Feynman Path Integral Formulation

5.1 Cosmological Wavefunctions 145
will need to be factored out. Or, equivalently, one needs to restrict the functional
integration to physical degrees of freedom.
To carry the program through in specific cases, one first needs a choice of suitable
background metric, then an explicit expression for the second variation of the action
around this metric, and finally a procedure for evaluating the contribution of the
quantum fluctuation around this background. With the metric written as in Eq. (5.10)
one finds for the second variation of the action for λ = 0
with
Î 2 [h] =− 1
32πG
∫
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 ∇ μ
is the covariant derivative with respect to the background metric g μν , and ¯h μν the
trace-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∇h
with h μν , thus vanishing on the boundary.
The path integral over the h μν variables in Eq. (5.11) suffers from the usual
problem 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 μν
h 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 divergence
in the integration over the quantum fluctuations h μν , one needs to restrict
the functional integration over physically distinct metrics. One way of doing this is
to introduce a gauge-fixing term, and the associated Faddeev-Popov determinant. A
possible 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 corresponding
Faddeev-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 is
C 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 and
Perry, 1978). But it runs into technical problems when boundaries are present, which
makes the procedure less transparent.