Feynman Path Integral Formulation

116 4 Hamiltonian and Wheeler-DeWitt Equation
and Φ[g ij ] some new wave functional to be determined. From the original Wheeler-
DeWitt equation one then obtains a new set of equations for the wavefunctional
Φ[g ij ],
{
−2iG ij,kl
δS
δg ij
δ
δ 2 S
− iG ij,kl
δg kl δg ij δg kl
−16πG · G ij,kl
δ 2
δg ij δg kl
+ Ĥ φ }
Φ[g ij (x)] = 0 . (4.77)
{
}
δ
2ig ij ∇ k + Ĥ φ i
Φ[g ij (x)] = 0 . (4.78)
δg jk
The next step consists in approximating (again, in analogy with the semiclassical
expansion in non-relativistic quantum mechanics), the solution by neglecting
second derivative terms δ 2 S/δg 2 and δ 2 /δg 2 terms in the equations for Φ[g ij ]
(Wheeler, 1964). The latter step is usually justified by regarding (or assuming) the
back-reaction of quantum matter on the gravitational field as small.
The resulting truncated Wheeler-deWitt equations then become, to first order in
δ/δg,
{
−2iG ij,kl
δS
δg ij
{
2ig ij ∇ k
}
δ
+ Ĥ φ Φ[g ij (x)] = 0
δg kl
}
δ
δg jk
+ Ĥ φ i
Φ[g ij (x)] = 0 .
(4.79)
Furthermore the wavefunction Φ[g ij (x)] is now evaluated along a solution of the
classical field equations g ij (x,t). This means that S[g ij ] is first determined from a
solution of the classical Hamilton-Jacobi equations
with [see Eq. (4.73)]
∂ t g ij = NG ij,kl
δS
δg kl
− 2(D i N j + D j N i ) , (4.80)
D i ≡− 2 i ∇ j
δ
δg ij
, (4.81)
after which the relevant derivatives δS/δg are inserted in Eq. (4.77).
To make further progress, one need to be more specific about the form of the
lapse (N) and shift (N i ) functions. One can show that Φ[g ij ] satisfies an evolution
equation of the type
∫
∂
∂t Φ(t) = d 3 δ
x ∂ t g ij (x)
δg ij (x) Φ[g mn] , (4.82)