Feynman Path Integral Formulation

148 5 Semiclassical Gravity
with TT quadratic action
Î 2 (π TT ,h TT )=− 1 ∫
16πG
(
)
d 4 x π ij
TT ∂ th TT
ij − H T [π TT ,h TT ]
. (5.27)
After assuming a canonical measure ∏(dpdq/2π ¯h) over the conjugate variables
π ij
TT
and hTT ij , and integrating out the momenta π ij
TT
, one finally obtains the simple
result
∫ {
P[a] = [dh TT
μν] exp − 1 ∫
d 4 x √ }
gh TT μν ∇
32πG
λ ∇ λ h TT
μν . (5.28)
Note that the last expression has been re-written in terms of purely TT components
h TT μν of the original four-metric perturbation in Eq. (5.10). Since physical perturbations
of the metric are known to be transverse-traceless (associated with a particle
of zero mass and spin two) the result is not surprising, and in fact has wider applicability
to the semiclassical expansion, beyond the simple choice for background
metric implicit in Eq. (5.21).
Formally, the Gaussian integration over the h TT variables gives
( )
P[a] =det −1/2 −∇ λ ∇ λ
4π lP 2 , (5.29)
μ2
where μ is a parameter with units of inverse length, and lP 2 = 16πG. The specific
power of μ appearing in this last expression actually depends on the details of the
measure [dh TT
μν], which will be discussed further below.
The determinant in Eq. (5.29) is fomally defined through an infinite product of
eigenvalues λ n of the Laplacian ∇ λ ∇ λ satisfying Dirichlet boundary conditions,
det
−∇ λ ∇ λ φ (n)
μν
)
(−μ 2 ∇ λ ∇ λ
= λ n φ (n)
μν (5.30)
→ ∏(μ 2 λ n ) . (5.31)
n
But the product is expected to be divergent and needs to be regularized. One way of
doing it is to define a zeta-function sum over eigenvalues (Hawking, 1976)
ζ (s) ≡ ∑
n
1
λ s n
, (5.32)
so that formally on obtains
( )
( )
− 1 2 logdet −∇ λ ∇ λ = − 1 2 trlog −∇ λ ∇ λ = − 1 2 ∑logλ n = 1 2 ζ ′ (0) , (5.33)
with ζ ′ (0) ≡ dζ /ds| s=0 . For the wave function P[a] itself one then has
n