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# Fermion-antifermion pairs in Bonnor-Melvin magnetic space-time with non-zero
cosmological constant

Abdullah Guvendi 1111E-mail<EMAIL_ADDRESS>(Corr. Auth.)
Omar Mustafa 2222E-mail<EMAIL_ADDRESS>
1 Department of Basic Sciences, Faculty of Science, Erzurum Technical
University, 25050, Erzurum, Turkey
2 Department of Physics, Eastern Mediterranean University, G. Magusa, north
Cyprus, Mersin 10 - Turkey

###### Abstract

We study the relativistic dynamics of fermion-antifermion pairs in the Bonnor-
Melvin magnetic (BMM) spacetime in non-zero cosmology. We focus on the
$(1+2)$-dimensional cylindrically symmetric BMM-spacetime background. Within
the context of such a magnetized universe, we rigorously investigate the
fully-covariant two-body Dirac equation. We derive the corresponding radial
part of the equation and obtain an exact closed form analytical solution for
the problem at hand. Moreover, we report some intriguing parallels with
relativistic and non-relativistic quantum oscillators in flat spaces. Notably,
our findings suggest a compelling correlation between the cosmological
constant and the energy spectrum of fermion-antifermion systems, hinting at
profound connections between quantum realms and cosmology.

Keywords: Fermion-antifermion pairs; Magnetized Universe; Non-zero cosmology;
Bonnor-Melvin Universe; Quantum electrodynamics

##  1 Introduction

In the pursuit of understanding the dynamics of interacting fermions within
the framework of relativistic quantum mechanics, the investigation involves
the exploration of many-body equations constructed phenomenologically. These
equations typically encompass the free Dirac Hamiltonian for each fermion
along with interaction potentials. However, a notable challenge arises from
relative time problems stemming from retardation effects, often disregarded in
these phenomenological equations, despite their inherent involvement in a
many-time problem. Complications also arise in selecting interaction
potentials and determining the total angular momentum of fermion-formed
systems within these equations. Conventionally, interaction potentials are
simplified as either a single boson or photon exchange potential. Moreover,
these phenomenologically established many-body equations fail to maintain full
covariance in curved spaces. Contrasting this approach, non-relativistic
quantum mechanics traditionally relies on one-time equations comprising
individual particle free Hamiltonians and an interaction term, with wave
functions dependent on spatial coordinates for each particle. The introduction
of the Dirac equation prompted research into formulating a comprehensive two-
body Dirac equation, notably initiated by Breit [1], incorporating two free
Dirac Hamiltonians and a modified interaction potential, akin to the Darwin
potential in electrodynamics. However, limitations arise under conditions of
long-range interactions or high particle velocities due to retardation
effects. Subsequent attempts by Bethe and Salpeter, leveraging quantum field
theory, encountered relative time problems, necessitating approximations like
instantaneous interactions to solve for interacting fermions [2]. Decades
later, Barut proposed a fully covariant many-body Dirac equation encompassing
spin algebra and the most general electric and magnetic potentials [3].
Barut’s equation, characterized by a spin algebra involving Kronecker products
of Dirac matrices, results in a $16\times 16$ dimensional matrix equation in
$(3+1)$-dimensions. Achieving separation between angular and radial components
involves employing group theoretical methods. Nevertheless, solving the
subsequent set of $16$ radial equations remains an unsolved challenge even for
familiar systems like one-electron atoms or unstable systems such as
positronium. The challenge arises due to the interconnection of radial
equations, resulting in pairs of coupled second-order wave equations -a hurdle
that demands attention. However, in the past few years, there’s been evidence
indicating that the Barut equation is entirely solvable when dealing with low-
dimensional systems or systems exhibiting particular dynamical symmetries,
irrespective of whether they’re in flat or curved spaces. This equation has
seen remarkable progress in its application, marking significant breakthroughs
[4, 5, 6, 7]. Hereby, in the current methodical proposal, our investigation
seeks to elucidate the behavior of fermion-antifermion pairs in a magnetized
universe (magnetized Bonnor-Melvin universe), specifically addressing the
impact of non-zero cosmological constant.

