Feynman Path Integral Formulation

296 8 Numerical Studies
10
8
6
1Ν
4
2
0 0.2 0.4 0.6 0.8 1
z ⩵
d 2
d 1
Fig. 8.9 Universal gravitational exponent 1/ν as a function of the dimension. The abscissa is
z =(d − 2)/(d − 1), which maps d = 2toz = 0andd = ∞ to z = 1. The larger circles at d = 3and
d = 4 are the lattice gravity results, interpolated (continuous curve) using the exact lattice results
1/ν = 0ind = 2, and ν = 0atd = ∞ [from Eq. (7.159)]. The two curves close to the origin are the
2 + ε expansion for 1/ν to one loop (lower curve) and two loops (upper curve). The lower almost
horizontal line gives the value for ν expected for a scalar field theory, for which it is known that
ν = 1ind = 2andν = 1 2
in d ≥ 4.
the definition in Eq. (8.32), or even from the correlation of Wilson lines associated
with the propagation of two heavy spinless particles. The outcome of such large
scale numerical calculations is eventually a determination of the quantities ν, k c =
1/8πG c and A ξ from first principles, to some degree of numerical accuracy.
In either case one expects the scaling result of Eq. (8.76) close to the fixed point,
which we choose to rewrite here in terms of the inverse correlation length m ≡ 1/ξ
m = Λ A m |k − k c | ν . (8.77)
Note that in the above expression the correct dimension for m (inverse length) has
been restored by inserting explicitly on the r.h.s. the ultraviolet cutoff Λ. Herek
and k c are of course still dimensionless quantities, and correspond to the bare microscopic
couplings at the cutoff scale, k ≡ k(Λ) ≡ 1/[8πG(Λ)]. A m is a calculable
numerical constant, related to A ξ in Eq. (8.50) by A m = A −1 . It is worth pointing
ξ
out that the above expression for m(k) is almost identical in structure to the one for
the non-linear σ-model in the 2 + ε expansion, Eq. (3.36) and in the large N limit,
Eqs. (3.59), (3.60) and (3.64). It is of course also quite similar to 2 + ε result for
continuum gravity, Eq. (3.121).
The lattice continuum limit corresponds to the large cutoff limit taken at fixed m,
Λ → ∞ , k → k c , m fixed , (8.78)