#  Investigations on Effective Electromagnetic and Gravitational Scenarios
Doctoral Thesis

Marcelo Granzotto Campos  Supervisor: Prof. José Abdalla Helayël-Neto
Supervisor: Prof. Leonardo Ospedal Prestes Rosas  Centro Brasileiro de
Pesquisas Físicas - CBPF

(June 2022)

##  Agradecimentos

À Amada/Ao Amado Ser, Consiência, Absoluto, que é amor, que é verdade, que
tudo me deu e a todas, todos e tudo habita, agradeço pela experiência
vivenciada, pelas pessoas que trouxeste até mim ao longo desse caminho, pelas
situações proporcionadas, pelos aprendizados de toda ordem realizados, por
tudo que ouvi, vi e saboreei, por cada momento, cada passo e cada respiração.

Minha amada mãe Maristela, obrigado pelo amor e carinho presentes em cada
instante. Suas palavras são emanações de Deus que se manifestam através do seu
ser, pois provêm do fundo do seu coração. Suas palavras me fortaleceram
durante esses oito anos de jornada, sempre me incentivaram e me reconfortaram.

Meu amado pai José Antônio, com o seu amor e esforço, o senhor se dedicou
durante sua vida a assegurar um ensino libertador para mim. Veja-se como
grande responsável pelo alcançado pelo seu filho, que se materializa através
das linhas escritas nessa Tese.

Minha amada tia Dorita, que transcendeu desse plano recentemente, obrigado
pelo carinho e amor com os quais, em incontáveis vezes, nos recebeu em sua
casa e nos proporcionou momentos tão agradáveis, felizes e acolhedores.

Meu amado tio Orestes e minhas amadas primas-irmãs Andrea, Patrícia e
Fernanda, juntos à tia Dorita, vocês estão em memórias repletas de felicidades
e amor no meu coração. Tudo o que vivenciamos foi envolvido por esses
sentimentos divinos, ainda que em circunstâncias difíceis, essas foram
suavizadas pelo amor e pelo carinho. Obrigado a vocês. Amada Fernanda,
obrigado pelo generoso auxílio com a tradução para a língua inglesa.

Minha amada Tatiana, você caminhou ao meu lado por toda essa estrada,
compartilhou e conviveu cada curva, vale e cume. Por meio do amor e carinho
você me apoiou e renovou, ajudando-me a seguir em frente. Agradeço a você por
tanto e à/ao Ser por cruzar nossas existências nessa passagem terrena.

Amados Professores, Orientadores e amigos Helayël e Leo, obrigado por
alicerçarem as bases dessa realização. Professor Helayël, o senhor foi a luz
na minha trajetória do Doutorado. Com toda a sua suavidade, paciência, bondade
e empatia abraçou a minha vida acadêmica desde o primeiro dia que fui ao CBPF
conversar com o senhor para pleitear uma vaga sob sua preciosa Orientação,
antes mesmo de ter ingressado na instituição. Ser seu orientando e aluno e
receber seu valioso conhecimento científico e de vida é incalculável. Obrigado
por tudo o que o senhor me proporcionou dentro e fora de sala de aula.
Professor Leo, sua dedicação ao aluno e o seu tato para abordar as diversas
situações que se interpõem são tesouros que você carrega em seu coração.
Obrigado por cada diálogo, ensinamento, ajuda e paciência empreendidos nesses
anos e pela amizade desenvolvida.

Aos amados Professores Accioly e Mário Junior, agradeço-lhes pelas
contribuições construtivas e enriquecedoras agregadas ao projeto de Doutorado.

Ao amado Professor Sebastião, obrigado por nos enriquecer academicamente e
enquanto pessoas através dos cursos ministrados e das conversas amigáveis
sobre temas científicos e sociais.

Aos meus amados amigos de décadas (ou seriam séculos?) Bruno Homem, Daniel
Prieto, Luiz Direito, Rafael Chapinoti, Raphael Cadete, Ricardo Brunet, Ruben
Leon e Victor Fonseca, que são presentes ininterruptamente no meu coração e
constituem o refúgio no qual atraco com segurança.

Às amadas servidoras e aos amados servidores do CBPF, através do Fabiano como
o representante do serviço de guarda, da dona Maria como a representante pelo
serviço de limpeza, do Francisco Leonardo pelo pessoal de apoio, da Bete e
Cláudia Vinhas pelo administrativo e secretariado, a cada trabalhador não
citado diretamente, obrigado por se dedicarem ao pleno funcionamento da
instituição. Obrigado pelas portas abertas, por nos receberem, por limparem os
espaços, por assegurarem a iluminação, por suprirem os materiais utilizados,
por realizarem as devidas manutenções nas instalações e equipamentos, pelos
serviços administrativos, por cada hora de trabalho entregue, de modo a
possibilitar o desenvolvimento da ciência em nosso país.

Obrigado às amadas brasileiras e aos amados brasileiros que me transportaram,
pavimentaram as ruas e cimentaram as calçadas por onde caminhei, que limparam
esses caminhos, que plantaram os alimentos que me nutriram, que cozinharam,
que construíram a moradia e demais instalações que estive, que extraíram os
materiais em sua forma bruta, processaram eles, armazenaram os mesmos, às e
aos que costuraram as roupas vestidas, às e aos que me vacinaram, às e aos que
me serviram nos postos de saúde e hospitais, às e aos porteira(o)s e demais
funcionária(o)s dos prédios que habitei (Auguto, Pedro, seu Valdir, José,
Valteír, David, seu Severino, Seu Gildo, Dona Maria), às e aos que garantem
energia, gás e água e a toda sorte de serviço e trabalho que compuseram essa
imensa interdependência social e proveram os meios para realização desse
pequeno fruto científico brasileiro. Dedico esse às trabalhadoras, aos
trabalhadores e aos marginalizados, pois, na verdade, delas e deles é esse
trabalho. Eu apenas concluo algo que começou muito antes com elas e eles.
Dessas trabalhadoras, desses trabalhadores e marginalizados, foi furtada a
possibilidade de estarem em minha posição e por isso eu me comovo (e me
envergonho), pois ocupo um lugar de privilégios e de exclusividades, sabotado
e negado a elas e eles em grande maioria. Como posso dormir em paz se desfruto
de condições que elas e eles não desfrutam e, no entanto, são igualmente
merecedora(e)s e digna(o)s de tal. Saibam que fiz o melhor trabalho que pude.
Espero que de algum meio ou de alguma forma isso possa se reverter em algum
benefício sensível para vocês. Seguirei meu caminho servindo de coração a
vocês, da melhor maneira possível.

