Feynman Path Integral Formulation

8.9 Renormalization Group and Lattice Continuum Limit 295Method ν −1 in d = 3 ν −1 in d = 4lattice 1.67(6) -lattice - 2.98(7)2 + ε 1.6 4.4truncation 1.2 2.666exact ? 1.5882 3Table 8.1 Direct determinations of the critical exponent ν −1 for quantum gravitation, using variousanalytical 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 ultravioletcutoff, and ν −1 = 2.98(7) which is used for comparison in Table 8.1. In threedimensions the numerical results are consistent with the universality class of the interactingscalar field. The same set of results are compared graphically in Fig. 8.9and Fig. 8.10 below.The direct numerical determinations of the critical point k c = 1/8πG c in d = 3and d = 4 space-time dimensions are in fact quite close to the analytical predictionof the lattice 1/d expansion given previously in Eq. (7.150),k cλ 1−2/d0= 21+2/dd 3[ Γ (d)√d + 1] 2/d. (8.75)The above expression gives for a bare cosmological constant λ 0 = 1 the estimatek 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 becompared 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 LimitThe discussion in the previous sections points to the existence of a phase transitionin the lattice gravity theory, with divergent correlation length in the vicinity of thecritical point, as in Eq. (8.50)ξ (k)∼k→k cA ξ |k c − k| −ν . (8.76)As described previously, the existence of such a correlation length is usually inferredindirectly by scaling arguments, from the presence of singularities in the free energyF latt = − 1 V lnZ latt as a function of the lattice coupling k. Equivalently, ξ could havebeen computed directly from correlation functions at fixed geodesic distance using