Feynman Path Integral Formulation

9.3 Poisson’s Equation and Vacuum Polarization Cloud 309Fig. 9.1 A virtual gravitoncloud surrounds the pointsource of mass M, leading toan anti-screening modificationof the static gravitationalpotential. This antiscreeningeffect of vacuum fluctuationsis quite natural in gravity,since the larger the cloud is,the stronger the gravitationalforce is expected to be.M4π(k 2 + μ 2 ) → 4π(k 2 + μ 2 )⎡( ) 1⎣ m2 2ν1 + a 0k 2 + m 2+ ...⎤⎦ , (9.17)where the limit μ → 0 should be taken at the end of the calculation.Given the running of G from either Eqs. (9.2) or (9.1) in the large k limit, the nextstep is naturally an attempt at finding a solution to Poisson’s equation with a pointsource at the origin, so that one can determine the structure of the quantum correctionsto the static gravitational potential in real space. There are in principle twoequivalent ways to compute the potential φ(r), either by inverse Fourier transformof Eq. (9.16), or by solving Poisson’s equation Δφ = 4πρ with the source term ρ(r)given by the inverse Fourier transform of the correction to G(k 2 ), as given belowin Eq. (9.20). The zero-th order term then gives the standard Newtonian −MG/rterm, while the correction in general is given by a rather complicated hypergeometricfunction. But for the special case ν = 1/2 the Fourier transform of Eq. (9.17)is easy to do, the integrals are elementary and the running of G(r) so obtained isparticularly transparent,(G(r) =G ∞ 1 − a )0e −mr , (9.18)1 + a 0where we have set G ∞ ≡ (1+a 0 )G and G ≡ G(0). G therefore increases slowly fromits value G at small r to the larger value (1 + a 0 )G at infinity. Fig. 9.1 illustratesthe anti-screening effect of the virtual graviton cloud. Fig. 9.2 gives a schematicillustration of the behavior of G as a function of r.Another possible procedure to obtain the static potential φ(r) is to solve directlythe radial Poisson equation for φ(r). This will give a density ρ(r) which can laterbe used to generalize to the relativistic case. In the a 0 ≠ 0 case one needs to solveΔφ = 4πρ, or in the radial coordinate for r > 0