##  Abstract

The work aims effective and low-dimensional systems. Some different contexts
involving gravitational and electromagnetic interactions are investigated. The
electromagnetic one approaches bosonic and fermionic Effective Quantum Field
Theories non-minimally coupled in three spacetime dimensions submitted to the
expansion of Foldy-Wouthuysen Transformation, what generates
(non-)relativistic corrections. A study of the effects of an external
electromagnetic field derived from the Maxwell-Chern-Simons Electrodynamics on
the obtained interactions are executed, as well as the impact produced by the
dimensional reduction on expanded higher dimensional fermionic system in
comparison to the low-dimensional one. In the scenario of gravitational
effective model, scalar and fermionic particle scatterings reveal inter-
particles interactions beyond monopole-monopole, leading to velocity and spin
contributions, and the results are compared to a modified Electrodynamics
effective model. A non-perturbative model resourcing to Casual Dynamics
Triangulation data is adopted to serve as consistency check of the potentials
resultants. Low-dimensional Maxwell-Higgs effective models with modified
kinetic terms are studied, submitting them to a Bogomol’nyi prescription-type
for calculation of inferior (non-trivial) bound energy and the self-dual
equations. Vortex solutions for gauge field non-specified by an ansatz are
achieved and their topological feature detailed.

Keywords: low-dimensional phenomenology; non-minimum couplings; quantum
gravity; vortices.

##  Resumo

O trabalho é voltado para sistemas efetivos e de baixa dimensionalidade.
Alguns diferentes contextos são investigados, incluindo Teorias de Campos
Quânticos Efetivos bosônicos e fermiônicos não-minimamente acoplados em três
dimensões espaco-temporais submetidos a expansão da Transformação de Foldy-
Wouthuysen, o que gera correções (não-)relativísticas. Uma análise dos efeitos
de um campo eletromagnético externo derivado da Eletrodinâmica de Maxwell-
Chern-Simons sobre as interações obtidas é considerada, da mesma forma que os
impactos produzidos por uma redução dimensional sobre o sistema fermiônico de
dimensionalidade maior em comparação com o de baixa dimensionalidade. No
cenário de modelo efetivo gravitacional, espalhamentos de partículas escalares
e fermiônicas revelam interações entre partículas além de monopolo-monopolo,
levando a contribuições de velocidade e spin, e os resultados são comparados
aos de um modelo efetivo de Eletrodinâmica modificada. Um modelo não-
perturbativo recorrendo a dados de Triangulação Dinâmica Causal é adotado para
servir como cheque de consistência dos potenciais resultantes. Modelos
efetivos de Maxwell-Higgs de baixa dimensão com termos cinéticos modificados
são estudados, submetendo-os a uma prescrição tipo Bogomol‘nyi para cálculo de
limite mínimo (não-trivial) de energia e as equações auto-duais. Soluções de
vórtices para campo de calibre não especificado através de palpite são
alcançadas e as características topológicas deles detalhadas.

Palavras-chave: fenômenos planares; acoplamentos não-mínimos; gravidade
quântica; vórtices.

##  Presentation and Contextualization

###  A Tribute to Paulo Freire - One Brazilian Exponent: Educator,
Intellectual and Militant

“Se não amo o mundo,

se não amo a vida,

se não amo os homens,

não me é possível o diálogo.”

(Paulo Freire)

The thesis starts with a brief and unassuming tribute to Paulo Reglus Neves
Freire, the brilliant Brazilian Professor Paulo Freire, in his birth
centenary. He was born in Recife, capital of Pernambuco state, in 1921,
September $19^{th}$. He developed and elaborated a revolutionary methodology
of education, centered in the idea of promoting the liberty of the students by
the education, considering the education process as an exchanging of knowledge
among teachers and students, developing a critical conscience in the students
by self-inquiring and the capacity of reading the world that is around them,
moving them away from the oppressed condition. In parallel, criticizing the
conventional educational methodology (untitled by him as "bank account
methodology") which assumes the students as an empty box ("bank account"),
inspired in the Aristotle and Locke‘s philosophy of the "tabula rasa", where
is deposited the information, so the students are only docile recipients of
what is brought to them as ultimate truth. He used to say: "it is about to
learn how to read the reality to then be able to rewrite this reality." His
endeavor to teach poor people how to read and write became a inspiration in
Latin America and Africa. Paulo Freire and some mates, from 1960 to 1962,
worked in the Brazilian North East region alphabetizing adults. In the famous
case of Angicos, a small city in the Rio Grande do Norte state, where they
alphabetized 300 persons in 40 hours, he registered:

"Three hundred persons were alphabetized in Angicos in less than 40 hours. Not
only alphabetized. 300 persons were raising awareness in Angicos. Three
hundred persons were learning how to read and write, and arguing Brazilian
problems (…) Themes as regional and national development, base reformulation,
which is among them the constitutional, nationalism, imperialism, profit
remittance abroad, illiterate’s vote, "colonelism", socialism (…) were debated
with the participants of the Circles. We had the opportunity to watch some of
those debates. Impressed me the attitude of decision they revealed (…) -
"Ma’am do you know what is exploitation? \- asked certain visitor to one of
the Circles participant (…) - Maybe you, whom is a rich young person \- said
her - do not know. Me, who is a poor woman, know what exploitation is." "I
make shoes, and now I discover that I have the same value of the versed who
makes books." "The land only lives because the peasant works." "The union
results in force: if the designer draws the building, it is the workman who
knows the brick that builds it, and both forces united that make the
progress." "

