Feynman Path Integral Formulation

5.1 Cosmological Wavefunctions 145will need to be factored out. Or, equivalently, one needs to restrict the functionalintegration to physical degrees of freedom.To carry the program through in specific cases, one first needs a choice of suitablebackground metric, then an explicit expression for the second variation of the actionaround this metric, and finally a procedure for evaluating the contribution of thequantum fluctuation around this background. With the metric written as in Eq. (5.10)one finds for the second variation of the action for λ = 0withÎ 2 [h] =− 132πG∫d 4 x √ gh μν G μν (5.12)G μν = ∇ λ ∇ λ ¯h μν + g μν ∇ λ ∇ σ ¯h λσ − ∇ μ ∇ λ ¯h λ ν − ∇ ν ∇ λ ¯h λ μ , (5.13)for a background metric satisfying R μν = 0, and up to total derivatives. Here ∇ μis the covariant derivative with respect to the background metric g μν , and ¯h μν thetrace-reversed metric perturbation,¯h μν = h μν − 1 2 g μν h μ ν . (5.14)There is no surface contribution in Î 2 [h], since it would have to be of the form h∇hwith h μν , thus vanishing on the boundary.The path integral over the h μν variables in Eq. (5.11) suffers from the usualproblem of configuration over-counting due to the gauge freedom in the metric h.Specifically, the action Î 2 [h] is invariant under local gauge variations of h μνh μν → h μν + ∂ μ ξ ν + ∂ ν ξ μ , (5.15)such that the gauge function ξ μ vanishes on the boundary. In order to avoid a divergencein the integration over the quantum fluctuations h μν , one needs to restrictthe functional integration over physically distinct metrics. One way of doing this isto introduce a gauge-fixing term, and the associated Faddeev-Popov determinant. Apossible gauge condition would be)f ν = ∇ μ(h μν − β g μν h λ λ, (5.16)with f ν some prescribed vector function on the background manifold. The correspondingFaddeev-Popov determinant would then involve a differential operator,determined by the derivative of the gauge condition with respect to the gauge parameter.For the gauge condition in Eq. (5.16) the relevant operator C μν isC μν (h) ξ ν = −∇ λ ∇ λ ξ μ − R μν ξ ν +(2β − 1)∇ μ ∇ ν ξ ν . (5.17)The above procedure works well with manifolds without boundaries (Gibbons andPerry, 1978). But it runs into technical problems when boundaries are present, whichmakes the procedure less transparent.