At last, the potentials dependencies on the spin-orbit and spin-spin
contributions conserve close form. When $V^{(s=1/2)}_{\textrm{spin-orbit}}(r)$
(3.2c) is compared to $V^{(s=1/2)}_{\textrm{EM,spin-orbit}}(r)$ (3.8c), the
rising of the equivalent terms are observed, which differ in factors of
proportionality but maintain the major arrangement based on the spins
$\vec{S}_{1}$ and $\vec{S}_{2}$ participation. One may extend the later
analysis to the spin-spin sector, confirming the similarity of the eqs. (3.2d)
and (3.8d). Noting that, for the gravitational scenario, only the term
proportional to $\mathcal{P}_{\mu\nu\alpha\beta}^{(2)}$ contributes to this
sector, remembering it is associated to $Q^{(2)}(q^{2})$ in the propagator
(2.21) and the same $Q^{(2)}$ is part of the integral $I_{0}^{(2)}(r)$.

###  3.3 Form Factors Motivated by Quantum Gravity Model

At this section the developed theory is submitted to analyses when applied to
a model. The central reference that supports this model is an article by Knorr
and Saueressig [5], where the authors, through a non-pertubative method,
established a reverse engineering procedure to determine a quantum effective
action for gravity under low-energy conditions.

The matching-template formalism, as named by the authors, begun with an
effective action in terms of parameters, that, in their case, is composed by
one local and one non-local parts, respectively,

$\displaystyle\Gamma^{\textmd{local}}=\frac{1}{16\pi G_{N}}\int
d^{4}x\,\sqrt{-g}\,\big{(}2\Lambda-R\big{)},$ (3.9a)

$\displaystyle\Gamma^{\textmd{NL}}=\frac{-1}{96\pi G_{N}}\int
d^{4}x\,\sqrt{-g}\,\big{(}b^{2}\,R\,\Box^{-2}R+\tilde{b}^{2}\,C_{\mu\nu\alpha\beta}\,\Box^{-2}C^{\mu\nu\alpha\beta}\big{)},$
(3.9b)

with $R$ being the regular Riemannian curvature scalar (see, for instance,
[229]) and $C_{\mu\nu\alpha\beta}$ the Weyl tensor, eq. (2.19). The criteria
to definition of the (non)local parts were the construction of an effective
quantum action for gravity and make a connection with cosmological scales, in
a way that the authors required the Einstein-Hilbert action and
diffeomorphism-invariant contributions quadratic in the curvature tensors and
presence of two inverse powers of the Laplacian $\Box^{-2}$ (keeping in mind
that the spacetime coordinate system is the Euclidean one). The Ricci tensor
$R_{\mu\nu}$ was absent, once is possible to write it in terms of the
curvature scalar and the Weyl tensor. For the determination of the parameters
$b$, $\tilde{b}$, $G_{N}$ and $\Lambda$, they worked with a methodology based
on foliation structure [234, 235]. This demanded the calculation of two-point
autocorrelation functions derived from fluctuations of 3-volume around a
toroidal geometry background [236, 237]. The algebra indicated no contribution
of the mass-type parameter $\tilde{b}$, leaving only $b$. By matching the
analytical approach with the lattice data from Monte Carlo simulation within
Causal Dynamics Triangulation (CDT) (see the ref. [238] for a wide review),
the authors could reverse engineering the couplings through an establishment
of a bind between continuum and lattice. The final non-local gravitational
model

$\displaystyle\Gamma=\frac{2}{\kappa^{2}}\int
d^{4}x\,\sqrt{-g}\Big{(}2\Lambda-R-\frac{b^{2}}{6}R\,\Box^{-2}R\Big{)}\,$
(3.10)

is the central part of the Maggiore-Mancarella cosmological model [239] (for a
review of extended gravity models for cosmological scales involving non-local
terms, see refs. [240, 241]), in which the authors studied a case with
undefined number of spatial dimensions, hence, achieved an automatically dark
energy manifestation and self-accelerating background evolution. In ref. [242]
the authors studied a very similar non-local gravity model focusing in low
energy cosmological dynamics, but altering the Einstein equation
$G_{\mu\nu}-m^{2}\tilde{g_{\mu\nu}}=8\pi GT_{\mu\nu}$, where
$\tilde{g_{\mu\nu}}\approx g_{\mu\nu}\Box^{-1}R$. The inclusion of the non-
local term with a mass parameter $m$ generates a dynamical dark energy
component, which reproduces the observed dark energy density with no
introduction of the cosmological constant. According to the ref. [243], the
terms $R\,\Box^{-2}R$ and
$C_{\mu\nu\alpha\beta}\Box^{-2}C^{\mu\nu\alpha\beta}$ appeared due to the
decoupling phenomena in a renormalization group analysis. The Weyl tensor was
also considered for cosmological perturbation and background evolution levels
in non-local gravity in the ref. [244], indeed, performing an action in the
mold of eq. (2.18).

For completeness of the present study, the Weyl tensor term is also
considered. Putting in practice the form factors methodology, one takes the
sum of the effective actions $\Gamma^{\textrm{local}}$ and
$\Gamma^{\textrm{NL}}$, eqs. (3.3), as the guide to the analyses and defines

$\displaystyle F(\Box)=-\frac{\rho_{0}}{\Box^{2}}\,\qquad\textmd{and}\qquad
W(\Box)=-\frac{\rho_{2}}{\Box^{2}}\,,$ (3.11)

adopting $\rho_{0}$ and $\rho_{2}$ as positive parameters motivated by $b^{2}$
and $\tilde{b}^{2}$ from eq. (3.9b) and imposing any restriction on them. It
seems worthy to clarify that the reconstruction program content in ref. [5] is
apart from the current example and the main contribution to it, besides the
precious research delivered to the community and enlargement of its knowledge,
is to supply with an effective action applicable (and not a toy model) to
reproduce observed data. One more note should be done, the effective actions
eqs. (3.3) and (3.10) are structured on Euclidean signature, and a
transformation to Lorentzian one demands a Wick rotation, e.g. ref. [245],
what is not addressed inside this work. From the form factors (3.11) are
determined the integrals $I_{0}^{(s)}(r)$, $I_{1}^{(s)}(r)$ and
$I_{ij}^{(s)}(r)$ participants of the potentials eqs. (2.29) and (2.39)

$\displaystyle
I^{(s)}_{1}(r)=\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}\,\frac{1}{\vec{q}^{\,2}+\mu_{s}^{2}}\,e^{i\vec{q}\cdot\vec{r}}\,=\frac{e^{-\mu_{s}r}}{4\pi
r},$ (3.12a)

$\displaystyle
I^{(s)}_{0}(r)=\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}\,\frac{\vec{q}^{\,2}}{\vec{q}^{\,2}+\mu_{s}^{2}}\,e^{i\vec{q}\cdot\vec{r}}=\delta^{3}(\vec{r})-\mu_{s}^{2}\,\frac{e^{-\mu_{s}r}}{4\pi
r},$ (3.12b)

$\displaystyle I_{ij}^{(s)}(r)$
$\displaystyle=\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}\,\frac{\vec{q}_{i}\vec{q}_{j}}{\vec{q}^{\,2}+\mu_{s}^{2}}\,e^{i\vec{q}\cdot\vec{r}}$
$\displaystyle=\frac{1}{3}\delta_{ij}\delta^{3}(\vec{r})+\bigg{[}(1+\mu_{s}r)\delta_{ij}-(3+3\mu_{s}r+\mu_{s}^{2}r^{2})\frac{x_{i}x_{j}}{r^{2}}\bigg{]}\frac{e^{-\mu_{s}r}}{4\pi
r^{3}}\,,$ (3.12c)

where is defined $\mu_{s}^{2}=2(\rho_{s}-\Lambda)$ and $(\rho_{s}-\Lambda)>0$,
what assures that the non-local form factors introduce mass-type terms in the
graviton propagator. At last, the particular gravitational potentials, derived
from the integrals above (eqs. (3.3)), are absent of Dirac deltas. Firstly, it
is presented the case for spin-0 external particles:

$\bullet$ Monopole-monopole sector

$\displaystyle V_{\textmd{mon-mon}}^{(s=0)}(r)=$
$\displaystyle-\frac{\kappa^{2}m_{1}m_{2}}{24\pi\,r}\Bigg{[}\left(e^{-\mu_{2}r}-\frac{1}{4}e^{-\mu_{0}r}\right)$
$\displaystyle-\frac{1}{8}\bigg{(}\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\bigg{)}\left(\mu_{2}^{2}\,e^{-\mu_{2}r}+\frac{5}{4}\mu_{0}^{2}\,e^{-\mu_{0}r}\right)\Bigg{]}\,.$
(3.13a)

$\bullet$ Velocity-velocity sector

$\displaystyle V_{\textrm{vel-
vel}}^{(s=0)}(r)=-\frac{\kappa^{2}m_{1}m_{2}}{24\pi
r}\,\vec{p}\,^{2}\Bigg{[}\bigg{(}\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\bigg{)}\left(e^{-\mu_{2}r}+\frac{1}{8}e^{-\mu_{0}r}\right)+\frac{3}{m_{1}m_{2}}e^{-\mu_{2}r}\Bigg{]}\,.$
(3.13b)

In the sequence, for spin-1/2 external particles:

$\bullet$ Monopole-monopole sector

$\displaystyle V_{\textmd{mon-
mon}}^{(s=1/2)}(r)=-\frac{\kappa^{2}m_{1}m_{2}}{24\pi\,r}\left(e^{-\mu_{2}r}-\frac{1}{4}e^{-\mu_{0}r}\right)\,.$
(3.14a)

$\bullet$ Velocity-velocity sector

$\displaystyle V_{\textrm{vel-
vel}}^{(s=1/2)}(r)=-\frac{\kappa^{2}m_{1}m_{2}}{24\pi
r}\,\vec{p}\,^{2}\Bigg{[}\bigg{(}\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\bigg{)}\left(e^{-\mu_{2}r}+\frac{1}{8}e^{-\mu_{0}r}\right)+\frac{3}{m_{1}m_{2}}e^{-\mu_{2}r}\Bigg{]}\,.$
(3.14b)

$\bullet$ Spin-orbit sector

$\displaystyle V_{\textrm{spin-orbit}}^{(s=1/2)}(r)$
$\displaystyle=\frac{\kappa^{2}m_{1}m_{2}}{24\pi
r^{3}}\Bigg{[}\bigg{(}\frac{1}{m_{1}^{2}}\vec{S}_{1}\cdot\vec{L}+\frac{1}{m_{2}^{2}}\vec{S}_{2}\cdot\vec{L}+\frac{3(\vec{S}_{1}+\vec{S}_{2})\cdot\vec{L}}{2\,m_{1}m_{2}}\bigg{)}\,(1+r\mu_{2})e^{-\mu_{2}r}$
$\displaystyle\,+\frac{1}{8}\bigg{(}\frac{1}{m_{1}^{2}}\vec{S}_{1}\cdot\vec{L}+\frac{1}{m_{2}^{2}}\vec{S}_{2}\cdot\vec{L}\bigg{)}\,(1+r\mu_{0})e^{-\mu_{0}r}\Bigg{]}\,.$
(3.14c)

$\bullet$ Spin-spin sector

$\displaystyle V_{\textrm{spin-spin}}^{(s=1/2)}(r)=$
$\displaystyle-\frac{\kappa^{2}m_{1}m_{2}}{32\pi\,r^{3}}\Bigg{[}\frac{\vec{S}_{1}\cdot\vec{S}_{2}}{m_{1}m_{2}}\,(1+r\mu_{2}+r^{2}\mu_{2}^{2})e^{-\mu_{2}r}$
$\displaystyle-3\frac{(\hat{r}\cdot\vec{S}_{1})\,(\hat{r}\cdot\vec{S}_{2})}{m_{1}m_{2}}\,(1+r\mu_{2}+\frac{1}{3}r^{2}\mu_{2}^{2})e^{-\mu_{2}r}\Bigg{]}\,.$
(3.14d)

The equations are accompanied by exponential factors that generate a damping
effect on the potentials sectors in general. This is consequence of the mass-
type term presents in the graviton propagator. Concentrating on the respective
equations (3.13a) and (3.14a) for $V_{\textmd{mon-mon}}^{(s=0)}(r)$ and
$V_{\textmd{mon-mon}}^{(s=1/2)}(r)$, one observes that they are proportional
to $r^{-1}$, what reaffirms the presence of the universal term. Furthermore,
both monopole sectors – for spin-0 and -1/2 – carry the static limit potential
(see eqs. (3.3) and (3.4) to review the definitions and result), nonetheless,
the spin-0 expression still contains an extra component, as already commented,
in a manner that the latter referred potential is written as

$\displaystyle V_{\textmd{mon-
mon}}^{(s=0)}(r)=V_{\textmd{stat.}}^{(s=0)}(r)+\Delta V_{\textmd{mon-
mon}}^{(s=0)}(r)\,,$ (3.15a)

where

$\displaystyle
V_{\textmd{stat.}}^{(s=0)}(r)=-\frac{\kappa^{2}m_{1}m_{2}}{24\pi\,r}\left(e^{-\mu_{2}r}-\frac{1}{4}\,e^{-\mu_{0}r}\right)$
(3.15b)

and

$\displaystyle\Delta V_{\textmd{mon-
mon}}^{(s=0)}(r)=\frac{\kappa^{2}m_{1}m_{2}}{192\pi\,r}\bigg{(}\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\bigg{)}\left(\mu_{2}^{2}\,e^{-\mu_{2}r}+\frac{5}{4}\mu_{0}^{2}\,e^{-\mu_{0}r}\right)\,,$
(3.15c)

in which the suppressed behaviour of $\Delta V_{\textmd{mon-mon}}^{(s=0)}(r)$
is passive of being demonstrated.

Assuming the Newtonian potential is a suitable theory to describe interactions
within the bound of the Solar System perimeter, being aware that the
experimental data evidence points out that the Newtonian theory, for large
distances, presents deviation from the observed data within the galaxy and
extra-galaxy scales [246, 247, 248, 249]. To define the Solar System range or
area is not a trivial task, however, considering, in rough way, its radius
($r_{s}$) as the semi-major axis of the Pluto’s orbit, one finds approximately
39 A.U. [250]. Rescuing the Pioneer anomaly [251, 252] that registered from
Pioneer 10 and 11 spacecrafts the presence of a small anomalous blue-shifted
frequency among the distances of 20 to 70 A.U. from the Sun and kept an
unsolved question, it is adopted the distance $r_{s}\approx$ 10 A.U., whereby
the Newtonian physics is assumed valid. Since is provided that
$\mu_{i}r_{s}\ll 1$, then $\mu_{i}\ll 10^{-25}\textrm{MeV}$, what incurs in
$(\mu_{i}^{2}/m_{j}^{2})\rightarrow{0}\,$ – even to particle masses (order
$\sim\textrm{MeV}$) – and vanishes the spin-0 extra term in monopole potential
sector, recovering the Newtonian one.

The velocity sector keeps the same $r$-dependency order ($r^{-1}$) as the
monopole one. As already argued, under the static limit scenario, where
$\,\vec{p}\rightarrow{\vec{0}}$ and $\,m_{i}\rightarrow{\infty}$, this sector
is suppressed in both types of particles. The same happens upon the non-
relativistic limit, once the characteristic factor of the sector,
$\,\vec{p}^{\,\,2}/m_{i}m_{j}$, tends to a value far smaller than the unity,
what practically annuls the velocity sector influence in these gravitational
potentials.