In the begin of 1964, he was named the general coordinate of the National Plan
of Alphabetization during João Goulart‘s Presidency. But, few months later,
with the military political coup, he was arrested for seventy days and, then,
exiled. During the exiling in Chile he published his first book in Brazil
("Education: the Practice of the Freedom"), organized alphabetization planning
for Africa countries, worked to the Movement of Agrarian Reform of the
Christian Democracy and wrote in 1968 his most famous work - Pedagogy of
Oppressed -, which was translated for over than 40 languages, third most cited
reference in social science in the world. In 1979, he was amnestied, but
returned to Brazil in 1980, affiliating to the Workers Party (Partido dos
Trabalhadores) and assuming the Secretary of Education of São Paulo city from
1989 to 1991. Received 30 titles of Doctor of Honoris Causa and 6 in memoriam.

We selected "The Concept of Technology", one excelsior Brazilian literature
piece of his Master, Professor Álvaro Borges Vieira Pinto, which Paulo Freire
used to refer to him as the "Brazilian Master", (in an independent translation
version):

"The role of the philosophers belonging to underdeveloped environment in the
comprehension of their world, of the reasons of such conditions and in the
propositions of guidelines and political and cultural actions capable of
transforming the ambient reality is decisive. To do it so, however, is
imperative, as initial point, to comprehend what signify being a philosopher
in a poor and dependent country. The first requirement consists in to accept
that it can not signify the same thing being a philosopher in a developed,
imperialist and autonomous country and in one which vegetates in
underdevelopment, in the ignorance of the literate knowledge and in the
absence of sovereignty and capacity of determination and management of its own
existence process as particular historical being. In the underdeveloped world
and in the largest extension illiterate, the philosopher, to think
authentically the reality, must be illiterate. It does not mean, evidently, to
ignore the ability of reading and writing - nevertheless, we well know that it
is not exclusively this lack what constitutes the illiteracy - but because
firstly, in the attempting of conceiving and interpreting the world in its
real conditions, among them is included being a world of illiterates. One will
consider odd the culture accumulation and the multiple cogitations, past and
present, known through the study of books, a subsidiary source, although
indispensable, to the formation of the conscience of oneself. One will have to
learn much more from what sees than from what reads. The philosophic
conscience will only be legitimate if explains the state of its environment,
not as an external passive reflection, even truthful, but by the apprehension
of the social being essence in which the thinker is part of. The philosopher
must identify himself with the illiterate masses, constitute the figure
apparently paradoxical of the illiterate literate, to reach the foundations
which support his thinking with maximized possibilities of legitimacy."

After this masterful piece, which refreshes in our minds and hearts the role
of the philosopher, as well as it extends to professors, scientists, teachers
and a vast number of positions (possibly to all one) in the society, in an
underdeveloped and dependent country, we initiate the contextualization and
presentation of the thesis.

###  Back to Physics

Maybe looking to mathematics as a tool conceded by God to the humanity to
describe the world in which it is inserted, the ordinary physical world, one
could accept that, as any regular tool, it accomplishes the finality what was
designed for, but, in parallel, has its limitations, mainly performing a task
different than the one it is project to. Thereby, for example, thinking the
irrational feature of the pi number, one could interpret it as result of the
tool (mathematical) limitation to describe the world of ideas, once the tool
is made for the non-ideal world description. Additionally, maybe the infinity
and continuum are essences of the ideas Universe (or mental world), and, if
so, is comprehensive to expect some deviations produced by the tool, taking
into account its foundations is in a discrete counting base, and artifices
like the Dirac‘s delta to comply with both Nature features. Therefore,
imprecision occurs in attempt to cover a wide spectra of energy with a single
theory. Effective theories restricts the covered energy range, what
circumvents the obstacle of parametrization of the infinity and permits a more
suitable application of the mathematics. Besides, the continuum is
sufficiently reproduced, specially in theories with presence of higher
spacetime derivatives, which delineates more accurately this feature,
decreasing the intrinsic "lack" characteristic of the discrete system of
counting (what is aggravated, for instance, by the description of the
interaction system by point particles exchanging a mediated point particle).
In a concurrent way, theories covering undefined spacetime dimensionality or
carrying high dimension, when dimensional reduced, tends to diverge from one
in low dimension due to the algebra inaccuracy to manipulate the continuum and
infinity. Then, a tool, intrinsically based on discrete counting, describing
continuity and infinity, two inseparable properties, is vulnerable to generate
divergent and imprecise theories, mainly the ones that try to cover wide
ranges, say, of energy, of space, of dimensions. Thus, theories restricted in
its range action are an alternative to address both Natures imposition, in a
way that the inaccuracies of the tool are not sensible, or at least,
observable to the practiced measurements. After some reflection and to
divagate about Nature aspects, the present thesis works on effective and low
dimension theories analysing particles interactions in some scenarios, where
is investigated gravitation in semi-classical arrangement, Maxwell
electrodynamics and extensions.

The chapter 1 opens the debate about the gratifying work [1] developed with
Leonardo P.R. Ospedal, which is centred in effective low-dimensional systems
in bosonic and fermionic context, non-minimally coupled. The expectation is,
through a Foldy-Wouthuysen Transformation, to unfold these systems in
Hamiltonian formalism and emerge (non)relativistic corrections. It is also
questioned the influence of an external electromagnetic field on the
interaction terms constituent of the Hamiltonian originated from the
transformation, in this way, is proposed to alter the Maxwell Electrodynamics
by the Maxwell-Chern-Simons one in the fermionic scenario. A second
questioning is raised, in which is inquired the impacts on the interactions
formulation when a Foldy-Wouthuysen transformed higher dimension system is
dimensional reduced, and a comparative evaluation is carried out confronting
it with the previous low-dimensional result. This chapter closes with some
lines of conclusions and comments regarding to its content.

