# Klein-Gordon particles in a quasi-pointlike global monopole spacetime and a
Wu-Yang magnetic monopole: invariance and isospectrality.

Omar Mustafa<EMAIL_ADDRESS>Department of Physics, Eastern
Mediterranean University, G. Magusa, north Cyprus, Mersin 10 - Turkey.

###### Abstract

Abstract: We use two correlated metric functions and transform/deform the
pointlike global monopole (PGM) spacetime metric into a quasi-PGM (QPGM)
spacetime one. We study Klein-Gordon (KG) particles (manifestly introduced by
the non-minimal coupling form of the operator
$\tilde{D}_{\mu}=D_{\mu}+\mathcal{F}_{\mu}$, with $\mathcal{F}_{\mu}$
$\in\mathbb{R}$, and $D_{\mu}=\partial_{\mu}-ieA_{\mu}$) in such QPGM
spacetime background. We show that the KG-particles in the QPGM spacetime are
isospectral and invariant with KG-particles in the PGM spacetime, provided
that the two metric functions are correlated. The Wu-Yang magnetic monopole
(WYMM) is also included in the process. We report the effects of both
monopoles PGM/QPGM and WYMM on the spectroscopic structure of two models, KG-
oscillators and KG-Coulomb particles in such spacetime background. Moreover,
we discuss KG-Coulomb particles and shifted KG-oscillators in a PGM spacetime
and a WYMM, where such models are manifestly introduced by some Lorentz scalar
interaction potentials $S(r)$. The current study is not only of fundamental
observability nature but also of pedagogical interest in quantum gravity.

PACS numbers: 05.45.-a, 03.50.Kk, 03.65.-w

Keywords: Klein-Gordon (KG) oscillators, KG-Coulomb particles, Shifted KG-
oscillators, Quasi-pointlike global monopole, Wu-Yang magnetic monopole,
isospectrality and invariance.

##  I Introduction

During the phase transition in the early universe, Grand Unified Theories have
predicted several kinds of topological defects that are manifestly introduced
by the topology of the vacuum manifold Re1 ; Re2 ; Re021 . The effects of such
topological defects include (but not restricted to) modifications in the
spectroscopic structures of the quantum mechanical systems and have,
therefore, stimulated research attention in many areas of physics. For
example, in condensed matter physics and gravitation, and within the
geometrical theory of topological defects, linear defects are associated with
dislocations (i.e., torsions) and disclinations (curvatures) Re0211 ; Re0212 .
The topological defects may include cosmic string Re021 ; Re3 ; Re4 ; Re5 ;
Re6 , domain walls Re2 ; Re021 , and pointlike global monopole (PGM) Re7 .
Global monopoles formed as a consequence of a spontaneous global symmetry
breaking (i.e., global $O(3)$ symmetry is broken to $U\left(1\right)$) behave
like elementary particles, where most of their energies are concentrated near
the monopole core Re7 . Global monopoles are zero-dimensional (pointlike)
objects that do not introduce gravitational interactions but they modify the
geometry of spacetime Re3 ; Re6 ; Re7 ; Re8 . They are spherically symmetric
topological defects that admit the general static metric

$ds^{2}=-B\left(r\right)\,dt^{2}+A\left(r\right)\,dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}\right).$
(1)

In their study, Barriola and Vilenkin Re7  have shown that the PGM,
effectively, exerts no gravitational force. The space around and outside the
PGM has a solid deficit angle that deflects all light. Moreover, they have
found that the correlation between $B\left(r\right)$ and $A\left(r\right)$ is
given by

$B\left(r\right)=A\left(r\right)^{-1}=1-8\pi G\eta^{2}-\frac{2GM}{r},$ (2)

where $M$ is a constant of integration and in the flat space $M\sim M_{core}$
( $M_{core}$ is the mass of the PGM core). By neglecting the mass term and
rescaling the variables $r$ and $t$ Re7 , one may rewrite the PGM metric as

$ds^{2}=-dt^{2}+\frac{1}{\alpha^{2}}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}\right),$
(3)

where $0<\alpha^{2}=1-8\pi G\eta^{2}\leq 1$, $\alpha$ is a PGM parameter that
depends on the energy scale $\eta$, and $G$ is the gravitational constant Re7
; Re8 ; Re81 ; Re9 ; Re10 . This has stimulated research attention so that PGM
effects are studied, for example, within $f\left(R\right)$ theory Re81 ,
vacuum polarization effects in the presence of a Wu-Yang magnetic monopole
(WYMM) Re101 ; Re1011 ; Re102 , gravitating magnetic monopole Re103 , Dirac
and Klein-Gordon (KG) oscillators Re11 , Schrödinger oscillators Re9 , KG
particles with a dyon, magnetic flux and scalar potential Re8 , bosons in
Aharonov-Bohm flux field and a Coulomb potential Re131 , Schrödinger particles
in a Kratzer potential Re132 , Schrödinger particles in a Hulthėn potential
Re1321  , scattering by a monopole Re133 , Schrödinger particles in a Hulthėn
plus Kratzer potential Re1331 , KG-oscillators and AB-effect Re1332 , to
mention a few. In fact, the influence of topological defects in spacetime on
the spectroscopic structure of quantum mechanical systems have been a subject
of research attention over the years. The harmonic oscillator is studied in
the context of Dirac and Klein-Gordon (KG) Re11 ; Re12 ; Re121 ; Re13 ; Re14 ;
Re15 ; Re16 ; Re17 ; Re18 ; Re19 ; Re20 ; Re21 ; Re211 ; Re22 ; Re23 ; Re24 ;
Re25 ; Re26 ; Re27 ; Re271 ; Re272  in different spacetime backgrounds.

On the other hand, the WYMM Re101 ; Re1011 ; Re102  is a theoretical concept
that is yet to be observed as a physical particle in nature. However, it
remains an interesting model for modern theoretical particle physics and
topological properties in gauge theories. It would, therefore, be interesting
to study KG-particles in a PGM and a quasi-point-like global monopole
spacetime (QPGM) and a WYMM. QPGM spacetime is the notion we shall adopt
throughout to refer to a deformed/transformed (through a point canonical
transformation) PGM spacetime. Such a study would facilitate not only exact
solvability of quantum gravity models, but would also be useful for
pedagogical interest in quantum gravity. This should represent the motivation
of the current methodical proposal. To the best of our knowledge, no such
study have ever been done before.

