Feynman Path Integral Formulation

8.7 Physical and Unphysical Phases 285
χ R (k)
∼
k→k c
A χR |k c − k| −(2−dν) . (8.54)
At a critical point the fluctuation χ is in general expected to diverge, corresponding
to the presence of a divergent correlation length. From such averages one can
therefore in principle extract the correlation length exponent ν of Eq. (8.50) without
having to compute a correlation function.
An equivalent result, relating the quantum expectation value of the curvature to
the physical correlation length ξ , is obtained from Eqs. (8.50) and (8.53)
R(ξ )
∼
k→k c
ξ 1/ν−4 , (8.55)
again up to an additive constant. Matching of dimensionalities in this last equation
is restored by inserting an appropriate power of the Planck length l P = √ G on the
r.h.s..
One can relate the critical exponent ν to the scaling behavior of correlations at
large distances. The curvature fluctuation is related to the connected scalar curvature
correlator at zero momentum
∫ ∫ √ √ dx dy < gR(x) gR(y) >c
χ R (k) ∼
< ∫ dx √ g >
. (8.56)
A divergence in the above fluctuation is then indicative of long range correlations,
corresponding to the presence of a massless particle. Close to the critical point one
expects for large separations l 0 ≪|x − y| ≪ξ a power law decay in the geodesic
distance, as in Eq. (8.31),
< √ gR(x) √ gR(y) > ∼
|x−y|→∞
1
|x − y| 2n . (8.57)
Inserting the above expression in Eq. (8.56) and comparing with Eq. (8.54) determines
the n as n = d − 1/ν. A priori one cannot exclude to possibility that some
states acquire a mass away from the critical point, in which case the correlation
functions would have the behavior of Eq. (8.32) for |x − y|≫ξ .
8.7 Physical and Unphysical Phases
An important alternative to the analytic methods in the continuum is an attempt
to solve quantum gravity directly via numerical simulations. The underlying idea
is to evaluate the gravitational functional integral in the discretized theory Z by
summing over a suitable finite set of representative field configurations. In principle
such a method given enough configurations and a fine enough lattice can provide an
arbitrarily accurate solution to the original quantum gravity theory.
In practice there are several important factors to consider, which effectively limit
the accuracy that can be achieved today in a practical calculation. Perhaps the most