The chapters 2 and 3, which are part of the work [2] in collaboration with
Gustavo P. de Brito, Judismar T. Guaitolini Jr., Leonardo P.R. Ospedal and Kim
P.B. Veiga, move the effective theories inspection to the perspective of
gravitational interaction. At this site, the physics is brought to the four
spacetime dimensions and a gravitation model, with higher derivative terms
containing form factors, is the starting point to calculation of inter-
particle potentials beyond the monopole-monopole interaction, considering
scalars and fermions. Then, the chapters are two parts of one body, where the
chapter 2 is essentially theoretical-methodological approach presentation and
the potentials resulted for scalar and fermionic scattering. In chapter 3 is
discussed the latter chapter results and pushed them into some analysis.
Firstly, comparing them to results of a modified effective Electrodynamics
inter-particle potentials, published in the paper ref. [3] of "Republic of
Diracstan" collaborators and presented in the PhD Thesis [4]. Secondly is
taking a non-perturbative model subsidized by Casual Dynamics Triangulation
data [5] to verify the consistency of the potential results. This chapter is
finalized with partial conclusions and perspectives.

The last chapter 4 is an investigation in progress, dedicated to low-
dimensional Maxwell-Higgs models with modified kinetic term. The modification
aggregates a strong electromagnetic background field to the systems. The
Bogomol’nyi type prescription is adopted to express the inferior (non-trivial)
bound energy and the self-dual equations. Hence, the results are obtained in
vortex configuration, promoting an evaluation about the topological feature of
them. Some effort to work with an unspecified gauge field is done, in a manner
that, it is attempted to determine it naturally by the evolution of the
calculations. It has been developed few variations of the models and progress
have been made on it. At this stage is commented in the partial conclusions
section the overall results so far and works on progress.

The general conclusions concerning the PhD program and incoming possibilities
are synthesized in the Finals Conclusions topic 5. A Foldy-Wouthuysen
Transformation review and Fourier transform integrals belonged to potential
expressions in Chapters 2 and 3 are in the appendixes A and B.

###### Contents

  1. Agradecimentos
  2. Abstract
  3. Resumo
  4. Presentation and Contextualization
    1. A Tribute to Paulo Freire - One Brazilian Exponent: Educator, Intellectual and Militant
    2. Back to Physics
  5. 1 Effective Theories and Non-minimal Couplings in Low-dimensional Systems
    1. 1.1 Introduction
      1. 1.1.1 A Brief Historical Review
      2. 1.1.2 Foldy-Wouthuysen Transformation and  Non-minimal Couplings
    2. 1.2 Non-minimal Electromagnetic Coupling
    3. 1.3 Low-dimensional Systems
      1. 1.3.1 Scalar Field
      2. 1.3.2 Fermionic Field
    4. 1.4 Dimensional Reduction Analysis
    5. 1.5 Partial Conclusion
  6. 2 Beyond Monopole-Monopole Gravitational Interactions
    1. 2.1 Introduction
    2. 2.2 Linearized Gravity
    3. 2.3 The Effective Action: A Brief Review
    4. 2.4 Non-relativistic Potentials
      1. 2.4.1 Spin-0 External Particles
      2. 2.4.2 Spin-1/2 External Particles
  7. 3 Spin and Velocity Corrections in Gravitational Potentials
    1. 3.1 Scalar and Fermionic Gravitational Potentials: Aspects and Comparisons
    2. 3.2 Comparison with Electromagnetic Potentials
    3. 3.3 Form Factors Motivated by Quantum Gravity Model
    4. 3.4 Partial Conclusion
  8. 4 Modified Maxwell-Higgs Model in an Effective Scenario  Work in Progress
    1. 4.1 Introduction
    2. 4.2 Maxwell-Higgs Model with a Strong Electromagnetic Background Field
      1. 4.2.1 The Model and Field Equations
      2. 4.2.2 Energy and Bogomol’nyi Type Equations
      3. 4.2.3 Topological Vortices Solutions
    3. 4.3 Partial Conclusion
  9. 5 Concluding Considerations
  10. A Foldy-Wouthuysen Transformation Review
  11. B Integrals

##  Chapter 1 Effective Theories and Non-minimal Couplings in Low-dimensional
Systems

###  1.1 Introduction

####  1.1.1 A Brief Historical Review

The low dimensional systems are active participants in the history of
theoretical and experimental physics researches. Over the years, physicists
developed a brilliant group work in such arrangements, understanding its
phenomenological particularities, grubbing new lands, formulating theories,
discovering properties in materials and creating new ones.

It seems opportune to recapitulate some remarkable novels that compound the
mentioned history. Here is recorded few moments of a period incredibly vast of
fruitful research productions. There is no intention of covering minutely the
theory and/or the history, even it was the case, the author has no knowledge
enough to accomplish it. Sailing through the superconductors and superfluids
sea, aiming to reach topological phases of matter, one returns to the second
decade of twentieth century, precisely at 1911, when Heike Kamerlingh Onnes
[6] studied the electric resistance of solid mercury at the cryogenic
temperature of the liquid helium and observed that its electric property
disappeared in such condition. Paul Dirac largely contributed to elucidated
the conceit, interpretation and importance of symmetry to Quantum Mechanics.
His perspective converges to the principle of emergence in condensed matter,
which states that the properties of a material are mainly determined by how
particles are organized in the material. One of his works in 1926 [7] and a
later one in 1931 [8] guided the formulation of symmetry and topological
perspectives, which probably influenced Landau later. In early of 1930, Lev
Landau, a prominent theorist, for describing superconductivity, published his
theory [9] to describe quantizations of cyclotron orbits of charged particles,
resulting in degenerated and discrete energy values, called Landau levels.
Three years later, Meissner and Ochsenfeld [10] found out that magnetic field
could not deeply penetrate in lead cylinders cooled below a critical
temperature, but only in an extreme thin region on surface (the London length
[11]), what was named the Meissner effect. London brothers in 1935 proposed a
phenomenological Electrodynamics model [11] to describe superconductivity
based on Meissner findings and on two fluids theory of Hendrik Casimir e
Cornelis Gorter [12]. Landau resurged in 1937 with the theory of second order
phase transitions, introducing the concept of an order parameter that
increases, starting from zero, for systems at the critical temperature and
lower ones [13]. He realized that the difference among the phases (or orders)
is due to the fact they have different symmetries, thus a phase transition is
a change of the symmetry. Before the closure of thirty´s, Pyotr Kapitza and,
independently, J.F. Allen and A.D. Misener, with the papers published in
sequence [14, 15], discovered the superfluid 4He, in which is supposed that
the transition to superfluid occurs via Bose-Einstein condensation [16, 17],
and presents properties like zero viscosity.