The $r$-dependency assumes different orders in the spin-orbit and spin-spin
sectors. In the spin-orbit (3.14c) there are terms in $r^{-1}$ and $r^{-2}$,
noticing that the angular momentum $\vec{L}$ is proportional to $r$. In its
turn, $V_{\textmd{spin-spin}}^{(s=1/2)}(r)$ contains orders from $r^{-1}$ to
$r^{-3}$. In these two cases, the $r^{-1}$ terms are suppressed by the same
reasons that leaded the extra monopole term $\Delta V_{\textmd{mon-
mon}}^{(s=0)}(r)$ to be suppressed, namely,
$(\mu_{i}^{2}/m_{j}^{2})\rightarrow{0}\,$.

Therefore, it is notable the predominance of $V_{\textmd{mon-
mon}}^{(s)}(r)\sim r^{-1}$, restricted to the static one
$V_{\textmd{stat.}}^{(s=0)}(r)$ in the spin-0 case (see eqs. (3.3)), over the
others in the long-range assumption. Things are quite different when the
interactions of short-range distances are analysed. Firstly, taking the spin-0
potential, one observes that

$\bullet$ Monopole-monopole sector

$\displaystyle V_{\textmd{mon-mon}}^{(s=0)}(r)\sim$
$\displaystyle-\frac{\kappa^{2}m_{1}m_{2}}{32\pi\,r}\bigg{(}1-\sum_{i=0,2}\,\sum_{j=1,2}c_{ij}\frac{\mu_{i}^{2}}{m_{j}^{2}}\bigg{)}\,.$
(3.16a)

$\bullet$ Velocity-velocity sector

$\displaystyle V_{\textrm{vel-
vel}}^{(s=0)}(r)\sim-\frac{\kappa^{2}m_{1}m_{2}}{32\pi
r}\,\vec{p\,\,}^{2}\sum_{i,j=1,2}\frac{c_{ij}}{m_{i}m_{j}}\,.$ (3.16b)

and, then, the gravitational potential for spin-1/2 externals particles

$\bullet$ Monopole-monopole sector

$\displaystyle V_{\textmd{mon-
mon}}^{(s=1/2)}(r)\sim-\frac{\kappa^{2}m_{1}m_{2}}{32\pi\,r}\,.$ (3.17a)

$\bullet$ Velocity-velocity sector

$\displaystyle V_{\textrm{vel-
vel}}^{(s=1/2)}(r)\sim-\frac{\kappa^{2}m_{1}m_{2}}{32\pi
r}\,\vec{p\,\,}^{2}\sum_{i,j=1,2}\frac{c_{ij}}{m_{i}m_{j}}\,.$ (3.17b)

$\bullet$ Spin-orbit sector

$\displaystyle V_{\textrm{spin-orbit}}^{(s=1/2)}(r)$
$\displaystyle\sim\frac{\kappa^{2}m_{1}m_{2}}{32\pi
r^{3}}\sum_{i,j=1,2}\bigg{(}\frac{c_{ij}}{m_{i}m_{j}}\vec{S}_{i}\cdot\vec{L}\bigg{)}\,.$
(3.17c)

$\bullet$ Spin-spin sector

$\displaystyle V_{\textrm{spin-spin}}^{(s=1/2)}(r)\sim$
$\displaystyle-\frac{\kappa^{2}m_{1}m_{2}}{32\pi\,r^{3}}\Bigg{(}\frac{\vec{S}_{1}\cdot\vec{S}_{2}}{m_{1}m_{2}}\Bigg{)}\,.$
(3.17d)

in which, $c_{ij}$ stands for particular constants to each one of the eqs.
(3.3) and (3.3). The potentials lose their damping effect, which tends to the
unity as $r$ goes to zero. The expressions presented above (eqs. (3.3) and
(3.3)) are the major contributions for the gravitational potentials in both
external particles under short-range restrictions. In spin-0 situation, the
monopole sector, eq. (3.16a), still holds as the predominant term, since the
velocity part, eq. (3.16b), is suppressed in the non-relativistic limit.

The behaviour of the potential in the static limit (eq. (3.3)) is straight
forward. The condition $\mu_{i}r_{s}\ll 1$ is guaranteed because $r$ already
tends to zero. Considering the static limit premise
$m_{j}\rightarrow{\infty}\,$, one has $(\mu_{i}^{2}/m_{j}^{2})\ll 1\,$, what
rises the static potential
$V_{\textrm{stat.}}^{(s=0)}(r)=-\frac{\kappa^{2}m_{1}m_{2}}{32\pi r}\,$. In
contrast, taking a look at the spin-1/2 potential equations (3.3), one notices
an inversion of the relevance among the sectors when compared to long-range
results manifested in eqs. (3.3). The spin interaction potential, expressed in
eq. (3.17d), assumes the leading contribution due to its order $r^{-3}$,
followed by $V_{\textrm{spin-orbit}}^{(s=1/2)}(r)$ (eq. (3.17c)), which
carries an order $r^{-2}$. The monopole-monopole term equation (3.17a) tends
naturally to the Newtonian potential, but occupying a suppressed position in
the gravitational potential under analysis. In the short-range panorama, one
concludes that the leading gravitational potential terms for spin-0 and
spin-1/2 external particles are absent of the form factors described in eq.
(3.11). This finding may express the infrared nature of them.

###  3.4 Partial Conclusion

The analysis aims a quantum linearized gravity model, sustained by effective
field theory, structured with form factors associated to squared curvature
terms, from which is obtained quantum corrections to non-relativistic inter-
particle potentials involving scattering of scalars and spinors. The form
factors are capable of revealing the relevant information attributed to the
graviton propagator in a framework of metric perturbative fluctuations around
a flat background. The work contemplates tree level vertices and graviton
propagator, for which is obtained quantum corrections. The research is passive
of further investigations including loop corrections.

The comparison of spin-0 and $-\frac{1}{2}$ potential sectors, points the
presence of the universal potential contribution in both monopole-monopole
interactions (eqs. (3.1a) and (3.2a)) and a presence of extra sub-leading
correction for the scalar one. The velocity-velocity sectors (3.1b) and (3.2b)
are identical for both particles. The spin-orbit one (3.2c) manifests
contributions of both form factors $F(\Box)$ and $W(\Box)$, while the spin-
spin (3.2d) only from $W(\Box)$. In spite of the particularities for each
sector, it is observed that, in the static limit (eq. (3.3)), both
gravitational potentials (for spin-0 and $-\frac{1}{2}$ particles) converge to
$V_{mon-mon}^{(s=1/2)}(r)$ (eqs. (3.2a) and (3.4)).

An investigation confronting the potentials results of the currently
gravitational system with the modified effective electrodynamics [3] unveils
similar structures in each sector contribution of the spin-0 and -1/2 inter-
particles potentials.

The results are submitted to a specific model, in order to evaluate its
performance. The model is grounded in a non-perturbative method combined to
Casual Dynamics Triangulation to establish a quantum gravity effective action,
which subsidizes the information to determine the form factors. The non-
relativistic potentials for spin-0 and -1/2 particles in long-range regime
present exponential formulation $\sim e^{-\mu r}$ characterizing the damping
effect and the universal term leading the contribution for both scenarios. In
short-range distances the damping effect disappears, anyway the major spin-0
contribution keep coming from the universal term in monopole sector. To
spin-1/2, the spin-spin sector assumes the main contribution, once it is
proportional to $1/r^{3}$ and the monopole-monopole becomes a suppressed
correction.

Besides the possibility of expanding the calculations of the quantum
corrections present in this study beyond the tree-level propagator and
vertices, there is also the opportunity of developing the spin-1 inter-
particle potential in the non-relativistic scattering process exchanging a
graviton. In ref. [211], the authors achieved contributions of velocity, spin
and involving quadrupole nature for massive spin-1 scattered particle, what
endorses the motivation to explore this theme in the future.

##  Chapter 4 Modified Maxwell-Higgs Model in an Effective Scenario
Work in Progress

###  4.1 Introduction

Since the phenomenological formulation of superconductivity by Landau and
Ginzburg in 1950 [18] and its extension for Type-II superconductors by
Abrikosov [26], the investigation on topological systems presenting vorticity
has been increased and reached several areas of physics, specially in
condensate matter, particles physics and cosmology. In the begin of the 70’s,
Nielsen and Olesen [253] generalized the Landau-Ginzburg-Abrikosov description
through a relativistic Abelian Maxwell-Higgs model revealing a quantized
magnetic flux electrically neutral and obtained approximate solutions for the
equations of motion. Three years later, Bogomol’nyi [254] introduced an
algebraic manipulation that, stipulating some specific values to the potential
coupling constant – the Bogomol’nyi-Prasad-Somerfield (BPS) limit [255]–, was
capable of converting the second order differential equations of the system in
first ones. Along the history of physics, planar topological defects have been
unfolded in a large variety of research branches. Some of these branches
widely developed are the fractional statistics and anyons (see, for instance,
[39, 40, 256, 257]). In condensed matter systems are observed giant vortices
[258, 259, 260]. Supersymmetric models [261] and cosmic strings [262] integer
prominent branches as well, while topological arrangements are directly
related to the early Universe period during phases transitions [263].
Topological theories involving different dimensionalities with unusual kinetic
terms are explored in the context of inflationary phase of the Universe [264,
265, 266], dark [267, 268] and tachyonic matter [269], gravitational waves
[270], alternatives proposals for Yang-Mills theory in infrared regime [271],
strongly interacting particles [272], solitons mapped by Hopf index [273, 274,
275] and defects in D-dimensional systems [276].

The vortex description has multi faces. Diversified formulations structuring
models to describe the phenomenology of topological defects has been developed
since Nielsen and Olesen in 1973. The Maxwell-Higgs model in Abelian version
was exactly solved [277], adopted for two votices interaction [278], extended
to generate the Chern-Simons term from spontaneous symmetry breaking [153],
altered to include non-conventional kinetic terms (see, for example, [279,
280] and references therein), added Born-Infeld non-linear Electrodynamics
(see [281] and references therein), generalized such that presented compact
vortices [282] and non-minimally coupled in non-Abelian version [283]. Models
formed just by Higgs for two-component superconductors [284] and associated to
Born-Infeld [285] are derivations from the 70’s too.

In the manufacturing process of construction of models, the Chern-Simons term
stands out due to the possibility of obtaining vortex solutions in Lagrangian
possessing its presence summed to the Higgs (kinetic and potential scalar
terms). These vortex solutions carry magnetic flux and, contrarily to the
Nielsen-Olesen solution, electric charge. In this way, theorists have been
explored models in different configurations with the Chern-Simons term. One of
them is the pure Chern-Simons theory [286], with multivortex solution [287],
supporting topological and non-topological solitons solutions [288], there is
also non-relativistic model [289] and non-minimally coupled in Abelian [161]
and non-Abelian [290] theory generating non-topological magnetic flux. Other
configurations investigated are the Maxwell-Chern-Simons-Higgs models [291],
with addition of two neutral fields [292, 293] to enable the development of
Bogomol’nyi prescription [254], in non-Abelian theory [294, 295], carrying
non-minimal coupling (see refs. [296, 297, 298] and references therein) and
structured in generalized models with unconventional kinetic term [299] and
modified Maxwell term [300, 301]. There are researches segments involving
supersymmetry [149], sigma model (see, for instance, refs. [302, 303, 304]),
Lorentz symmetry violation [305] (see also references therein from paper 21 to
24), fermionic theory [306, 307] and diversified geometry [308] (see
references therein).

The present work aims to analyse an Abelian Maxwell-Higgs model, minimally
coupled and involved by a strong electromagnetic background field. A cosmic
star like a neutron star (pulsars or magnetars) could assume the background
field source role, perturbing the vacuum structure, spontaneously breaking its
symmetry. This perturbation is manifested in the Lagrangian through a
modification in the (Minkowski) metric of the kinetic scalar term, similarly
practiced in the gravitational theory in eq. (2.1), originating a new metric
which absorbs the field strength representing the Maxwell Electrodynamics of
the background field. It is taken into account the scenario with the new
metric constant in relation to space-time derivatives. The Bogomol’nyi [254]
equations are determined in the energy ground state condition and vortex
solutions are implemented, what leads to quantized magnetic flux coming from
the propagating electromagnetic wave. The chapter starts with the Lagrangian
containing the constant metric, in a matter that firstly is developed the
field equations and achieved the energy (density) of the system (Sec. 4.2.1).
Then, in subsection 4.2.2, the energy equation is diagonalized, the
Bogomol’nyi equations are calculated and the Bogomol’nyi limits obtained. In
the sequence 4.2.3, the scalar and vector fields are achieved, where is
analysed their structure in terms of quantization, anisotropy and continuity.
The charge of the system is also found. One last analysis around the energy in
the lower bound is executed, where, through the equations system formed by the
self-dual type expressions, are formulated the gauge potential $a(r)$ and the
Higgs potential. The partial conclusions in Sec. 4.3 close the chapter.

###  4.2 Maxwell-Higgs Model with a Strong Electromagnetic Background Field

####  4.2.1 The Model and Field Equations

The first model proposed is the Maxwell-Higgs Lagrangian (density), similar to
the one studied, for example, in refs. [253, 277], containing the
particularity of modified metric in relation to the kinetic scalar term, what
could classify it as theory of modified kinetic term as pointed above (refs.
[279, 299] and references therein). The Maxwell term representing the
background field responds for the modification in the Minkowski metric,
equivalently to practice in linearized gravitation (eq. (2.1)), inciting the
interpretation of the background field as perturbation of elevate intensity in
the system. As intensity as capable of perturbing the propagating
electromagnetic wave and the vacuum and promoting alteration in the physical
properties of the propagating wave as frequency, incurring in birefringence or
some anisotropy, as well as collapsing the vacuum through a spontaneous
symmetry breaking, what could alter the masses of the gauge field. Such a
strong field occurs at the proximity of neutron stars, which could suggest
a(n) (ultra)superconducting behavior and topological defects formation.

The conventions adopted are: $\eta_{\mu\nu}=(+,-,-)$,
$x_{\mu}=(x_{0},x_{i})=(t,-\vec{x})$ and natural unities $c=\hbar=1$,
observing that latin index means spatial components and greek space-time ones.
Along the text, $\varphi$ stands for a charged complex scalar field,
$f_{\mu\nu}$ is the usual field strength for the gauge field
$a_{\mu}=(\phi,-\vec{a})$,
$f_{\mu\nu}=\partial_{\mu}\,a_{\nu}-\partial_{\nu}\,a_{\mu}$, and the
covariant derivative associated to the minimal coupling is identified by
$D_{\mu}$ where $D_{\mu}\varphi=(\partial_{\mu}+ig\,a_{\mu})\varphi$ with $g$
being a constant. $G_{\mu\nu}$ is a symmetric tensor playing the role of
effective modified Minkowski metric

$G_{\mu\nu}=\eta_{\mu\nu}+\zeta F_{\mu\kappa}F^{\kappa}_{\,\,\,\nu}\,\,,$
(4.1)

where the modification is due to the presence of $\zeta
F_{\mu\kappa}F^{\kappa}_{\,\,\,\nu}$, which represents the field strength for
an external strong electromagnetic background field, and $\zeta$ a constant.
This term carries no mass dimension, following $\eta_{\mu\nu}$. In first
moment, $G_{\mu\nu}$ is considered to be space-time constant. At last, the
Higgs potential $U$ is left unspecified. The first Maxwell-Higgs model
evaluated is

$\displaystyle\mathcal{L}=-\frac{1}{4}f_{\mu\nu}^{\,2}+G_{\mu\nu}D^{\mu}\varphi^{*}D^{\nu}\varphi-U\,.$
(4.2)

The Lagrangian conserves the gauge invariance in relation to the
transformations

$\displaystyle\varphi(x)\rightarrow\varphi(x)e^{i\Lambda(x)}\,,\,\varphi^{*}(x)\rightarrow\varphi^{*}(x)e^{-i\Lambda(x)}\,,\,a_{\mu}(x)\rightarrow
a_{\mu}(x)-\frac{1}{g}\partial_{\mu}\Lambda(x)\,.$ (4.3)

The field equations that arise from (4.2) are

$\displaystyle ig\,G_{\mu}^{\,\,\,\,\nu}\,(\varphi
D^{\mu}\varphi^{*}-\varphi^{*}D^{\mu}\varphi)+\partial_{\mu}f^{\mu\nu}=0\,,$
(4.4a)
$\displaystyle\,-G_{\mu\nu}\,D^{\mu}D^{\nu}\varphi=\delta_{\varphi^{*}}U\,,$
(4.4b)
$\displaystyle\,-G_{\mu\nu}\,D^{\mu}D^{\nu}\varphi^{*}=\delta_{\varphi}U\,,$
(4.4c)

noticing that $\delta_{\varphi^{(*)}}U$ denotes the functional variational
principle applied on the Higgs potential in relation to the scalar field
$\varphi^{*}$ in (4.4b) and $\varphi$ in (4.4c).

