Feynman Path Integral Formulation

300 8 Numerical Studies
guide, the gradual removal of such a cutoff would then plunge the theory back into
a degenerate two-dimensional, and therefore physically unacceptable, geometry.
8.10 Curvature Scales
As can be seen from Eqs. (3.79) and (8.21) the path integral for pure quantum gravity
can be made to depend on the gravitational coupling G and the cutoff Λ only: by a
suitable rescaling of the metric, or the edge lengths in the discrete case, one can set
the cosmological constant to unity in units of the cutoff. The remaining coupling G
should then be viewed more appropriately as the gravitational constant in units of
the cosmological constant λ.
The renormalization group running of G(μ) in Eq. (8.84) involves an invariant
scale ξ = 1/m. At first it would seem that this scale could take any value, including
very small ones based on the naive estimate ξ ∼ l P , which would preclude any
observable quantum effects in the foreseeable future. But the result of Eqs. (8.62)
and (8.63) suggest otherwise, namely that the non-perturbative scale ξ is in fact
related to curvature. From astrophysical observation the average curvature is very
small, so one would conclude from Eq. (8.63) that ξ is very large, and possibly
macroscopic. But the problem with Eq. (8.63) is that it involves the lattice Ricci
scalar, a quantity related curvature probed by parallel transporting vectors around
infinitesimal loops with size comparable to the lattice cutoff Λ −1 . What one would
like is instead a relationship between ξ and quantities which describe the geometry
on larger scales.
In many ways the quantity m of Eq. (8.80) behaves as a dynamically generated
mass scale, quite similar to what happens in the non-linear σ-model case
[Eq. (3.60)], or in the 2 + ε gravity case [Eq. (3.118)]. Indeed in the weak field
expansion, presumably appropriate for slowly varying fields on very large scales, a
mass-like term does appear, as in Eq. (1.79), with μ 2 = 16πG|λ 0 |≡2|λ| where λ
is the scaled cosmological constant. From the classical field equation R = 4λ one
can relate the above λ, and therefore the mass-like parameter m, to curvature, which
leads to the identification
λ obs ≃ 1
ξ 2 , (8.86)
with λ obs the observed small but non-vanishing cosmological constant.
A further indication that the identification of the observed scaled cosmological
constant with a mass-like - and thefore renormalization group invariant - term makes
sense beyond the weak field limit can be seen for example by comparing the structure
of the three classical field equations
R μν − 1 2 g μν R + λ g μν
= 8πGT μν
∂ μ F μν + μ 2 A ν = 4πe j ν
∂ μ ∂ μ φ + m 2 φ = g 3! φ 3 , (8.87)