The postulation of a phenomenological model for description of
superconductivity, dividing the superconductors in two classes according to
their behaviour in a magnetic field, was proposed by Vitaly Ginzburg and Lev
Landau in 1950 [18]. The different phases of matter are classified by a pair
of groups, which are the symmetry group of the system and the unbroken
symmetry group of the equilibrium state. Through a parameter – Ginzburg-Landau
parameter –, which transforms non-trivially under symmetry transformation, the
superconductors are distinguished in the ones which superconductivity and
strong magnetic field coexist and the ones which not. Still this year, a
German and a North American physicists, Hebert Fröhlich [19] and John Bardeen
[20], independently elaborated the idea of the superconductor state as a
result of the interaction among the electrons and the crystal atoms vibrations
(phonons) – the further interaction electron-phonon. After six years, Leon
Cooper described the process of formation of electrons pairs strongly coupled
to each other – Cooper pairs – due to their interaction with the crystals
lattice of a Fermi gas [21]. Finally, in 1957, the celebrated BCS theory [22]
– Bardeen, Cooper and Schrieffer –, a theory for microscopic
superconductivity, was published and could completely explain the
superconductivity resourcing to the Cooper pairs, which, with helium atoms,
are spherically symmetric objects, forming isotropic superfluids on
condensation. It is important to remark that, in the same period, Nikolay
Bogoliubov also explained superconductivity through the Bogoliubov
transformations [23, 24] and Philip Anderson presented his theory introducing
the pseudospins [25]. Motivated by the Ginzburg-Landau theory, still in 1957,
the young soviet physicist A. A. Abrikosov predicted vortices forming a
lattice in superconductors under strong magnetic field and their core
overlapping, what suppresses the order parameter allover the superconductor
material [26] and, hence, allows the coexistence of the field and
superconductivity. Therefore, covering an non-previewed situation by the
Meissner effect and confirming the new type of superconductor, the anisotropic
one named type-II, already appeared in Ginzburg-Landau theory and non-
contemplated by BCS theory. Later, it was showed by Gor’kov [27] that the
Ginzburg-Landau equations are passive of being deduced in an appropriated
limit from BCS theory. The invention of MOSFET – metal-oxide-semiconductor
field-effect transistor – in 1959 permitted closer analysis of the electrons
behavior and quantum effects in a nearly ideal two-dimensional space gas, once
conducted electrons travel in a thin surface layer.

In the begin of sixties, the graduating student Brian Josephson, inspired by
Giaever tunnel experiments [28], predicted new phenomena in superconductors –
the Josephson effects – [29], for instance, a Cooper pair tunneling through a
thin insulating barrier.

Ten years later (1972), the anisotropic superfluidity in 3He was discovered by
David Lee, Douglas Osheroff and Robert Richardson [30]. The condensation of
Cooper pairs formed by 3He atoms are "quasiparticles" in a way that it is an
emergent phenomenon involving many particles and oppositely to what occurs in
4He atoms, 3He generates fermions. A breakthrough in anisotropic superfluid
theory was revealed by Anthony Leggett in that year [31] (see also [32] for
reviewing), when he demonstrated that several breaks of symmetries can
manifest simultaneously in condensed matter. In fact, the pairs in 3He are
organized in spin-triplet states what incurs in broken of the rotational
symmetry in spin space, equivalently, the spontaneous symmetry break of
orbital rotation is provoked by the anisotropy of the Cooper pairs wave
functions and the gauge symmetry is broken as well, then, three symmetries are
broken in this superfluid. Three years passed to the Japanese researches
Tsuneya Ando, Yukio Matsumoto and Yasutada Uemura predicted the quantization
of the Hall conductance in planar system [33]. Another Japanese team of
researches, Jun-ichi Wakabayashi and Shinji Kawaji, observed the quantum Hall
effect in MOSFETs [34]. In three spacetime dimensions, theoretical particles
may not be submitted to Fermi-Dirac or Bose-Einstein statistics, but to both
at the same time, continuously intermingling between them as Jon M. Leinaas
and Jan Myrheim detailed in [35].

In the beginning of 1980, Klaus von Klitzing, working with samples of silicon-
based MOSFET elaborated by Michael Pepper and Gerhard Dorda, verified the
existence of exactly quantized Hall resistance, rising the current integer
quantum Hall effect [36]. The Hall resistance manifested itself in plateaus as
function of the electron and magnetic flux densities, what determine the
filling factor. Once the factor assumes integer values, degenerated electron
energy levels or Landau levels are formed in a two-dimensional space electron
gas. One year after, Robert Laughlin proposed a mental experiment [37] to
explain the exactness of Hall conductance based on gauge invariance. Disposing
of lower temperatures and magnetic fields of higher insensitivity, Störmer and
Tsui, manipulating gallium arsenide heterostructures developed by Arthur
Gossard, observed a Hall resistance plateau three times higher than the
highest measured by Klitzing and they noted fractional values for the physical
property. In this sense, in 1982 the fractional quantum Hall effect in a
quantum fluid of electrons in three spacetime dimensions arrangement was
observed [38]. Frank Wilczek published two papers [39, 40], being
chronologically one before and the other after the discovery of fractional
quantum Hall effect, where he expatiated the fractional statistics of
quasiparticle in two space dimensions and coined the term "anyon". In the next
year, Laughlin presented a phenomenological explanation to the fractional
quantum Hall effect [41], where fractionally charged quasiparticles are
created. Arovas, Schrieffer and Wilczek explicitly deduced the statistic of
quasiparticles proving that it is governed by fractional statistics and,
hence, that the particles presented in such planar systems are anyons [42]. In
1986, Bednorz and Müller working with a new class of ceramics of copper
oxides, encountered that its electric resistance was zero at the temperature
around –238°C (35.1 K) [43], the starting point in superconductivity research
in high temperatures.