A notable example of a fascinating curved space arises from the Bonnor-Melvin
universe, an exact solution within the framework of the Einstein-Maxwell
equations. Initially describing a static, cylindrically symmetric
(electro)magnetic field embedded in its gravitational counterpart [11, 12,
13], this solution was extended to encompass a non-zero cosmological constant
($\Lambda$), resulting in a magnetized, static, and cylindrically symmetric
universe model [8]. The inclusion of $\Lambda$ significantly influences the
overall geometry of spacetime, altering the behavior of the universe on a
large scale. In particular, it impacts the expansion or contraction dynamics
and gravitational behavior within the cylindrical structure described by the
Bonnor-Melvin solution. The interaction between the cosmological constant and
magnetic fields, as depicted in this unique solution of general relativity,
sheds light on fundamental aspects of physics within Einstein’s gravity
theory. Moreover, understanding the evolution of vector fields in the presence
of a magnetic field is crucial for investigating phenomena like
magnetohydrodynamics, especially relevant in astrophysical contexts, such as
plasma behavior in space. Exploring such magnetized curved spacetime can offer
insights into early universe magnetic fields and their evolutionary paths,
providing a foundation for theories about their origin and development on
cosmological scales.

On the other hand, the presence of cylindrical symmetry implies that quantum
particle dynamics remain invariant under Lorentz boosts (rotation-free Lorentz
transformations) in the $\hat{z}$, significantly simplifying the problem at
hand. This simplification allows for a clearer comprehension of dynamics and
often streamlines calculations [14, 15, 16, 17, 18]. Relativistic quantum
systems exhibiting dynamical symmetry refer to those governed by quantum
mechanics, displaying specific symmetries within the framework of special
relativity. These symmetrical behaviors in quantum systems play a pivotal role
in understanding their properties, often leading to equations or Hamiltonians
possessing invariant properties under particular transformations. The concept
of dynamical symmetry facilitates the mathematical description of these
systems, aiding in solving intricate quantum problems and predicting
observable quantities. This understanding not only identifies patterns and
predicts behavior but also underpins various applications, ranging from
fundamental physics research to technological advancements in fields like
materials science and beyond [4, 6, 15, 16].

In this study, our focus revolves around investigating the intricate dynamics
of fermion-antifermion pairs within a magnetized universe embedded in three
dimensions and characterized by a non-zero cosmological constant $\Lambda$.
Building upon the foundational work delineated in [8], which established a
four-dimensional magnetic universe governed by $\Lambda$, our research delves
into a space-time configuration manifesting cylindrical symmetry. Notably,
this configuration preserves the quantum field dynamics’ invariance under
Lorentz boosts along the $z$-direction, thereby facilitating a streamlined
examination in $(1+2)$-dimensions. Our approach commences by deducing the
radial wave equation governing bi-local fields, subsequently yielding its
analytical solution through the utilization of special functions. Our
investigation underscores a pivotal revelation: the cosmological constant
$\Lambda$ exerts a profound influence on the eigenvalue solutions,
underscoring their significant role in shaping the characteristics of these
fermion-antifermion systems.

This manuscript is organized into several key sections addressing the dynamics
of fermion-antifermion systems within a magnetized universe under non-zero
cosmological conditions. Section 2 lays the foundation by introducing the
covariant two-body Dirac equation, initiating the derivation process for a
wave equation applicable to such systems. Following this, in section 3, a non-
perturbative wave equation is presented, accompanied by analytical results
pertinent to this specialized scenario. Subsequently, a comprehensive overview
of the outcomes will be presented, followed by an in-depth analysis and
exploration of the findings. Throughout this manuscript, we will use the
natural units.

