Feynman Path Integral Formulation

8.4 Invariant Correlations at Fixed Geodesic Distance 279
having the two given vertices as endpoints. This can be done at least in principle by
enumerating all paths connecting the two points, and then selecting the shortest one.
Equivalently it can be computed from the scalar field propagator, as in Eq. (6.145).
Consequently physical correlations have to be defined at fixed geodesic distance
d, as in the following correlation between scalar curvatures
∫
<
∫
dx
dy √ gR(x) √ gR(y) δ(|x − y|−d) > . (8.26)
Generally these do not go to zero at large separation, so one needs to define the
connected part, by subtracting out the value at d = ∞. These will be indicated in the
following by the connected c average, and we will write the resulting connected
curvature correlation function at fixed geodesic distance compactly as
G R (d) ∼ < √ gR(x) √ gR(y) δ(|x − y|−d) > c . (8.27)
One can define several more invariant correlation functions at fixed geodesic distance
for other operators involving curvatures (Hamber, 1994). The gravitational
correlation function just defined is similar to the one in non-Abelian gauge theories,
Eq. (3.146).
In the lattice regulated theory one can define similar correlations, using for example
the correspondence of Eqs. (6.38) or (6.110) for the scalar curvature
√ gR(x) → 2 ∑ δ h A h . (8.28)
h⊃x
If the deficit angles are averaged over a number of contiguous hinges h sharing a
common vertex x, one is naturally lead to the connected correlation function
G R (d) ≡ < ∑ δ h A h ∑ δ h ′A h ′ δ(|x − y|−d) > c , (8.29)
h⊃x h ′ ⊃y
which probes correlations in the scalar curvatures. In practice the above lattice correlations
have to be computed by a suitable binning procedure: one averages out all
correlations in a geodesic distance interval [d,d + Δd] with Δd comparable to one
lattice spacing l 0 . See Fig. 8.1. Similarly one can construct the connected correlation
functions for local volumes at fixed geodesic distance
G V (d) ≡ < ∑ V h ∑ V h ′ δ(|x − y|−d) > c . (8.30)
h⊃x h ′ ⊃y
In general one expects for the curvature correlation either a power law decay, for
distances sufficiently larger than the lattice spacing l 0 ,
< √ gR(x) √ gR(y) δ(|x − y|−d) > c ∼
d ≫ l 0
1
d 2n , (8.31)