In a superficial manner, the core of the theoretical and experimental
unfolding was illustrated. At this stage, it is convenient to branch the
historical review in anyons, quantum spin liquid and topological phases of
matter.

The hunting for (Abelian and non-Abelian) anyons expended over forty years
since Leinaas and Myrheim predicted theoretically the fractional statistics
particles and Störmer and Tsui achieved the superconductivity phase in
heterostructures and observed fractional quantum Hall effect. The construction
of adequate instrumentation for measuring anyons has demanded great endeavour.
Experimental attempts involving single-particles interferometers [44, 45, 46,
47, 48, 49, 50, 51] were made, but the results contained noises from Coulomb
blockage and Aharanov-Bohm interference. Current noise from fractional
quasiparticles pointed out that they carry fractional charge [52, 53]. For
non-Abelian anyons [54] some indirect evidences of this state were measured
[55, 56]. In 2020, two different teams of researches announced the detection
of anyons by micro-interferometers [57, 58].

Philip Anderson [59] suggested the existence of a new phase of matter, the
quantum liquid (the current quantum spin liquid), with a lattice formed by a
frustrated geometry of the spins disposition, what would result in a
superposition of singlet pairs of spins, named resonating valence bond. This
model failed, however, in 1987, moved by the Bednorz and Müller discovery
[43], he connected the resonating valence bond with Cooper pairs [60] and
originated the concept of spin-charge separation, in which the electron is
transformed in two quasiparticles: a spinon with neutral charge and spin-1/2
and a holon electrically charged and spin-0. In the same year, topology was
linked to quantum spin liquid by Kalmeyer and Laughlin [61], introducing the
‘chiral spin liquid’ state, and by Kivelson, Rokhsar and Sethna [62]. Two
years after, Wen proposed the modern notion of topological order, describing
it by Chern-Simons theory [63]. Around 2000, Senthil and Fisher pointed out
that spinons could present fractional quantum number [64, 65]. In 2003, Kitaev
published his toric code model [66], interrelating a two-dimensional anyonic
quantum system with quantum computation. Only passed three years and he
developed the honeycomb model disposing of quantum spin liquid ground states
and a solvable Hamiltonian [67]. Recently, in 2015, the first signatures of
Majorana fermions in two-dimensional materials were detected, and the results
matching with Kitaev honeycomb model for quantum spin liquids [68] (see also
ref. [69]). Quantum spin liquid is at the forefront current research, as
discussed in refs. [70, 71, 72].

In the third branch, Kosterlitz and Thouless, in 1972, came across with the
determinant role of topological defects (vortices) in two dimensional solids
phase transition [73, 74]. Ten years later, Thouless, Kohmoto, Nightingale and
den Nijs explained the quantization of the Hall conductance of electron gases
in two space dimensions in the absence of external magnetic field using
topological concepts [75], specified by an integer topological invariant –
TKNN invariant also called the first Chern number. Since that, passed six
years, Haldane [76] suggested in a model of graphene [77] that a quantum Hall
effect – with energy bands not following Landau levels – may also result from
breaking of time-reversal symmetry in a three spacetime dimensional system
with no net magnetic flux through it. The mentioned Chern topological index of
Thouless et al. was interpreted as a topological property of bands in two
dimensional insulators without time reversal symmetry. This phase of matter
was named Chern insulators and took twenty five years to be observed in
particular thin films by Chang et al. [78].

In 2004 Murakami, Nagaosa and Zhang theoretically predicted the spin Hall
insulators [79], describing some classes of band insulators presenting finite
spin-Hall conductivity but no charged current. After one year, Kane and Mele
[80, 81], also working on a graphene model, but preserving the time-reversal
symmetry, proposed a new insulating phase, the further called quantum spin
Hall insulator. Their model contained a $\mathcal{Z}_{2}$ topological index
which reflects the constraints imposed on the electronic states by the time-
reversal symmetry, indicating that it is a symmetry-related topological
property. Independently, few months later, the Bernevig and Zhang’s two-
dimensional semiconductors model [82] also proposed the quantized spin Hall
effect. The same former duo and Hughes elaborated a two space dimensional
model to produce a $\mathcal{Z}_{2}$ topological phase, what resulted in a
prediction of generation of quantum spin Hall effect in semiconductor quantum
wells [83]. In less than one year, electronic transport measurements executed
by König et al. [84] confirmed that a thin layer of mercury and tellurium
compound is a topological insulator. In 2007, Novoselov et al. reported the
detection of integer quantum Hall effect in graphene at room temperatures
[85]. Haldane and Raghu [86] described the unidirectional electromagnetic
waves propagation effect in photonic crystals analogous to quantum Hall
effect. Since this description, photonic topological phases [87] has been
aroused the attention of physicists, with the capacity to provide robust
unidirectional channels for light propagation [88]. Photonic topological
insulators [89], similar to quantum spin Hall states, were recently proposed
in planar meta-surfaces [90, 91]. See also ref. [92] for a more detailed
material about the paragraph contents.

For last, a relatively recent predicted material by Wang et al. [93, 94] (see
also [95, 96, 97, 98]) is the organic topological insulator, moving into a new
research area. Organic materials are recognized by their abundant electronic
and condensed structures, although, their relatively weak interactions are a
challenge to produce highly ordered structures for identification of this type
of insulator.

This introduction tries to minimally elucidate how wide is the research in
low-dimensional systems and how deeply it composes the forefront of physics
researches. The number of ramified areas under investigation increases, so
that, this simple introduction is far away from encompass all the branches of
low dimensionality studying.

