#### Scalar pair production in a magnetic field in de Sitter universe

Eur. Phys. J. C
Scalar pair production in a magnetic field in de Sitter universe
Mihaela-Andreea Ba˘loi 1
Cosmin Crucean ; 0
Diana Popescu 0
0 West University of Timis ̧oara , V. Parvan Ave. 4, 300223 Timisoara , Romania
1 Politehnica University of Timis ̧oara , V. Parvan Ave. 2, 300223 Timisoara , Romania
The production of scalar particles by the dipole magnetic field in de Sitter expanding universe is analyzed. The amplitude and probability of transition are computed using perturbative methods. A graphical study of the transition probability is performed obtaining that the rate of pair production is important in the early universe. Our results prove that in the process of pair production by the external magnetic field the momentum conservation law is broken. We also found that the probabilities are maximum when the particles are emitted perpendicular to the direction of magnetic dipole momentum. The total probability is computed and is analysed in terms of the angle between particles momenta.
1 Introduction
The problem of particle generation in magnetic field in
Minkowski space-time was first studied by Heisenberg and
Euler [
1
] and this paper represents the basis for the today
well known Schwinger effect. This paper proves for the first
time how to solve the Dirac equation when a magnetic field
is present, and these results represent the basis for the more
recent studies that imply particle production in the presence
of external fields [
2–9
] in curved backgrounds. An important
result was obtained recently in [9], where the minimally
coupled Klein–Gordon equation was solved in four dimensions
in a de Sitter geometry. Further in [
9
] the rate of pair
production was computed with the help of Bogoliubov method and
the Minkowski limit was addressed. Another result which
uses the nonperturbative treatment of scalar pair production
in magnetic field on Robertson–Walker universe was
discussed in [
10
], where the density number was computed using
the Bogoliubov coefficients. In [
10
] the de Sitter case is not
contained since the scale factor a(t ) was chosen to have a
linear time dependence. The result of [
10
] has proved how to
work out the density number when a magnetic field is present
in Robertson–Walker spacetime. While the vast majority of
nonperturbative computations are done in two dimensions, in
[
10
] the computations were made in four dimensions. These
results need to be completed with a perturbative treatment in
the case of scalar field since the perturbative method allows
a complete study of the case when the gravitational fields are
strong in comparison with the magnetic fields.
In this paper we will study the problem of scalar pair
production in dipolar magnetic field on de Sitter geometry. We
will use the perturbative methods presented in [
11–21
], for
computing the amplitude/probability of scalar pair
production in magnetic field on de Sitter expanding universe. The
problem of particle production when a magnetic field and a
gravitational field are present seems to receive little attention
in literature because of the technical difficulties in solving
the field equations in the presence of magnetic fields. The
perturbative approach to the problem of fermions production
in magnetic field on de Sitter geometry was used in [
13
], and
the main results prove that the probability of pair production
is nonvanishing only in strong gravitational fields. However
for a complete picture of the problem of pair production in
gravitational fields one needs to combine the nonperturbative
results [
2–8
], with the perturbative treatment of this
problem. This will imply as we mentioned above that the
nonperturbative computations to be extended in four dimensions,
since only in these cases we will have a complete picture
on how the field equations with external field will look like
and how these equations can be solved. However some time
ago is was argued [
12
], that the production due to the field
interactions should also be taken into consideration. Despite
of this observation, the perturbative approach only recently
received attention and the results are based on the
calculations of QED transition amplitudes that generates particle
production in a curved background [
13,14,20
]. The problem
of space expansion that generate particle production was first
discussed in [
22
], and important results were also obtained
in [
4,5
]. Since the de Sitter space-time could describe our
universe, we believe that it is important to study the problem
of scalar particle production in magnetic fields in this
geometry by using both perturbative and nonperturbative methods.
The first step in our study will be the computation of the first
order QED transition amplitude that generates scalar
particles in the field of a magnetic dipole. Then we will present
the main steps required for computing the total probability
for the process of pair production in magnetic field.