We organize our paper in the following manner. In section 2, we derive the KG-
particles’ equation in a QPGM and a 4-vector potential
$A_{\mu}=\left(0,0,A_{\varphi},0\right)$. Where we use a point canonical
transformation so that the PGM metric would eventually indulge two correlated
metric functions that makes the notion QPGM spacetime unavoidable in the
process. We use our findings for the KG-particles in a QPGM spacetime and
include the WYMM in section 3. Hereby, we show that the KG-particles in a QPGM
spacetime and the WYMM are isospectral and invariant with KG-particles in the
regular PGM spacetime and a WYMM. In section 4, we discuss KG-oscillators in a
QPGM spacetime and a WYMM and report the effects of both monopoles on the
spectroscopic structure of the KG-oscillators. In section 5, we discuss KG-
Coulomb particles in a QPGM spacetime and a WYMM, where the Coulombic nature
of such KG-particles is introduced by the non-minimal coupling form of the
operator $\tilde{D}_{\mu}=D_{\mu}+\mathcal{F}_{\mu}$, with $\mathcal{F}_{\mu}$
$\in\mathbb{R}$, and $D_{\mu}=\partial_{\mu}-ieA_{\mu}$ is the gauge-covariant
derivative that admits minimal coupling. The KG-oscillators and KG-Coulomb
particles in such spacetime background are not only of fundamental
observability nature but also of pedagogical interest in quantum gravity. For
both KG-oscillators and KG-Coulomb particles in a QPGM spacetime and a WYMM,
we use two sets of the metric functions. Moreover, we discuss KG-Coulomb
particles (in section 6) and a shifted KG-oscillators (in section 7) in a PGM
spacetime and a WYMM, where the interactions’ nature of which is introduced by
some Lorentz scalar interaction potentials $S(r)$. We conclude in section 8.

##  II KG-particles in a QPGM spacetime background

Consider KG-particles interacting with a PGM with a spacetime metric given by
(3) and subjected to a point canonical transformation (PCT) in the form of

$r=\int\sqrt{f\left(\rho\right)}d\rho=\sqrt{q\left(\rho\right)}\rho\Leftrightarrow\sqrt{f\left(\rho\right)}=\sqrt{q\left(\rho\right)}\left[1+\frac{q^{\prime}\left(\rho\right)}{2q\left(\rho\right)}\rho\right].$
(4)

Then the PGM metric (3) transforms into

$ds^{2}=-dt^{2}+\frac{f\left(\rho\right)}{\alpha^{2}}d\rho^{2}+q\left(\rho\right)\,\rho^{2}\left[d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}\right].$
(5)

At this point, one should notice that $q\left(\rho\right)$ and
$f\left(\rho\right)$ are two correlated functions and are dimensionless
positive-valued scalar multipliers, where $q\left(\rho\right)=1\Rightarrow
f\left(\rho\right)=1$ recovers the PGM metric (3). This would not change the
radial direction nor the range of the radial coordinates $\rho$ or $r$ ( i.e.,
$r\in\left[0,\infty\right]\Longrightarrow\rho\in\left[0,\infty\right]$). They
are commonly called metric functions so that
$q\left(\rho\right),f\left(\rho\right)\in\mathbb{R}$.

Consequently, the corresponding deformed/transformed metric tensor is

$g_{\mu\nu}=\left(\begin{tabular}[]{cccc}$-1$&$0$&$0$&$0$\\\
$0$&$\;f\left(\rho\right)/\alpha^{2}$&$0$&$0\vskip 6.0pt plus 2.0pt minus
2.0pt$\\\ $0$&$0$&$q\left(\rho\right)\,\rho^{2}$&$0\vskip 6.0pt plus 2.0pt
minus 2.0pt$\\\
$0$&$0$&$0$&$q\left(\rho\right)\,\rho^{2}\sin^{2}\theta$\end{tabular}\right);\;\mu,\nu=t,\rho,\theta,\varphi,$
(6)

which implies

$\det\left(g_{\mu\nu}\right)=g=-\frac{f\left(\rho\right)}{\alpha^{2}}q\left(\rho\right)^{2}\,\rho^{4}\sin^{2}\theta,$

and

$g^{\mu\nu}=\left(\begin{tabular}[]{cccc}$-1$&$0$&$0$&$0$\\\
$0$&$\;\alpha^{2}/f\left(\rho\right)$&$0$&$0\vskip 6.0pt plus 2.0pt minus
2.0pt$\\\ $0$&$0$&$1/q\left(\rho\right)\,\rho^{2}$&$0\vskip 6.0pt plus 2.0pt
minus 2.0pt$\\\
$0$&$0$&$0$&$1/q\left(\rho\right)\,\rho^{2}\sin^{2}\theta$\end{tabular}\right).$
(7)

The KG-equation then reads

$\left(\frac{1}{\sqrt{-g}}\tilde{D}_{\mu}\sqrt{-g}g^{\mu\nu}\tilde{D}_{\nu}+\zeta
R\right)\,\Psi\left(t,\rho,\theta,\varphi\right)=\left(m_{\circ}+S\left(\rho\right)\right)^{2}\,\Psi\left(t,\rho,\theta,\varphi\right),$
(8)

where $\tilde{D}_{\mu}=D_{\mu}+\mathcal{F}_{\mu}$ is in a non-minimal coupling
form with $\mathcal{F}_{\mu}$ $\in\mathbb{R}$,
$D_{\mu}=\partial_{\mu}-ieA_{\mu}$ is the gauge-covariant derivative that
admits minimal coupling, $A_{\nu}=\left(0,0,A_{\varphi},0\right)$ is the
4-vector potential, $S\left(\rho\right)$ is the the Lorentz scalar potential,
$m$ is the rest mass energy (i.e., $m\equiv mc^{2}$, with $\hbar=c=1$ units to
be used through out this study), $\zeta$ is an arbitrary coupling constant,
and the scalar curvature $R$ reads
$R=R_{\nu}^{\nu}=2\left(1-\alpha^{2}\right)/q\left(\rho\right)\,\rho^{2}$ .
One should notice that the term $\mathcal{F}_{\mu}$ in the non-minimal
coupling form of $\tilde{D}_{\mu}$ is used to incorporate the KG-oscillators
for $\mathcal{F}_{\mu}=\left(0,\mathcal{F}_{\rho},0,0\right)$,
$\mathcal{F}_{\rho}\equiv\mathcal{F}_{\rho}\left(\rho\right)$ (c.f., e.g.,
Re0281 ; Re0282 , and the same recipe can be used for KG-Coulombic particles
as well). This would, in turn, yield

$\displaystyle\left\\{-\partial_{t}^{2}+\frac{\alpha^{2}}{\sqrt{f\left(\rho\right)}q\left(\rho\right)\rho^{2}}\,\left(\partial_{\rho}+\mathcal{F}_{\rho}\right)\frac{q\left(\rho\right)\rho^{2}}{\sqrt{f\left(\rho\right)}}\,\left(\partial_{\rho}-\mathcal{F}_{\rho}\right)+\right.\frac{2\zeta\left(1-\alpha^{2}\right)}{q\left(\rho\right)\,\rho^{2}}+\frac{1}{q\left(\rho\right)\rho^{2}}\left[\frac{1}{\sin\theta}\,\partial_{\theta}\,\sin\theta\,\partial_{\theta}\right.$
$\displaystyle\left.\left.+\frac{1}{\sin^{2}\theta}\left(\partial_{\varphi}-ieA_{\varphi}\right)^{2}\right]\right\\}\,\Psi\left(t,\rho,\theta,\varphi\right)=\left(m_{\circ}+S\left(\rho\right)\right)^{2}\,\Psi\left(t,\rho,\theta,\varphi\right).$
(9)

This equation describes KG-particles in a QPGM (a deformed/transformed PGM if
you wish) spacetime and a 4-vector potential
$A_{\nu}=\left(0,0,A_{\varphi},0\right)$. In the forthcoming sections, we
shall consider some interesting characterizations of such KG-systems that are
of fundamental importance in the fields of theoretical particle physics and
quantum gravity.