##  2 Two-body Dirac equation

Within this section of the manuscript, we commence by presenting the
fundamental structure of the covariant two-body Dirac equation in a $(2+1)$\-
dimensional curved space. Subsequently, we derive the specific form of this
equation for a fermion pair in a magnetized $(2+1)$-dimensional magnetized
universe under non-zero cosmology. The generalized form of the covariant two-
body equation in $(2+1)$-dimensions manifests as follows [4, 5, 6, 7]

$\displaystyle\left[\mathcal{H}^{f}\otimes\gamma^{t^{\overline{f}}}+\gamma^{t^{f}}\otimes\mathcal{H}^{\overline{f}}\right]\Psi(x^{\mu^{f}},x^{\mu^{\overline{f}}})=0,$
$\displaystyle\mathcal{H}^{f}=\left\\{\gamma^{\mu^{f}}\not{\mathcal{D}}_{\mu}^{f}+im\textbf{I}_{2}\right\\},\mathcal{H}^{\overline{f}}=\left\\{\gamma^{\mu^{\overline{f}}}\not{\mathcal{D}}_{\mu}^{\overline{f}}+im\textbf{I}_{2}\right\\},$
$\displaystyle\not{\mathcal{D}}_{\mu}^{f}=\partial_{\mu}^{f}-\Gamma^{f}_{\mu},\quad\not{\mathcal{D}}_{\mu}^{\overline{f}}=\partial_{\mu}^{\overline{f}}-\Gamma^{\overline{f}}_{\mu},$
(2.1)

in which $x^{\mu}$ represents the coordinates within this curved space,
$f\,(\overline{f})$ denotes fermions (antifermions), $\gamma^{\mu}$ stand as
space-dependent Dirac matrices, $m$ signifies the rest mass of individual
particles, $\Psi(x^{\mu^{f}},x^{\mu^{\overline{f}}})$ signifies the bi-local
spinor field that depends on the spacetime position vectors
($x^{\mu^{f}},x^{\mu^{\overline{f}}}$) of the fermions, $\textbf{I}_{2}$
denotes the 2-dimensional identity matrix, and $\Gamma{\mu}$ represents the
spinorial affine connections for each Dirac field. The magnetic universe (with
a non-zero cosmological constant) can be described by the following line
element

$\displaystyle
ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=dt^{2}-dr^{2}-\diamondsuit(r)^{2}d\phi^{2},$
$\displaystyle\diamondsuit(r)=\sigma\,\sin\left(\sqrt[\
]{2\Lambda}\,r\right).$ (2.2)

The relationship between $\sigma$ (an integration constant) and $\Lambda$ is
connected to the magnetic field. This magnetic field is defined by $H=\sqrt[\
]{\Lambda}\sigma\sin(\sqrt[\ ]{2\Lambda}r)$, as found in [8]. In this context,
Greek indices are used to denote coordinates within the curved space-time,
represented as $x^{\mu}=t,r,\phi$. The Gaussian curvature ($\mathcal{K}$) of
this particular space-time background can be calculated as
$\mathcal{K}=-\frac{\diamondsuit_{,rr}}{\diamondsuit}=2\Lambda$, where the
subscript ,r signifies derivative with respect to $r$. Next, the derivation of
the space-dependent Dirac matrices involves employing the expression:
$\gamma^{\mu}=e^{\mu}_{k}\gamma^{k},(k=0,1,2.)$. The quantities $e^{\mu}_{k}$
correspond to the inverse tetrad fields, while $\gamma^{k}$ symbolize the
space-independent Dirac matrices. The Latin indices within this context denote
the coordinates within a flat spacetime. The selection of $\gamma^{k}$ relies
on the familiar Pauli matrices ($\sigma^{x},\sigma^{y},\sigma^{z}$),
determined by the signature ($+,-,-$) of the specific line element. This
ensures $\gamma^{0}=\sigma^{z}$, $\gamma^{1}=i\sigma^{x}$, and
$\gamma^{2}=i\sigma^{y}$, where $i=\sqrt[\ ]{-1}$ [5, 6]. The determination of
tetrad fields involves utilizing the expression
$g_{\mu\nu}=e^{k}_{\mu}e^{l}_{\nu}\tilde{\eta}_{kl}$ [5, 6]. Here,
$\tilde{\eta}_{kl}$ signifies the flat Minkowski tensor characterized by a
signature of $(+,-,-)$. The covariant metric tensor $g_{\mu\nu}$ is
$g_{\mu\nu}=\textrm{diag}(1,-1,-\diamondsuit^{2})$ and its inverse becomes
$g^{\mu\nu}=\textrm{diag}(1,-1,-\diamondsuit^{-2})$. The determination of the
inverse tetrad fields ($e^{\mu}_{k}$) is feasible through the relationship
$e^{\mu}_{k}=g^{\mu\nu}e^{l}_{\nu}\tilde{\eta}_{kl}$ in which $e^{l}_{\nu}$
are tetrad fields that can be determined through
$g_{\mu\nu}=e^{k}_{\mu}e^{l}_{\nu}\tilde{\eta}_{kl}$. Accordingly, we have