The origin of magnetic fields in the Universe is a
subject that has been approached by many authors [
23–27,38
],
mostly motivated by the astronomical observations which
show that, the galaxies and galaxy clusters have a proper
magnetic field of weak intensity. The vast majority of the
papers propose technical solutions of accurate measurements
of the large scales magnetic fields, but an interesting idea that
is analysed in the literature is that, these magnetic fields own
their origin in the early stages of Universe evolution.
The paper begins with an introduction in the problem of
pair production in magnetic field. In section two we
compute the amplitude/probability of pair production in magnetic
field. Section three is dedicated to the graphical study of our
analytical results and limit cases, while in section four we
compute the total probability. Our conclusions are presented
in section five.
2 Probability calculation
We start this section by mentioning that we work in the chart
with conformal time that covers only half of the whole de
Sitter manifold. More precisely we consider the chart with
conformal time tc ∈ (−∞, 0), which covers the expanding
portion of de Sitter manifold [
29
]:
ds2 = dt 2 − e2ωt d x 2 = (ω1tc)2 (dtc2 − d x 2),
where ω is the expansion factor and ω > 0, while the
conformal time is defined as tc = − ω1 e−ωt .
In de Sitter expanding geometry the free Klein–Gordon
equation has analytical solutions as it was shown in Refs.
[
22,30
]. In this paper we will use the exact solutions of the
Klein–Gordon equation with a defined momentum [
22,30
]:
1
f p (x ) = 2
π (−ωtc)3/2
ω (2π )3/2 e−πμ/2 Hi(μ1) (− p tc) ei p·x , (2)
where Hμ1(z) is the Hankel function of first kind, p = | p| is
the momentum modulus. We also use the notations:
μ =
m
k2 − 49 , k = ω ,
(1)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
The form of the potential that produces the dipole
magnetic field is known from electrodynamics [
31,32
]:
AM =
M × x
x 3 ,
| |
where M is the magnetic dipole moment. The expression of
the field of a magnetic dipole can be obtained as the curl of
the vector potential [
31,32
]:
BM = ∇ × AM =
3x (M · x )x−5 M(x · x ) .
| |
The expression of A in de Sitter geometry is established
by using the conformal invariance of Maxwell equations.
The metric given in Eq. (1) is conformal with the Minkowski
metric, i.e. gμν = ημν . Then knowing that AM is the vector
potential in Minkowski space, then the vector potential in de
Sitter geometry is [
33
]:
Aμ =
−1 AμM ,
where = (ωtc)−2 as can be seen from the line element
given in Eq. (1), and is the conformal factor transformation.
Taking now A0(x ) = 0, we obtain for the potential vector:
A(x ) =
Mx×3 x (ωtc)2.
| |
The transition amplitude of scalar pair production in
external field, defined in the first order of perturbation theory in
Coulomb gauge is [
15,19
]:
Ai→ f = −e
−g(x ) f p∗ (x ) ↔∂i f p∗ (x ) Ai (x )d4x ,
where √−g(x ) = (ωtc)−4.
We must specify that we work in Coulomb gauge i.e.
∇i (√−g Ai ) = 0, and our amplitude is gauge invariant since
the transformations Ai → Ai + ∂i , leave the amplitude
unchanged [
36
].
Amplitude calculation can be done by using the
fundamental solutions given in (2) and (7). Then the four
dimensional integral is split in a temporal integral and a spatial
integral. By taking the bilateral derivative and then solving
the spatial integral we obtain [
13
]:
with m > 3ω/2. The fundamental solutions of negative
frequencies are obtained by complex conjugation f p∗(x ).
d3x
x
x 3
| |
e−i( p+ p )x = −
4π i ( p + p )
| p + p |2 .