##  III KG-particles in a QPGM spacetime and a WYMM

Let us now consider that the KG-particles are in a PGM spacetime Eq. (3) and
under the influence of a WYMM. Wu and Yang Re101  have introduced a magnetic
monopole that is free of strings of singularities around it Re8 ; Re101 ;
Re102 ., and have defined the 4-vector potential $A_{\mu}$ in two regions,
$R_{A}$ and $R_{B}$ covering the whole space, outside the magnet monopole, and
overlap in $R_{AB}$ so that

$R_{A}:0\leq\theta<\frac{\pi}{2}+\delta\vskip 6.0pt plus 2.0pt minus 2.0pt,\;$ | $\;\;r>0,\;\;$ | $0\leq\varphi<2\pi,$
---|---|---
$R_{B}:\frac{\pi}{2}-\delta<\theta\leq\pi,\;$ | $\;\;r>0,\;\;$ | $0\leq\varphi<2\pi,\vskip 6.0pt plus 2.0pt minus 2.0pt$
$R_{AB}:\frac{\pi}{2}-\delta<\theta<\frac{\pi}{2}+\delta,$ | $\;\;r>0,\;$ | $0\leq\varphi<2\pi,$
(10)

where $0<\delta\leq\pi/2$. Moreover, the 4-vector potential has a non-
vanishing component in each region given by

$A_{\varphi}=sg-g\cos\theta=\left\\{\begin{tabular}[]{ll}$A_{\varphi,A}=g\left(1-\cos\theta\right),$&for
$s=1\vskip 6.0pt plus 2.0pt minus 2.0pt$\\\
$A_{\varphi,B}=-g\left(1+\cos\theta\right),$&for $s=-1$\end{tabular}\right.,$
(11)

where, $A_{\varphi,A}$ and $A_{\varphi,B}$ are correlated by the gauge
transformation Re8 ; Re102

$A_{\varphi,A}=A_{\varphi,B}+\frac{i}{e}S\,\partial_{\varphi}\,S^{-1}\,;\;S=e^{2i\sigma\varphi},\;\sigma=eg.$
(12)

Under such settings, the substitution of
$\Psi\left(t,\rho,\theta,\varphi\right)=e^{-iEt}\psi\left(\rho\right)Y_{\sigma,\ell,m}\left(\theta,\varphi\right)$
would allow separability of equation (9) to imply

$\left\\{E^{2}+\frac{\alpha^{2}}{\sqrt{f\left(\rho\right)}q\left(\rho\right)\rho^{2}}\left(\partial_{\rho}+\mathcal{F}_{\rho}\right)\,\frac{q\left(\rho\right)\rho^{2}}{\sqrt{f\left(\rho\right)}}\,\,\left(\partial_{\rho}-\mathcal{F}_{\rho}\right)-\frac{\tilde{\lambda}}{q\left(\rho\right)\,\rho^{2}}\right\\}\psi\left(\rho\right)=\left(m_{\circ}+S\left(\rho\right)\right)^{2}\psi\left(\rho\right),$
(13)

where $\tilde{\lambda}=\lambda+2\xi\left(1-\alpha^{2}\right)$, and

$\left\\{\frac{1}{\sin\theta}\partial_{\theta}\sin\theta\;\partial_{\theta}+\frac{1}{\sin^{2}\theta}\left[\partial_{\varphi}-ieA_{\varphi}\right]^{2}\right\\}Y_{\sigma\ell
m}\left(\theta,\varphi\right)=-\lambda Y_{\sigma\ell
m}\left(\theta,\varphi\right).$ (14)

Obviously, we need the eigenvalues of Eq.(14) to solve for Eq.(13). In so
doing, we use

$Y_{\sigma\ell m}\left(\theta,\varphi\right)=e^{i\left(m+s\sigma\right)}\
\Theta_{\sigma\ell m}\left(\theta\right),$ (15)

where $m=0,\pm 1,\pm 2,\cdots$ is the magnetic quantum number, to obtain

$\left\\{\frac{1}{\sin\theta}\partial_{\theta}\sin\theta\;\partial_{\theta}-\frac{\left(m+\sigma\cos\theta\right)^{2}}{\sin^{2}\theta}\right\\}\Theta_{\sigma\ell
m}\left(\theta\right)=-\lambda\Theta_{\sigma\ell m}\left(\theta\right).$ (16)

At this point, one should notice that Eq.(16) is $s$-independent so that
$\Theta_{\sigma\ell m}\left(\theta\right)=\left[\Theta_{\sigma\ell
m}\left(\theta\right)\right]_{A}=\left[\Theta_{\sigma\ell
m}\left(\theta\right)\right]_{B}$ as readily observed by Wu-Yang Re101 .

Next, we use $x=\cos\theta$ and re-write Eq.(16) as

$\left\\{\left(1-x^{2}\right)\,\partial_{x}^{2}-2x\,\partial_{x}-\frac{\left(m+\sigma\,x\right)^{2}}{1-x^{2}}\right\\}\Theta_{\sigma\ell
m}\left(\theta\right)=-\lambda\Theta_{\sigma\ell m}\left(\theta\right).$ (17)

Which, with the substitution

$\Theta_{\sigma\ell
m}\left(\theta\right)=\left(1-x\right)^{\beta/2}\left(1+x\right)^{\gamma/2}\,P_{\sigma\ell
m}\left(x\right)\,;\;\beta=|m|+\sigma,\;\gamma=|m|-\sigma,$ (18)

would yield that

$\,P_{\sigma\ell
m}\left(x\right)=C\;M\left(\frac{1}{2}+|m|\pm\sqrt{\sigma^{2}+\lambda+\frac{1}{4}},|m|-\sigma+1,\frac{1}{2}\left(1+x\right)\right).$
(19)

However, we are interested in a finite and bounded solution for quantum
states. This would manifestly require that the confluent hypergeometric series
$M\left(a,b,z\right)$ be truncated into a polynomial of order $n\geq 0$ using
the condition

$\frac{1}{2}+|m|\pm\sqrt{\sigma^{2}+\lambda+\frac{1}{4}}=-n\Longleftrightarrow\sigma^{2}+\lambda=\left(n+|m|\right)\left(n+|m|+1\right);\;n,|m|\,\geq
0.$ (20)

One should reminded that when $\sigma=0$ (i.e., no Wu-Yang monopole effect)
our $\lambda=\ell\left(\ell+1\right)$, where $\ell=0,1,2,\cdots$ is the
angular momentum quantum number. Hence, without any lose of generality of the
solution, one may safely consider that $\ell=n+|m|$ and take
$\lambda=\ell\left(\ell+1\right)-\sigma^{2}$. The validity of such assumptions
is motivated by the fact that for $\sigma=0$ and $\alpha=1$ (i.e., flat
Minkowski spacetime) one should retrieve the textbook value
$\lambda=\ell\left(\ell+1\right)$. Consequently, our $\tilde{\lambda}$ in
Eq.(13) now reads