####  4.2.2 Energy and Bogomol’nyi Type Equations

Moving toward the energy-momentum tensor $\Theta^{\mu}_{\,\,\,\nu}$, it
presents the structure

$\displaystyle\Theta^{\mu}_{\,\,\,\,\nu}=G^{\,\mu\kappa}(D_{\kappa}\varphi^{*}D_{\nu}\varphi+D_{\kappa}\varphi
D_{\nu}\varphi^{*})-f^{\mu\kappa}f_{\nu\kappa}-\eta^{\mu}_{\,\,\,\nu}\mathcal{L}\,.$
(4.5)

The component $\Theta^{0}_{\,\,\,0}$ is passive of being algebraically
manipulated. The energy assumes the equation

$\displaystyle\mathcal{E}$ $\displaystyle=$ $\displaystyle\int
d^{\,2}x(1+\zeta E_{i}^{\,2})D_{0}\varphi^{*}D_{0}\varphi+\,[(1-\zeta
B^{\,2})\,\delta_{ij}+\zeta E_{i}E_{j}]\,D_{i}\varphi^{*}D_{j}\varphi$ (4.6)
$\displaystyle+\frac{1}{2}(e_{i}^{\,2}+b^{\,2})+U\,,$

which is exempted of crossed spatio-temporal terms. $E_{i}$ and $B$ stand for
the electric and magnetic background fields, respectively, as well as $e_{i}$
and $b$ for the propagating wave fields. This configuration suggests a matrix
diagonalization trial through rotation of the elements. The eigenvalues, which
compound the diagonalized Hamiltonian (density), are $1+\zeta E^{2}$ and $1$,
turning the equation (4.6) in

$\displaystyle\mathcal{E}$ $\displaystyle=$ $\displaystyle\int
d^{\,2}x(1+\zeta\vec{E}^{\,2})\bar{D}_{0}\varphi^{*}\bar{D}_{0}\varphi+(1+\zeta\vec{E}^{\,2})\,\bar{D}_{1}\varphi^{*}\bar{D}_{1}\varphi+\bar{D}_{2}\varphi^{*}\bar{D}_{2}$
(4.7) $\displaystyle+$ $\displaystyle\int
d^{\,2}x\frac{1}{2}(\vec{e}^{\,\,2}+b^{\,2})+U\,,$

observing that $\bar{D}_{\mu}$ means the covariant derivative expressed in the
rotated base. Along the diagonalization calculus, an intrinsic constrain
emerges: $\zeta B^{2}=0$. Then, the equality (dropping the "bar" sign)

$\displaystyle-\big{|}\big{(}aD_{0}\,+bD_{1}\,+cD_{2}\big{)}\,\varphi\big{|}^{2}+\big{|}\big{(}aD_{0}\,+bD_{1}\big{)}\,\varphi\big{|}^{2}+\big{|}\big{(}aD_{0}\,+cD_{2}\big{)}\,\varphi\big{|}^{2}$
$\displaystyle+\big{|}\big{(}bD_{1}\,+cD_{2}\big{)}\,\varphi\big{|}^{2}=a^{2}D_{0}\,\varphi^{*}D_{0}\,\varphi+b^{2}\,D_{1}\,\varphi^{*}D_{1}\,\varphi+c^{2}D_{2}\,\varphi^{*}D_{2}\,\varphi$
(4.8)

is adopted in eq. (4.7), where $a$, $b$ and $c$ are constants. In this case,
the usual relation ($D_{i}\varphi D_{i}\varphi^{*}\sim|(D_{1}\pm
iD_{2})\varphi|^{2}$, few examples in [282, 288]) is rewritten in a form
including the time component. Noteworthy of the left side of eq. (4.8), which
is supported by triangular inequality – establishes that: being $\vec{M}$ and
$\vec{N}$ vectors, $|\vec{M}|+|\vec{N}|\geq|\vec{M}+\vec{N}|\,$ (see, for
instance, ref. [309]) – and it guarantees that the right side of eq. (4.8) is
equal or bigger than zero (even tough $D_{0}$ being a scalar). Managing the
second integral of the right side of eq. (4.7) to determine the Bogomol’nyi
type equations and the inferior energy bound, one finds for the
electromagnetic field of the free propagating wave and Higgs potential that

$\displaystyle\int d^{\,2}x\,\,\frac{1}{2}(\vec{e}^{\,\,2}+b^{2})+U$
$\displaystyle=$ $\displaystyle\int
d^{\,2}x\,\,\frac{1}{2}\big{[}(|\vec{e}\,|\pm b)\pm\sqrt{2U}\big{]}^{2}\mp\int
d^{\,2}x\,\,(|\vec{e}\,|\pm b)\sqrt{2U}$ (4.9a) $\displaystyle\mp$
$\displaystyle\int d^{\,2}x\,\,|\vec{e}\,|\,b\,$ or $\displaystyle\int
d^{\,2}x\,\,\frac{1}{2}(\vec{e}^{\,\,2}+b^{2})+U$ $\displaystyle=$
$\displaystyle\int d^{\,2}x\,\ \frac{1}{2}(|\vec{e}\,|\pm\sqrt{U})^{2}\mp\int
d^{\,2}x\,\,|\vec{e}\,|\sqrt{U}$ (4.9b) $\displaystyle+$ $\displaystyle\int
d^{\,2}x\,\,\frac{1}{2}(b\pm\sqrt{U})^{2}\mp\int d^{\,2}x\,\,b\sqrt{U}\,.$

The liberty of choosing the minus and plus signals in equations (4.9)
guarantees to each term be bigger than zero and, consequently, the energy be
positive. In order to obtain the ground energy value from equation eq. (4.7),
taking into account eqs. (4.8) and (4.9), are determined the self-dual
expressions

$\displaystyle
D_{0}\,\varphi^{*}D_{0}\,\varphi=-(D_{1}\,\varphi^{*}D_{1}\,\varphi+\,\frac{1}{1+\zeta\vec{E}^{\,\,2}}\,D_{2}\,\varphi^{*}D_{2}\,\varphi)\,,$
(4.10a) $\displaystyle(|\vec{e}\,|\pm b)-\sqrt{2U}=0\,\,\,\text{for eq.
\eqref{campo_em_Higgs1}}$ (4.10b) or
$\displaystyle|\vec{e}\,|-\sqrt{U}=0\,\,\,\text{and}\,\,\,b-\sqrt{U}=0\,\,\,\text{for
eq. \eqref{campo_em_Higgs2}}\,,$ (4.10c)

and the Bogomol’nyi limit-type equation in the saturated scenario (eq. (4.10))
is

$\displaystyle\mathcal{E}^{\prime}\geq+\sqrt{2U}(\Phi_{e}\pm\Phi_{b})\mp\int
d^{\,2}x\,|\vec{e}\,|b\,\,\,\,\text{for}\,\,\eqref{campo_em_Higgs1}\,,$
(4.11a) or
$\displaystyle\mathcal{E}^{\prime\prime}\geq+\sqrt{U}(\Phi_{e}+\Phi_{b})\,\,\,\,\text{for}\,\,\eqref{campo_em_Higgs2},$
(4.11b)

with $\Phi_{b}$ representing the magnetic flux $\int d^{\,2}x\,\,b$. One
observes in both equations (4.11) the appearance of the electric flux $\,\int
d^{\,2}x\,|\vec{e}\,|\,\equiv\,\Phi_{e}\,$, which is not commonly obtained
and, depending on $a_{0}$, it will or not be quantized. Besides, in eq.
(4.11a), one of the energy parcel‘s is originated from the Poynting vector
$\,\int d^{\,2}x\,|\vec{e}\,|\,b\,$, where both are unexpected. No stationary
condition is imposed and the gauge choice is opened, so far. The possibility
of expressing in two formulations (4.9) the same energy term $\big{(}\int
d^{\,2}x\,\,\frac{1}{2}(\vec{e}^{\,\,2}+b^{2})+U\big{)}$ opens an alternative
to define the energy lower bound in function of the Poynting vector (4.11a).
The self-dual type equations (eqs. (4.10b) and (4.10c)) should converge in
some moment. Then, demanding
$\mathcal{E}^{\prime}=\mathcal{E}^{\prime\prime}$, the Poynting vector becomes
$\,\mp\int
d^{\,2}x\,|\vec{e}\,|\,b\,=\sqrt{U}\,[(\Phi_{e}+\Phi_{b})-\sqrt{2}(\Phi_{e}\pm\Phi_{b})]\,$.
So, the convergence of the energy formulations $\mathcal{E}^{\prime}$ and
$\mathcal{E}^{\prime\prime}$ enables to express the Poynting vector as
function of the magnetic and electric fluxes.

####  4.2.3 Topological Vortices Solutions

The proposition of this section is to find vortices solutions to the Maxwell-
Higgs model in scene, starting from generalized (reads initially unspecified)
vector field carrying, in principle, radial and angular degrees of freedom in
a polar coordinate system. The expectation is to determine algebraically
$a_{0}$ and $\vec{a}$ from the field equations. Thus, is defined

$\displaystyle a_{0}(x,y)\rightarrow a_{0}(r,\theta)$ (4.12a) $\displaystyle
a(x,y)\rightarrow a(r,\theta)$ (4.12b)

where, in principle, the gauge potentials $a_{0}$ and $a$ are assumed
dependent on both coordinates ($r$ and $\theta$). Aiming vortex solution, is
chosen for the scalar field a recurrent ansatz (e.g., refs. [161, 277, 286,
310])

$\displaystyle\varphi(r,n,\theta)=f(r,n)e^{in\theta}$ (4.13)

with $n$ being an integer and $f(r,n)$ a function of the radius $r$ and the
integer $n$. In this context, maintaining the raised constraint $\zeta
B^{2}=0$ along the whole subsequent analysis, the temporal component of the
field equation (4.4a) given by the charge (density) $J^{0}\equiv
ig\,G_{\mu}^{\,\,\,\,0}\,(\varphi
D^{\mu}\varphi^{*}-\varphi^{*}D^{\mu}\varphi)$, the so called Gauss equation,
becomes

$\displaystyle\Big{(}\nabla^{2}-(1+\zeta\vec{E}^{2})2g^{2}f^{2}\Big{)}\,a_{0}=0\,,$
(4.14)

where is restricted to the stationary case ($\partial_{0}\,a_{\mu}=0$ and
$\partial_{0}\,\varphi=0$ ). These constrains are adopted from this point on.
The eq. (4.14) is algebraically resolvable setting
$\big{(}\frac{1}{r}\frac{\partial^{2}}{\partial\theta^{2}}+\frac{\partial}{\partial
r}\big{)}a_{0}=0$, resulting in an exponential form to
$a_{0}(r,n,\theta)=l\,e^{-\chi(r,n,\theta)}\,$ where $l$ is a constant and the
function $\chi(r,n,\theta)$ allows a few different combinations of terms
involving $r$ and $\theta$. Some matching varying the power degree of the
radius $r$ and considering $\chi$ a pure real or pure imaginary, were tried.
In general, in the limit $r\rightarrow 0$, they presented a divergent or
vanishing behaviour. In a way that the chosen option is
$\chi(r,n,\theta)=\chi(r,n)+in\theta$, where $\chi(r,n)$ is a general function
of $r$ and $n$. This choice naturally leads to $\chi(r,n)\sim n^{2}\ln{r}$,
with $f$ from eq. (4.14) being resolved explicit and presenting an unexpected
quantization $n$,

$\displaystyle a_{0}(r,n,\theta)=l\,r^{n^{2}}e^{-in\theta}\,,$ (4.15a)
$\displaystyle
f(r,n)=\sqrt{\frac{n^{2}(n^{2}-1)}{2g^{2}(1+\zeta\vec{E}^{\,2})}}\frac{1}{r}\,\,\,,$
(4.15b) $\displaystyle\lim_{r\rightarrow
0}a_{0}=0\,\,\,\text{and}\,\,\,\lim_{r\rightarrow\infty}a_{0}=\infty\,,$
$\displaystyle\lim_{r\rightarrow
0}f=\infty\,\,\,\text{and}\,\,\,\lim_{r\rightarrow\infty}f=0\,,$ (4.15c)

observing that $l$ absorbs the constant emergent from the integral calculation
of $\chi(r,n)$. The scalar potential $a_{0}$ points out an anisotropic
solution dependent on $\theta$. It also contains a topological feature
manifesting quantization and a particular direct proportionality to $r$ and
$n$. The limit of $a_{0}$ is divergent for long ranges, in other hand, the
function $f(r,n)$ detains a singularity at $r\rightarrow 0$, which reflects in
the scalar field $\varphi$ (4.13). In generalized models including effects of
magnetic permeability [282] (which is an extension of multivortex study in
ref. [287]) is obtained resembling quadratic dependence on $n$ for scalar
functions like $f(r)$. Analysing the current (density) – Ampère-Maxwell
equation –, extracted from eq. (4.4a),

$\displaystyle\Big{(}\nabla^{2}\delta_{ij}-\partial_{i}\partial_{j}+2f^{2}g^{2}(\eta_{ij}+\zeta
E_{i}E_{j})\Big{)}a_{j}=-2f^{2}gn(\eta_{ij}+\zeta
E_{i}E_{j})\partial_{j}\theta$ (4.16)

in which are explicit the individualized contributions of the vector potential
components $a_{1}$ and $a_{2}$. Particularizing to the already mentioned
conditions of stationary scenario, one finds that, firstly for $i=1$,

$\displaystyle\bigg{(}\sin{\theta}\frac{\partial}{\partial
r}-\cos{\theta}\frac{1}{r}\frac{\partial}{\partial\theta}\bigg{)}\bigg{(}\frac{\partial
a}{\partial r}+\frac{1}{r}a\bigg{)}$
$\displaystyle=2gf^{2}\Big{(}ga+\frac{n}{r}\Big{)}\Big{(}\sin{\theta}+\zeta
E_{1}(\cos{\theta}\,E_{2}-\sin{\theta}\,E_{1})\Big{)}\,,$ (4.17)

and for $i=2$

$\displaystyle\bigg{(}\cos{\theta}\frac{\partial}{\partial
r}+\sin{\theta}\frac{1}{r}\frac{\partial}{\partial\theta}\bigg{)}\bigg{(}\frac{\partial
a}{\partial r}+\frac{1}{r}a\bigg{)}$
$\displaystyle=2gf^{2}\Big{(}ga+\frac{n}{r}\Big{)}\Big{(}\cos{\theta}+\zeta
E_{2}(\sin{\theta}\,E_{1}-\cos{\theta}\,E_{2})\Big{)}\,.$ (4.18)

