Feynman Path Integral Formulation

118 4 Hamiltonian and Wheeler-DeWitt Equation
Other approaches to the issue of boundary conditions and the construction of
the wave functional can be found, among others, in (Linde, 1998), where an anti-
Wick rotation t = iτ is suggested, which might improve convergence issues involving
the gravitational conformal mode, but causes irreparable damage in the
non-gravitational path integrals, and in (Vilenkin, 1998) where tunneling minisuperspace
models are examined and considered as phenomenologically viable, based
on claims that the Hartle-Hawking wafe functions might tend to disfavor inflationary
evolution. A very recent reference addressing these issues is (Hartle, Hawking
and Hertog, 2008).
4.10 Minisuperspace
The quantum state of a gravitational system is described, in the Wheeler-deWitt
framework just introduced, by a wave function Ψ which is a functional of the threemetric
g ij and the matter fields φ. In general the latter could contain fields of arbitrary
spins, but here we will consider for simplicity just one single component scalar
field φ(x).
The wavefunction Ψ will then obey the zero energy Schrödinger-like equation of
Eqs. (4.72) and (4.73),
δ
{−16πG 2
· G ij,kl − 1 √
( )
g
3
R − 2λ + √ }
gT 00 (∂/∂φ,φ) Ψ[g ij ,φ] =0 ,
δg ij δg kl 16πG
(4.87)
with inverse supermetric
G ij,kl = 1 2 g−1/2 ( g ik g jl + g il g jk − g ij g kl
)
, (4.88)
and momentum constraint
{
}
δ
i∇ i − 1 √
δg
2 gT
0 j (∂/∂φ,φ)
ij
Ψ[g ij ,φ] =0 . (4.89)
Then quantum state described by Ψ is a functional on the infinite dimensional manifold
W consisting of all positive definite metrics g ij (x) and matter fields φ(x) on a
spacelike three-surface S. On this space there is a natural metric Γ (N)
∫ d
ds 2 3 xd 3 x ′ [
[δg,δφ]=
G ij,kl (x,x ′ ) δg ij δg kl (x ′ )+ √ ]
gδ 3 (x−x ′ ) δφ(x)δφ(x ′ ) ,
N(x)
(4.90)
where
G ij,kl (x,x ′ )=G ij,kl (x)δ 3 (x − x ′ )
G ij,kl (x) = 1 √
[
]
2 g g ik (x)g jl (x)+g il (x)g jk (x) − 2g ij (x)g kl (x) ,
(4.91)