Feynman Path Integral Formulation

Chapter 9
Scale Dependent Gravitational Couplings
9.1 Renormalization Group and Scale Dependence of G
Non-perturbative studies of quantum gravity suggest the possibility that gravitational
couplings might be weakly scale dependent due to nontrivial renormalization
group effects. This would introduce a new gravitational scale, unrelated to Newton’s
constant, required in order to parametrize the gravitational running in the infrared
region. If one is willing to accept such a scenario, then it seems difficult to find
a compelling theoretical argument for why the non-perturbative scale entering the
coupling evolution equations should be very small, comparable to the Planck length.
One possibility is that the relevant non-perturbative scale is related to the curvature
and therefore macroscopic in size, which could have observable consequences. One
key ingredient in this argument is the relationship, in part supported by Euclidean
lattice results combined with renormalization group arguments, between the scaling
violation parameter and the scale of the average curvature.
9.2 Effective Field Equations
To summarize the results of the previous section, the result of Eq. (8.84) implies for
the running gravitational coupling in the vicinity of the ultraviolet fixed point
⎡
⎤
( ) 1
G(k 2 )=G c
⎣ m
2 2ν
1 + a 0 + O[(m 2
k 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 quantity
G 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 small
as long as k 2 ≫ m 2 , which in real space corresponds to the “short distance” regime
r ≪ ξ .
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