Equations (4.17) and (4.18) reveal a potential source of anisotropy due to the
background electric field components ${E}_{1}$ and ${E}_{2}$ presented in both
equations. In Amperè-Maxwell equation, the stationary scenario transfers
exclusively the source of the current to the magnetic field, which in this
case is $b=\big{(}\frac{\partial}{\partial r}+\frac{1}{r}\big{)}a$. There are
two extra approaches to solve the differential equations system formed by eqs.
(4.17) and (4.18), which consist in impose a magnetic field $b$ only in
function of $r$ or $\theta$, that means $\frac{\partial b}{\partial\theta}=0$
or $\frac{\partial b}{\partial r}=0$, respectively. In both situations,
$a(r,\theta)$ results in

$\displaystyle a(r,n)=-\frac{n}{g}\frac{1}{r}\,\hat{\theta}\,\,.$ (4.19)

The solution leads $b$ to zero, has a topological feature with vorticity $n$
and is dependent only on $r$. It also presents the expected angular direction
of movement $\hat{\theta}$, perpendicular to the radius direction $\hat{r}$.
Noticing the bound limits in (4.19), when $r\rightarrow\infty$ it has a
convergent behaviour with $a=0$, while it diverges in $r\rightarrow 0$.
Situations of singularities in $r\rightarrow 0$ for $a(r)\sim\frac{1}{r}$ are
commonly prevented inserting a general function at the gauge potential
expression, which controls the limit bound (see some examples in ref. [298],
in the context of a Maxwell-Chern-Simons model coupled to an external
background charge, in ref. [299], which treats a generalized Maxwell-Chern-
Simons-Higgs model with modified Maxwell and kinetic terms and in ref. [288],
where is demonstrated that Chern-Simons solitons support topological and non-
topological solutions). At the same time, exists in literature a case of non-
vanishing gauge potential at large distances [300]. An observation in relation
to these examples is the fact that they start from an ansatz
$a(r)\sim\frac{1}{r}$, while a very similar result was obtained naturally from
the calculation of the field equations in the present work. A third option is
to solve, probably numerically, the expression resultant from the manipulation
and summing of eqs. (4.17) and (4.18),

$\Omega\,\Bigg{(}\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}-\bigg{(}1+\frac{1}{\Omega}\bigg{)}\frac{1}{r^{2}}\Bigg{)}a=\frac{n}{g}\frac{1}{r^{3}}\,\hat{\theta}\,\,\,$
(4.20)

with

$\Omega(\theta,n,E)=\frac{1+\tan^{2}{\theta}}{2g^{2}\tau^{2}(1+\tan^{2}{\theta}+\zeta(\tan{\theta}\,E_{1}-E_{2})^{2})}\,,$

observing that $\tau(r,n)=rf(r,n)$ (4.15b). The algebraic resolution is
feasible imposing $\frac{\partial a}{\partial
r}-\bigg{(}1+\frac{1}{\Omega}\bigg{)}\frac{a}{r}=0$, so that

$a(r,\theta,n,E)=\frac{\Omega}{\Omega+1}\,\frac{n}{g}\frac{1}{r}\,\hat{\theta},$

in an explicitly way

$a(r,\theta,n,E)=\frac{1+\zeta\vec{E}^{\,2}}{1+\zeta\vec{E}^{\,2}+\big{(}1+\zeta(E_{1}\sin{\theta}-E_{2}\cos{\theta})^{2}\big{)}n^{2}(n^{2}+1)}\,\frac{n}{g}\frac{1}{r}\,\hat{\theta}\,\,.$
(4.21)

$a(r,\theta,n,E)$ conserves the main structure manifested in (4.19), however
the gauge potential is also in function of $\theta$ and $E$. Beyond that, it
carries vorticity more elaborated and anisotropy because of $\theta$ and $E$
components dependency.

Turning the attention to the charge $Q$, it reveals an electrically neutral
flux

$Q=\int\vec{r}\,d\vec{r}d\theta\,2\,g^{2}(1+\zeta\vec{E}^{2})f^{2}a_{0}\,=0\,,$
(4.22)

in which $a_{0}$ is given by (4.15a). Paul and Khare [153] also found neutral
vortices from a model containing Chern-Simons term generated by spontaneous
symmetry breaking and a line of investigation developed by a Brazilian group
works with neutral vortices as consequence of the gauge-fixing adopted (see
refs. [280, 299] just as a start point for a long researching line), noticing
that the particular investigation of this group on generalized Maxwell-Higgs
models [282] resulted in ones with neutral charge and, in spite of it, with
existence of quantized magnetic flux, as observed in the present study.
Additionally, in this thesis there is the presence of the electric flux too.

A last analysis is an inspection on the energy functional equation (4.7)
expressed in terms of polar coordinates

$\displaystyle\mathcal{E}$
$\displaystyle=\int\,r\,drd\theta\,\,\,(1+\zeta\vec{E}^{\,\,2})\Big{(}\frac{\tau}{r}\,ga_{0}\Big{)}^{2}$
(4.23)
$\displaystyle+2\frac{\tau^{2}}{r^{4}}+\frac{\tau^{2}}{r^{2}}\Big{(}-\frac{1}{r^{2}}+\frac{n^{2}}{r^{2}}+g^{2}a^{2}+\frac{2n}{r}ga\Big{)}$
$\displaystyle+\zeta\vec{E}^{\,\,2}\bigg{[}\frac{\tau^{2}}{r^{4}}+\frac{\tau^{2}}{r^{2}}\sin^{2}{\theta}\Big{(}-\frac{1}{r^{2}}+\frac{n^{2}}{r^{2}}+g^{2}a^{2}+\frac{2n}{r}ga\Big{)}\bigg{]}$
$\displaystyle+\frac{1}{2}\bigg{[}\bigg{(}\frac{1}{r}\frac{\partial
a_{0}}{\partial\theta}\bigg{)}^{2}+\bigg{(}\frac{\partial a_{0}}{\partial
r}\bigg{)}^{2}+\bigg{(}\frac{\partial a}{\partial
r}+\frac{a}{r}\bigg{)}^{2}\bigg{]}+U\,,$

what evidences the topological structure associated to $\tau$ and to the gauge
potentials $a_{0}$ and $a$, endorsing the quantized characteristic of the
energy. The dependency on $\theta$ indicates an anisotropy in the energy
value. Moving towards the calculation of the Bogomol’nyi limit (4.11), one
rescues the self-dual relations (4.10) in polar coordinates,

$\displaystyle-$
$\displaystyle(1+\zeta\vec{E}^{\,\,2})\,a_{0}^{2}=\bigg{(}a^{2}+\frac{2n}{g}\frac{a}{r}\Big{)}(1+\zeta\vec{E}^{\,\,2}\sin^{2}{\theta})$
(4.24a) $\displaystyle+$
$\displaystyle\frac{1}{g^{2}r^{2}}\big{(}n^{2}-1\big{)}(1+\zeta\vec{E}^{\,\,2}\sin^{2}{\theta})+\frac{1}{g^{2}r^{2}}(2+\zeta\vec{E}^{2})\,,$
where one has for eqs. (4.9a) and (4.10a)
$\displaystyle\bigg{[}\bigg{(}\frac{1}{r}\frac{\partial
a_{0}}{\partial\theta}\bigg{)}^{2}+\bigg{(}\frac{\partial a_{0}}{\partial
r}\bigg{)}^{2}\bigg{]}^{\frac{1}{2}}\pm\bigg{(}\frac{\partial a}{\partial
r}+\frac{a}{r}\bigg{)}-\sqrt{2U}=0$ (4.24b) or for eqs. (4.9b) and (4.10b)
$\displaystyle\bigg{[}\bigg{(}\frac{1}{r}\frac{\partial
a_{0}}{\partial\theta}\bigg{)}^{2}+\bigg{(}\frac{\partial a_{0}}{\partial
r}\bigg{)}^{2}\bigg{]}^{\frac{1}{2}}=\frac{\partial a}{\partial
r}+\frac{a}{r}=\sqrt{U}\,.$ (4.24c)

With the equations above is possible to calculate $U$. The self-dual
expression (4.24a) enables to set the gauge potential $a_{0}$ in function of
$a$ and vice versa, which in association with eq. (4.24b) or (4.24c) forms a
equations system and provide material enough to calculate $a_{0}$, $a$ and $U$
explicitly in the inferior bound energy regime. Then, one taking (4.24a)
demonstrates that

$\displaystyle a(r,\theta,n)=-\frac{n}{g}\frac{1}{r}\,\hat{\theta}\pm
H(r,\theta,n)\,\hat{\theta}$ (4.25a) where $\displaystyle
H(r,\theta,n)=\bigg{[}\frac{1}{g^{2}r^{2}}\bigg{(}1-\frac{1}{1+\zeta\vec{E}^{\,\,2}\sin^{2}{\theta}}\bigg{)}+\bigg{(}a_{0}^{2}-\frac{1}{g^{2}r^{2}}\bigg{)}\bigg{(}\frac{1+\zeta\vec{E}^{\,\,2}}{1+\zeta\vec{E}^{\,\,2}\sin^{2}{\theta}}\bigg{)}\bigg{]}^{\frac{1}{2}},$
(4.25b)

and substituting eq. (4.25a) in (4.24c), one writes that

$\mp\bigg{(}\frac{\partial H}{\partial r}+\frac{H}{r}\bigg{)}=\sqrt{U}\,.$
(4.26)

The results presented in (4.25) and (4.26) are in function of $a_{0}$.
Remember that, working with the expression for the electric field in eq.
(4.24c), $a_{0}$ may be determined probably numerically due to the complexity
of the equations system. The eq. (4.25a) preserves a similar structure to eqs.
(4.19) and (4.21), but has no dependency on the background electric field. It
contains an anisotropy generated by $\theta$ and the singularity when
$r\rightarrow 0$. All the features just described for $a(r,\theta,n)$ are
propagated to the Higgs potential $U$ in eq. (4.26), which means that it
carries anisotropy and is quantized in the minimum energy bound scenario. And
finally, once obtained the potential $U$, one is in condition of calculating
the energy lower bound $\mathcal{E^{\prime}}$ (4.11a) or
$\mathcal{E^{\prime\prime}}$ (4.11b). Observing that the magnetic flux (and
probably the electric too) generator of $\mathcal{E^{\prime}}$ and
$\mathcal{E^{\prime\prime}}$ is quantized.

This closes the present section involving the Maxwell-Higgs model with a
strong electromagnetic background field with constant metric. Some comments
about the results achieved along the section are exposed in the Partial
Conclusions 4.3. In the next section is proposed a similar model to the first
one, differing by a non-constant metric.

###  4.3 Partial Conclusion

The exploration through the two Maxwell-Higgs models unveiled some aspects of
the topological physics around these systems. Initially, it seems worthy to
briefly debate about the methodology applied to approach the cases, which has
few variations in relation to the usually one adopted [254]. The
diagonalization of the energy density equation (4.6) allows to write it in a
simplified arrangement (4.7) carrying the differential temporal component and
naturally manifesting an anisotropy associated to the energy (see eq. (4.7) or
(4.23)). It should be added that the triangular inequality (4.8) permits to
express the differential terms from the diagonalized eq. (4.7) in definite
positive ones, keeping the temporal component, and assures that the equation
(4.8) is equal or bigger than zero. This procedure generates equations (4.10)
(and (4.24) in polar coordinates) alike to self-dual ones, what enables to
determine the energy lower bound of the system (4.11). Then, the methodology
described may be an alternative to calculate this energy value, which has no
requesting of stationary condition or imposition of the gauge-fixing to
achieve the energy expression. However, so far, it is tested just in one model
and certainly there is room for improvements or even discarding it. Besides,
the diagonalization of the energy density is a vulnerable step depending on
the complexity of the Hamiltonian and maintain differential equations of
second order. The unexpected constraint on the background magnetic field
$\zeta B^{2}=0$, emerged from the diagonalization of eq. (4.6), simplified the
calculation, although imposed a severe limitation to physical systems,
probably turning it in an unfeasible proposition as model for an neutron star
scenario.

Some comments related to the field formulations should be done, once one of
the purposes of the study is to determine algebraically the vector field from
the field equations, assuming, initially , undefined $a_{0}$ and $a$, both
containing radial and angular degrees of freedom. Starting with the function
$f(r)$, representing the radial component of the field $\varphi$, it is
quantized and presents a divergent behaviour when $r\rightarrow 0$. Such
difficult could be circumvented defining boundary conditions for $f(r)$ as
practiced in refs. [312] and [313] (in magnetic monopoles scenario), for
instance, or redefining $\varphi$. Another option, which diverges from one of
the premises of the working, is to consider $a_{0}(r)$, eliminating the
angular degree of freedom, and recalculating the Gauss equation (4.14). In
this case, $f(r)$ is undefined and $a_{0}(r)$ is in function of $f(r)$ (a
similar result is also observed for an Abelian Chern-Simons-Higgs model in
ref. [312]). The obtained exponential and the $\theta$ dependency resulting
for the gauge scalar potential are unusual in the literature. Moving the
debate to the results to $a(r)$, the structure $\sim\frac{1}{r}$ of the
solutions in eqs. (4.19), (4.21) and (4.25a) is generally adopted as ansatz in
diversified context of the literature as magnetic monopoles [255], Maxwell-
Chern-Simons model [301], sigma model [303], Lorentz violation [310], are just
few examples. In all the scenarios described, the gauge potential $a$ has
vorticity and a singularity for short distances. The particular case of
$a(r,\theta,n,E)$ (4.21) (and also $f(r)$ (4.15b)) presents a richer vorticity
structure manifested in generalized Maxwell-Higgs models [282]. The former
gauge potential carries an anisotropy originated by the angular dependency and
by the background electric field components $E_{1}$ and $E_{2}$. The
background field anisotropy is not manifested in $a(r,\theta,n)$ (4.25a), what
incurs in an energy lower bound expression (4.25) absent of them.
Nevertheless, one consideration must be done: both $a(r,\theta,n)$ and the
energy $\mathcal{E}$ have an anisotropy source from the presence of the
$\sin{\theta}$ in their equations.

The equations (4.9) revealed that, besides the contribution of a magnetic
flux, the one from an electric flux and a possibility of writing the lower
energy bound in terms of the Poynting vector (4.11a). In other hand, the
system in (4.22) pointed out a neutral charge. The potential $U$ is
algebraically calculated from the self-dual type equations (4.24) and from the
results for $a_{0}$ (not calculated here) and $a$ (4.25a) in the context of
the inferior energy bound. The result is unstable when $r\rightarrow 0$, what
is frequently controlled by imposition of boundary conditions and betaking
auxiliary functions [279, 280, 314]. It inherit the anisotropy and the
quantized characteristic from $a(r,\theta,n)$.

The motivation is to let the mathematics points whether the $\theta$ is
feasible or not. In general line, to work with generalized $a_{0}$ and $a$
tends to a complicated equation system, what imposes a hard task to find the
solutions to the equations (which frequently are determined only numerically).
At the same time, work with this assumption of generalized gauge potentials,
whenever possible, allows the system speak "by itself" and to manifest its
inner formation.

##  Chapter 5 Concluding Considerations

Throughout the PhD program, an extremely deep theoretical foundation was laid
by the researchers, postdocs and guest lectures in our scientific
coordination, where a great variety of courses that approached the forefront
themes in physics have been delivered. During this period, I collaborated in a
workgroup to produce chapters 2 and 3, and paper [2]. The research for chapter
1 and the publication of [1] was conducted with another researcher; whilst
chapter 4 was individually produced. This work took form in a complete
experience of multifaceted production scenario, collaboration, and working
environment. The group of researchers working at the Great "Republic of
Diracstan" of CBPF also promoted an inclusive and participative workspace,
conducting internal debates, collaborative studies, and experiences, as well
as knowledge sharing, which were continuously supported and encouraged by the
staff members. The opportunities and frequently encouraged chances of
presentation of ongoing or concluded works, equally enriched the supervised
researcher’s development.