####  1.1.2 Foldy-Wouthuysen Transformation and
Non-minimal Couplings

The Foldy-Wouthuysen (FW) Transformation, which is also called Pryce-Tani-
Foldy-Wouthuysen Transformation [99, 100, 101, 102], is detailed in appendix
A. This transformation enables to diagonalize the Hamiltonian in an
approximate way or, in some cases, in an exact form (check some examples in
[103, 104, 105, 106, 107]). The Foldy and Wouthuysen methodology, initially
developed to set the Dirac equation in a comparable form to the Pauli
Hamiltonian [100], is applied to describe particles with different spin
configurations [107, 108, 109, 110, 111, 112], as well as to analyse
fundamental and excited states of atomic nuclei [113], some interactions with
gravitational and torsion fields [114, 115, 116, 117, 118]. It was also
considered in scenarios with Lorentz symmetry violation [119, 120, 121, 122],
accelerated frames [123, 124], topological defects [125, 126] and topological
insulators [127, 128]. There are different approaches to the FW
transformation, in a way that, in the present work is adopted the non-
relativistic method, explained in the Appendix A. For details on relativistic
methods and their validity, see [129] and references therein.

The FW transformation is also a methodology suitable for systems in different
spacetime dimensions rather than four. In a seminal paper by Binegar [130],
where he structured the irreducible representations of the Poincaré group in
three-dimensions, the author also contemplated the correspondent FW
transformation for a free particle. Here, it is considered the effects of non-
minimal electromagnetic couplings in interactions of three-dimensional
systems. The inclusion of these couplings contributes, among others, to
effects such as the Aharonov-Casher phase [131], quantum Hall effect [132,
133] and high-temperature superconductivity [134]. Moreover, the Chern-Simons
term and non-minimal couplings in three dimensions allow spinless particles to
acquire anomalous magnetic moments [135, 136]. Therefore, investigations on FW
transformation in electromagnetic systems non-minimally coupled have potential
to reveal new interactions and relativistic corrections to the scientific
community and contribute to the literature.

This chapter starts with an introductory section about non-minimal coupling,
contemplating definitions and conventions. The section 1.3 contains the
resultant interactions of scalar and fermionic effective systems in low-
dimensional scenario and submitted to the Foldy-Wouthuysen transformation. It
is also analysed the impacts due to substitution of the Maxwell
Electrodynamics by the Maxwell-Chern-Simons one, which are structuring the
external electromagnetic field in the fermionic case, and some comments around
this modification and their consequences are elaborated. In section 1.4 is
evaluated the influence of a dimensional reduction on a fermionic higher
dimensional system FW transformed when compared to the low-dimensional result
of the antecedent section. This chapter, in section 1.5, finishes with the
conclusions and perspectives.

###  1.2 Non-minimal Electromagnetic Coupling

The local presence of an electromagnetic field at the vicinity of a charged
field naturally provokes changes in the equation structures. These changes are
manifested through several theories. One manifestation emerges from the local
transformations of the charged field, demanding the insertion of the Abelian
gauge field in order to assure the Lagrangian invariance due to the
transformation and promoting the redefinition of the regular derivative to the
covariant one due to the addition of the minimal coupling (see, for instance,
[137]). In the Hamiltonian formalism [138, 139], one observes the modification
of the canonical moment with the presence of the gauge vector potential, as
well as the gauge scalar potential, both yielding from Helmholtz Theorem
[140], representing the consequences of the external electromagnetic field
presence acting on a charged particle. A third manifestation comes from the
Topological Theory [141], where the electric and magnetic fields are agents
responsible for varying the velocity of the particle in direction and module,
implying a similar behaviour of gravitation geometry. Therefore, the
appearance and a possible interpretation to the minimal coupling as an affine
connection gains relevance and a common origin for electromagnetism and
gravity. Then, the minimal coupling in four spacetime dimensions,

$D_{\mu}=\partial_{\mu}+iqA_{\mu}\,,$ (1.1)

alters the conventional derivative $\partial_{\mu}$ to the covariant one
$D_{\mu}$, which carries the gauge potential $A^{\mu}\equiv A_{\,\bf
a}^{\mu}X^{\bf a}$, where $A_{\bf a}^{\mu}$ are fields associated to each
generator $X^{\bf a}$ of the group symmetry. In the present case, the Abelian
one, it works with the simplest scenario of $U(1)$ group, $X^{\bf a}=1$ and
one has $A^{\mu}=(\phi,\vec{A}\,)$. The coupling constant $q$ assumes the
electrical charge dimension in a system settled in natural unity $c=\hbar=1$.

Whether the system under analyses is a planar one, the dual
$\widetilde{F}_{\mu}$ of the field strength
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is a rank-1 tensor,
bringing it to a vector condition
$\widetilde{F}_{\mu}\equiv\frac{1}{2}\epsilon_{\mu\nu\kappa}F^{\nu\kappa}$,
with $\epsilon_{\mu\nu\kappa}$ representing the Levi-Civita symbol obeying the
convention $\epsilon^{012}=\epsilon_{012}=+1$ and the Minkowski metric
$\eta_{\mu\nu}=\textrm{diag}(+,-,-)$. Hence, this particular property of the
planar configuration opens the possibility to write a generalized prescription
to the covariant derivative

$\mathcal{\mathfrak{D}}_{\mu}=\partial_{\mu}+iqA_{\mu}+ig\widetilde{F}_{\mu}\,,$
(1.2)

with the introduction of the non-minimal coupling represented by the dual
tensor $\widetilde{F}_{\mu}$ and the new coupling constant $g$, which performs
the anomalous magnetic dipole moment and points out the feature of effective
field theory, once it acquires negative mass dimension, $[g]=-1/2$, implying
in a non-renomalizable theory. It is worthy to observe that the non-minimal
coupling conserves the gauge invariance of the system. Throughout the
development of the calculations is defined the dual vector
$\widetilde{\vec{f}}_{i}=\epsilon_{ij}\vec{f}_{j}$. In this context, it is
implicit $\epsilon_{ij}\equiv\epsilon_{0ij}$, where Latin indexes denote the
purely spatial sector $i,j=(1,2)$, and, consequently, $\widetilde{\vec{f}}$ is
perpendicular to $\vec{f}$.

