Feynman Path Integral Formulation

18 1 Continuum Formulation+ c 1 {R − 1 2 hα αR + h α β Rβ α − 1 8 Rhα αh β β + 1 4 Rhα β hβ α− h ν β hβ αR α ν + 1 2 hα αh ν β Rβ ν − 1 4 ∇ νh α β ∇ν h β α+ ∇ ν h α α∇ ν h β β − 1 2 ∇ β h α α∇ μ h β μ + 1 2 ∇α h ν β ∇ νh β α}] ,(1.94)up to total derivatives. Here ∇ μ denotes a covariant derivative with respect to themetric g μν .Forg μν = η μν the above expression coincides with the weak fieldLagrangian contained in Eqs. (1.7) and (1.67), with a cosmological constant termadded, as given in Eq. (1.55).To this expression one needs to add the gauge fixing and ghost contributions, aswas done in Eq. (1.67). The background gauge fixing term used is− 1 2 C2 μ = − 1 2√ g(∇α h α μ − 1 2 ∇ μh α α)(∇ β h βμ − 1 2 ∇μ h β β ) , (1.95)with a corresponding ghost LagrangianL ghost = √ g ¯η μ (∂ α ∂ α η μ − R μ αη α ) . (1.96)The integration over the h μν field can then be performed with the aid of the standardGaussian integral formula∫ln [dh μν ] exp{− 1 2h · M(g) · h − N(g) · h}= 1 2 N(g) · M−1 (g) · N(g) − 1 2trlnM(g)+const. , (1.97)leading to an effective action for the g μν field. In practice one is only interested inthe divergent part, which can be shown to be local. Specific details of the functionalmeasure over metrics [dg μν ] are not deemed to be essential at this stage, as in perturbationtheory one is only doing Gaussian integrals, with h μν ranging from −∞ to+∞. In particular when using dimensional regularization one uses the formal rule∫d d k =(2π) d δ (d) (0)=0 , (1.98)which leads to some technical simplifications but obscures the role of the measure.In the flat background field case g μν = η μν , the functional integration over theh μν fields would have been particularly simple, since then one would be usingh μν (x)h αβ (x ′ ) → = G μναβ (x,x ′ ) , (1.99)with the graviton propagator G(k) given in Eq. (1.77). In practice, one can use theexpected generally covariant structure of the one-loop divergent partΔL g ∝ √ g ( α R 2 + β R μν R μν) , (1.100)with α and β some real parameters, as well as its weak field form, obtained from