Feynman Path Integral Formulation

9.3 Poisson’s Equation and Vacuum Polarization Cloud 309
Fig. 9.1 A virtual graviton
cloud surrounds the point
source of mass M, leading to
an anti-screening modification
of the static gravitational
potential. This antiscreening
effect of vacuum fluctuations
is quite natural in gravity,
since the larger the cloud is,
the stronger the gravitational
force is expected to be.
M
4π
(k 2 + μ 2 ) → 4π
(k 2 + μ 2 )
⎡
( ) 1
⎣ m
2 2ν
1 + a 0
k 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 next
step is naturally an attempt at finding a solution to Poisson’s equation with a point
source at the origin, so that one can determine the structure of the quantum corrections
to the static gravitational potential in real space. There are in principle two
equivalent ways to compute the potential φ(r), either by inverse Fourier transform
of 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 below
in Eq. (9.20). The zero-th order term then gives the standard Newtonian −MG/r
term, while the correction in general is given by a rather complicated hypergeometric
function. 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 is
particularly transparent,
(
G(r) =G ∞ 1 − a )
0
e −mr , (9.18)
1 + a 0
where we have set G ∞ ≡ (1+a 0 )G and G ≡ G(0). G therefore increases slowly from
its value G at small r to the larger value (1 + a 0 )G at infinity. Fig. 9.1 illustrates
the anti-screening effect of the virtual graviton cloud. Fig. 9.2 gives a schematic
illustration of the behavior of G as a function of r.
Another possible procedure to obtain the static potential φ(r) is to solve directly
the radial Poisson equation for φ(r). This will give a density ρ(r) which can later
be used to generalize to the relativistic case. In the a 0 ≠ 0 case one needs to solve
Δφ = 4πρ, or in the radial coordinate for r > 0