Feynman Path Integral Formulation

1.4 Feynman Rules 13and the graviton Feynman propagator in d dimensions is then found to be of theformD μναβ (k)= η μαη νβ + η μβ η να −d−2 2 η μν η αβk 2 , (1.77)with a suitable iε prescription to correctly integrate around poles in the complexk space. Equivalently the whole procedure could have been performed from thestart with an Euclidean metric η μν → δ μν and a complex time coordinate t = −iτwith hardly any changes of substance. The simple pole in the graviton propagatorat d = 2 serves as a reminder of the fact that, due to the Gauss-Bonnet identity, thegravitational Einstein-Hilbert action of Eq. (1.64) becomes a topological invariantin two dimensions.Higher order correction in h to the Lagrangian for pure gravity then determineto order h 3 the three-graviton vertex, to order h 4 the four-graviton vertex, and soon. Because of the √ g and g μν terms in the action, there are an infinite number ofvertices in h.Had one included a cosmological constant term as in Eq. (1.55), which can alsobe expressed in terms of the matrix V as√ g = 1 +12h μμ − 1 2 h αβV αβμν h μν + O(h 3 ) , (1.78)then the expression in Eq. (1.74) would have readL 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 theexchange of a particle of mass μ 2 = −λ 0 κ 2 . The term linear in h can be interpretedas a uniform constant source for the gravitational field. But one needs to be quitecareful, since for non-vanishing cosmological constant flat space g μν ∼ η μν is nolonger a solution of the vacuum field equations and the problem becomes a bit moresubtle: one needs to expand around the correct vacuum solutions in the presence ofa λ-term, which are no longer constant.Another point needs to be made here. One peculiar aspect of perturbative gravityis that there is no unique way of doing the weak field expansions, and one canhave therefore different sets of Feynman rules, even apart from the choice of gaugecondition, 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 todefine 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 fromthe g μν ’s to the g μν (x) √ g(x)’s involves a Jacobian, which can be taken to be onein dimensional regularization. There is one obvious advantage of this expansion