$\displaystyle e^{0}_{t}=1,\quad e^{1}_{r}=1,\quad e^{2}_{\phi}=\diamondsuit,$
$\displaystyle e^{t}_{0}=1,\quad e^{r}_{1}=1,\quad
e^{\phi}_{2}=\diamondsuit^{-1}.$ (2.3)

Here, it can be seen that the tetrad fields satisfy the orthogonality and
orthonormality conditions. Also, now, we need to obtain the affine spin
connections by using the relation:
$\Gamma_{\eta}=\frac{1}{8}g_{\mu\lambda}\left(e^{q}_{\nu,\eta}e^{\lambda}_{q}-\Gamma_{\nu\eta}^{\lambda}\right)\left[\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}\right]$
where $\Gamma_{\nu\eta}^{\lambda}$ symbols can be calculated by using the
expression given by
$\Gamma_{\nu\eta}^{\lambda}=\frac{1}{2}g^{\lambda\epsilon}\left\\{\partial_{\nu}g_{\eta\lambda}+\partial_{\eta}g_{\epsilon\nu}-\partial_{\epsilon}g_{\nu\eta}\right\\}$
[5, 6]. Thereby, one can find that

$\displaystyle\Gamma^{r}_{\phi\phi}=-\diamondsuit\diamondsuit_{,r},\quad\Gamma^{\phi}_{r\phi}=\frac{\diamondsuit_{,r}}{\diamondsuit}.$

Finally, the outcomes regarding the generalized Dirac matrices and the non-
vanishing component of the affine spin connections can be summarized as
follows333Here, one should note that $\gamma^{t}$ and $\gamma^{r}$ are
independent from the $r$, but the others must be considered according to
radial coordinates of $f$ and $\overline{f}$.

$\displaystyle\gamma^{t}=\sigma^{z},\quad\gamma^{r}=i\sigma^{x},\quad\gamma^{\phi}=\frac{i}{\diamondsuit}\sigma^{y},$
$\displaystyle\Gamma_{\phi}=\frac{i}{2}\diamondsuit_{,r}\sigma^{z}.$ (2.4)

It might be advantageous to express the covariant two-body equation
explicitly, taking the form of $\hat{\Upsilon}\Psi=0$ in which
$\hat{\Upsilon}$ is

$\displaystyle\gamma^{t^{f}}\otimes\gamma^{t^{\overline{f}}}\left\\{\partial_{t}^{f}+\partial_{t}^{\overline{f}}\right\\}+\gamma^{r^{f}}\otimes\gamma^{t^{\overline{f}}}\partial_{r}^{f}+\gamma^{t^{f}}\otimes\gamma^{r^{\overline{f}}}\partial_{r}^{\overline{f}}$
$\displaystyle+\gamma^{\phi^{f}}\otimes\gamma^{t^{\overline{f}}}\partial_{\phi}^{f}+\gamma^{t^{f}}\otimes\gamma^{\phi^{\overline{f}}}\partial_{\phi}^{\overline{f}}$
$\displaystyle+im\left\\{\textbf{I}_{2}\otimes\gamma^{t^{\overline{f}}}+\gamma^{t^{f}}\otimes\textbf{I}_{2}\right\\}$
$\displaystyle-\left\\{\gamma^{\phi^{f}}\Gamma_{\phi}^{f}\otimes\gamma^{t^{\overline{f}}}+\gamma^{t^{f}}\otimes\gamma^{\phi^{\overline{f}}}\Gamma_{\phi}^{\overline{f}}\right\\},$
(2.5)