Observing the theoretical contentment of the present construction, the low
dimensional effective theories with non-minimal coupling explored in the
chapter 1 are a wide territory. In the context of the particular study
approached in this thesis, there are room for investigation of terms in higher
orders of $\mathcal{O}(1/m)$, for diversification of the electrodynamics which
describes the external field and for testing the impact of different
dimensional reduction methodologies. The calculation of higher order terms in
fermionic case, apart from revealing new relativistic interactions, is a
consistent checking for the validity of the dimensional reduction result
obtained in eq. (1.36) and for verification of the matter relevance
($\vec{J}\neq\vec{0}$) in orders beyond $\mathcal{O}(1/m^{2})$ in non-minimal
relativistic interactions in eq. (1.23) when the external electromagnetic
field is governed by Maxwell-Chern-Simons electrodynamics (see eqs. (1.26)).
Still in the chapter 4, now turning to the scalar field configuration,
considering the surging of a Chern-Simons term when a non-minimal Abelian
Higgs model is submitted to a spontaneous symmetry breaking [153], maybe to
find out the Hamiltonian density of this resultant Lagrangian and FW transform
it, comparing the results with the one achieved in eq. (1.14), or even better
for the situation containing terms in order superior to
$\mathcal{O}(1/m^{3})$, will permit to establish some notices. This analysis
is extended to an inspection which quests if the critical Hamiltonian $H_{c}$
in eq. (1.28) is maintained in the breaking symmetry scenario mentioned.

In chapters 2 and 3 around effective gravitational quantum theory structured
with higher derivatives, naturally comes the perspective of advancing the
results considering vertices and/or graviton propagator beyond the tree level.
The work developed in the present thesis was a first step in the inter-
particle potentials investigation among bosonic and fermionic particles
mediated by graviton, in a sense that the calculations involving one-loop
quantum corrections are the next step. Other perspective is to develop similar
inter-particles potential study involving massive spin-1 particles under
gravitational scattering, partially worked by Holstein and Ross [211]. There
is one more investigation, the inter-particle potential involving the
scattering of two anti-fermions or one fermion and one anti-fermion, which
could reveal, theoretically, whether the gravitational potentials for fermions
are the same that to anti-fermions (considering that their rest energy (mass)
are numerically equal).

An invitation to address a reflection about the interpretation of gravity is
proposed. Does the gravity emanate from matter (as a charge) or it is there
yet, independently from the matter? Maybe, this is the bifurcation where one
way (the emanation concept) leads to interpret gravity as a discrete point
(virtual) particle and analyzes it as the mediator of inter-particle
potentials between massive (or energy holder) particles. The other way leads
to the geometrical consideration (General Relativity), a continuum
interpretation of the field, which is deformed in presence of energy. This
impasse among discrete and continuum, commented in the presentation section of
the thesis, maybe summarize the difficult in conciliate the General Relativity
and the Standard Model. It seems a materialism tendency to work with point
particles. The Green functions, fundamentally settle in the delta‘s Dirac,
perform this connection/switch from discrete to continuum description of the
Universe in a great accuracy. Even though, the struggle to conciliate
unitarity and renormalizability is patent. Then, one could inquiry whether the
electromagnetism field is a geometric continuum occupying the Universe, which
deforms in a presence of an electric charge, reading the gauge field as an
affine connection.

The chapter 4 is fully opened in relation to the directions definitions. A
second model with the same Lagrangian expression but carrying a non-constant
metric ($\partial_{\kappa}\bar{G}_{\mu\nu}\neq 0$) is proposed. The algebraic
development of the energy-momentum tensor resulted in a non-conservative
system. The condition to attend the Noether’s theorem is demanding that
$-(D_{\kappa}\,\bar{G}_{\,\mu\nu})D^{\mu}\varphi^{*}D^{\nu}\varphi=0$, what
violates the premise of the model (non-constant metric). There are other two
local gauge invariant models to be explored, in which the modified
(non-)constant metric acts on the kinetic and Maxwell terms
$\,(\mathcal{L}\sim
G_{\mu\nu}\big{(}-\frac{1}{4}f^{\mu\kappa}f_{\kappa}^{\,\,\,\nu}+D^{\mu}\varphi^{*}D^{\nu}\varphi\big{)}-U)$.
In these cases, it is evaluated the effect of the strong electromagnetic
background field on the propagating wave as well. The analysis of the
Lagrangian with constant metric is already on the road with the energy density
equation calculated. In this scenario, the constraint $\zeta B^{2}=0$ is
present, once the index structure of
$G_{\mu\nu}D^{\mu}\varphi^{*}D^{\nu}\varphi$ is the same of the models
investigated in this work. A possible alternative to detour it, is a
modification in the index arrangement $\,\mathcal{L}\sim
G_{\mu\kappa}\big{(}-\frac{1}{4}f^{\,\kappa\xi}f_{\xi}^{\,\,\,\mu}+D^{\kappa}\varphi^{*}D^{\mu}\varphi\big{)}-U$.
The exploration of these models will probably rise insights about the
structure of the gauge scalar and vector potentials, pointing their vorticity
and topological behaviour. In these cases, the diagonalization methodology
applied in the thesis (eq. (4.7)) keeps validated, as well as the triangular
inequality (eq. (4.8)), once the kinematic scalar sector is unmodified. Then,
the anisotropy manifested (see, for instance, eq. (4.23)) will be present in
these two models. The modifications will appear in the Bogomol’nyi type
equations (see eqs. (4.9) and (4.10)) and the field equations due to the
metric tensor associated to the Maxwell term.

One of the most relevant lessons and thought-provoking experiences during the
PhD program was the publication of the paper [1], which was co-produced with
Professor Leonardo Ospedal. We selected its publication to be in a free-of-
charge journal, so the Brazilian people would not be encumbered with fees.
However, upon the publication of our paper, we confirmed that we were not
granted access to the journal and in order for us to be able to access our own
work we would need to pay for it.

In our group of researchers, we follow the practices of publishing every
single work, from the preliminary versions up to the final manuscript, in the
arXiv repositor and, at the same time, is awaited to submit the results to the
appreciation of the community. This procedure grants open access for the
community to the whole research developed by the group. Such attitude is
widely adopted by researchers, transforming arXiv in a place where is possible
to find the forefront and breakthrough works, like the one uploaded by the
Russian mathematician Grigori Perelman [315] which awarded him the Fields
Medal in 2006, but he never published it in any magazine or journal or even
accepted the award, or like the complete Field Theory book Fields by Warren
Siegel [316], which material he just published in arXiv. Therefore, one can
see arXiv as a start point of the emancipation process from the private
publication companies.

Maybe we should raise the question: are private scientific magazines and
journals really indispensable for the promotion of science? It seems important
to promote this debate in the scientific community involving researchers and
public scientific institutions (and, immediately thereafter, the State and
government) with the aim to structure a centralized social system of paper
review, analysis and publication that are independent from these private
journals and magazines, which, in a general way, set a monopolistic and
domineering stage in the world of science.

Researchers must have awareness of this situation in order not to become
alienated. Those who are basically, and sometimes almost exclusively, focused
on the betterment of their own curriculum vitae (which clearly is a tool of
classification, distinction, and social status), may not realize how harmful
these publication and impact factor processes are on scientific community,
which in turn, subjects itself and abides before the magazines and journals
system (that is inherently bind to the productivity system). As a result,
providing to the oppressive hands the fruit of our hard labour and the
societal investment, granting them the control of the global scientific work.

One of the roles of the researcher is to question and argue if the environment
in which they and their labor activities enter are ethic. We should not
unquestionably insert and immerse ourselves in a pre-built structure, which is
presented and sold as the best and/or only one existent. We need to be aware
of the social context of our country and understand that we (as researchers,
professors, middle and high economic classes) live a privileged condition, in
which the vast majority of our people will not have access to the scientific
community due to the sequestration of social inclusion, sequestration of
opportunity, sequestration of racial equality, sequestration of gender
equality, sequestration of education, sequestration of basic constitutionals
rights. A solid foundation of inequality supports the world of researchers and
assures our position. We have to step out of this self-centered sense of
accomplishment where we believe our achievements are exclusively a result of
our own merits and realize that the positions we occupy as researchers only
exist because they have been supported by the foundational work of the less
fortunate and working class, which find themselves very distant from any
possibility and chance of frequenting and accessing "our" positions, academic
spaces and of spaces in science in general, those who, due to their level of
disadvantage, frequently cannot merely comprehend the specificity of the craft
of a researcher. We must unite ourselves, work towards modifying this
injustice and we cannot consent and connive with this enormous loot of our
nation and to our work as nation. Science, Nature, politics and social
principles work and walk together.

##  Appendix A Foldy-Wouthuysen Transformation Review

It expends seventy years Leslie Foldy and Siegfried Wouthuysen contemplated
the literature with the transformation methodology that takes their names
[100]. In this paper, which was curiously followed by a José Lopes Leite paper
[317] in the sequence of the magazine, the authors presented an approach to
diagonalize Hamiltonians in approximate and, in particulars cases, exact way.
The FW Transformation is described in several Chemistry and Physic text books,
e.g. [158, 157, 318], thus it is possible to find a wide number of discussions
and developed applications. With the spirit of facilitate and make more
comfortable the reading of the Thesis, the FW transformation procedure is
detailed below, in a way to possibility an eventual consultation by the
reader.

The central idea of the FW transformation is to interpret it as an unitary
transformation on a Hamiltonian $H$ which permits to decouple equations in a
linear system, in order to avoid mixed terms and, in parallel, has the
potential to reveal new terms and to establish corrections of higher order.
Considering the Wigner principle that demands the eigenfunctions be
(anti)unitarity transformed in a way to assure the conservation of the
probability amplitudes of the eigenstates, one establishes a canonical unitary
transformation on an eigenfunction $\psi$

$\psi^{\prime}=e^{-iS}\psi\,,$ (A.1)

where $S$ is a Hermitian operator. Then, substituting the expression (A.1) in
a Schröndiger Equation

$\displaystyle i\partial_{t}(e^{-iS}\psi)=H(e^{-iS}\psi)$ $\displaystyle
ie^{-iS}\partial_{t}\psi+(i\partial_{t}\,e^{-iS})\psi=H(e^{-iS}\psi)$
$\displaystyle(e^{iS}\,\times)\,\,\,\,\,\,\,\,i\partial_{t}\psi=-e^{iS}(i\partial_{t}\,e^{-iS})\psi+e^{iS}H(e^{-iS}\psi)$
$\displaystyle i\partial_{t}\psi=e^{iS}(H-i\partial_{t})\,e^{-iS}\psi\,,$
(A.2)

one obtains the Hamiltonian transformed $H^{\prime}$

$H^{\prime}=e^{iS}(H-i\partial_{t})e^{-iS}\,.$ (A.3)

The apparent exponential in eq. (A.3) is passive of being expanded through the
Baker–Hausdorff identity

$e^{A}\,B\,e^{-A}=B+[A,B]+\frac{1}{2}[A,[A,B]]+\frac{1}{6}[A,[A,[A,B]]]+...+\frac{1}{n!}[A,...[A,[A,B]]...]\,,$
(A.4)

with $[A,B]$ meaning the commutator of the operators $A$ and $B$, resulting in

$H^{\prime}=H+i[S,H]+\frac{i^{2}}{2}[S,[S,H]]+\frac{i^{3}}{6}[S,[S,[S,H]]]+...-(\dot{S}+\frac{i}{2}[S,\dot{S}]+\frac{i^{2}}{6}[S,[S,\dot{S}]]\,...\,),$
(A.5)

noticing that the dotting above the operators, like $\dot{S}$, stands for the
time derivative of the operator $S$. The substitution of the Hamiltonian $H$
in the eq. (A.5) is preceded by one step. The Hamiltonian must be organized in
a particular form

$H=\beta m+\mathcal{E}+\mathcal{O}\,,$ (A.6)

where $\beta$, $\mathcal{E}$ and $\mathcal{O}$ are operators which attend to
the (anti)commutation relations $\beta\mathcal{E}=\mathcal{E}\beta$ and
$\beta\mathcal{O}=-\mathcal{O}\beta$, with $\mathcal{E}$ meaning "even"
operators and $\mathcal{O}$ "odd" ones. This arrangement (in eq. (A.6)) is
fundamental to achieve the diagonilized Hamiltonian, once the products
$\mathcal{E}^{2}$ and $\mathcal{O}^{2}$ result in even operators, while
$\mathcal{E}\mathcal{O}$ and $\mathcal{O}\mathcal{E}$ in odd ones. $\beta$ is
usually represented as a diagonal matrix and $m$ stands for the mass. The
generator $S$ is defined as

$S=-i\,\frac{\beta\mathcal{O}}{2m}\,,$ (A.7)

in which the presence of $\mathcal{O}$ performs the conversion of odd
operators in even ones, diagonalizing the Hamiltonian order by order, where
the orders are in terms of the mass inverse. Substituting eqs. (A.6) and (A.7)
in eq. (A.5), one obtains the Hamiltonian

$H^{\prime}=\beta
m+\mathcal{E}+\beta\frac{\mathcal{O}^{2}}{2m}-\frac{1}{8m^{2}}[\mathcal{O},[\mathcal{O},\mathcal{E}]+i\mathcal{\dot{O}}]-\beta\frac{\mathcal{O}^{4}}{8m^{3}}+\frac{\beta}{2m}[\mathcal{O},\mathcal{E}]-\frac{\mathcal{O}^{3}}{3m^{2}}+\frac{i\beta\mathcal{\dot{O}}}{2m}...\,,$
(A.8)

in which the highest order of odd terms is $m^{-1}$. Therefore, $H$ is
diagonalized until order $m^{0}$ and written as $H^{\prime}=\beta
m+\mathcal{E}^{\prime}+\mathcal{O}^{\prime}$, in a way that the terms present
in eq. (A.8) are passive of being regrouped in one even operator
$\mathcal{E}^{\prime}$, holding the odd powers of $\mathcal{O}$, and one odd
$\mathcal{O}^{\prime}$

$\displaystyle\mathcal{E}^{\prime}=\mathcal{E}+\beta\frac{\mathcal{O}^{2}}{2m}-\frac{1}{8m^{2}}[\mathcal{O},[\mathcal{O},\mathcal{E}]+i\mathcal{\dot{O}}]-\beta\frac{\mathcal{O}^{4}}{8m^{3}}...\,,$
(A.9a)

$\displaystyle\mathcal{O}^{\prime}=\frac{\beta}{2m}[\mathcal{O},\mathcal{E}]-\frac{\mathcal{O}^{3}}{3m^{2}}+\frac{i\beta\mathcal{\dot{O}}}{2m}...\,,$
(A.9b)

and, consequently, a new $S$ operator is determined

$\displaystyle S^{\prime}=-i\,\frac{\beta\mathcal{O}^{\prime}}{2m}\,.$ (A.9c)

Once defined the new operators $\mathcal{E}^{\prime}$, $\mathcal{O}^{\prime}$
and $\mathcal{S}^{\prime}$ \- eqs. (A) -, the Hamilotnian $H^{\prime}$ is
submitted to the unitary transformation in eq. (A.3), what results in a
Hamiltonian $H^{\prime\prime}$ diagonalized until order $m^{-1}$

$H^{\prime\prime}=\beta
m+\mathcal{E^{\prime}}+\beta\frac{\mathcal{O^{\prime}}^{2}}{2m}-\frac{1}{8m^{2}}[\mathcal{O^{\prime}},[\mathcal{O^{\prime}},\mathcal{E^{\prime}}]+i\mathcal{\dot{O^{\prime}}}]-\beta\frac{\mathcal{O^{\prime}}^{4}}{8m^{3}}+\frac{\beta}{2m}[\mathcal{O^{\prime}},\mathcal{E^{\prime}}]-\frac{\mathcal{O^{\prime}}^{3}}{3m^{2}}+\frac{i\beta\mathcal{\dot{O^{\prime}}}}{2m}...\,,$
(A.10)

indeed, the odd term with the lowest order in $H^{\prime\prime}$,
$\,\frac{\beta}{2m}[\mathcal{O^{\prime}},\mathcal{E^{\prime}}]\,$, is a term
of order $O(1/m^{2})$. Then, from $H^{\prime\prime}$, new even, odd and
$\mathcal{S}$ terms are defined, namely $\mathcal{E}^{\prime\prime\prime}$,
$\mathcal{O}^{\prime\prime\prime}$ and $\mathcal{S}^{\prime\prime\prime}$, and
the process may be repeated $n+1$ times till the odd components achieve the
desired $(n+1)^{th}$ order to $m^{-1}$, resulting in a Hamiltonian
diagonalized up to order $O(1/m^{n})$.