The transition amplitude can be expressed using the relation
between Hankel functions and Bessel K functions [
34,35
]:
e
Ai→ f = − 2π 3| p + p |2 (M × ( p + p )) · ( p − p )
0
∞
d zz K−iμ(i pz)K−iμ(i p z)
(10)
where we pass in the temporal integral to the new variable of
integration z = e−ωt /ω = −tc. The computation of the
temporal integral leads to the final expression for the transition
amplitude corresponding to the process of pair production
in dipolar magnetic field in terms of unit step functions θ ,
Euler gamma functions and Gauss hypergeometric
functions 2 F1:
e
Ai→ f = − 4π 3| p + p |2 (M × ( p + p )) · ( p − p )
θ ( pp−2 p ) fk
p
p
+
θ ( pp−2 p) fk
p
p
.
(11)
p
The functions fk p
defined above are:
fk
p
p
(1 − i μ) (1 + i μ) 2 F1 1, 1
p
p
2
p
p
iμ
(1 + i μ) (−i μ)
(1 − i μ) (i μ) .
(12)
×
=
=
p
p
−iμ
−i μ; 2; 1 −
1 −
p
p
+
1
p 2
p
−iμ
The probability of pair production is obtained by taking the
square modulus of the amplitude:
2 e2
P = |Ai→ f | = 16π 6| p + p |4 |(M × ( p + p )) · ( p − p )|2
×
θ ( pp−4 p ) fk
p
p
2
+
θ ( pp −4 p) fk
p
p
2
.
(13)
Now let us discuss the physical consequences of our
calculation. Firstly, we observe that the momentum conservation law
is broken in the process of pair production by the dipole field.
This is a consequence of the presence of the external field in
de Sitter background [
13,20
]. Secondly from the
probability expression results that the fermions could not be emitted
on the direction of magnetic field because the probability is
vanishing due to the vectorial product M × ( p + p ). The
most probable transitions will be those in which the scalar
particles will be emitted perpendicular on the direction of the
magnetic field, result obtained also in [13], for fermions.
Considering the above mentioned observations, let us fix
the magnetic dipole moment on the e3 direction such that
M = M e3. Then because of the vectorial product we can
consider the momenta vectors p, p are in the (1, 2) plane,
such that M ⊥ p and M ⊥ p. Taking the polar coordinates
for p , p :
p1 = p cos β;
p2 = p sin β
p1 = p cos ϕ; p2 = p sin ϕ,
we obtain that the angle between momenta vectors p and
p is just β − ϕ. Using the angular analysis above, the final
expression for the probability of scalar pair production turns
out to be:
e2M2 ( p2 + p 2 − 2 pp cos(β − ϕ))
P = 16π 6 ( p2 + p 2 + 2 pp cos(β − ϕ))
2 2
×
θ ( pp−4 p ) fk
p
p
+
θ ( pp−4 p) fk
p
p
(14)
.
(15)
From the probability equation we observe that the momenta
vectors p, p of the produced particles, could be on the same
direction while their orientations could be opposite or the
same.
3 Graphical results and limit cases
The probability equation (15) is analysed in terms of
parameter k = m/ω, by making the plots for given momenta ratios
p/ p and given angle β −ϕ between momenta vectors. These
graphs are obtained by using MAPLE. We must specify that
our computations for probability of pair production, were
done in the case m/ω > 3/2 and that is reflected in our
graphs in terms of parameter k and we considered for
modulus of magnetic moment the value M = 1.
From Figs. 1, 2, 3 and 4 we observe that the probability
drops to zero for large values of parameter k. The probability
is nonvanishing only for k ∈ (1.51, 2), where the gravity is
still strong and we can conclude that this process is possible
only in strong gravitational fields. Another observation is that
the probability of pair production is sensibly larger when
the angle between momenta vectors approaches π and the
momenta ratio p/ p are close to unity as seen from Figs. 1
and 4.
The above stated conclusions can be confirmed if we plot
the probability equation (15) as function of the angle β − ϕ
between momenta vectors. In these graphs k and p/ p are
fixed (Figs. 5, 6).