and it is obvious in the Eq. (2.4) that
$\gamma^{\phi}\Gamma_{\phi}=-\frac{i}{2}\frac{\diamondsuit_{,r}}{\diamondsuit}\sigma^{x}$.
It’s pertinent to mention that our aim is to analyze the relative motion of
the considered pair in the magnetized universe affected by non-zero
cosmological constant $\Lambda$. To accomplish this, separating the center of
mass motion coordinates ($R$) and relative motion coordinates ($r$) becomes
imperative, the process facilitated by the expressions elucidated in the
following [4]:

$\displaystyle r_{\mu}=x_{\mu}^{f}-x_{\mu}^{\overline{f}},\quad
R_{\mu}=\frac{x_{\mu}^{f}}{2}+\frac{x_{\mu}^{\overline{f}}}{2},\quad
x_{\mu}^{f}=R_{\mu}+\frac{r_{\mu}}{2},$ $\displaystyle
x_{\mu}^{\overline{f}}=R_{\mu}-\frac{r_{\mu}}{2},\quad\partial_{x_{\mu}}^{f}=\frac{\partial_{R_{\mu}}}{2}+\partial_{r_{\mu}},$
$\displaystyle\partial_{x_{\mu}}^{\overline{f}}=\frac{\partial_{R_{\mu}}}{2}-\partial_{r_{\mu}},$
(2.6)

and thus
$\partial_{x_{\mu}}^{f}+\partial_{x_{\mu}}^{\overline{f}}=\partial_{R_{\mu}}$.
Now, it is crucial to emphasize that the system’s energy relates to the time
coordinate of the center of mass motion (see Eq. (2.5)). This association
arises due to the presence of the
$\partial_{t}^{f}+\partial_{t}^{\overline{f}}$ term, resulting in
$\partial_{t}^{f}+\partial_{t}^{\overline{f}}=\partial_{R_{t}}$. Assuming the
center of mass for the pair remains stationary at the spatial origin, we can
effectively decompose the spinor $\Psi$,
$\Psi\left(t,r,\phi\right)=\textrm{e}^{-i\omega
t}\textrm{e}^{is\phi}\left(\psi_{1}\,\psi_{2}\,\psi_{3}\,\psi_{4}\right)^{T}$,
with $T$ representing the spinor’s transpose depending on the relative radial
coordinate $r$, and where $s$ signifies the combined spin of two fermions.
Consequently, following some algebraic arrangements, one can deduce the
subsequent equation(s)

$\displaystyle\omega\chi_{1}-\tilde{m}\chi_{2}-\frac{2s}{\sigma\sin\left(ar\right)}\chi_{3}+2\left[\partial_{r}+a\cot\left(ar\right)\right]\chi_{4}=0,$
(2.7)

where

$\displaystyle\chi_{2}=\frac{\tilde{m}}{\omega}\chi_{1},\quad\chi_{3}=\frac{2\,s}{\omega\,\sigma\,\sin\left(ar\right)}\chi_{1},\quad\chi_{4}=\frac{2}{\omega}\left[\partial_{r}+a\,\cot\left(ar\right)\right]\chi_{1}$

provided

$\displaystyle\chi_{1}(r)=\psi_{1}(r)+\psi_{4}(r),\quad\chi_{2}(r)=\psi_{1}(r)-\psi_{4}(r),$
$\displaystyle\chi_{3}(r)=\psi_{2}(r)+\psi_{3}(r),\quad\chi_{4}(r)=\psi_{2}(r)-\psi_{3}(r),$

and $\tilde{m}=2m$, $a=\sqrt[\ ]{\Lambda/2}$.

