Eq. (8.31) is valid, and the “long distance” regime
l 0 ≪ d ≪ ξ , (8.33)
ξ ≪ d ≪ L , (8.34)
where Eq. (8.32) is appropriate. Here l 0 = √ < l 2 > is the average lattice spacing,
and L = V 1/4 the linear size of the system.
Recently the issue of defining diffeomorphism invariant correlations in quantum
gravity has been re-examined from a new perspective (Giddings, Marolf and Hartle,
2006).