Feynman Path Integral Formulation

8.11 Gravitational Condensate 303where the condensate is, according to Eq. (8.79), non-analytic at G = G c . A gravitonvacuum condensate of order ξ −1 ∼ 10 −30 eV is of course extraordinarily small comparedto the QCD color condensate (Λ MS≃ 220MeV) and the electro-weak Higgscondensate (v ≃ 250GeV ). One can pursue the analogy with non-Abelian gaugetheories further by stating that the quantum gravity theory cannot provide a valuefor the non-perturbative curvature scale ξ : it needs to be fixed by some sort of phenomenologicalinput, either by Eqs. (8.84) or by (8.86). But one important messageis that the scale ξ in those two equations is one and the same.Can the above physical picture be used to provide further insight into the natureof the phase transition, and more specifically the value for ν? We will mention herea simple geometric argument which can be given to support the exact value ν = 1/3for pure gravity (Hamber and Williams, 2004). First one notices that the vacuumpolarization induced scale dependence of the gravitational coupling G(r) as givenin Eq. (8.84) implies the following general structure for the quantum corrected staticgravitational potential,V (r) =−G(r) mM ≈−G(0) mM [[1 + c(r/ξ ) 1/ν + O (r/ξ ) 2/ν]] , (8.93)rrfor a point source of mass M located at the origin and for intermediate distancesl p ≪ r ≪ ξ . One can visualize the above result by stating that virtual graviton loopscause an effective anti-screening of the primary gravitational source M, giving riseto a quantum correction to the potential proportional to r 1/ν−1 . But only for ν = 1/3can the additional contribution be interpreted as being due to a close to uniform massdistribution surrounding the original source, of strengthρ ξ (M) = 3cM4πξ 3 . (8.94)Such a simple geometric interpretation fails unless the exponent ν for gravitationis exactly one third. In fact in dimensions d ≥ 4 one would expect based on thegeometric argument that ν = 1/(d − 1) if the quantum correction to the gravitationalpotential arises from such a virtual graviton cloud. These arguments rely ofcourse on the lowest order result V (r) ∼ ∫ d d−1 ke ik·x /k 2 ∼ r 3−d for single gravitonexchange in d > 3 dimensions.Equivalently, the running of G can be characterized as being in part due to a tinynon-vanishing (and positive) non-perturbative gravitational vacuum contribution tothe cosmological constant, withλ 0 (M) = 3cMξ 3 , (8.95)and therefore an associated effective classical average curvature of magnitudeR class ∼ Gλ 0 ∼ GM/ξ 3 . It is amusing that for a very large mass distribution M,the above expression for the curvature can only be reconciled with the naive dimensionalestimate R class ∼ 1/ξ 2 , provided for the gravitational coupling G itself onehas G ∼ ξ /M, which is reminiscent of Mach’s principle and its connection with theLense-Thirring effect (Lense and Thirring, 1918; Sciama, 1953; Feynman, 1962).