In the present work is considered the Hamiltonian diagonalized until the order
$O(1/m^{3})$

$H_{FW}\approx\beta
m+\mathcal{E}+\beta\frac{\mathcal{O}^{2}}{2m}-\frac{1}{8m^{2}}[\mathcal{O},[\mathcal{O},\mathcal{E}]+i\mathcal{\dot{O}}]-\beta\frac{\mathcal{O}^{4}}{8m^{3}}\,.$
(A.11)

This formulation for the diagonalized Hamiltonian is the one applied to all
cases investigated in the present Thesis. Since the scalar to fermionic
systems or from three to four spacetime dimensions, the general structure of
the $H_{FW}$ is that expressed in (A.11). What varies is the definition of the
operators and the dimensional adjustments, basically manifested through
matrices dimensionality and their algebra.

##  Appendix B Integrals

The integrals indicated along the chapters 2 and 3 are

$I^{(a)}_{n}(r)=\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}\,\frac{e^{i\vec{q}\cdot\vec{r}}}{(\vec{q}^{\,2})^{n}\,Q_{a}}\,,$
(B.1)

$I^{(a)}_{ij}(r)=\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}\,\frac{e^{i\vec{q}\cdot\vec{r}}}{\vec{q}^{\,2}\,Q_{a}}\,\vec{q}_{i}\vec{q}_{j}\,,$
(B.2)

where $n\in\mathbb{N}$ and $a=0,2$. For Fourier transform of the type eq.
(B.1) is possible to solve its angular part passing the function
$Q_{a}(\vec{q}^{2})$ to the spherical coordinate system, leaving only radial
dependence (see, for instance, ref. [319]). This artifice enables one to
recast some integrals, for example:

$\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}\,\frac{e^{i\vec{q}\cdot\vec{r}}}{\vec{q}^{\,2}\,Q_{a}}\,i\vec{A}\cdot\vec{q}=\vec{A}\cdot\vec{\nabla}\left[I^{(a)}_{1}(r)\right]=\vec{A}\cdot\frac{\vec{r}}{r}\,\frac{d}{dr}\,I^{(a)}_{1}(r)\,,$
(B.3)

being $\vec{A}$ a vector independent of $\vec{q}$.

