Feynman Path Integral Formulation

7.6 Discrete Gravity in the Large-d Limit 265
Here we have further allowed for the possibility that the average lattice spacing
l 0 = 〈l 2 〉 1/2 is not equal to one (in other words, we have restored the appropriate
overall scale for the average edge length, which is in fact largely determined by the
value of λ 0 ).
The average lattice spacing l 0 can easily be estimated from the following argument.
The volume of a general equilateral simplex is given by Eq. (7.133), multiplied
by an additional factor of l0 d . In the limit of small k the average volume of a
simplex is largely determined by the cosmological term, and can therefore be computed
from
< V >= − ∂ ∫
log [dl 2 ]e −λ 0V (l 2) , (7.148)
∂λ 0
with V (l 2 )=( √ d + 1/d!2 d/2 )l d ≡ c d l d . After doing the integral over l 2 with measure
dl 2 and solving this last expression for l0 2 one obtains
[ ] 2/d
l0 2 = 1 2 d!2 d/2
λ 2/d √ , (7.149)
d d + 1
0
(which, for example, gives l 0 = 2.153 for λ 0 = 1 in four dimensions, in reasonable
agreement with the actual value l 0 ≈ 2.43 found near the transition point).
This then gives for λ 0 = 1 the estimate k c = √ 3/(16·5 1/4 )=0.0724 in d = 4, to
be compared with k c = 0.0636(11) obtained in (Hamber, 2000) by direct numerical
simulation in four dimensions. Even in d = 3 one finds again for λ 0 = 1, from
Eqs. (7.147) and (7.149), k c = 2 5/3 /27 = 0.118, to be compared with k c = 0.112(5)
obtained in (Hamber and Williams, 1993) by direct numerical simulation.
Using Eq. (7.149) inserted into Eq. (7.147) one obtains in the large d limit for
the dimensionless combination k/λ (d−2)/d
0
k c
λ 1−2/d
0
= 21+2/d
d 3
[ Γ (d)
√
d + 1
] 2/d
. (7.150)
To summarize, an expansion in powers of 1/d can be developed, which relies on a
combined use of the weak field expansion. It can be regarded therefore as a double
expansion in 1/d and ε, valid wherever the fields are smooth enough and the geometry
is close to flat, which presumably is the case in the vicinity of the lattice critical
point at k c .
A somewhat complementary 1/d expansion can be set up, which does not require
weak fields, but relies instead on the strong coupling (small k = 1/8πG, or large
G) limit. As such it will be a double expansion in 1/d and k. Its validity will be
in a regime where the fields are not smooth, and in fact will involve lattice field
configurations which are very far from smooth at short distances.
The general framework for the strong coupling expansion for pure quantum gravity
was outlined in the previous section, and is quite analogous to what one does in
gauge theories (Balian, Drouffe and Itzykson, 1975). One expands Z latt in powers
of k as in Eq. (7.68)