Feynman Path Integral Formulation

32 1 Continuum Formulation
∂ω
= − 1 ∂t a (3β 3 + ωβ 2 )+...
∂ ˜λ = 1
∂t
2 κ4 β 5 + 2˜λκ 2 β 4 + ... (1.161)
with the dots indicating higher loop corrections. Here t is the logarithm of the relevant
energy scale, t =(4π) −2 ln(μ/μ 0 ), with μ a momentum scale q 2 ≈ μ 2 , and μ 0
some fixed reference scale. It is argued furthermore by the quoted authors that only
the quantities β 2 , β 3 and the combination κ 4 β 5 +4˜λκ 2 β 4 are gauge independent, the
latter combination appearing in the renormalization group equation for ˜λ(t) (thisisa
point to which we shall return later, as it follows quite generally from the properties
of the gravitational action, and therefore from the gravitational functional integral,
under a field rescaling, see Sect. 3.5).
The perturbative scale dependence of the couplings a(μ), b(μ) and ˜λ(μ) follows
from integrating the three differential equations in Eq. (1.161). The first renormalization
group equation is easily integrated, and shows the existence of an ultraviolet
fixed point at a −1 = 0; the one-loop result for the running coupling a is simply given
by a(t)=a(0)+β 2 t,or
a −1 (μ)
16π
∼
2
μ→∞ β 2 ln(μ/μ 0 ) , (1.162)
with μ 0 a reference scale. It suggests that the effective higher derivative coupling
a(μ) increases at short distances, but decreases in the infrared regime μ → 0. But
one should keep in mind that the one loop results are reliable at best only at very
short distances, or large energy scales, t → ∞. At the same time these results seem
physically reasonable, as one would expect curvature squared terms to play less of
a role at larger distances, as in the classical theory.
The scale dependence of the other couplings is a bit more complicated. The equation
for ω(t) exhibits two fixed points at ω uv ≈−0.0229 and ω ir ≈−5.4671; in
either case this would correspond to a higher derivative action with a positive R 2
term. It would also give rise to rapid short distance oscillations in the static potential,
as can be seen for example from Eq. (1.153) and the definition of m 0 = μ/ √ 2b.
The equation for ˜λ(t) gives a solution to one-loop order ˜λ(t) ∼ const.t q with
q ≈ 0.91, suggesting that the effective gravitational constant, in units of the cosmologial
constant, decreases at large distances. The experimental value for Newton’s
constant ¯hG/c 3 =(1.61624×10 −33 cm) 2 and for the scaled cosmological constant
Gλ 0 ∼ 1/(10 28 cm) 2 is such that the observed dimensionless ratio between the
two is very small, G 2 λ 0 ∼ 10 −120 . In the present model is seems entirely unclear
how such a small ratio could arise from perturbation theory alone.
At short distances the dimensionless coupling ˜λ ∼ λ 0 G 2 seems to increases
rapidly, thus partially invalidating the conclusions of a weak field expansion around
flat space, which are based generally on the assumption of small G and λ 0 .Atthe
same time, the fact that the higher derivative coupling a grows more rapidly in the
ultraviolet than the coupling ˜λ can be used retroactively at least as a partial justification
for the flat space expansion, in which the cosmological and Einstein terms are