##  3 Analytical solutions for the wave equation

Here, we present a non-perturbative wave equation governing a fermion-
antifermion pair within a magnetized universe affected by non-zero
cosmological constant. Our objective is to seek a closed form analytical
solution for this specific wave equation. The Eq. (2.7) leads to the
subsequent wave equation manifestation

$\displaystyle\chi_{1_{,rr}}(r)+2a\cot\left(ar\right)\chi_{1_{,r}}(r)+\left[\frac{\epsilon}{4}-a^{2}-\frac{s^{2}}{\vartheta(r)^{2}}\right]\chi_{1}(r)=0,$

where $\epsilon=\omega^{2}-\tilde{m}^{2}$ and
$\vartheta(r)=\sigma\sin\left(ar\right)$. This equation, with the change of
variable $y=\left[1+\cos\left(ar\right)\right]/2$, would yield

$y\left(y-1\right)\chi_{1,yy}\left(y\right)+\left(3y-\frac{3}{2}\right)\chi_{1,y}\left(y\right)-\left(\frac{\epsilon}{4a^{2}}-1+\frac{s^{2}}{4a^{2}\sigma^{2}y\left(y-1\right)}\right)\chi_{1}\left(y\right)=0.$
(3.1)

We may now use the substitution

$\chi_{1}\left(y\right)=y^{\beta/2}\left(y-1\right)^{\gamma/2}\chi\left(y\right),$
(3.2)

where,

$\beta=-\frac{1}{2}+\frac{1}{2}\sqrt[\
]{1+\frac{4s^{2}}{a^{2}\sigma^{2}}},\;\gamma=-\frac{1}{2}-\frac{1}{2}\sqrt[\
]{1+\frac{4s^{2}}{a^{2}\sigma^{2}}}$ (3.3)

to obtain

$y\left(y-1\right)\chi_{,yy}\left(y\right)+\left(2y+\gamma-\frac{1}{2}\right)\chi_{,y}\left(y\right)+\frac{1}{4}\left(1-\frac{\epsilon}{a^{2}}\right)\chi\left(y\right)=0.$
(3.4)

Obviously, this equation is in the form of the Gauss’s hypergeometric
differential equation [9, 10]. that admits a solution in the form of

$\displaystyle\chi\left(y\right)$ $\displaystyle=$
$\displaystyle\mathcal{N}\,M\left(\left[\frac{1}{2}+\frac{1}{2}\sqrt[\
]{\frac{\epsilon}{a^{2}}},\frac{1}{2}-\frac{1}{2}\sqrt[\
]{\frac{\epsilon}{a^{2}}}\right],\frac{1}{2}\sqrt[\
]{1+\frac{4s^{2}}{a^{2}\sigma^{2}}}+1,y\right)$ (3.5) $\displaystyle=$
$\displaystyle\mathcal{N}\,M\left(\frac{1}{2}\pm\frac{1}{2}\sqrt[\
]{\frac{\epsilon}{a^{2}}},\frac{1}{2}\sqrt[\
]{1+\frac{4s^{2}}{a^{2}\sigma^{2}}}+1,y\right).$

where $\mathcal{N}$ is the normalization constant (the determination of which
is far beyond the scope of the current study). Next, we have to truncate the
hypergeometric power series into a polynomial of order $n\geq 0$ (to secure
finiteness and square integrability of the radial wave function) by requiring
that

$\frac{1}{2}\pm\frac{1}{2}\sqrt[\
]{\frac{\epsilon}{a^{2}}}=-n\Longleftrightarrow\epsilon=\omega^{2}-\tilde{m}^{2}=a^{2}\left(2n+1\right)^{2}.$
(3.6)

Consequently,

$\omega_{n}=\pm\sqrt[\
]{\tilde{m}^{2}+a^{2}\left(2n+1\right)^{2}};\,n=0,1,2,\cdots.$ (3.7)

However, it could be interesting to know that a power series expansion in the
form of

