Feynman Path Integral Formulation

30 1 Continuum Formulation
consistent with the assumption that the two higher derivative couplings a and b are
large, since in such a limit one is close to flat space. One-loop radiative corrections
then show that the theory is asymptotically free in the higher derivative couplings a
and b (Julve and Tonin, 1978; Fradkin and Tseytlin, 1981; Avramidy and Barvinsky,
1985).
The calculation of one-loop quantum fluctuation effects proceeds in a way that
is similar to the pure Einstein gravity case. One first decomposes the metric field
as a classical background part g μν (x) and a quantum fluctuation part h μν (x) as in
Eq. (1.92), and then expands the classical action to quadratic order in h μν , with
gauge fixing and ghost contributions added, similar to those in Eqs. (1.95) and
(1.96), respectively. The first order variation of the action of Eq. (1.137) gives the
field equations for higher derivative gravity in the absence of sources,
∂I
∂g μν = 1 √ g(R μν
κ 2 − 1 2 gμν R)+ 1 2 λ √
0 gg
μν
+a √ g [ 2 3 (1 + ω)R(Rμν − 1 4 gμν R)
+ 1 2 gμν R αβ R αβ − 2R μανβ R αβ + 1 3 (1 − 2ω)∇μ ∇ ν R
−✷R μν + 1 6 (1 + 4ω)gμν ✷R]=0 , (1.154)
where we have set for the ratio of the two higher derivative couplings ω = b/a.
The second order variation is done similarly. It then allows the Gaussian integral
over the quantum fields to be performed using the formula of Eq. (1.97). One then
finds that the one-loop effective action, which depends on g μν only, can be expressed
as
Γ = 1 2 trlnF mn − trlnQ αβ − 1 2 trlncαβ , (1.155)
with the quantities F nm and Q αβ defined by
F nm =
δ 2 I
δg m δg n + δχ α δχ
δg m cαβ β
δg n
Q αβ = δχ α
δg m ∇m β . (1.156)
A shorthand notation is used here, where spacetime and internal indices are grouped
together so that g m = g μν (x). χ α are a set of gauge conditions, c αβ is a nonsingular
functional matrix fixing the gauge, and the ∇ i α are the local generators of the group
of general coordinate transformations, ∂ i α f α = 2g α(μ ∇ ν) f α (x).
Ultimately one is only interested in the divergent part of the effective one-loop
action. The method of extracting the divergent part out of the determinant (or trace)
expression in Eq. (1.155) is similar to what is done, for example, in QED to evaluate
the contribution of the fermion vacuum polarization loop to the effective action.
There, after integrating out the fermions, one obtains a functional determinant of
the massless Dirac operator ̸D(A) in an external A μ field,