Feynman Path Integral Formulation

1.5 One-Loop Divergences 19
R 2 = ∂ 2 h μ μ∂ 2 h α α − 2∂ 2 h μ μ∂ α ∂ β h αβ + ∂ α ∂ β h αβ ∂ μ ∂ ν h μν
R αβ R αβ = 1 4 (∂ 2 h μ μ∂ 2 h α α + ∂ 2 h μα ∂ 2 h μα − 2∂ 2 h μ μ∂ α ∂ β h αβ
−2∂ α ∂ ν h ν μ∂ α ∂ β h μβ + 2∂ μ ∂ ν h μν ∂ α ∂ β h αβ ) ,
(1.101)
[compare with Eq. (1.45)], combined with some suitable special choices for the
background metric, such as g μν (x) =η μν f (x), to further simplify the calculation.
This eventually determines the required one-loop counterterm for pure gravity to be
ΔL g =
√ g
8π 2 (d − 4)
( 1
120 R2 + 7 20 R μνR μν )
. (1.102)
For the simpler case of classical gravity coupled invariantly to a single real quantum
scalar field one finds
√ g 1 (
ΔL g =
12
8π 2 R 2 + R μν R μν) . (1.103)
(d − 4) 120
The complete set of one-loop divergences, computed using the alternate method of
the heat kernel expansion and zeta function regularization 4 close to four dimensions,
can be found in the comprehensive review (Hawking, 1977) and further references
therein. In any case one is led to conclude that pure quantum gravity in four dimensions
is not perturbatively renormalizable: the one-loop divergent part contains
local operators which were not present in the original Lagrangian. It would seem
therefore that these operators would have to be added to the bare L , so that a consistent
perturbative renormalization program can be developed in four dimensions.
As a result, the field equations become significantly more complicated due to the
presence of the curvature squared terms (Barth and Christensen, 1983).
There are two interesting, and interrelated, aspects of the result of Eq. (1.102).
The first one is that for pure gravity the divergent part vanishes when one imposes
the tree-level equations of motion R μν = 0: the one-loop divergence vanishes onshell.
The second interesting aspect is that the specific structure of the one-loop
divergence is such that its effect can actually be re-absorbed into a field redefinition,
g μν → g μν + δg μν
δg μν ∝ 7 20 R μν − 11
60 Rg μν , (1.104)
which renders the one-loop amplitudes finite for pure gravity. Unfortunately this
hoped-for mechanism does not seem to work to two loops, and no additional miraculous
cancellations seem to occur there. At two loops one expects on general grounds
4 The zeta-function regularization (Ray and Singer, 1971; Dowker and Critchley, 1976; Hawking,
1977) involves studying the behavior of the function ζ (s) =∑ ∞ n=0(λ n ) −s ,wheretheλ n ’s are the
eigenvalues of the second order differential operator M in question. The series will converge for
s > 2, and can be used for an analytic continuation to s = 0,whichthenleadstotheformalresult
log(detM)=log∏ ∞ n=0 λ n = −ζ ′ (0).