Feynman Path Integral Formulation

124 4 Hamiltonian and Wheeler-DeWitt Equation
y
Ψ( x,
y)
x
Fig. 4.2 Minisuperspace wavefunction Ψ(x,y) for the problem in Eq. (4.112), gravity with higher
derivative terms, in the region y > |x|.
In the following we will consider for simplicity only the case σ = 3, corresponding
to non-relativistic matter. The classical Friedman equations for λ = 0give
ȧ 2 + k − 8πG
3 a3 ρ(a) =0 , (4.115)
with k = 0,±1, and subject to some initial conditions at t = t 0 . The above equation
can be regarded as a classical one-dimensional mechanics problem, with an inverted
parabolic potential V (a) =
2 k − 8πG
3 a3 ρ(a). Introducing, as before, the canonical
momentum derived from the appropriate classical Lagrangian one finds for the classical
Hamiltonian
H = −Ḡa −1 p 2 − ¯ka+ a 3 ρ(a) =0 , (4.116)
with Ḡ = 2π 3 G and ¯k =
8πG 3 . After setting p2 /a = a −q+1 pa q p, with q a parameter
introduced to describe an operator ordering ambiguity, and replacing p →−i∂/∂a
one obtains for the Wheeler-DeWitt equation
{α 1 ∂ 2
a 2 ∂a 2 + qα 1 ∂
a 3 ∂a − k + 8πG }
3 a2 ρ(a) Ψ(a) =0 , (4.117)
with α ≡ 4Ḡ 2 = 16π 2 G 2 /9. Then for the choice q = 1 one can re-write the equation
as a one-dimensional zero-energy stationary state Schrödinger-like problem,
{ 1 ∂
a 2 ∂a a ∂
∂a − k α a + 8πG }
3α a3 ρ(a) Ψ(a) =0 , (4.118)
which in the non-relativistic matter case (ρ(a) =M/a 3 ) corresponds to onedimensional
quantum motion with potential V ef f (a) =(k/α)a 2 − β a, with β ≡
8πGM/3α > 0. The shape of the potential for k = 1isaninvertedU going through