$\tilde{\lambda}=\ell\left(\ell+1\right)+2\zeta\left(1-\alpha^{2}\right)-\sigma^{2}=L\left(L+1\right),$
(21)

where $L$ may very well be denoted as an irrational angular momentum quantum
number. Moreover, the so called Wu and Yang Re8 ; Re101  monopole harmonics
$Y_{\sigma\ell m}\left(\theta,\varphi\right)$ are now given by

$Y_{\sigma\ell
m}\left(\theta,\varphi\right)=\left\\{\begin{tabular}[]{ll}$e^{i\left(m+\sigma\right)\varphi}\,\left(1-x\right)^{\beta/2}\left(1+x\right)^{\gamma/2}\,P_{\sigma\ell
m}\left(x\right);\;\vskip 6.0pt plus 2.0pt minus 2.0pt$&in region $R_{A}$\\\
$e^{i\left(m-\sigma\right)\varphi}\,\left(1-x\right)^{\beta/2}\left(1+x\right)^{\gamma/2}\,P_{\sigma\ell
m}\left(x\right);\vskip 6.0pt plus 2.0pt minus 2.0pt$&in region
$R_{B}$\end{tabular}\right..$ (22)

As a result, our equation (13) reduces, with
$\mathcal{F}_{\rho}\equiv\mathcal{F}_{\rho}\left(\rho\right)$, to

$\left\\{E^{2}+\frac{\alpha^{2}}{\sqrt{f\left(\rho\right)}q\left(\rho\right)\rho^{2}}\,\left(\partial_{\rho}+\mathcal{F}_{\rho}\left(\rho\right)\right)\,\frac{q\left(\rho\right)\rho^{2}}{\sqrt{f\left(\rho\right)}}\,\,\left(\partial_{\rho}-\mathcal{F}_{\rho}\left(\rho\right)\right)-\frac{L\left(L+1\right)}{q\left(\rho\right)\,\rho^{2}}\right\\}\psi\left(\rho\right)=\left(m_{\circ}+S\left(\rho\right)\right)^{2}\psi\left(\rho\right).$
(23)

Let us now oversimplify this equation using our PCT in Eq. (4) so that
$\partial_{r}=\partial/\partial r=\partial\rho/\partial r\,\
\partial_{\rho}=\partial_{\rho}/\sqrt{f\left(\rho\right)}$ and
$r=\sqrt{q\left(\rho\right)}\rho$ to obtain

$\left\\{\tilde{E}^{2}+\frac{1}{r^{2}}\left[\partial_{r}+\mathcal{F}_{r}\left(r\right)\right]\,r^{2}\left[\partial_{r}-\mathcal{F}_{r}\left(r\right)\right]-\frac{\tilde{L}\left(\tilde{L}+1\right)}{r^{2}}-\frac{1}{\alpha^{2}}\left(m_{\circ}+S\left(r(\rho)\right)\right)^{2}\right\\}\psi\left(r(\rho)\right)=0,$
(24)

where $\mathcal{F}_{r}\left(r\right)=$
$\mathcal{F}_{\rho}\left(\rho\right)/\sqrt{f\left(\rho\right)}$ is used. This
equation (24) would, with $\psi\left(r\right)=R\left(r\right)/r$, read

$\left\\{\partial_{r}^{2}-\frac{\tilde{L}\left(\tilde{L}+1\right)}{r^{2}}-\left[\frac{2}{r}\mathcal{F}_{r}\left(r\right)+\mathcal{F}_{r}^{{}^{\prime}}\left(r\right)+\mathcal{F}_{r}\left(r\right)^{2}\right]-2\tilde{m}\frac{S\left(r\right)}{\alpha}-\frac{S\left(r\right)}{\alpha^{2}}^{2}+\Lambda\right\\}R\left(r\right)=0,$
(25)

where

$\Lambda=\tilde{E}^{2}-\tilde{m}^{2},\;\tilde{E}=\frac{E}{\alpha},\;\tilde{m}=\frac{m_{\circ}}{\alpha},$
(26)

and $\tilde{L}\left(\tilde{L}+1\right)=L\left(L+1\right)/\alpha^{2}$ to imply

$\tilde{L}=-\frac{1}{2}+\frac{\sqrt{\alpha^{2}+4L\left(L+1\right)}}{2\alpha}=-\frac{1}{2}+\frac{\sqrt{\alpha^{2}+4\left[\ell\left(\ell+1\right)+2\zeta\left(1-\alpha^{2}\right)-\sigma^{2}\right]}}{2\alpha}.$
(27)

Obviously, the result in Eq.(27) converges to $\tilde{L}=\ell$ for $\alpha=1$
and $\sigma=0$ (i.e., no effects of the PGM or the WYMM) as should be. More
interestingly, our KG-particles in a QPGM spacetime, Eq, (23), are isospectral
and invariant with those in a PGM spacetime, Eq.(24), under our correlated
metric functions (through the PCT of Eq.(4)). Yet, in a straightforward
manner, one may show that Eq.(24) is nothing but the KG-particles in the
regular PGM spacetime (3). In the following section, we shall consider some
illustrative examples on such systems.

##  IV KG-oscillators in a QPGM/PGM spacetime and a WYMM at $S(r)=0$ and
$\mathcal{F}(r)=\eta r$

In this section, we shall consider KG-oscillators in a QPGM spacetime under
the influence of a WYMM at $S\left(r\right)=0$. In this case, the substitution
of
$\mathcal{F}_{\rho}\equiv\mathcal{F}_{\rho}\left(\rho\right)=\eta\sqrt{f\left(\rho\right)q\left(\rho\right)}\rho$
in Eq. (23) would yield $\mathcal{F}_{r}\left(r\right)=$
$\mathcal{F}_{\rho}\left(\rho\right)/\sqrt{f\left(\rho\right)}=\eta\,r$ in Eq.
(24). Consequently, the KG-oscillators in a QPGM spacetime and under the
influence of a WYMM are obtained so that the KG-equation (25), with
$S\left(r\right)=0$, now reads

$\left\\{\partial_{r}^{2}-\frac{\tilde{L}\left(\tilde{L}+1\right)}{r^{2}}-\eta^{2}r^{2}+\tilde{\Lambda}\right\\}R\left(r\right)=0,\;\tilde{\Lambda}=\Lambda-3\eta.$
(28)

Our KG-oscillators in a QPGM spacetime and a WYMM, Eq. (28), admit exact
textbook solution in the form of confluent hypergeometric series so that

$R\left(r\right)=C\,e^{-\eta
r^{2}/2}\,r^{\tilde{L}+1}\,M\left(\frac{3}{4}+\frac{\tilde{L}}{2}-\frac{\tilde{\Lambda}}{4\eta},\tilde{L}+\frac{3}{2},\eta
r^{2}\right).$ (29)