Finally we conclude that there are higher probabilities for
pair production processes that have as result particles with
the momenta ratio close to unity and with the momenta on
the same direction but opposite as orientation.
Let us study now the Minkowski limit for amplitude
and probability. First we observe from our graphs that for
k → ∞ our probability is vanishing. This result is in
fact the Minkowski limit which is obtained when k = ∞.
Indeed in the Minkowski scalar QED [
37
] the first order
processes of pair production in external fields are not allowed
Fig. 4 P as a function of k for p/ p = 0.001 solid line and p/ p =
0.003 the point line. Angle β − ϕ = π
by the simultaneous energy–momentum conservation. In
de Sitter case the translational invariance with respect to
time is lost and as a consequence the amplitudes of
particle production in the first order of perturbation theory are
nonvanishing.
Further let us verify that our analytical results confirm the
graphical predictions that the probability is vanishing in the
Minkowski limit. For large k, the parameter μ k . Then by
using the Eqs. (27) and (28) from Appendix the function fk
that defines the amplitude can be brought for large k to the
form:
fk>>1
2π i e−πk
1 −
p 2
p
p
p
−ik
−
p
p
ik
.
(16)
The probability of pair production will then be
proportional with:
P ∼ e−2πk ,
which clearly will be negligible for large k and will vanish
for k = ∞. Here we remark that the probability is
proportional with the factor e−2πk , which is the same factor found in
[
6,7
], where the problem of pair production in external fields
on de Sitter geometry was studied by using a
nonperturbative approach [
6,7
]. We must mention that a nonperturbative
calculation of pair production in magnetic field on de Sitter
geometry has been done in [
9
].
4 The total probability
In this section we will compute the total probability of scalar
pair production in the field of the magnetic dipole on de
Sitter geometry. If one use the nonperturbative approach to
the problem of particle production in external field then the
number of particles is evaluated by integrating after the final
momenta and because of the approximations made for the
out modes the momenta dependence in the density number
of particles is lost. This has as a consequence the fact that the
momenta integrals are solved using a cut-off method. Here
we want to present the computation of the final momenta
integrals in the case of perturbative approach of the
subject, where the momenta dependence in the final
amplitude and probability is preserved. The total probability is
obtained integrating the probability equation (15) after the
final momenta p, p :
Ptot =
P d3 p d3 p .
Since the graphical results suggest that the probability is
significative in strong gravitational field we will rewrite the
equation for function fk pp such that the parameter m/ω
will be considered in the interval m/ω ∈ (1.51, 2). We
consider for the present calculation the case when p > p with
the observation that the ratio of the momenta will be taken
in the interval p/ p ∈ (0.5, 0.9). Then for parameters
specified above, the fk functions that define the probability, can
be brought in the following form by using Eqs. (25) and (26)
from Appendix:
fμ
p
p
2π μ p
= sinh(π μ) ( p + p )
.
Further we present only the case p > p since for p > p the
computations are similar. The integration after final momenta
is taken such that p ∈ (0, ∞) and p ∈ (0, pB ), where
(17)
(18)
(19)
P B = p + e A, is the particle momentum in the presence of
the dipole magnetic field B. Here the potential vector could be
expressed in terms of magnetic dipole moment using Eq. (4),
since we take M = MMxex331,,thAe3n=the0.cTomhipsomneenthtsodoffoAr cauret-:
tAin1g=off−thM|ex|mx32o,mAe2nt=um was also used in [
7
], and in this way
| |
the number of particles was obtained as a finite quantity. An
important remark is that one of the momenta integrals can be
solved without any approximations and this is a consequence
of the fact that in the perturbative calculation the probability
is a function which depends on the particle momenta (see
Eq. (15)). This also opens the way for using other
regularization methods for obtaining the total probability.
