Feynman Path Integral Formulation

Chapter 9Scale Dependent Gravitational Couplings9.1 Renormalization Group and Scale Dependence of GNon-perturbative studies of quantum gravity suggest the possibility that gravitationalcouplings might be weakly scale dependent due to nontrivial renormalizationgroup effects. This would introduce a new gravitational scale, unrelated to Newton’sconstant, required in order to parametrize the gravitational running in the infraredregion. If one is willing to accept such a scenario, then it seems difficult to finda compelling theoretical argument for why the non-perturbative scale entering thecoupling evolution equations should be very small, comparable to the Planck length.One possibility is that the relevant non-perturbative scale is related to the curvatureand therefore macroscopic in size, which could have observable consequences. Onekey ingredient in this argument is the relationship, in part supported by Euclideanlattice results combined with renormalization group arguments, between the scalingviolation parameter and the scale of the average curvature.9.2 Effective Field EquationsTo summarize the results of the previous section, the result of Eq. (8.84) implies forthe running gravitational coupling in the vicinity of the ultraviolet fixed point⎡⎤( ) 1G(k 2 )=G c⎣ m2 2ν1 + a 0 + O[(m 2k 2 /k 2 ) 1 ν ] ⎦ , (9.1)with m = 1/ξ , a 0 > 0 and ν ≃ 1/3. Since ξ is expected to be very large, the quantityG c in the above expression should now be identified with the laboratory scale value√Gc ∼ 1.6 × 10 −33 cm. Quantum corrections on the r.h.s. are therefore quite smallas long as k 2 ≫ m 2 , which in real space corresponds to the “short distance” regimer ≪ ξ .305