Feynman Path Integral Formulation

8.9 Renormalization Group and Lattice Continuum Limit 295
Method ν −1 in d = 3 ν −1 in d = 4
lattice 1.67(6) -
lattice - 2.98(7)
2 + ε 1.6 4.4
truncation 1.2 2.666
exact 1.5882 3
Table 8.1 Direct determinations of the critical exponent ν −1 for quantum gravitation, using various
analytical and numerical methods in three and four space-time dimensions.
previously in Sects. (3.5) and (3.6), respectively. The lattice model of Eq. (6.91)
in four dimensions gives for the critical point G c ≈ 0.626 in units of the ultraviolet
cutoff, and ν −1 = 2.98(7) which is used for comparison in Table 8.1. In three
dimensions the numerical results are consistent with the universality class of the interacting
scalar field. The same set of results are compared graphically in Fig. 8.9
and Fig. 8.10 below.
The direct numerical determinations of the critical point k c = 1/8πG c in d = 3
and d = 4 space-time dimensions are in fact quite close to the analytical prediction
of the lattice 1/d expansion given previously in Eq. (7.150),
k c
λ 1−2/d
0
= 21+2/d
d 3
[ Γ (d)
√
d + 1
] 2/d
. (8.75)
The above expression gives for a bare cosmological constant λ 0 = 1 the estimate
k c = √ 3/(16·5 1/4 )=0.0724 in d = 4, to be compared with the numerical result k c =
0.0636(11) in (Hamber, 2000). Even in d = 3 one has k c = 2 5/3 /27 = 0.118, to be
compared with the direct determination k c = 0.112(5) from (Hamber and Williams,
1993). These estimates are compared below in Fig. 8.10.
8.9 Renormalization Group and Lattice Continuum Limit
The discussion in the previous sections points to the existence of a phase transition
in the lattice gravity theory, with divergent correlation length in the vicinity of the
critical point, as in Eq. (8.50)
ξ (k)
∼
k→k c
A ξ |k c − k| −ν . (8.76)
As described previously, the existence of such a correlation length is usually inferred
indirectly by scaling arguments, from the presence of singularities in the free energy
F latt = − 1 V lnZ latt as a function of the lattice coupling k. Equivalently, ξ could have
been computed directly from correlation functions at fixed geodesic distance using