As we point out in the previous section the pair of particles
will be emitted perpendicular to the direction of dipole
magnetic field. Let us consider the case when the momenta of the
scalar particles have the same orientation β − ϕ = 0. By
fixing the angle between the momenta vectors we obtain the total
probability when the particles are emitted on given directions
and it will be sufficiently to solve only the momenta
modulus integrals in this case. Using now Eq. (19) and integrating
after the final momenta p , p in the above mentioned limits,
we obtain the total probability in the case β − ϕ = 0 by using
Eq. (29) from Appendix:
(20)
Ptot = 24πe42sMinh2μ2(2π μ) P 2B .
The total probability can be computed using the same method
for other values of the angle between momenta vectors β − ϕ
(see Eq. (30) from Apendix) and the results can be found
bellow :
e2M2μ2
Ptot = 24π 4 sinh2(π μ)
⎪⎨⎪⎪⎪⎪⎧⎪⎪⎪⎪⎪⎩ ((((229170+π−π√89√4π3√3√π√−33√3−−33+)3√322)√P3P−3+26B2)6B)PP2B2B ffffoooorrrr ββββ −−−− ϕϕϕϕ ==== 25ππππ//36//63 (21)
A numerical calculation proves that the total probabilities
increase when the angle between the momenta vectors β − ϕ
approaches π , result that confirms our graphical analysis
from the previous section. A remarkable property of the
total probability is that it becomes negligible as the factor
k = m/ω takes larger values, due to the dependence on the
factor sinh−2(π μ). This result proves that the process of pair
production in dipolar magnetic field was important only in
early universe when the gravitational fields were very strong.
In the Minkowski limit the total probability is vanishing
proving that the phenomenon of pair production in dipolar
magnetic field is allowed as a perturbative process only in strong
gravitational fields.
Another aspect concerning the problem of particle
production in de Sitter expanding space-time is related to the
spectator gauge fields which appear in magnetogenesis or in
anisotropic inflation. These spectator fields [
38
] are present
when one study the phenomenon of pair production in a
time dependent electric field in de Sitter spacetime or the
Schwinger effect [
9
]. The result obtained in [
38
] suggest that
there are some effects that were not taken into consideration
until now, such as the postulation of a time dependent current
which could cause the violation of the weak energy
condition and the second law of thermodynamics. This subject
deserves further investigations since in the case of quantum
field theory on de Sitter space-time, the energy operator is
different from that of the flat case both in the theory of free
Maxwell field and in the theory of fields interactions, as was
shown in [
14
]. Using the Noether theorem, the energy
operator written using the normal ordering, in terms of the electric
and magnetic field strengths E , B, for the free Maxwell field
turn out to be [
14
]:
H =
d3x : (−ωtc) 21 (E 2 + B2)
ω
+ωx · E ∧ B − 2 ∂tc ( A 2) : .
(22)
The above expression is completely different from that of the
flat space energy operator, because there appears terms that
are specific to the de Sitter geometry. An extended discussion
about this subject can be found in [
14,33
].
In the present paper we restrict to compute the amplitude
of transition and probability for the process of pair production
in the field of a magnetic dipole, using the QED formalism
developed in [
14,39
]. Our calculations are valid only in early
universe when the gravitational fields were strong in
comparative with the magnetic fields and present the first derivation
of the total probability for the process of pair production
in magnetic field using a perturbative method. The result
obtained in [9] study the problem of scalar pair production in
strong magnetic fields using a nonperturbative method. The
interesting idea is now to combine the two methods using the
formalism developed in [
3
] for obtaining the contribution of
both cosmological particle production and perturbative
particle production. That will require some work to be done to
translate the calculations in terms of density number of
particles, as was shown in [
12
].
