Feynman Path Integral Formulation

4.11 Solution of Simple Minisuperspace Models 123
“light cone” y ≈|x|, and one possible choice is Ψ = 1. Presumably a better procedure
would be an approximate determination in this region of Ψ via an Euclidean
functional integral. The qualitative behavior of the solutions in general depends crucially
on whether one is in the region V < 0 (where the solution Ψ increases exponentially)
or in the region V > 0 (where the solution for Ψ is oscillatory). From
the nature of the solution and its semiclassical correspondence it has been argued
that the behavior close to y = |x| (a = 0 in the original variable) is consistent with
a minimum radius or “bounce” at V = 0 (Hawking and Wu, 1984; Ochiai and Sato,
2000).
How do the properties of the solutions to the Wheeler-DeWitt equation applied to
minisuperspace models depend on additional terms that might enter into the original
gravitational action To answer this question, one could consider the addition of
higher derivative terms, such as
Î = − 1 ∫
d 4 x √ g [ R − α C μνρσ C μνρσ + β R 2] + s.t. , (4.111)
16πG
where C μνρσ is the Weyl tensor, and “s.t.” refers to surface terms (Hawking and
Luttrell, 1984a,b). Following a procedure analogous to the one discussed in the
previous case, one obtains for a homogeneous isotropic universe the following
Wheeler-DeWitt equation for Ψ(a,R), or, more conveniently, for Ψ(x,y)
[
1 ∂
2
2 ∂y 2 − ∂ 2 ]
∂x 2 + V (x,y) Ψ(x,y) =0 , (4.112)
now with “potential” V
V (x,y) =x 2 − y 2 + gx 2 (x − y) 2 , (4.113)
with g ≡ 1/18πβ, and variables x and y given by x = 2βaR and y = 2a(1 + βR),
where R is the scalar curvature.
As in the previous example, one is interested in a solution for a > 0 which corresponds
to y > x, with boundary condition Ψ(x = y)=1(seeFig.4.2).Butthisis
not sufficient to determine the Cauchy data, and one has to make some additional
guess on what a semiclassical wavefunction might look like in the vicinity of x = y
or x = −y, which suggests
Ψ(y = −x) =exp(−a 4 /36πβ) . (4.114)
These conditions are then in principle sufficient to determine the wavefunction
Ψ(x,y).
Physically the oscillatory behavior of the wavefunction then suggests the existence
of small amplitude oscillations superimposed on an overall expansion or
contraction. These oscillations in radius would cause particle creation in any matter
fields present, which in turn would damp the oscillation.
As one last illustrative example consider a minisuperspace model for a universe
filled with matter (or radiation) of uniform density, such that ρ(a)=M/a σ where
σ = 3 (matter) or 4 (radiation); σ = 0 would correspond to a pure vacuum energy.