Nevertheless, the radial function $R\left(r\right)$ should be finite and
square integrable in order to be an admissible solution for a quantum
mechanical system. This can be achieved by using the condition that the
confluent hypergeometric series $M\left(a,b,z\right)$ is truncated into a
polynomial of order $n_{r}=0,1,2,\cdots$ so that

$\frac{3}{4}+\frac{\tilde{L}}{2}-\frac{\tilde{\Lambda}}{4\eta}=-n_{r}\Longleftrightarrow\tilde{\Lambda}=2\eta\left(2n_{r}+\tilde{L}+\frac{3}{2}\right).$
(30)

In a straightforward manner, one may use Eq.(26) to obtain the energies of the
KG-oscillators in a QPGM spacetime and a WYMM as

$E=\pm\sqrt{m_{\circ}^{2}+4\eta\alpha^{2}\left(n_{r}+\frac{1}{4\alpha}\sqrt{\alpha^{2}+4\ell\left(\ell+1\right)+8\zeta\left(1-\alpha^{2}\right)-\sigma^{2}}+\frac{5}{4}\right)}.$
(31)

This result is in exact accord with that reported in Eq. (38) of Bragança et
al. Re11  when the WYMM is switched off (i.e., $\sigma=0$) and with our
$\eta=m_{\circ}\omega$ of Bragança et al. Re11 .

Figure 1:  The energy levels Eq.(31) of the KG-oscillators in a PGM/QPGM
spacetime and a Wu-Yang magnetic monopole for $n_{r}=1$, $\ell=0,1,2,3$, (a)
for $\alpha=1$, (b) for $\alpha=0.5$, and (c) for $\alpha=0.05$.

Figure 2: The energy levels Eq.(31) of the KG-oscillators in a PGM/QPGM
spacetime and a Wu-Yang magnetic monopole for $n_{r}=1$, $\ell=0,1,2,3$, (a)
for $\zeta=0.5$, (b) for $\zeta=5$, and (c) for $\zeta=15$.

In Figures 1, 2, and 3, we plot the energy levels, Eq.(31), against the
oscillator frequency $\eta$. Figures 1(a),1(b), and 1(c), are plotted for
$\alpha=1$, $\alpha=0.5$, and $\alpha=0.05$, respectively, to observe the
effect of the PGM/QPGM parameter $\alpha$ (where we take $\zeta=0$, the
coupling constant of the scalar curvature $R_{\mu}^{\mu}$ , and $\sigma=0$ for
the WYMM strength) on the energy levels of KG-oscillators in a PGM?QPGM
spacetime and a WYMM at $S\left(r\right)=0$,
$\mathcal{F}_{r}\left(r\right)=\eta\,r$. It can be clearly observed that as
$\alpha$ decreases from $\alpha=1$ towards $\alpha\sim 0$, the energy levels
are destined to cluster on $E=\pm m_{\circ}$, as is obvious from the
asymptotic tendency of (31) at $\alpha\sim 0$. Moreover, as the PGM parameter
$\alpha$ increases from above zero value towards $\alpha=1$ (i.e., flat
Minkowski spacetime) the particles/anti-particles’ energies increase with an
increasing frequency $\eta$. In Figures 2(a), 2(b), and 2(c) we plot the
energy levels Eq.(31) against the oscillator frequency $\eta$ at a fixed
$\alpha=0.5$ and $\sigma=0$ for $\zeta=0.5$, $\zeta=5$, and $\zeta=15$,
respectively. Hereby, we observe that not only the energies for the particles
and anti-particles increase with increasing coupling constant $\zeta$ of the
scalar curvature $R_{\mu}^{\mu}$, but we also observe eminent clustering of
the energy levels.

Figure 3:  The energy levels Eq.(31) of the KG-oscillators in a PGM/QPGM
spacetime and a Wu-Yang magnetic monopole for $n_{r}=1$, $\ell=0,1,2,3$, (a)
for $\sigma=0.5$, (b) for $\sigma=1$, and (c) for $\sigma=1.8$.

In Figures 3(a), 3(b), and 3(c), we plot the energy levels Eq.(31) against the
oscillator frequency $\eta$ at the fixed values $\alpha=0.5$ and $\zeta=0.5$
for $\sigma=0.5$, $\sigma=1$, and $\sigma=1.8$, respectively. It is clear that
the separation between the energy levels increases with increasing $\sigma$.

At this point, nevertheless, one should be reminded that the KG-oscillators in
Eq.(28) (i.e., KG-oscillators in a PGM spacetime Eq.(3) and a WYMM) are
isospectral and share the same energy levels, Eq.(31), with those in Eq.(23)
(i.e., KG-oscillators in a QPGM spacetime in Eq.(5) and a WYMM). Two
illustrative examples are in order.

###  IV.1 Exponential type metric function $q\left(\rho\right)=e^{a\rho^{2}}$

Let us take a positive-valued (non-zero) deformation/transformation function
$q\left(\rho\right)=e^{a\rho^{2}}$ to yield, through (4),
$f\left(\rho\right)=$ $\left(1+a\rho^{2}\right)e^{a\rho^{2}}$. This would,
with
$\mathcal{F}_{r}\left(r\right)=\eta\,r=\eta\sqrt{q\left(\rho\right)}\rho=$
$\mathcal{F}_{\rho}\left(\rho\right)/\sqrt{f\left(\rho\right)}$, consequently
imply that
$\mathcal{F}_{\rho}\left(\rho\right)=\eta\left(\rho+a\rho^{3}\right)e^{a\rho^{2}}$.
Our QPGM spacetime (5) now reads

$ds^{2}=-dt^{2}+\frac{\left(1+a\rho^{2}\right)e^{a\rho^{2}}}{\alpha^{2}}d\rho^{2}+e^{a\rho^{2}}\,\rho^{2}\left[d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}\right],$
(32)

which consequently implies that the KG-particles in equation (23) are now
described by

$\displaystyle\left\\{\frac{\alpha^{2}}{\sqrt{\left(1+a\rho^{2}\right)}e^{3a\rho^{2}/2}\rho^{2}}\,\left(\partial_{\rho}+\eta\left(\rho+a\rho^{3}\right)e^{a\rho^{2}}\right)\,\frac{e^{a\rho^{2}/2}\rho^{2}}{\sqrt{\left(1+a\rho^{2}\right)}}\,\,\left(\partial_{\rho}-\eta\left(\rho+a\rho^{3}\right)e^{a\rho^{2}}\right)\right.$
$\displaystyle\left.-\frac{L\left(L+1\right)}{e^{a\rho^{2}}\,\rho^{2}}+\left[E^{2}-m_{\circ}^{2}\right]\right\\}\psi\left(\rho\right)=0.$
(33)

This is nothings but the KG-oscillators in a PGM spacetime and a WYMM as
described by (28) and admits the exact solvability in (29) and (31). That is,
the two KG-systems in Eq. (28) and (33) are invariant and isospectral.