## References

  * [1] L.P.R. Ospedal and M.G. Campos, Mod. Phys. Lett. A 36 (2021) 2150099; arXiv:2012.06297v2 [hep-th].
  * [2] G.P. de Brito, M.G. Campos, L.P.R. Ospedal and K.P.B. Veiga, Phys. Rev. D 102 (2020) 084015; arXiv:2006.12824v2 [hep-th].
  * [3] G.P. de Brito, P.C. Malta and L.P.R. Ospedal, Phys. Rev. D 95 (2017) 016006; arXiv:1612.01181v1 [hep-th].
  * [4] L.O.P. Rosas, Efeitos de Spin, Velocidade e Dimensionalidade em Potenciais Inter-partículas Associados a Modelos de Gauge, PhD Thesis, Centro Brasileiro de Pesquisas Físicas - CBPF (2017).
  * [5] B. Knorr and F. Saueressig, Phys. Rev. Lett. 121 (2018) 161304; arXiv: 1804.03846 [hep-th].
  * [6] H.K. Onnes, KNAW Proceedings 13 II (1911) 1274.
  * [7] P.A.M. Dirac, Proc. R. Soc. Lond. A 112 (1926) 661.
  * [8] P.A.M. Dirac, Proc. R. Soc. Lond. A 133 (1931) 60.
  * [9] L.D. Landau, Zeit. Phys. 64 (1930) 629.
  * [10] W. Meissner and R. Ochsenfeld, Die Naturw. 21 (1933) 787; translation in: A.M. Forrest, Eur J. Phys. 4 (1983) 117.
  * [11] F. London and H. London, Proc. Roy. Soc. A 149 (1935) 71.
  * [12] C.J. Gorter and H.B.G. Casimir, Phys. 1 (1934) 306.
  * [13] L.D. Landau, Zh. Eksp. Teor. Fiz. 7 (1937) 19; translation in: Ukr J. Phys. 53 (2008) 25.
  * [14] P. Kapitza, Nat. 141 (1938) 74.
  * [15] J.F. Allen and A.D. Misener, Nat. 141 (1938) 75.
  * [16] S.N. Bose, Z. Physik 26 (1924) 178; translation in: J. Astrophys. Astr. 15 (1994) 3.
  * [17] A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. 1 (1925) 3; translation in: The Collected Papers of Albert Einstein, Vol. 14, Princeton University Press, (1989).
  * [18] V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20 (1950) 1064; translation in: V.L. Ginzburg, L.D. Landau, On the Theory of Superconductivity, Springer, 2009.
  * [19] H. Fröhlich, Phys. Rev. 79 (1950) 845.
  * [20] J. Bardeen, Phys. Rev. 80 (1950) 567.
  * [21] L.N. Cooper, Phys. Rev. 104 (1956) 1189.
  * [22] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175.
  * [23] N.N Bogoliubov, J. Exptl. Theoret. Phys. 34 (1958) 58.
  * [24] N.N Bogoliubov, J. Exptl. Theoret. Phys. 34 (1958) 73.
  * [25] P.W. Anderson, Phys. Rev. 112 (1958) 1900.
  * [26] A.A. Abrikosov, Jour. Phys. Chem. Sol., 2 (1957) 199.
  * [27] L.P. Gor’kov, J. Exptl. Theoret. Phys. 36 (1959) 1918.
  * [28] I. Giaever, Phys. Rev. Lett. 5 (1960) 464.
  * [29] B.D. Josephson, Phys. Lett. 1 (1962) 251.
  * [30] Q.D. Osheroff, R.C. Richardson and D.M. Lee, Phys. Rev. Lett. 28 (1972) 885.
  * [31] A.J. Leggett, Phys. Rev. Lett. 29 (1972) 1227.
  * [32] A.J. Leggett, Rev. Mod. Phys. 47 (1975) 331.
  * [33] T. Ando, Y. Matsumoto and Y. Uemura, J. Phys. Soc. Jap. 39 (1975) 279.
  * [34] J. Wakabayashi and S. Kawaji, J. Phys. Soc. Jap. 44 (1978) 1839.
  * [35] J.M. Leinaas and J. Myrheim, Nuo. Cim. B 37 (1977) 1.
  * [36] K.v. Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45 (1980) 494.
  * [37] R.B. Laughlin, Phys. Rev. B 23 (1981) 5632.
  * [38] D.C. Tsui, H.L. Stormer and A.C. Gossard, Phys. Rev. Lett. 48 (1982) 1559.
  * [39] F. Wilczek, Phys. Rev. Lett. 48 (1982) 1144.
  * [40] F. Wilczek, Phys. Rev. Lett. 49 (1982) 957.
  * [41] R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395
  * [42] D. Arovas, J.R. Schrieffer and F. Wilczek, Phys. Rev. Lett. 53 (1984) 722.
  * [43] J.G Bednorz and K.A. Müller, Zeit. Phys. B 64 (1986) 189.
  * [44] C.de C. Chamon, D.E. Freed, S.A. Kivelson, S.L. Sondhi and X.G. Wen, Phys. Rev. B 55 (1997) 2331; arXiv:0911.0794 [cond-mat.mes-hall].
  * [45] K.T. Law, D.E. Feldman and Y. Gefen, Phys. Rev. B 74 (2006) 045319; arXiv:cond-mat/0506302 [cond-mat.mes-hall].
  * [46] F.E. Camino, W. Zhou and V.J. Goldman, Phys. Rev. Lett. 98 (2007) 076805; arXiv:cond-mat/0610751v1 [cond-mat.mes-hall].
  * [47] N. Ofek et al., Proc. Natl. Acad. Sci. 107 (2010) 5276; arXiv:0911.0794v3 [cond-mat.mes-hall].
  * [48] D.T. McClure, W. Chang, C.M. Marcus, L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 108 (2012) 256804; arXiv:1112.0538v1 [cond-mat.mes-hall].
  * [49] R.L. Willett, C. Nayak, K. Shtengel, L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 111 (2013) 186401; arXiv:1301.2639 [cond-mat.mes-hall].
  * [50] J. Nakamura et al., Nat. Phys. 15 (2019) 563; arXiv:1901.08452 [cond-mat.mes-hall].
  * [51] R.L. Willett et al., arXiv:1905.10248v1 [cond-mat.mes-hall].
  * [52] R. de-Picciotto et al., Nat. 389 (1997) 162; arXiv:cond-mat/9707289 [cond-mat.mes-hall].
  * [53] L. Saminadayar, D.C. Glattli, Y. Jin and B. Etienne, Phys. Rev. Lett. 79 (1997) 2526; arXiv:cond-mat/9706307.
  * [54] G. Moore and N. Read, Nucl. Phys. B 360 (1991) 362.
  * [55] M. Banerjee et al., Nat. 559 (2018) 205; arXiv:1710.00492 [cond-mat.mes-hall].
  * [56] Y. Kasahara et al., Nat. 559 (2018) 227; arXiv:1805.05022 [cond-mat.str-el].
  * [57] H. Bartolomei et al., Sci. 368 (2020) 173; arXiv:2006.13157 [cond-mat.mes-hall].
  * [58] J. Nakamura, S. Liang, G.C. Gardner and M.J. Manfra, Nat. Phys. 16 (2020) 931; arXiv:2006.14115 [cond-mat.mes-hall].
  * [59] P. Anderson, Mater. Res. Bull. 8 (1973) 153.
  * [60] P. Anderson, Sci. 235 (1987) 1196.
  * [61] V. Kalmeyer and R. Laughlin, Phys. Rev. Lett. 59 (1987) 2095.
  * [62] S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Phys. Rev. B 35 (1987) 8865.
  * [63] X.G. Wen, Phys. Rev. B 40 (1989) 7387.
  * [64] T. Senthil and M.P.A. Fisher, Phys. Rev. B 62 (2000) 7850; arXiv:cond-mat/9910224 [cond-mat.str-el].
  * [65] T. Senthil and M.P.A. Fisher, Phys. Rev. Lett. 86 (2001) 292; arXiv:cond-mat/0006481 [cond-mat.supr-con].
  * [66] A.Y. Kitaev, Ann. Phys. 303 (2003) 2; arXiv:quant-ph/9707021.
  * [67] A.Y. Kitaev, Ann. Phys. 321 (2006) 2; arXiv:cond-mat/0506438 [cond-mat.mes-hall].
  * [68] A. Banerjee et al., Nat. 15 (2016) 733; arXiv:1504.08037 [cond-mat.mtrl-sci].
  * [69] A. Banerjee et al., Sci. 356 (2017) 1055; arXiv:1609.00103 [cond-mat.mtrl-sci].
  * [70] P. Laurell and S. Okamoto, Nat. 5 (2020) 2; arXiv:1906.07579 [cond-mat.str-el].
  * [71] M.G. Yamada, Nat. 5 (2020) 82; arXiv:2004.06257 [cond-mat.str-el].
  * [72] H. Liu, J. Chaloupka and G. Khaliullin, Phys. Rev. Lett. 125 (2020) 047201; arXiv:2002.05441 [cond-mat.str-el].
  * [73] J.M. Kosterlitz and D.J. Thouless, J.Phys. C 5 (1972) L124.
  * [74] J.M. Kosterlitz and D.J. Thouless, J.Phys. C 6 (1973) 1181.
  * [75] D.J. Thouless, M. Kohmoto, M.P. Nightingale and M. den Nijs, Phys. Rev. Lett. 49 (1982) 405.
  * [76] F.D.M. Haldane, Phys.Rev. Lett. 61 (1988) 2015.
  * [77] K.S. Novoselov et al., Proc. Natl. Acad. Sci. 102 (2005) 10451; arXiv:cond-mat/0503533 [cond-mat.mtrl-sci].
  * [78] C.-Z. Chang et al., Sci. 340 (2013) 167; arXiv:1605.08829 [cond-mat.mes-hall].
  * [79] S. Murakami, N. Nagaosa and S.-C. Zhang, Phys. Rev. Lett. 93 (2004) 156804; arXiv:cond-mat/0406001 [cond-mat.mes-hall].
  * [80] C.L. Kane and E.J. Mele, Phys. Rev. Lett. 95 (2005) 146802; arXiv:cond-mat/0506581 [cond-mat.mes-hall].
  * [81] C.L. Kane and E.J. Mele, Phys. Rev. Lett. 95 (2005) 226801; arXiv:cond-mat/0411737 [cond-mat.mes-hall].
  * [82] B.A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 96 (2006) 106802; arXiv:cond-mat/0504147 [cond-mat.mes-hall].
  * [83] B.A. Bernevig, T.L. Hughes and S.-C. Zhang, Sci. 314 (2006) 1757; arXiv:cond-mat/0611399 [cond-mat.mes-hall].
  * [84] M. König et al., Sci. 318 (2007) 766; arXiv:0710.0582 [cond-mat.mes-hall].
  * [85] K.S. Novoselov et al., Sci. 315 (2007) 1379; arXiv:cond-mat/0702408 [cond-mat.mes-hall].
  * [86] F.D.M. Haldane and S. Raghu, Phys. Rev. Lett. 100 (2008) 013904; arXiv: cond-mat/0503588 [cond-mat.mes-hall].
  * [87] L. Lu, J.D. Joannopoulos and M. Soljačić, Nat. 8 (2014) 821; arXiv:1408.6730 [physics.optics].
  * [88] T. Ozawa et al., Rev. Mod. Phys. 91 (2019) 015006; arXiv:1802.04173 [physics.optics].
  * [89] A.B. Khanikaev et al., Nat. 12 (2013) 233; arXiv:1204.5700 [cond-mat.mtrl-sci].
  * [90] M.A. Gorlach et al., Nat. 9 (2018) 909.
  * [91] M. Honari-Latifpour and L. Yousefi, Nanophot. 8 (2019) 799.
  * [92] S. Gupta and A. Saxena, The Role of Topology in Materials, Vol. 189, Springer, 2018.
  * [93] Z.F. Wang, Z. Liu and F. Liu, Nat. 4 (2013) 1471; arXiv:1304.5582 [cond-mat.mtrl-sci].
  * [94] Z.F. Wang, N.H. Su, F. Liu, Nano Lett. 13 (2013) 2842.
  * [95] Z. Liu, Z.-F. Wang, J.-W. Mei, Y.-S. Wu and F. Liu, Phys. Rev. Lett. 110 (2013) 106804; arXiv:1210.1826 [cond-mat.mtrl-sci].
  * [96] Z.-F. Wang, Z. Liu and F. Liu, Phys. Rev. Lett. 110 (2013) 196801; arXiv:1302.1088 [cond-mat.mes-hall].
  * [97] X. Ni, W. Jiang, H. Huang, K.-H. Jin and F. Liu, Nanoscale 10 (2018) 11901.
  * [98] L. Hernández-López et al., Nanoscale 13 (2021) 5216.
  * [99] M.H.L. Pryce, Proc. R. Soc. Lond. A 195 (1948) 62.
  * [100] L.L. Foldy and S.A. Wouthuysen, Phys. Rev. 78 (1950) 29.
  * [101] S. Tani, Prog. Theor. Phys. 6 (1951) 267.
  * [102] L.L. Foldy, Phys. Rev. 87 (1952) 688.
  * [103] K.M. Case, Phys. Rev. 95 (1954) 1323.
  * [104] E. Eriksen, Phys. Rev. 111 (1958) 1011.
  * [105] A.G. Nikitin, J. Phys. A 31 (1998) 3297.
  * [106] A.J. Silenko, J. Math. Phys. 44 (2003) 2952; arXiv:math-ph/0404067.
  * [107] A.J. Silenko, Phys. Rev. D 89 (2014) 121701(R); arXiv:1404.4953 [quant-ph].
  * [108] A.J. Silenko, Theor. Math. Phys. 156 (2008) 1308; arXiv:0810.1604 [math-ph].
  * [109] A.J. Silenko, Phys. Rev. D 87 (2013) 073015; arXiv:1303.6574 [hep-ph].
  * [110] A.J. Silenko, Phys. Rev. D 88 (2013) 045004; arXiv:1305.6378 [math-ph].
  * [111] A.J. Silenko, Phys. Rev. D 98 (2018) 025014; arXiv:1712.08625 [gr-qc].
  * [112] J.I. Kapusta, E. Rrapaj and S. Rudaz, Phys. Rev. D 102 (2020) 125028; arXiv:2009.12010 [hep-th].
  * [113] Y. Guo and H. Liang, Phys. Rev. C 101 (2020) 024304; arXiv:2001.00117 [nucl-th].
  * [114] L.H. Ryder and I.L. Shapiro, Phys. Lett. A 247 (1998) 21; arXiv:hep-th/9805138.
  * [115] Y.N. Obukhov, Phys. Rev. Lett. 86 (2001) 192; arXiv:gr-qc/0012102.
  * [116] A. Accioly and H. Blas, Phys. Rev. D 66 (2002) 067501; arXiv:gr-qc/0201015; Int. J. Theor. Phys. 43 (2004) 47.
  * [117] A.J. Silenko and O.V. Teryaev, Phys. Rev. D 71 (2005) 064016; arXiv:gr-qc/0407015.
  * [118] Y.N. Obukhov, A.J. Silenko and O.V. Teryaev, Phys. Rev. D 80 (2009) 064044; arXiv:0907.4367 [gr-qc]; Phys. Rev. D 84 (2011) 024025; arXiv:1106.0173 [hep-th]; Phys. Rev. D 88 (2013) 084014; arXiv:1308.4552 [gr-qc]; Phys. Rev. D 90 (2014) 124068; arXiv:1410.6197 [hep-th].
  * [119] H. Belich and K. Bakke, Int. J. Mod. Phys. A 30 (2015) 1550136.
  * [120] B. Gonçalves, Y.N. Obukhov and I.L. Shapiro, Phys. Rev. D 80 (2009) 125034; arXiv:0908.0437 [hep-th].
  * [121] B. Gonçalves, M.M.Dias Júnior and B.J. Ribeiro, Phys. Rev. D 90 (2014) 085026; arXiv: 1406.5477 [hep-th]; Phys. Rev. D 99 (2019) 096015; arXiv: 1812.11657 [hep-th].
  * [122] Z. Xiao, Phys. Rev. D 98 (2018) 035018; arXiv:1808.02599 [hep-ph].
  * [123] F.W. Hehl and W.T. Ni, Phys. Rev. D 42 (1990) 2045.
  * [124] D. Chowdhury and B. Basu, Ann. Phys. 329 (2013) 166; arXiv:1301.5431 [cond-mat.mes-hall]; Ann. Phys. 339 (2013) 358; arXiv:1309.1376 [cond-mat.mes-hall]; Physica B 448 (2014) 155; arXiv:1404.2376 [cond-mat.mes-hall].
  * [125] J.-H. Wang, K. Ma and K. Li, Phys. Rev. A 87 (2013) 032107; arXiv:1301.4363 [cond-mat.mes-hall].
  * [126] D. Chowdhury and B. Basu, Phys. Rev. D 90 (2014) 125014; arXiv:1411.1818 [cond-mat.mes-hall].
  * [127] D.G. Rothe et al, New J. of Phys. 12 (2010) 065012; arXiv:1002.2904 [cond-mat.mes-hall].
  * [128] O.F. Dayi, M. Elbistan and E. Yunt, Ann. Phys. 327 (2012) 935; arXiv: 1109.4808 [math-ph].
  * [129] A.J. Silenko, Phys. Rev. A 91 (2015) 022103; arXiv:1501.02052 [math-ph].
  * [130] B. Binegar, J. Math. Phys. 23 (1982) 1511.
  * [131] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53 (1984) 319.
  * [132] D. Yoshioka, The Quantum Hall Effect, Springer-Verlag Berlin Heidelberg, 2002.
  * [133] Z.F. Ezawa, Quantum Hall Effects - Field Theoretical Approach and Related Topics, 2nd ed., World Scientific, 2008.
  * [134] N.E. Mavromatos and A. Momen, Mod. Phys. Lett. A 13 (1998) 1765; arXiv:hep-th/9802119.
  * [135] I.I. Kogan, Phys. Lett. B 262 (1991) 83.
  * [136] J. Stern, Phys. Lett. B 265 (1991) 119.
  * [137] L.H. Ryder, Quantum Field Theory, Cambridge University Press, 1996.
  * [138] C.C.-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics, Vol. 01 and 02, Wiley-Interscience Publication, 1977.
  * [139] L.D. Landau and E.M. Lifshitz, Mechanics - Course of Theoretical Physics, Vol. 01, Elsevier, 2007.
  * [140] R. Fitzpatrick, Maxwell’s Equations and the Principles of Electromagnetism, Infinity Science Press LLC, 2008.
  * [141] M. Nakahara, Geometry, Topology and Physics, Institute of Physics Publishing, 2003.
  * [142] J.K. Jain, Phys. Rev. Lett. 63 (1989) 199.
  * [143] R.C. Paschoal and J.A. Helayël-Neto, Phys. Lett. A 313 (2003) 412; arXiv:hep-th/0302014.
  * [144] M.E. Carrington and G. Kunstatter, Phy. Rev. D 51 (1995) 1903.
  * [145] F.A.S. Nobre and C.A.S. Almeida, Phys. Lett. B 455 (1999) 213; arXiv:hep-th/9904159.
  * [146] N. Itzhaki, Phys. Rev. D 67 (2003) 065008; arXiv:hep-th/0211140.
  * [147] D. Dalmazi, Phys. Rev. D 70 (2004) 065021; arXiv:hep-th/0405196.
  * [148] D. Dalmazi and E.L. Mendonça, J. Phys. A 39 (2006) 11091; arXiv:hep-th/0608212.
  * [149] H.R. Christiansen, M.S. Cunha, J.A. Helayël-Neto, L.R.U. Manssur and A.L.M.A. Nogueira, Int. J. Mod. Phys. A 14 (1999) 147; arXiv:hep-th/9802096.
  * [150] R.C. Paschoal, J.A. Helayël-Neto and L.P.G. de Assis, Phys. Lett. A 349 (2006) 67; arXiv:hep-th/0407109.
  * [151] H. Belich, L.P. Colatto, T. Costa-Soares, J.A. Helayël-Neto and M.T.D. Orlando, Eur. Phys. J. C 62 (2009) 425; arXiv:0806.1253 [hep-th].
  * [152] S. Ghosh, Mod. Phys. Lett. A 20 (2005) 1227; arXiv:hep-th/0407086.
  * [153] S.K. Paul and A. Khare, Phys. Lett. B 193 (1987) 253; S.K. Paul and A. Khare, Phys. Lett. B 196 (1987) 571, Erratum.
  * [154] M.E. Carrington and G. Kunstatter, Phy. Rev. D 50 (1994) 2830; arXiv:hep-ph/9403375.
  * [155] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485.
  * [156] J.J. Sakurai and J. Napolitano, Modern Quantum Mechanics, Cambridge University Press, 2017.
  * [157] F. Schwabl, Advanced Quantum Mechanics, Springer, 2008.
  * [158] C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill Inc, 1980.
  * [159] S. Deser, R. Jackiw and S. Templeton, Ann. Phys. 140 (1982) 372; S. Deser, R. Jackiw, S. Templeton, Ann. Phys. 185 (1988) 406, (erratum); S. Deser, R. Jackiw, S. Templeton, Ann. Phys. 281 (2000) 409, (reprinted).
  * [160] M. Nowakowski, E.A. Paschos and J.M. Rodríguez, Eur. J. Phys. 26 (2005) 545; arXiv:physics/0402058 [physics.ed-ph].
  * [161] M. Torres, Phys. Rev. D 46 (1992) R2295.
  * [162] Y. Georgelin and J.C. Wallet, Mod. Phys. Lett. A 7 (1992) 1149.
