Feynman Path Integral Formulation

1.4 Feynman Rules 13
and the graviton Feynman propagator in d dimensions is then found to be of the
form
D μναβ (k)= η μαη νβ + η μβ η να −
d−2 2 η μν η αβ
k 2 , (1.77)
with a suitable iε prescription to correctly integrate around poles in the complex
k space. Equivalently the whole procedure could have been performed from the
start with an Euclidean metric η μν → δ μν and a complex time coordinate t = −iτ
with hardly any changes of substance. The simple pole in the graviton propagator
at d = 2 serves as a reminder of the fact that, due to the Gauss-Bonnet identity, the
gravitational Einstein-Hilbert action of Eq. (1.64) becomes a topological invariant
in two dimensions.
Higher order correction in h to the Lagrangian for pure gravity then determine
to order h 3 the three-graviton vertex, to order h 4 the four-graviton vertex, and so
on. Because of the √ g and g μν terms in the action, there are an infinite number of
vertices in h.
Had one included a cosmological constant term as in Eq. (1.55), which can also
be expressed in terms of the matrix V as
√ g = 1 +
1
2
h μμ − 1 2 h αβV αβμν h μν + O(h 3 ) , (1.78)
then the expression in Eq. (1.74) would have read
L 0 = λ 0 (1 + κ 1 2 hα α)+ 1 2 h αβ V αβμν (∂ 2 + λ 0 κ 2 )h μν , (1.79)
with κ 2 = 16πG. Then the graviton propagator would have been remained the same,
except for the replacement k 2 → k 2 −λ 0 κ 2 . In this gauge it would correspond to the
exchange of a particle of mass μ 2 = −λ 0 κ 2 . The term linear in h can be interpreted
as a uniform constant source for the gravitational field. But one needs to be quite
careful, since for non-vanishing cosmological constant flat space g μν ∼ η μν is no
longer a solution of the vacuum field equations and the problem becomes a bit more
subtle: one needs to expand around the correct vacuum solutions in the presence of
a λ-term, which are no longer constant.
Another point needs to be made here. One peculiar aspect of perturbative gravity
is that there is no unique way of doing the weak field expansions, and one can
have therefore different sets of Feynman rules, even apart from the choice of gauge
condition, depending on how one chooses to do the expansion for the metric.
For example, the structure of the scalar field action of Eq. (1.65) suggests to
define instead the small fluctuation graviton field h μν (x) via
˜g μν (x) ≡ g μν (x) √ g(x)=η μν + Kh μν (x) , (1.80)
with K 2 = 32πG (Faddeev and Popov, 1974; Capper et al, 1973). Here it is h μν (x)
that should be referred to as “the graviton field”. The change of variables from
the g μν ’s to the g μν (x) √ g(x)’s involves a Jacobian, which can be taken to be one
in dimensional regularization. There is one obvious advantage of this expansion