5 Concluding remarks
We present in this paper the first perturbative approach to the
problem of scalar pair production in a magnetic field in a de
Sitter geometry. The main result of our paper is related to
the fact that the first order transition amplitude and
probability are nonvanishing only in strong gravitational fields that
corresponds in our case to the early universe conditions. Our
study proves that the particles will be most probably emitted
perpendicular to the magnetic field direction. This conclusion
shows that there are differences comparatively with the
problem of pair production in electric field where the particles are
emitted parallel with the electric field direction [
6, 7, 15
]. In
the Minkowski limit the amplitude and probability are
vanishing since in Minkowski scalar QED [
37
] this process is
forbidden by the energy–momentum conservation.
The problem of particle production when a gravitational
field and a magnetic field are present seems to receive little
attention in literature. This study is important since it is well
known that black holes and neutron starts have also strong
magnetic fields and it is fundamental to understand the
problem of fields interactions in these geometries. For further
study it will be important to combine the nonperturbative
and perturbative results related to the problem of pair
production in magnetic field on de Sitter spacetime for obtaining
a complete picture of this phenomenon. This can be done by
using the formalism proposed in [
3
], and we hope that in a
future study to present these results.
Acknowledgements This work was supported by a grant of the
Ministry of National Education and Scientific Research, RDI Programme
for Space Technology and Advanced Research - STAR, project number
181/20.07.2017.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix
Here we present the main steps for computing the amplitude
of pair production in magnetic field. Using the relation that
connects Hankel functions and Bessel K functions [
34, 35
]:
2i
π
H (1,2)(z) = ∓
ν
e∓iπ ν/2 Kν (∓i z),
we arrive at integrals of the type [
35
]:
0
∞
×
d zz−λ Kμ(az)Kν (bz) =
1 − λ + μ + ν
2
1 − λ + μ − ν
2
2−2−λa−ν+λ−1b ν
(1 − λ)
1 − λ − μ + ν
2
1 − λ − μ − ν
2
(23)
(25)
(26)
(28)
(29)
(30)
2 F1 1 − λ +2 μ + ν , 1 − λ −2 μ + ν
Re(a + b) > 0 , Re(λ) < 1 − | Re(μ)| − | Re(ν)|.
,
(24)
when calculating the total
probabil
The momenta integrals for computing the final formulas
for the total probability are:
π
(1 − z) (z) = sin(π z) .
0
0
0
0
0
∞
∞
=
∞
∞
∞
=
d p
( p − p )2 1
( p + p )4 = 3 p
.
d p ( p2 + p 2 −
( p + p )2( p2 + p 2 +
2π √3 − 3√3 − 6
3 p (√3 − 2) ,
d p ( p2 + p 2 − pp )
( p + p )2( p2 + p 2 + pp ) =
√3 pp )
√3 pp )
d p
( p2 + p 2 + pp )
( p + p )2( p2 + p 2 − pp ) =
√3 pp )
√3 pp )
d p ( p2 + p 2 +
( p + p )2( p2 + p 2 −
10π √3 − 3√3 + 6
3 p (√3 + 2) .
and the next property of the natural logarithm:
y − 1
y
≤ ln y ≤ y − 1 , y > 0,
with the mention that in our case we can use that ln y y −1.
