Feynman Path Integral Formulation

3.5 The Gravitational Case 85
with ω is a constant, and thus the renormalization properties of G 0 have no physical
meaning for this theory. This simply follows from the fact that √ gR is homogeneous
in g μν , which is quite different from the Yang-Mills case. The situation changes
though when one introduces a second dimensionful quantity to compare to. In the
pure gravity case this contribution is naturally supplied by the cosmological constant
term proportional to λ 0 ,
L = − 1
16πG 0
√ gR + λ0
√ g . (3.78)
Under a rescaling of the metric as in Eq. (3.77) one has
L = − 1
16πG 0
ω d/2−1 √ g ′ R ′ + λ 0 ω d/2 √ g ′ , (3.79)
which is interpreted as a rescaling of the two bare couplings
G 0 → ω −d/2+1 G 0 , λ 0 → λ 0 ω d/2 , (3.80)
leaving the dimensionless combination G d 0 λ 0 d−2 unchanged. Therefore only the latter
combination has physical meaning in pure gravity. In particular, one can always
choose the scale ω = λ −2/d
0
so as to adjust the volume term to have a unit coefficient.
More importantly, it is physically meaningless to discuss separately the
renormalization properties of G 0 and λ 0 , as they are individually gauge-dependent
in the sense just illustrated. These arguments should clarify why in the following it
will be sufficient at the end to just focus on the renormalization properties of one
coupling, such as Newton’s constant G 0 .
In general it is possible at least in principle to define quantum gravity in any d > 2
(Weinberg, 1979). There are d(d + 1)/2 independent components of the metric in
d dimensions, and the same number of algebraically independent components of
the Ricci tensor appearing in the field equations. The contracted Bianchi identities
reduce the count by d, and so does general coordinate invariance, leaving d(d −3)/2
physical gravitational degrees of freedom in d dimensions. At the same time, four
space-time dimensions is known to be the lowest dimension for which Ricci flatness
does not imply the vanishing of the gravitational field, R μνλσ = 0, and therefore the
first dimension to allow for gravitational waves and their quantum counterparts,
gravitons.
In a general dimension the position space tree-level graviton propagator of the
linearized theory, given in k-space in Eq. (1.77), can be obtained by Fourier transform
and is proportional to
∫
d d k 1 eik·x Γ ( )
d−2
2
k 2 =
4π d/2 (x 2 . (3.81)
) d/2−1
The static gravitational potential is then proportional to the spatial Fourier transform
∫
V (r) ∝ d d−1 eik·x
k
k 2 ∼ 1 , (3.82)
rd−3