A Non-empty Bouncing Milne Model for the Universe
13
Quantization of a classical system like the one above means introduction of a wave
function ( ˆ
a) and requiring that it satisfies [38]
i
∂∂
∂t
= H c = N H.
(18)
To ensure that time reparametrization invariance is not lost at the quantum level, the
conventional practice is to ask that the wave function is annihilated by the operator
version of H, i.e.
H = 0.
(19)
Equation (19) is called the Wheeler–DeWitt equation. It is analogous to a zero energy
Schrodinger equation, in which the dynamical variable ˆ
a and its conjugate momentum ˆ
π a are replaced by the corresponding operators. The wave function is defined
on the minisuperspace with just one coordinate ˆ
a and we expect it to provide information regarding the evolution of the universe. We may note here that the wave
function is independent of time; it is a stationary solution in the minisuperspace.
The Wheeler–DeWitt equation for our case can be written by making the operator
replacements for ˆ
π a and ˆ
a in H. However, finding the operator corresponding to
ˆ
π
2
a / ˆ
a is problematic due to an operator ordering ambiguity. We shall adopt the most
commonly used form
ˆ
π
2
a
ˆ
a
→ −ˆ a
−r −1 ∂
∂ ˆ
a
ˆ
a
r ∂
∂ ˆ
a
,
(20)
where the choice of r is arbitrary and is usually made according to convenience.
Using this expression with r = −1, we obtain the Wheeler–DeWitt equation for the
complexified Milne model as
d
2
d ˆ
a 2 −
1
ˆ
a
d
d ˆ
a
+
9π
2
4G 2 ˆ
a
2
= 0.
(21)
This equation has an exact solution
(a) ∝ exp
±i
3π
4G
ˆ
a
2
.
(22)
Whether this solution corresponds to the classical evolution of the model can be
determined by drawing the de Broglie–Bohm trajectories for this wave function
[19]. Identifying ≡ R exp(i S), one can draw these quantum trajectories by using
the de Broglie equation of motion
ˆ
π a =
∂ S
∂ ˆ
a
.
(23)
13
Quantization of a classical system like the one above means introduction of a wave
function ( ˆ
a) and requiring that it satisfies [38]
i
∂∂
∂t
= H c = N H.
(18)
To ensure that time reparametrization invariance is not lost at the quantum level, the
conventional practice is to ask that the wave function is annihilated by the operator
version of H, i.e.
H = 0.
(19)
Equation (19) is called the Wheeler–DeWitt equation. It is analogous to a zero energy
Schrodinger equation, in which the dynamical variable ˆ
a and its conjugate momentum ˆ
π a are replaced by the corresponding operators. The wave function is defined
on the minisuperspace with just one coordinate ˆ
a and we expect it to provide information regarding the evolution of the universe. We may note here that the wave
function is independent of time; it is a stationary solution in the minisuperspace.
The Wheeler–DeWitt equation for our case can be written by making the operator
replacements for ˆ
π a and ˆ
a in H. However, finding the operator corresponding to
ˆ
π
2
a / ˆ
a is problematic due to an operator ordering ambiguity. We shall adopt the most
commonly used form
ˆ
π
2
a
ˆ
a
→ −ˆ a
−r −1 ∂
∂ ˆ
a
ˆ
a
r ∂
∂ ˆ
a
,
(20)
where the choice of r is arbitrary and is usually made according to convenience.
Using this expression with r = −1, we obtain the Wheeler–DeWitt equation for the
complexified Milne model as
d
2
d ˆ
a 2 −
1
ˆ
a
d
d ˆ
a
+
9π
2
4G 2 ˆ
a
2
= 0.
(21)
This equation has an exact solution
(a) ∝ exp
±i
3π
4G
ˆ
a
2
.
(22)
Whether this solution corresponds to the classical evolution of the model can be
determined by drawing the de Broglie–Bohm trajectories for this wave function
[19]. Identifying ≡ R exp(i S), one can draw these quantum trajectories by using
the de Broglie equation of motion
ˆ
π a =
∂ S
∂ ˆ
a
.
(23)