14
M. V. John
In the present case of quantum cosmology, while using (16), this equation of motion
simply reads
−
3π
2G
˙ ˆ
a ˆ
a = ±
3π
2G
ˆ
a
or
˙ ˆ
a = ±1,
(24)
which is the same classical equation (1). Hence, here we have the same classical and
quantum trajectories, implying identical behaviour as in the case of free particles
described by plane waves in ordinary quantum mechanics.
A notable feature here is the appearance of the factor
√
2G/3π in the wave
function (22). This has value very nearly equal to the Planck length. In all quantum
gravity theories, a natural length scale is the Planck length. Hence, one can deduce that
the value of a 0 , the imaginary constant appearing in the scale factor of the complexextended Milne model is also of this value. In turn, this is the minimum radius of the
bouncing, real, coasting model. Thus, we see that the nonsingular behaviour of the
present coasting model is due to the quantum effects at the earliest moments within
the Planck time.
3.2 Negative Energy Density
One can see that the complex extension of Milne model, which led to the nonempty, flat model with a =
a
2
0 + t 2 , naturally brings in a negative energy which
causes the bounce in the real model, and this is what relieves the model from the
singularity problem. As can be expected, this negative energy has an equation of
state corresponding to relativistic matter. There are speculations on a universe driven
by a Casimir energy, which is negative [39]. Casimir first showed, on the basis of
relativistic quantum field theory, that between two parallel perfect plane conductors
separated by a distance l, there is a renormalized energy E = π
2
/720l
4 per unit
area [40]. For a static universe and for a massless scalar field, it was calculated that
Casimir energy has a density [41].
ρ casimir = −
0.411505
4πa 4 .
(25)
If we accept the value of a 0 =
√
2G/3π , which is nearly equal to the Planck length,
our expression for negative energy (13) can be written as
ρ − = −
1
4πa 4 .
(26)
Thus, there is some strong ground to believe that the negative energy appearing in the
complex extension of Milne model is of the nature of Casimir energy. However, one
M. V. John
In the present case of quantum cosmology, while using (16), this equation of motion
simply reads
−
3π
2G
˙ ˆ
a ˆ
a = ±
3π
2G
ˆ
a
or
˙ ˆ
a = ±1,
(24)
which is the same classical equation (1). Hence, here we have the same classical and
quantum trajectories, implying identical behaviour as in the case of free particles
described by plane waves in ordinary quantum mechanics.
A notable feature here is the appearance of the factor
√
2G/3π in the wave
function (22). This has value very nearly equal to the Planck length. In all quantum
gravity theories, a natural length scale is the Planck length. Hence, one can deduce that
the value of a 0 , the imaginary constant appearing in the scale factor of the complexextended Milne model is also of this value. In turn, this is the minimum radius of the
bouncing, real, coasting model. Thus, we see that the nonsingular behaviour of the
present coasting model is due to the quantum effects at the earliest moments within
the Planck time.
3.2 Negative Energy Density
One can see that the complex extension of Milne model, which led to the nonempty, flat model with a =
a
2
0 + t 2 , naturally brings in a negative energy which
causes the bounce in the real model, and this is what relieves the model from the
singularity problem. As can be expected, this negative energy has an equation of
state corresponding to relativistic matter. There are speculations on a universe driven
by a Casimir energy, which is negative [39]. Casimir first showed, on the basis of
relativistic quantum field theory, that between two parallel perfect plane conductors
separated by a distance l, there is a renormalized energy E = π
2
/720l
4 per unit
area [40]. For a static universe and for a massless scalar field, it was calculated that
Casimir energy has a density [41].
ρ casimir = −
0.411505
4πa 4 .
(25)
If we accept the value of a 0 =
√
2G/3π , which is nearly equal to the Planck length,
our expression for negative energy (13) can be written as
ρ − = −
1
4πa 4 .
(26)
Thus, there is some strong ground to believe that the negative energy appearing in the
complex extension of Milne model is of the nature of Casimir energy. However, one