Feynman Path Integral Formulation

7.7 Mean Field Theory 269
ξ ∼ √ logT
∼
k→k c
|log(k c − k)| 1/2 . (7.159)
From the definition of the exponent ν, namely ξ ∼ (k c −k) −ν , the above result then
implies ν = 0 (i.e. a weak logarithmic singularity) at d = ∞.
It is of interest to contrast the result ν ∼ 0 for gravity in large dimensions
with what one finds for scalar (Wilson and Fisher, 1972; Wilson, 1973) and gauge
(Drouffe, Parisi and Sourlas, 1979) fields, in the same limit d = ∞. So far, known
results can be summarized as follows
scalar field ν = 1 2
lattice gauge field ν = 1 4
lattice gravity ν = 0 . (7.160)
It should be regarded as encouraging that the new value obtained here, namely ν = 0
for gravitation, appears to some extent to be consistent with the general trend observed
for lower spin, at least at infinite dimension. What happens in finite dimensions
The situation becomes much more complicated since the self-intersection
properties of the surface have to be taken into account. But a simple geometric argument
then suggests in finite but large dimensions ν = 1/(d − 1) (Hamber and
Williams, 2004).
7.7 Mean Field Theory
In this section we will describe briefly a simple mean-field approach to quantum
gravity, which contains some features observed in the numerical simulations. Write
for the local average curvature R,
R(k) = < ∫ d d x √ gR>
< ∫ d d x √ g > , (7.161)
an effective action (or effective potential) which entirely neglects any further effects
of the metric degrees of freedom,
I ef f (R) =(k − k c )V |R| + aV |R| λ , (7.162)
with as usual k ≡ 1/8πG and a > 0 some additional coupling; in the strong coupling
phase of gravity k < k c .
The above effective action is inspired by the analogy with the Landau theory for
order-disorder transitions (Landau and Lifshitz, 1980), and the term proportional to
a is supposed to represent, in some crude and effective theory way, the effects of
the interactions. Classically one has of course k c = 0, but fluctuations give rise to a