  * [163] X.-L. Qi, E. Witten and S.-C. Zhang, Phys. Rev. B 87 (2013) 134519; arXiv:1206.1407 [cond-mat.supr-con].
  * [164] D. Cocuroci, J.A. Helayël-Neto, M.J. Neves and L.P.R. Ospedal, Eur. Phys. J. C 75 (2015) 322; arXiv:1311.6735 [hep-th].
  * [165] A.J. Silenko and O.V. Teryaev, Phys. Rev. D 89 (2014) 041501; arXiv: 1311.5984 [gr-qc].
  * [166] L.P.R. Ospedal and J.A. Helayël-Neto, Phys. Rev. D 97 (2018) 056014; arXiv:1708.07410 [physics.gen-ph].
  * [167] E. Cavalcanti, C.A. Linhares, J.A. Lourenço and A.P.C. Malbouisson, Phys. Rev. D 100 (2019) 025008; arXiv:1904.10833 [hep-th].
  * [168] R. Turcati, C.A.B. Quintero, J.A. Helayël-Neto and E. Arias, Ann. der Phys. 532 (2020) 1900357; arXiv:1907.12196 [cond-mat.mes-hall].
  * [169] C. Castelnovo, R. Moessner and S.L. Sondhi, Nature 451 (2008) 42; arXiv: 0710.5515 [cond-mat.str-el].
  * [170] W. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714; arXiv: physics/0605038.
  * [171] M. Born and H. Infeld, Proc. R. Soc. Lond. A 144 (1934) 425.
  * [172] S.M. Carroll, G.B. Field and R. Jackiw, Phys. Rev. D 41 (1990) 1231.
  * [173] R. Gambini and J. Pullin, Phys. Rev. D 59 (1999) 124021; arXiv:gr-qc/9809038.
  * [174] J. Alfaro, H.A. Morales-Tecotl and L.F. Urrutia, Phys. Rev. D 65 (2002) 103509; arXiv:hep-th/0108061.
  * [175] T.-W. Chen and D.-W. Chiou, Phys. Rev. A 90 (2014) 012112; arXiv:1311.3432 [quant-ph].
  * [176] J. Scherk and J.H. Schwarz, Phys. Lett. B 57 (1975) 463.
  * [177] R. Mondal, M. Berritta and P.M. Oppeneer, Phys. Rev. B 94 (2016) 144419; arXiv:1609.07901 [cond-mat.mtrl-sci]; Phys. Rev. B 98 (2018) 214429; arXiv:1810.07438 [cond-mat.mtrl-sci].
  * [178] R. Mondal, M. Berritta, A.K. Nandy and P.M. Oppeneer, Phys. Rev. B 96 (2017) 024425; arXiv:1704.01559 [cond-mat.other].
  * [179] J.F. Donoghue, Phys. Rev. D 50 (1994) 3874; arXiv:gr-qc/9405057.
  * [180] H. Weyl, Ann. Phys. 59 (1919) 101.
  * [181] A. Eddington, The Mathematical Theory of Relativity, Cambridge Univ. Press, 1924.
  * [182] R. Utiyama, and B.S. DeWitt, J. Math. Phys. 3 (1962) 608.
  * [183] K.S. Stelle, Phys. Rev. D 16 (1977) 953.
  * [184] B.L. Voronov and I.V. Tyutin, Yad. Fiz. 39 (1984) 998.
  * [185] K.S. Stelle, Gen. Rel. Grav. 9 (1978) 353.
  * [186] D.A. Johnston, Nuc. Phys. B 297 (1988) 721.
  * [187] G. ’tHooft and M. Veltman, Ann. Inst. Henri Poincaré 20 (1974) 69.
  * [188] M.H. Goroff and A. Sagnotti, Nuc. Phys. B 266 (1986) 709.
  * [189] T.P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82 (2010) 451; arXiv:0805.1726 [gr-qc].
  * [190] E.A. Bergshoeff, O. Hohm and P.K. Townsend, Phys. Rev. Lett. 102 (2009) 201301; arXiv:0901.1766 [hep-th].
  * [191] I. Oda, J. High Energy Phys. 5 (2009) 064; arXiv:0904.2833 [hep-th].
  * [192] G.B. -Peixoto and I.L. Shapiro, Phys. Rev. D 71 (2005) 064005; arXiv:hep-th/0412249.
  * [193] B. Holdom and J. Ren, Int. J. Mod. Phys. D 25 (2016) 1643004; arXiv: 1605.05006 [hep-th].
  * [194] L. Modesto and L. Rachwał, Nuc. Phys. B 900 (2015) 147; arXiv:1503.00261 [hep-th].
  * [195] E.T. Tomboulis, arXiv:hep-th/9702146v1.
  * [196] J.F. Donoghue and G. Menezes, Phys. Rev. D 97 (2018) 126005; arXiv: 1804.04980 [hep-th].
  * [197] L. Modesto and I.L. Shapiro, Phys. Lett. B 755 (2016) 279; arXiv:1512.07600 [hep-th].
  * [198] L. Modesto, Nucl. Phys. B 909 (2016) 584; arXiv:1602.02421 [hep-th].
  * [199] I.J. Muzinich and S. Vokos, Phys. Rev. D 52 (1995) 3472; arXiv:hep-th/9501083.
  * [200] H.W. Hamber and S. Liu, Phys. Lett. B 357 (1995) 51; arXiv:hep-th/9505182.
  * [201] A.A. Akhundov, S. Bellucci and A. Shiekh, Phys. Lett. B 395 (1997) 16; arXiv:gr-qc/9611018.
  * [202] I.B. Khriplovich and G.G. Kirilin, J. Exp. Theor. Phys. 95 (2002) 981; arXiv:gr-qc/0207118.
  * [203] N.E.J. Bjerrum-Bohr, Phys. Rev. D 66 (2002) 084023. Corrected in arXiv:hep-th/0206236v3.
  * [204] N.E.J. Bjerrum-Bohr, J.F. Donoghue and B.R. Holstein, Phys. Rev. D 67 (2003) 084033. Erratum in Phys. Rev. D 71 (2005) 069903; arXiv:hep-th/0211072.
  * [205] S. Faller, Phys. Rev. D 77 (2008) 124039; arXiv:0708.1701 [hep-th].
  * [206] B.M. Barker, S.N. Gupta and R.D. Haracz, Phys. Rev. 149 (1966) 1027.
  * [207] I.B. Khriplovich and G.G. Kirilin, J. Exp. Theor. Phys. 98 (2004) 1063; arXiv:gr-qc/0402018.
  * [208] G.G. Kirilin, Nucl. Phys. B 728 (2005) 179; arXiv:gr-qc/0507070.
  * [209] M.S. Butt, Phys. Rev. D 74 (2006) 125007; arXiv:gr-qc/0605137.
  * [210] A. Ross and B.R. Holstein, J. Phys. A 40 (2007) 6973;.
  * [211] A. Ross and B.R. Holstein, arXiv:0802.0716[hep-ph].
  * [212] A. Ross and B.R. Holstein, arXiv:0802.0717[hep-ph].
  * [213] W.-T. Ni, Rep. Prog. Phys. 73 (2010) 056901; arXiv:0912.5057 [gr-qc].
  * [214] W.-T. Ni, Int. J. Mod. Phys. 40 (2016) 1660010; arXiv:1501.07696 [hep-ph].
  * [215] S.N. Gupta, Proc. Phys. Soc. A 65 (1952) 161.
  * [216] S.N. Gupta, Proc. Phys. Soc. A 65 (1952) 608.
  * [217] S.N. Gupta, Rev. Mod. Phys. 29 (1957) 3.
  * [218] I.J. Buchbinder, S.D. Odintsov and I.L. Shapiro, Effective Action in Quantum Theory, Institute of Physics Publishing Ltd, 1992.
  * [219] M. Maggiore, A Modern Introduction to Quantum Field Theory, Oxford University Press, 2005.
  * [220] T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar, Phys. Rev. Lett. 108 (2012) 031101; arXiv:1110.5249 [gr-qc].
  * [221] T. Biswas, A. Conroy, A.S. Koshelev and A. Mazumdar, Clas. Quant. Grav. 31 (2014) 015022; Corrigendum: Clas. Quant. Grav. 31 (2014) 159501; arXiv:1308.2319 [hep-th].
  * [222] A. Accioly, J. de Almeida, G.P. de Brito and W. Herdy, Phys. Rev. D 98 (2018) 064029; arXiv:1707.02083 [hep-th].
  * [223] P. van Nieuwenhuizen, Nuc. Phys. B 60 (1973) 478.
  * [224] A. Pais and G.E. Uhlenbeck, Phys. Rev. 79 (1950) 145.
  * [225] J.W. Moffat, Phys. Rev. D 41 (1990) 1177;
  * [226] A.O. Barvinsky, Phys. Lett. B 710 (2012) 12; arXiv:1107.1463 [hep-th].
  * [227] L. Modesto, Phys. Rev. D 86 (2012) 044005; arXiv:1107.2403 [hep-th].
  * [228] B.L. Giacchini and T.P. Netto, Eur. Phys. J. C 79 (2019) 217; arXiv:1806.05664 [gr-qc].
  * [229] M. Gasperini, Theory of Gravitational Interactions, Springer-Verlag, 2013.
  * [230] A. Accioly, J. Helayël-Neto, F.E. Barone and W. Herdy, Class. Quant. Grav. 32 (2015) 035021.
  * [231] A. Accioly, J. de Almeida, G.P. de Brito and W. Herdy, Phys. Rev. D 98 (2018) 064029; arXiv:1707.02083 [hep-th].
  * [232] N. Dorey and N.E. Mavromatos, Nucl. Phys. B 386 (1992) 614.
  * [233] L.P.R. Ospedal and J.A. Helayël-Neto, Phys. Rev. D 97 (2018) 056014; arXiv:1708.07410 [physics.gen-ph].
  * [234] J. Ambjørn and R. Loll, Nucl. Phys. B 536 (1998) 407; arXiv:hep-th/9805108.
  * [235] J. Ambjørn, J. Jurkiewicz, and R. Loll, Phys. Rev. Lett. 93 (2004) 131301; arXiv:hep-th/0404156.
  * [236] J. Ambjørn, J. Gizbert-Studnicki, A. Görlich, K. Grosvenor, and J. Jurkiewicz, Nucl. Phys. B 922 (2017) 226; arXiv:1705.07653 [hep-th].
  * [237] J. Ambjørn, J. Gizbert-Studnicki, A. Görlich, J. Jurkiewicz, and D. N’emeth, J. High Energy Phys. 06 (2018) 111; arXiv:1802.10434 [hep-th].
  * [238] J. Ambjørn, A. Goerlich, J. Jurkiewicz, and R. Loll, Phys. Rep. 519 (2012) 127; arXiv:1203.3591 [hep-th].
  * [239] M. Maggiore and M. Mancarella, Phys. Rev. D 90 (2014) 023005; arXiv: 1402.0448 [hep-th].
  * [240] E. Belgacem, Y. Dirian, S. Foffa and M. Maggiore, J. Cosm. Astrop. Phys. 03 (2018) 002; arXiv:1712.07066 [hep-th].
  * [241] E. Belgacem, Y. Dirian, A. Finke, S. Foffa and M. Maggiore, J. Cosm. Astrop. Phys. 04 (2020) 010; arXiv:2001.07619 [astro-ph.CO].
  * [242] S. Foffa, M. Maggiore and E. Mitsou, Int. J. Mod. Phys. A 29 (2014) 1450116; arXiv:1311.3435 [hep-th].
  * [243] E.V. Gorbar and I.L. Shapiro, J. High Energy Phys. 0302 (2003) 021; arXiv:hep-ph/0210388.
  * [244] G. Cusin, S. Foffa, M. Maggiore and M. Mancarella, Phys. Rev. D 93 (2016) 043006; arXiv:1512.06373 [hep-th].
  * [245] M. Visser, Arxiv:1702.05572v2 [gr-qc] (2017).
  * [246] M. Milgrom, Astroph. J. 270 (1983) 365.
  * [247] J.M. Overduin and P.S. Wesson, The Light/Dark Universe, World Scientific Publishing Co. Pte., 2008.
  * [248] B. Famaey and S.S. McGaugh, Liv. Rev. Relat. 15 (2012); arXiv:1112.3960 [astro-ph.CO].
  * [249] M. Hameeda et al, Phys. Dark Univ. 19 (2018) 137; arXiv:1712.08591 [gr-qc].
  * [250] J.G. Williams and S. Benson, Astron. J. 76 (1971) 2.
  * [251] J.D. Anderson et al, Phys. Rev. Lett. 81 (1998) 2858; arXiv:gr-qc/9808081.
  * [252] S.G. Turyshev and V.T. Toth, Liv. Rev. Relat. 13 (2010) 4; arXiv:1001.3686 [gr-qc].
  * [253] H.B. Nielsen and P. Olesen, Nucl. Phys. B 61 (1973) 45.
  * [254] E.B. Bogomol’nyi, Yad. Fiz. 24 (1976) 861; Sov. J. Nucl. Phys. 24 (1976) 449.
  * [255] M.K. Prasad and C.M. Sommerfield, Phys. Rev. Lett. 35 (1975) 760.
  * [256] F. Wilczek, Fractional Statistics and Anyon Superconductivity, World Scientific, 1996.
  * [257] A. Khare, Fractional Statistics and Quantum Theory, World Scientific, 2005.
  * [258] P.L. Marston and W.M. Fairbank, Phys. Rev. Lett. 39 (1977) 1208.
  * [259] P. Engels, I. Coddington, P.C. Haljan, V. Schweikhard, and E.A. Cornell, Phys. Rev. Lett. 90 (2003) 170405; arXiv:cond-mat/0301532 [cond-mat.soft].
  * [260] T. Cren, L. Serrier-Garcia, F. Debontridder, and D. Roditchev, Phys. Rev. Lett. 107 (2011) 097202.
  * [261] M.B. Hindmarsh and T.W.B. Kibble, Rep. Prog. Phys. 58 (1995) 477; arXiv:hep-ph/9411342.
  * [262] D. Tong, arXiv:hep-th/0509216v5.
  * [263] A. Vilenkin and E.P.S. Shellard, Cosmic Strings and Other Topological Defects, Cambridge University Press, 1994.
  * [264] E. Babichev, V. Mukhanov and A. Vikman, J. High Energy Phys. 02 (2008) 101; arXiv:0708.0561 [hep-th].
  * [265] J. Garriga and V. Mukhanov, Phys. Lett. B 458 (1999) 219; arXiv:hep-th/9904176.
  * [266] A.D. Rendall, Class. Quant. Grav. 23 (2006) 1557; arXiv:gr-qc/0511158.
  * [267] C. Armendariz-Picon and E.A. Lim, J. Cosmol. Astropart. Phys. 08 (2005) 007; arXiv:astro-ph/0505207.
  * [268] R.J. Scherrer, Phys. Rev. Lett. 93 (2004) 011301; arXiv:astro-ph/0402316.
  * [269] A. Sen, J. High Energy Phys. 07 (2002) 065; arXiv:hep-th/0203265.
  * [270] V. Mukhanov and A. Vikman, J. Cosmol. Astropart. Phys. 02 (2006) 004; arXiv:astro-ph/0512066.
  * [271] L. Faddeev and A.J. Niemi, Phys. Rev. Lett. 82 (1999) 1624; arXiv:hep-th/9807069.
  * [272] T.H.R. Skyrme, Proc. R. Soc. A 260 (1961) 127.
  * [273] D.A. Nicole, J. Phys. G 4 (1978) 1363.
  * [274] H. Aratyn, L.A. Ferreira, and A.H. Zimerman, Phys. Rev. Lett. 83 (1999) 1723; arXiv:hep-th/9905079.
  * [275] C. Adam, J. Sánchez-Guillén, R.A. Vázquez, and A. Wereszczyński, J. Math. Phys. 47 (2006) 052302; arXiv:hep-th/0602152.
  * [276] D. Bazeia, J. Menezes, and R. Menezes, Phys. Rev. Lett. 91 (2003) 241601; arXiv:hep-th/0305234.
  * [277] H.J. de Vega and F.A. Schaposnik, Phys. Rev. D 14 (1976) 1100.
  * [278] L. Jacobs and C. Rebbi, Phys. Rev. B 19 (1979) 4486.
  * [279] D. Bazeia, E. da Hora, C. dos Santos and R. Menezes, Eur. Phys. J. C, 71 (2011) 1833; arXiv:1201.2974 [hep-th].
  * [280] D. Bazeia, R. Casana, M.M. Ferreira Jr and E. da Hora, Eur. Phys. Lett. 109 (2015) 21001; arXiv:1502.05063 [hep-th].
  * [281] R. Casana, E. da Hora, D. Rubiera-Garcia and C. dos Santos, Eur. Phys. J. C 75 (2015) 380; arXiv:1507.08793 [hep-th].
  * [282] D. Bazeia, L. Losano, M.A. Marques, R. Menezes and I. Zafalan, Eur. Phys. J. C 77 (2017) 63; arXiv:1611.02110 [hep-th].
  * [283] L.F. Cugliandolo, G. Lozano and E.A. Schaposnik, Phys. Rev. D 40 (1989) 3440;.
  * [284] E. Babaev, J. Jäykkä and M. Speight, Phys. Rev. Lett. 103 (2009) 237002; arXiv:0903.3339 [cond-mat.supr-con].
  * [285] K. Shiraishi and S. Hirenzaki, Int. J. Mod. Phys. A 6 (1991) 2635; arXiv: 1810.02512 [hep-th].
  * [286] R. Jackiw and E.J. Weinberg, Phys. Rev. Lett. 64 (1990) 2234.
  * [287] J. Hong, Y. Kim, and P.Y. Pac, Phys. Rev. Lett. 64 (1990) 2230.
  * [288] R. Jackiw, K. Lee and E.J. Weinberg, Phys. Rev. D 42 (1990) 3488.
  * [289] R. Jackiw and S.-Y. Pi, Phys. Rev. Lett. 64 (1990) 2969.
  * [290] A. Antillón, J. Escalona, G. Germán and M. Torres, Phys. Lett. B 359 (1995) 327; arXiv:hep-th/9505115.
  * [291] S.K. Paul and A. Khare, Phys. Lett. B 174 (1986) 420.
  * [292] C. Lee, K. Lee and H. Min, Phys. Lett. B 252 (1990) 79.
  * [293] D. Bazeia, Phys. Rev. D 43 (1991) 4074.
  * [294] C.N. Kumar and A. Khare, Phys. Lett. B 178 (1986) 395.
  * [295] H.J. de Vega and F.A. Schaposnik, Phys. Rev. Lett. 56 (1986) 2564.
  * [296] A. Antillón, J. Escalona and M. Torres, Phys. Rev. D 55 (1997) 6327; arXiv:hep-th/9702118.
  * [297] M.A. Anacleto, A. Ilha, J.R.S. Nascimento, R.F. Ribeiro and C. Wotzasek, Phys. Lett. B 504 (2001) 268; arXiv:hep-th/0104152.
  * [298] F. Chandelier, Y. Georgelin, M. Lassaut, T. Masson and J.C. Wallet, Phys. Rev. D 70 (2004) 065016; arXiv:cond-mat/0405605 [cond-mat.other].
  * [299] I. Andrade, D. Bazeia, M.A. Marques and R. Menezes, Phys. Rev. D 102 (2020) 045018; arXiv:2005.09765 [hep-th].
  * [300] P.K. Ghosh, Phys. Lett. B 326 (1994) 264; arXiv:hep-th/9309104.
  * [301] P.K. Ghosh, Phys. Rev. D 49 (1994) 5458; arXiv:hep-th/9310096.
  * [302] K.C. Mendes, R.R. Landim and C.A.S. Almeida, Mod. Phys. Lett. A 20 (2005) 1005; arXiv:hep-th/0411211.
  * [303] M.S. Cunha, R.R. Landim and C.A.S. Almeida, Phys. Rev. D 74 (2006) 067701; arXiv:hep-th/0609064.
  * [304] A. Alonso-Izquierdo, W.G. Fuertes and J.M. Guilarte, J. High Energ. Phys. 02 (2015) 139; arXiv:1409.8419 [hep-th].
  * [305] R. Casana, M.M. Ferreira Jr., E. da Hora and C. Miller, Phys. Lett. B 718 (2012) 620; arXiv:1208.3908 [hep-th].
  * [306] R. Fridberg and T.D. Lee, Phys. Rev. D 15 (1977) 1694.
  * [307] B.S. Kim, H-j Lee and J.H. Yee, Phys. Rev. D 61 (1999) 025010; arXiv:hep-th/9903138.
  * [308] F. Ahmed, Commun. Theor. Phys. 72 (2020) 075102.
  * [309] R. Shankar, Principles of Quantum Mechanics, Plenum Press, $2^{\text{nd}}$ ed., 1994.
  * [310] R. Casana and L. Sourrouille, Phys. Lett. B 726 (2013) 488; arXiv:1307.1734 [hep-th].
  * [311] I. Andrade, D. Bazeia, M.A. Marques and R. Menezes, Phys. Rev. D 102 (2020) 025017; arXiv:2007.03138 [hep-th].
  * [312] S. Bolognesi and S.B. Gudnason, Nucl. Phys. B 805 (2008) 104; arXiv: 0711.3803 [hep-th].
  * [313] R. Casana, M.M. Ferreira Jr. and E. da Hora, Phys. Rev. D 86 (2012) 085034; arXiv:1109.2931 [hep-th].
  * [314] D. Bazeia, R. Casana, E. da Hora and R. Menezes, Phys. Rev. D 85 (2012) 125028; arXiv:1206.0998 [hep-th].
  * [315] G. Perelman, arXiv:math/0211159v1; arXiv:math/0303109v1;
arXiv:math/0307245v1 [math.DG].

  * [316] W. Siegel, Fields, 2005, arXiv:hep-th/9912205v3.
  * [317] J.L. Leite, Phys. Rev. 78 (1950) 36.
  * [318] M. Reiher and A. Wolf, Relativistic Quantum Chemistry - The Fundamental Theory of Molecular Science, Wiley-VCH, 2009.
  * [319] A.Accioly, J. Helayël-Neto, G. Correia, G. Brito, J. de Almeida and W. Herdy, Phys. Rev. D 93 (2016) 105042; arXiv:1605.04921v1 [hep-th].