The limit k >> 1 in the functions fk is obtained if we
use the following identity between hypergeometric functions
[
34, 35
]:
2 F1(a, b; c; z) =
(c) (c − a − b)
2 F1(a, b; a
(c − a) (c − b)
+b − c + 1; 1 − z)
+(1 − z)c−a−b (c) (a + b − c)
(a) (b)
2 F1(c − a, c − b; c − a − b + 1; 1 − z),
(27)
and the well known relations between gamma Euler functions
[
35
],
The function fk pp
ity is obtained if we use:
ai x = ei x ln a , a ∈ R
1. W. Heisenberg , H. Euler , Z. Phys . 98 , 714 ( 1936 )
2. S.P. Gavrilov , D.M. Gitman , Phys. Rev. D 87 , 125025 ( 2013 )
3. N.D. Birrel , P.C.W. Davies , Quantum Fields in Curved Space (Cambridge University Press, Cambridge, 1982 )
4. L. Parker , Phys. Rev. Lett . 21 , 562 ( 1968 )
5. L. Parker , Phys. Rev . 183 , 1057 ( 1969 )
6. V.M. Villalba , Phys. Rev. D 52 , 3742 ( 1995 )
7. J. Garriga , Phys. Rev. D 49 , 6343 ( 1994 )
8. J. Haro , E. Elizalde , J. Phys. A 41 , 372003 ( 2008 )
9. E. Bavarsad , S.P. Kim , C. Stahl , S.S. Xue , Phys. Rev. D 97 , 025017 ( 2018 )
10. K. Sogut , A. Havare , Nucl. Phys. B 901 , 76 ( 2015 )
11. D. Marolf , I.A. Morrison , M. Srednicki , Class. Quantum Gravity 30 , 155023 ( 2013 )
12. N.D. Birrel , P.C.W. Davies , L.H. Ford , J. Phys . A 13 , 961 ( 1980 )
13. C. Crucean , M.A. Ba ˘loi , Phys. Rev. D 93 , 044070 ( 2016 )
14. I.I . Cota˘escu, C. Crucean, Phys. Rev. D 87 , 044016 ( 2013 )
15. M. A. Ba˘loi, Mod . Phys. Lett. A 29 , 1450138 ( 2014 )
16. K.H. Lotze , Nucl. Phys. B 312 , 673 ( 1989 )
17. I.L. Buchbinder , E.S. Fradkin , D.M. Gitman , Fortschr. Phys. 29 , 187 ( 1981 )
18. I.L. Buchbinder , L.I. Tsaregorodtsev , Int. J. Mod. Phys. A 7 , 2055 ( 1992 )
19. C. Crucean , R. Racoceanu , A. Pop , Phys. Lett. B 665 , 409 ( 2008 )
20. C. Crucean, Phys. Rev. D 85 , 084036 ( 2012 )
21. K.H. Lotze , Class. Quantum Gravity 5 , 595 ( 1985 )
22. E. Schrödinger, Physica 6 , 899 ( 1939 )
23. A.H. Guth , Phys. Rev. D 23 , 347 ( 1981 )
24. L.F. Abbott , S.Y. Pi , Inflationary Cosmology (World Scientific, Singapore, 1986 )
25. M.S. Turner , L.M. Widrow , Phys. Rev. D 37 , 2743 ( 1988 )
26. G.B. Field , S.M. Carroll , Phys. Rev. D 62 , 103008 ( 2000 )
27. S. Kawati , A. Kokado , Phys. Rev. D 39 , 2959 ( 1989 )
28. S. Kawati , A. Kokado , Phys. Rev. D 39 , 3612 ( 1989 )
29. C.W. Misner , K.S. Thorne , J.A. Wheleer , Gravitation (W. H. Freeman and Company, New York, 1973 )
30. I.I . Cota˘escu, C. Crucean , A. Pop , Int. J. Mod. Phys. A 23 , 2563 ( 2008 )
31. J.D. Jackson , Classical Electrodynamics (Wiley, Hoboken, 1962 )
32. W. Greiner, Classical Electrodynamics (Springer, Berlin, 1998 )
33. I.I . Cota˘escu, C. Crucean, Prog. Theor. Phys . 124 , 1051 ( 2010 )
34. G.N. Watson , Theory of Bessel Functions (Cambridge University Press, Cambridge, 1922 )
35. I.S. Gradshteyn , I.M. Ryzhik , Table of Integrals, Series and Products (Academic Press, Cambridge, 2007 )
36. C. Crucean , M.A. Ba ˘loi , Phys. Rev. D 95 , 048502 ( 2017 )
37. S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, 1995 )
38. M. Giovannini , Phys. Rev. D 97 , 061301 ( 2018 )
39. C. Crucean , M.A. Ba ˘loi , Int. J. Mod. Phys. A 30 , 1550088 ( 2015 )