$\chi\left(y\right)=\sum\limits_{j=0}^{\infty}C_{j}\,y^{j+\upsilon},$ (3.8)

in Eq. (3.4), would result

$\sum\limits_{j=0}^{\infty}C_{j}\left\\{\left[\left(j+\upsilon\right)\left(j+\upsilon+1\right)+\mathcal{E}\right]\,y^{j+\upsilon}+\left(j+\upsilon\right)\left(\gamma+\frac{1}{2}-j-\upsilon\right)y^{j+\upsilon-1}\right\\}=0,$
(3.9)

where $\mathcal{E}=1/4-\epsilon/4a^{2}$. In a straightforward manner, one
obtains

$\displaystyle\sum\limits_{j=0}^{\infty}\left\\{C_{j}\left[\left(j+\upsilon\right)\left(j+\upsilon+1\right)+\mathcal{E}\right]\,+C_{j+1}\left(j+\upsilon\right)\left(\gamma-\frac{1}{2}-j-\upsilon\right)\right\\}y^{j+\upsilon}$
$\displaystyle+C_{0}\left[\upsilon\left(\gamma-\upsilon+\frac{1}{2}\right)\right]\,y^{\upsilon-1}=0.$
(3.10)

Which, in turn, suggests a two terms recursion relation

$C_{j}\left[\left(j+\upsilon\right)\left(j+\upsilon+1\right)+\mathcal{E}\right]\,+C_{j+1}\left(j+\upsilon\right)\left(\gamma-\frac{1}{2}-j-\upsilon\right)=0,$
(3.11)

and since $C_{0}\neq 0$

$C_{0}\neq
0\Longrightarrow\upsilon\left(\gamma-\upsilon+\frac{1}{2}\right)\Longrightarrow\upsilon=0,\;\upsilon=\gamma+\frac{1}{2}\text{.}$
(3.12)

It is clear that $\upsilon=\gamma+\frac{1}{2}$ renders the radial wave
function infinite and unbounded at $y=0$ (i.e., at $\cos\left(ar\right)=-1$).
We adopt $\upsilon=0$, therefore. Under such settings, our two terms recursion
relation now reads

$C_{j}\left[j\left(j+1\right)+\mathcal{E}\right]\,+C_{j+1}\left[j\left(\gamma-\frac{1}{2}-j\right)\right]=0.$
(3.13)

Our power series in Eq. (3.8) needs to be truncated into a polynomial of order
$n\geq 0$ by the requirement that $\forall j=n$ we set $C_{n+1}=0$.
Consequently,

$C_{n}\left[n\left(n+1\right)+\mathcal{E}\right]\,=0\Longrightarrow C_{n}\neq
0\Longrightarrow n\left(n+1\right)+\mathcal{E}=0.$ (3.14)

This result, along with $\mathcal{E}=1/4-\epsilon/4a^{2}$ and
$\epsilon=\omega^{2}-\tilde{m}^{2}$, would imply

$\frac{1}{4}-\frac{\epsilon}{4a^{2}}=n\left(n+1\right)\Longleftrightarrow\omega^{2}-\tilde{m}^{2}=a^{2}\left(2n+1\right)^{2}.$
(3.15)

This result is in exact accord with that in (3.6) and implies that in (3.7).
This condition leads to a specific frequency spectrum, which can be expressed
as follows:

$\displaystyle\omega_{n}=\pm\left\\{\tilde{m}^{2}+4a^{2}\tilde{n}^{2}\right\\}^{1/2},\quad\tilde{n}=n+1/2.$
(3.16)

Hereby, it is evident that the system’s energy remains unaffected by the total
spin of the fermion-antifermion pair, despite the explicit dependence of the
wave function on the total spin. The form of the wave equation represented by
Eq. (3.4) remains consistent even when $m=0$, enabling us to explore scenarios
involving massless pairs. In situations where $m$ is set to zero, the
frequency spectrum exhibits the pattern

