Feynman Path Integral Formulation

144 5 Semiclassical Gravity
Î[g μν ]=− 1 ∫
d 4 x √ g(R − 2λ) − 1 ∫
d 3 x √ g ij K , (5.7)
16πG M
8πG ∂M
where K is the trace of the extrinsic curvature on the boundary. Note that the inner
product between two wave functionals is obtained by gluing together two wave
functionals and integrating over the fields on their common boundary, which is located
on a spatial hypersurface,
∫
〈Ψ[g ij ,φ]|Ψ ′ [g ij ,φ]〉 = [dg ij ][dφ] ¯Ψ[g ij ,φ]Ψ ′ [g ij ,φ]
∫
{
=
M,M ′[dg μν][dφ] exp − 1¯h }
Î(g μν,φ) , (5.8)
is interpreted as an integral over all of space, half of it to the left of the spacelike
hypersurface (M), and half of it to the right (M ′ ).
To evaluate the path integral defining Ψ one option is to use the method of steepest
descent, which is equivalent to the semi-classical, or WKB, approximation and
produces an answer in the form of an expansion in powers of ¯h. In quantum field
theory such an expansion is equivalent to expanding in the number of loop diagrams
associated with perturbative Feynman diagrams. The leading term to the wave functional
Ψ is then the classical contributions, and the next correction is determined by
the quadratic quantum fluctuations around the chosen background. After the fluctuations
are integrated over via a Gaussian integral formula, one obtains a functional
determinant, whose effects then determine the leading quantum correction.
The wave functional will then take the form
Ψ[g ij ]=P[g ij ] exp { −Î cl (g ij ) } , (5.9)
where Î cl (g ij ) is the classical Euclidean action associated with the saddle point
(if there is more than one, an additional sum will be required), and P[g ij ] is a
prefactor whose form is determined by the expansion of the Euclidean action
to quadratic order and the subsequent functional integration. One sets for the
four-metric
g μν → ḡ μν = g μν + h μν , (5.10)
where h μν is a perturbation of the saddle-point four-metric g μν (which therefore
satisfies δÎ/δg μν = 0), vanishing on the boundary. The prefactor P in Eq. (5.9) is
then given formally by the integral,
∫
P[g ij ]= [dh μν ] exp { −Î 2 (h μν ) } , (5.11)
M
with Î 2 the contribution to the action Î quadratic in the metric perturbation h μν ,
and itself also a function of the background metric g μν . The integral over the h μν
fluctuations will involve zero modes from the gauge degrees of freedom, which