Feynman Path Integral Formulation

300 8 Numerical Studiesguide, the gradual removal of such a cutoff would then plunge the theory back intoa degenerate two-dimensional, and therefore physically unacceptable, geometry.8.10 Curvature ScalesAs can be seen from Eqs. (3.79) and (8.21) the path integral for pure quantum gravitycan be made to depend on the gravitational coupling G and the cutoff Λ only: by asuitable rescaling of the metric, or the edge lengths in the discrete case, one can setthe cosmological constant to unity in units of the cutoff. The remaining coupling Gshould then be viewed more appropriately as the gravitational constant in units ofthe cosmological constant λ.The renormalization group running of G(μ) in Eq. (8.84) involves an invariantscale ξ = 1/m. At first it would seem that this scale could take any value, includingvery small ones based on the naive estimate ξ ∼ l P , which would preclude anyobservable 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 factrelated to curvature. From astrophysical observation the average curvature is verysmall, so one would conclude from Eq. (8.63) that ξ is very large, and possiblymacroscopic. But the problem with Eq. (8.63) is that it involves the lattice Ricciscalar, a quantity related curvature probed by parallel transporting vectors aroundinfinitesimal loops with size comparable to the lattice cutoff Λ −1 . What one wouldlike is instead a relationship between ξ and quantities which describe the geometryon larger scales.In many ways the quantity m of Eq. (8.80) behaves as a dynamically generatedmass 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 fieldexpansion, presumably appropriate for slowly varying fields on very large scales, amass-like term does appear, as in Eq. (1.79), with μ 2 = 16πG|λ 0 |≡2|λ| where λis the scaled cosmological constant. From the classical field equation R = 4λ onecan relate the above λ, and therefore the mass-like parameter m, to curvature, whichleads to the identificationλ obs ≃ 1ξ 2 , (8.86)with λ obs the observed small but non-vanishing cosmological constant.A further indication that the identification of the observed scaled cosmologicalconstant with a mass-like - and thefore renormalization group invariant - term makessense beyond the weak field limit can be seen for example by comparing the structureof the three classical field equationsR 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)