$\displaystyle\omega_{n}=\sqrt[\ ]{2\Lambda}\tilde{n},$ (3.17)

mirroring the characteristic findings of a one-dimensional non-relativistic
quantum oscillator. It is worth noting that our findings bear a striking
resemblance to the characteristics exhibited by a spinless relativistic
quantum oscillator (specifically when $m\neq 0$) within flat spaces, while
also mirroring the well-documented outcomes of a non-relativistic quantum
oscillator (when $m=0$). The frequency spectrum described by Eq. (3.16)
incorporates the spacetime parameter $\Lambda$, which is associated with the
magnetic field, along with the quantum number $n$. Here, it is important to
highlight the significant role of the cosmological constant in cosmology,
especially concerning our comprehension of dark energy and its influence on
the universe’s evolution. The correlation between this constant and the energy
spectrum of fermion-antifermion systems suggests a potential profound
connection between the quantum domain and the wider scope of cosmology.
Additionally, Eq. (3.16) resembles the precise outcomes obtained for
relativistic oscillators (see also [4]). Within the realm of quantum field
theory, vacuum fluctuations resemble an expansive array of harmonic
oscillators spanning various frequencies. The energy stemming from these
fluctuations adds to the vacuum energy, potentially associated (at a
conceptual level) with the cosmological constant. Nonetheless, this
correlation is intricate, forming a significant facet of continual theoretical
exploration within quantum field theory, specifically in its implications for
cosmology.

Figure 1: Relativistic frequencies for varying values of the $\Lambda$. Here,
we take $m=1$.

##  4 Discussions

In this study, our exploration of a specific curved geometry within three-
dimensional space featuring a cosmological constant can be significantly
contributed to the broader scope of research in relativistic particle
dynamics. By focusing on the eigenvalue solutions for fermion-antifermion
pairs in a magnetized Bonnor-Melvin universe, characterized by a non-zero
cosmological constant, we have unveiled intriguing insights into the behavior
of fermion-antifermion pairs within this unique space-time configuration. The
investigation of (2+1)-dimensional metric has yielded surprising parallels
between relativistic fermionic behavior in this specialized setting and
solutions akin to the relativistic spinless oscillator in flat space (see also
[4] and Figure 1). Notably, our precise computations have revealed compelling
similarities between the dynamics of fermion-antifermion pairs in this
background and those encountered in relativistic and non-relativistic
oscillator frameworks (see Eq. (3.16)). This unexpected correspondence not
only sheds light on the intricate interplay between relativistic fermions and
the distinct geometry of the Bonnor-Melvin universe but also suggests an
intriguing connection between seemingly disparate physical phenomena.
Moreover, our examination of massless fermion-antifermion pairs within this
universe model has provided interesting results showing similarities to the
behavior of a non-relativistic one-dimensional quantum oscillator (see Eq.
(3.17)). This significant finding holds implications for comprehending
massless particles in various cosmological settings and urges further
investigation into the fundamental principles governing their behavior in such
intriguing space-time configurations. Our comprehensive analysis not only
highlights the nuances of fermionic dynamics in magnetized Bonnor-Melvin
universes with non-zero cosmological constants but also underscores unexpected
connections between relativistic fermionic behavior and well-established
phenomena in flat space. The existence of the cosmological constant $\Lambda$
in the obtained frequency spectra emphasizes a profound link between the
system’s energy and the fundamental nature of the universe’s evolution. The
presence of the cosmological constant within the frequency spectrum implies
that space-time dynamics, influenced by the $\Lambda$, intricately impact the
energy levels of fermion-antifermion pairs. This connection presents an
opportunity to glean insights into the interplay between quantum mechanics and
cosmology. The value of the cosmological constant significantly determines the
universe’s evolution rate and plays a pivotal role in cosmological models.
Within the realm of quantum field theory, vacuum fluctuations, akin to an
infinite collection of harmonic oscillators with different frequencies,
contribute to the vacuum energy. Conceptually linking this energy to the
cosmological constant remains a complex endeavor and is part of ongoing
theoretical research in quantum field theory, offering tantalizing
implications for our understanding of cosmological constants and their
relation to fundamental physical phenomena.

## Data availability

This manuscript has no associated data or the data will not be deposited.

## Conflicts of interest statement

No conflict of interest declared by the authors.

## Funding

No funding regarding this research.

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