###  IV.2 Power-law type metric function $q\left(\rho\right)=b\rho^{\upsilon}$

Following the same procedure as above, we obtain
$f\left(\rho\right)=\left(1+\upsilon/2\right)^{2}b\rho^{\upsilon},$
$\upsilon\neq 0,-2$ and $\mathcal{F}_{\rho}\left(\rho\right)=\eta
b\left(1+\upsilon/2\right)\rho^{\upsilon+1}$. At this point, one should notice
that the restrictions on $\upsilon$ are unavoidable to secure positive-valued
characterization for the deformation/transformation functions (metric
functions) $q\left(\rho\right)$ and $f\left(\rho\right)$. Our QPGM spacetime
of (5) now reads

$ds^{2}=-dt^{2}+\frac{\left(1+\upsilon/2\right)^{2}b\rho^{\upsilon}}{\alpha^{2}}d\rho^{2}+b\rho^{\upsilon+2}\left[d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}\right],$
(34)

and consequently yields that the KG-particles in equation (23) are now
described by

$\displaystyle\left\\{\frac{\alpha^{2}}{\left(1+\upsilon/2\right)b^{3/2}\rho^{3\upsilon/2+2}}\,\left(\partial_{\rho}+\eta
b\left(1+\upsilon/2\right)\rho^{\upsilon+1}\right)\,\frac{b^{1/2}\rho^{\upsilon/2+2}}{\left(1+\upsilon/2\right)}\,\,\left(\partial_{\rho}-\eta
b\left(1+\upsilon/2\right)\rho^{\upsilon+1}\right)\right.$
$\displaystyle\left.-\frac{L\left(L+1\right)}{b\rho^{\upsilon+2}}+\left[E^{2}-m_{\circ}^{2}\right]\right\\}\psi\left(\rho\right)=0.$
(35)

For both illustrative examples above, one should notice that the two KG-
oscillators in (33) and (35) in the QPGM spacetime of (5) are invariant and
isospectral with the KG-oscillators in (28) in the PGM spacetime of (3).

##  V KG-Coulombic particles in a QPGM spacetime and a WYMM with a Bessel-type
$\mathcal{F}_{r}\left(r\right)$

In this section, we consider KG-Coulomb like particles a QPGM spacetime and a
WYMM at $S\left(r\right)=0$. The substitution of a Bessel functions type

$\mathcal{F}_{r}=\mathcal{F}_{r}\left(r\right)=-\frac{A\,J_{2}\left(\sqrt{4Ar}\right)}{\sqrt{Ar}J_{1}\left(\sqrt{4Ar}\right)},$
(36)

in Eq.(25) would result

$\left\\{\partial_{r}^{2}-\frac{\tilde{L}\left(\tilde{L}+1\right)}{r^{2}}+\frac{A}{r}+\Lambda\right\\}R\left(r\right)=0.$
(37)

This is a Schrödinger-like radial equation that admits an exact textbook
solution

$R\left(r\right)=C\,e^{-i\sqrt{\Lambda}r}\,r^{\tilde{L}+1}\,\,U\left(\tilde{L}+1+\frac{iA}{\sqrt{\Lambda}},2\tilde{L}+2,2i\sqrt{\Lambda}r\right).$
(38)

We again use the condition that the confluent hypergeometric series
$U\left(a,b,z\right)$ is truncated into a polynomial of order
$n_{r}=0,1,2,\cdots$ so that

$\tilde{L}+1+\frac{iA}{\sqrt{\Lambda}}=-n_{r}\Longleftrightarrow\Lambda=-\frac{A^{2}}{\left(n_{r}+\tilde{L}+1\right)^{2}}\Longrightarrow
E=\pm\sqrt{m_{\circ}^{2}-\frac{\alpha^{2}A^{2}}{\left(n_{r}+\tilde{L}+1\right)^{2}}}.$
(39)

At this point, one should observe that The KG-Coulombic particles in (37) are
invariant and isospectral with those in (23). That is, they are invariant and
isospectral with the KG-particles in

$\left\\{\frac{1}{\sqrt{f\left(\rho\right)}q\left(\rho\right)\rho^{2}}\,\left(\partial_{\rho}+\mathcal{F}_{\rho}\left(\rho\right)\right)\,\frac{q\left(\rho\right)\rho^{2}}{\sqrt{f\left(\rho\right)}}\,\,\left(\partial_{\rho}-\mathcal{F}_{\rho}\left(\rho\right)\right)-\frac{\tilde{L}\left(\tilde{L}+1\right)}{q\left(\rho\right)\,\rho^{2}}+\Lambda\right\\}\psi\left(\rho\right)=0;\;\Lambda=\frac{E^{2}-m_{\circ}^{2}}{\alpha^{2}}.$
(40)

Hereby,
$\mathcal{F}_{\rho}\left(\rho\right)=\mathcal{F}_{r}\left(r\left(\rho\right)\right)\sqrt{f\left(\rho\right)}$
would read

$\mathcal{F}_{\rho}\left(\rho\right)=-\frac{A\,J_{2}\left(\sqrt{4Ar\left(\rho\right)}\right)}{\sqrt{Ar\left(\rho\right)}J_{1}\left(\sqrt{4Ar\left(\rho\right)}\right)}\sqrt{f\left(\rho\right)},$
(41)

Figure 4:  The energy levels Eq.(39) of the KG-Coulombic particles in a
PGM/QPGM spacetime and a Wu-Yang magnetic monopole for $n_{r}=1$,
$\ell=0,1,2,3$, (a) for $\alpha=1$, (b) for $\alpha=0.5$, and (c) for
$\alpha=0.25$.

Figure 5:  The energy levels Eq.(39) of the KG-Coulombic particles in a
PGM/QPGM spacetime and a Wu-Yang magnetic monopole for $n_{r}=1$,
$\ell=0,1,2,3$, (a) for $\zeta=0.5$, (b) for $\zeta=5$, and (c) for $\zeta=15$

Figure 6:  The energy levels Eq.(39) of the KG-Coulombic particles in a
PGM/QPGM spacetime and a Wu-Yang magnetic monopole for $n_{r}=1$,
$\ell=0,1,2,3$, (a) for $\sigma=0.5$, (b) for $\sigma=1$, and (c) for
$\sigma=1.8$.

where, for the deformation/transformation functions discussed in the preceding
section,

$r\left(\rho\right)=\left\\{\begin{tabular}[]{ll}$\rho\,e^{a\rho^{2}/2}$&for
$q\left(\rho\right)=e^{a\rho^{2}}\Longleftrightarrow$ $f\left(\rho\right)=$
$\left(1+a\rho^{2}\right)e^{a\rho^{2}}\vskip 6.0pt plus 2.0pt minus 2.0pt$\\\
$\sqrt{b}\rho^{1+\upsilon/2}$&for
$q\left(\rho\right)=b\rho^{\upsilon}\Longleftrightarrow
f\left(\rho\right)=\left(1+\upsilon/2\right)^{2}b\rho^{\upsilon}\vskip 6.0pt
plus 2.0pt minus 2.0pt$\end{tabular}\right..$ (42)

One should also notice that equation (40), along with (41), describes KG-
Coulombic particles in the QPGM spacetime of (5) and equation (37) describes
KG-Coulombic particles in the regular PGM spacetime (3).

In Figures 4, 5, and 6, we plot the energy levels Eq. (39) of the KG-Coulombic
particles in a PGM/QPGM spacetime and a WYMM. In Figures 4(a),4(b), and 4(c),
are plotted for $\alpha=1$, $\alpha=0.5$, and $\alpha=0.25$, respectively, to
observe the effect of global monopole parameter $\alpha$. We may, again,
observe that as the PGM parameter decreases from $\alpha=1$ (i.e., Minkowski
flat spacetime) the energy levels tend to cluster around $E=\pm m_{\circ}$.
This is obvious from equation (39) where a Taylor series expansion about
$\alpha=0$ would yield $E_{\alpha\rightarrow
0}=\pm\left[m_{\circ}+O\left(\alpha^{4}\right)\right].$ Moreover, the
separation between energy levels decreases (with a tendency of clustering)
with increasing WYMM parameter $\sigma$.

##  VI KG-Coulomb particles in a PGM spacetime and a WYMM at $S(r)=K/r$,
$\mathcal{F}_{r}(r)=0.$

It has been shown by de Mello and Furtado Re133  that a self-interaction on a
charged particle placed in a PGM spacetime is given by
$S\left(r\right)=K\left(\alpha\right)/r$, where $K\left(\alpha\right)$ is a
positive constant and $r$ is the distance to the PGM. For the sake of
simplicity of notations, let us consider that the Lorentz scalar potential
$S\left(r\right)=K/r$ and $\mathcal{F}_{r}\left(r\right)=0$ in (25) to obtain,
in a straightforward manner, that

$\left\\{\partial_{r}^{2}-\frac{\mathcal{L}\left(\mathcal{L}+1\right)}{r^{2}}-\frac{2B}{r}+\Lambda\right\\}R\left(r\right)=0,$
(43)

where, $B=\tilde{m}\mathcal{\tilde{K}},$ $\mathcal{\tilde{K}}=K/\alpha$ and

$\mathcal{L}\left(\mathcal{L}+1\right)=\tilde{L}\left(\tilde{L}+1\right)+\mathcal{\tilde{K}}^{2}\Longleftrightarrow\mathcal{L=-}\frac{1}{2}+\sqrt{\frac{1}{4}+\tilde{L}\left(\tilde{L}+1\right)+\mathcal{\tilde{K}}^{2}}.$
(44)

This system admits an exact textbook solution in the form of confluent
hypergeometric series and reads

$R\left(r\right)=C\,e^{-i\sqrt{\Lambda}r}\left(2i\sqrt{\Lambda}r\right)^{\mathcal{L}+1}\,\,U\left(\mathcal{L}+1-\frac{iB}{\sqrt{\Lambda}},2\mathcal{L}+2,2i\sqrt{\Lambda}r\right).$
(45)

Again, finiteness and square integrability of the radial function
$R\left(r\right)$ enforces the truncation of the confluent hypergeometric
series of the second kind $U\left(a,b,z\right)$ into a polynomial of order
$n_{r}$ so that

$\mathcal{L}+1-\frac{iB}{\sqrt{\Lambda}}=-n_{r}\Longleftrightarrow
i\sqrt{\Lambda}=-\frac{B}{\tilde{n}}\,\Longleftrightarrow\Lambda=-\frac{B^{2}}{\tilde{n}^{2}},$
(46)

where $\tilde{n}=n_{r}+\mathcal{L}+1$ is an irrational principle quantum
number (which collapses into the usual principle quantum number for $\alpha=1$
and $\sigma=0$). As a result, we get the energies and the radial function
$R\left(r\right)$ of such KG-Coulombic particles in a PGM spacetime and WYMM
as

$E=\pm\sqrt{m_{\circ}^{2}-\frac{\alpha^{2}B^{2}}{\tilde{n}^{2}}}=\pm\sqrt{m_{\circ}^{2}-\frac{m_{\circ}^{2}\,K^{2}}{\alpha^{2}\left(n_{r}+\mathcal{L}+1\right)^{2}}},$
(47)

and

$R\left(r\right)=\tilde{C}\exp(-\frac{Br}{\tilde{n}})\;r^{\mathcal{L}+1}\,\,U\left(-n_{r},2\mathcal{L}+2,2\frac{B}{\tilde{n}}r\right);\;B=\frac{m_{\circ}K}{\alpha^{2}},$
(48)

Figure 7:  The energy levels Eq. (48) of the KG-Coulombic particles in a PGM
spacetime and a Wu-Yang magnetic monopole for $n_{r}=1$, $\ell=0,1,2,3$, (a)
for $\alpha=1$, (b) for $\alpha=0.5$, and (c) for $\alpha=0.25$.

respectively, where $\mathcal{L}$ is given by Eq.(44). At this point, one
should observe that this result is in exact accord with that reported by de
Oliveira and de Mello Re8  (i.e., their result in Eq. (76) with their
$\alpha^{4}$ typo corrected into $\alpha^{2}$ as should be the case following
their procedure). We plot the effect of the PGM spacetime on the spectroscopic
structure of such a KG-Coulomb particle in Figure 7. We clearly observe that
as the PGM parameter decreases from $\alpha=1$ (i.e., Minkowski flat
spacetime) the separation between the energy levels increases (which is, in
fact, exactly an opposite effect of the PGM/QPGM spacetime of Figure 4) This
is obvious from the comparison between equation (39) and (48).

##  VII Shifted KG-oscillators in a PGM spacetime and a WYMM at
$\mathcal{F}_{r}\left(r\right)=0$ and $S\left(r\right)=\beta r$

A Lorentz scalar interaction potential $S\left(r\right)=\beta r$ is commonly
used in the study of mass spectroscopy of some quark-antiquark systems (cf.
e.g., Gunion and Li Re028 ). When this potential is substituted in Eq.(25) it
implies that

$\left\\{\tilde{E}^{2}+\partial_{r}^{2}-\frac{\tilde{L}\left(\tilde{L}+1\right)}{r^{2}}-\tilde{\beta}^{2}\left(\frac{m_{\circ}}{\beta}+r\right)^{2}\right\\}R\left(r\right)=0;\;\tilde{\beta}=\frac{\beta}{\alpha}.$
(49)

In this section we shall be interested in the case where

$\tilde{L}=0\Longleftrightarrow\sigma^{2}=\left(eg\right)^{2}=\ell\left(\ell+1\right)+2\zeta\left(1-\alpha^{2}\right).$
(50)

This would identify a critical correlation between the Wu-Yang monopole
strength $g$ in $\left(eg\right)^{2}//q\left(\rho\right)\,\rho^{2}$and the
scalar curvature
$R=R_{\nu}^{\nu}=2\left(1-\alpha^{2}\right)/q\left(\rho\right)\,\rho^{2}$ in
$2\zeta\left(1-\alpha^{2}\right)/q\left(\rho\right)\,\rho^{2}$, so that the
central repulsive core
$\tilde{L}\left(\tilde{L}+1\right)/q\left(\rho\right)\,\rho^{2}=\tilde{L}\left(\tilde{L}+1\right)/r^{2}$
is removed. Which would, in turn, allow the wave functions of the shifted KG-
oscillators to be finite and square integrable at the center of the PGM (i.e.,
at $r=0=\rho$). Under such settings, the Shifted KG-oscillators are described
by

$\left\\{\tilde{E}^{2}+\partial_{r}^{2}-\tilde{\beta}^{2}\left(\frac{m_{\circ}}{\beta}+r\right)^{2}\right\\}R\left(r\right)=0.$
(51)

We may now use the change of variable $z=$ $r+m_{\circ}/\beta$ and recast this
equation as

$\left\\{\tilde{E}^{2}+\partial_{z}^{2}-\tilde{\beta}^{2}z^{2}\right\\}R\left(z\right)=0,$
(52)

to imply

$R\left(r\right)=N\,{}{}\left(\frac{m_{\circ}}{\beta}+r\right)\,\exp\left(-\frac{\tilde{\beta}\left(\frac{m_{\circ}}{\beta}+r\right)^{2}}{2}\right){}\,M\left(\frac{3}{4}+\frac{\tilde{E}^{2}}{4\tilde{\beta}},\frac{3}{2},\tilde{\beta}\left(\frac{m_{\circ}}{\beta}+r\right)^{2}\right).$
(53)

We truncate the the confluent hypergeometric series $M\left(a,b,z\right)$ into
a polynomial of order $n_{r}$ so that

$\frac{3}{4}+\frac{\tilde{E}^{2}}{4\tilde{\beta}}=-n_{r},$ (54)

to obtain the shifted KG-oscillators energies as

$E=\pm\sqrt{2\alpha\beta\left(2n_{r}+\frac{3}{2}\right)}$ (55)

and the radial wavefunctions

$R\left(r\right)=\tilde{N}\,{}{}\left(m_{\circ}+\beta
r\right)\,\exp\left(-\frac{\alpha\left(m_{\circ}+\beta
r\right)^{2}}{2}\right){}M\left(-n_{r},\frac{3}{2},\alpha\left(m_{\circ}+\beta
r\right)^{2}\right).$ (56)

Evidently, at the critical values for the Wu-Yang magnetic monopole strength
$\sigma=eg$ given in Eq.(50) have made all $\ell$-states to effectively
cluster on $\ell=0$ states for a given radial quantum number $n_{r}$. That is,
all non-zero $\ell$-states have disappear from the spectra as is obvious from
Eq.(55). For example, for $\ell=0$ states (i.e., $S$-states) in (50) we have
$\sigma^{2}=\left(eg\right)^{2}=2\zeta\left(1-\alpha^{2}\right)$ that
identifies a critical correlation between the WYMM and the scalar curvature
$R$ and/or the PGM parameter $\alpha$ through
$2\zeta\left(1-\alpha^{2}\right)$. For $\ell=1$ states (i.e., $P$-states) we
have $\sigma^{2}=\left(eg\right)^{2}=2+2\zeta\left(1-\alpha^{2}\right)$ to
identify yet another correlation of the same kind, and so on so forth.

##  VIII Concluding remarks

In current study, we have used two correlated metric functions (correlated
deformation/transformation functions, if you wish) and transformed a PGM
spacetime into a QPGM spacetime, where the WYMM is involved in both cases.
Namely, we have studied KG-particles in a PGM/QPGM spacetimes and a WYMM. We
have shown that the KG-particles in a QPGM spacetime and a WYMM are
isospectral and invariant with KG-particles in the regular PGM spacetime and a
WYMM. This is documented in sections 2 and 3. Based on such findings, we use
the non-minimal coupling form of the operator
$\tilde{D}_{\mu}=D_{\mu}+\mathcal{F}_{\mu}$ (where, $\mathcal{F}_{\mu}$
$\in\mathbb{R}$, and $D_{\mu}=\partial_{\mu}-ieA_{\mu}$ is the gauge-covariant
derivative) of Moshinsky and Szczepaniak Re0281  and Mirza and Mohadesi Re0282
to discuss KG-oscillators in a QPGM/PGM spacetime and a WYMM and report the
effects of both on the spectroscopic structure of the KG-oscillators
(documented in Figures 1, 2, and 3). Two sets of correlated dimensionless
positive-valued metric functions are used in the process (documented in
section 4). Analogously, we show that a Bessel-like functional structure,
$\mathcal{F}_{r}$ in Eq. (36), would yield KG-Coulomb like particles in a
QPGM/PGM spacetime and a WYMM and report/discuss the effects of both monopoles
on their spectroscopic structure (as documented in Figures 4, 5, and 6).
Moreover, in the PGM spacetime and a WYMM, we discuss a KG-Coulomb and a
shifted KG-oscillator as manifestations/byproducts of the Lorentz scalar
potentials $S\left(r\right)=\mathcal{K}/r$ and
$S\left(r\right)=\mathcal{\beta}r$, respectively.

In connection with the KG-oscillators and KG-Coulomb like particles that are
introduced by the non-minimal coupling form of the operator
$\tilde{D}_{\mu}=D_{\mu}+\mathcal{F}_{\mu}$, in sections 4 and 5, we have
observed that as the PGM/QPGM parameter $\alpha$ decreases from $\alpha=1$
value towards $\alpha\sim 0$, the energy levels are destined to cluster on
$E=\pm m_{\circ}$. Whereas, the separations between the energy levels are
observed to increase with increasing WYMM strength $\sigma$ for the KG-
oscillators and decrease with increasing $\sigma$ for the KG-Coulombic
particles. However, for the KG-Coulombic particles introduced by the Lorentz
scalar potential $S(r)$, we have observed that decreasing the PGM parameter
$\alpha$ from $\alpha=1$ would increase the separation between energy levels.

On the PGM spacetime background side, the current methodical proposal suggests
that the quantum mechanical systems in the PGM spacetime

$ds^{2}=-dt^{2}+\frac{1}{\alpha^{2}}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}\right),$
(57)

have infinite number of quantum mechanical systems in the QPGM spacetime

$ds^{2}=-dt^{2}+\frac{f\left(\rho\right)}{\alpha^{2}}d\rho^{2}+q\left(\rho\right)\,\rho^{2}\left[d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}\right],$
(58)

that share the same energy levels, provided that the two metric functions
$f(\rho)$ and $q(\rho)$ are positive- valued dimensionless correlated (through
Eq. (4) ) multipliers. We have, hereinabove, tested this fact for scalar KG
-particles. In a recent work Re29 , we have also tested this for non-
relativistic Schrödinger particles in PGM/QPGM spacetime and have arrived at
the same conclusion. It would be, therefore, interesting to investigate this
fact for Dirac-particles (Dirac-oscillators in particular). Such
investigations should also include other quantum mechanical phenomena like
scattering, thermodynamical effects, etc.

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