The covariant derivative eq. (1.2) is rewritten in the electromagnetic
prescription

$\displaystyle i\partial_{t}\,\rightarrow\,\Pi^{0}\equiv
i\partial_{t}-q\phi+gB\,,$ (1.3a)

$\displaystyle\vec{p}\,\rightarrow\,\vec{\Pi}\equiv\vec{p}-q\vec{A}+g\widetilde{\vec{E}}\,,$
(1.3b)

it already reveals the anomalous magnetic dipole moment $gB$ and the
generalized canonical moment $\vec{\Pi}$. The contributions from the non-
minimal coupling, contrarily from the minimal one, are manifested in terms of
the electric and magnetic fields themselves. In time, remembering that, in
three spacetime dimensions, the electric and magnetic fields are given by
$\vec{E}=-\vec{\nabla}\phi-\partial_{t}\vec{A}$ and
$B=\vec{\nabla}\times\vec{A}\equiv\epsilon_{ij}\partial_{i}\vec{A}_{j}$.

The non-minimal coupling shows up in a wide variety of applications in the
literature. One relevant appearance is in the fractional quantum Hall effect
theme, precisely the magnetic field redefinition in Jain’s model for composite
fermionic objects [142] is effectively described by Helayël-Neto and Paschoal
[143] through means of a Maxwell–Chern–Simons (MCS) gauge field model non-
minimally coupled to matter. It is also possible to achieve fractional spin
resourcing to this kind of coupling term, which configures an alternative to
the Chern-Simons one [144, 145, 146]. Dalmazi (and Mendonça) analyzed the
influence of the non-minimal coupling of Pauli-type in static potentials for
planar scalar and fermionic Electrodynamics cases (see [147, 148], and
references therein). There are also investigations applying
$\widetilde{F}_{\mu}$ in supersymmetry [149, 150]. Moreover, in ref. [151],
the authors demonstrated that neutral bosonic particles acquire magnetic
properties when immersed in a scenario with Lorentz symmetry violation.
Additionally, a comparative evaluation demonstrates a similar behaviour among
inter-particle potentials in non-commutative Maxwell-Chern-Simons
Electrodynamics minimally coupled to matter and in the MCS with non-minimal
Pauli interaction [152].

The panorama about non-minimal Abelian gauge coupling and, derived from it,
non-minimal electromagnetic prescription articulated above paves the road to
explorer its application in some particular cases involving coupling to scalar
and fermionic matter fields. These cases, since they become structured in
Hamiltonians, are subjected to the FW transformation, what reveals
(non)relativistic corrections to the systems. In the following two sections
are developed these steps for scalar and fermionic fields, respectively.

###  1.3 Low-dimensional Systems

####  1.3.1 Scalar Field

The scalar field is taken firstly. It reveals a distinct particularity in this
arrangement of non-minimal coupling in planar system, that is the coupling of
the scalar particle to the magnetic field, probing, through a Pauli-type term,
magnetic dipole moment [135, 136]. Moreover, there is a possibility of
generating a pure Chern-Simons term by spontaneous symmetry breaking of a
generalized Abelian Higgs non-minimally coupled model [153, 154]. The
relevance of anyons in the phenomenology is notorious and its possibility of
description resourcing to scalar fields and non-minimal coupling turned
attentions to this kind of structure.

A simpler formulation is chosen: a scalar field carrying charge and matter
non-minimally coupled to the gauge field, i.e., the Klein Gordon Lagrangian
(density),

$\displaystyle\mathcal{L}_{KG}=\left(\mathcal{\mathfrak{D}}_{\mu}\varphi\right)^{\ast}\,\mathcal{\mathfrak{D}}^{\mu}\varphi-m^{2}\varphi^{\ast}\varphi\,.$
(1.4)

Manipulating eqs. (1.1) and (1.2) permits one to write the Lagrangian in terms
of the covariant derivative

$\displaystyle\mathcal{L}_{KG}$ $\displaystyle=$
$\displaystyle\left(D_{\mu}\varphi\right)^{\ast}D^{\mu}\varphi-m^{2}\varphi^{\ast}\varphi-g\,J^{\mu}\widetilde{F}_{\mu}+g^{2}\varphi^{\ast}\varphi\widetilde{F}_{\mu}^{2}\,,$
(1.5)

and to define the three-current

$J^{\mu}=i\left[\left(D^{\mu}\varphi\right)\varphi^{\ast}-\left(D^{\mu}\varphi\right)^{\ast}\varphi\right]\,.$
(1.6)

As anticipated, the coupling constant assumes the physical property of the
anomalous magnetic dipole moment and the term $g\,J^{\mu}\widetilde{F}_{\mu}$
evidences the occurrence of this effect coupled to the scalar field $\varphi$.
The term $g^{2}\varphi^{\ast}\varphi\widetilde{F}_{\mu}^{2}$ is characteristic
of the scalar system. Terms proportional to quadratic and cubic powers of
$\varphi^{\ast}\varphi$ are suitable to compose the Lagrangian (1.5),
nonetheless, they are out of the research scope due to the interest in terms
coupled to external electromagnetic field.

The FW procedure demands the Hamiltonian of the system under investigation. To
determine it for the case expressed through eq. (1.5), one starts from the
equation of motion

$(\mathcal{\mathfrak{D}}_{\mu}^{2}-m^{2})\,\varphi=0\,,$ (1.7)

which is passive of being written in two first-order equations with the
inclusion of the auxiliary field $\chi_{\mu}$

$\displaystyle m\chi_{\mu}=\mathcal{\mathfrak{D}}_{\mu}\varphi\,,$ (1.8a)
$\displaystyle\mathcal{\mathfrak{D}}^{\mu}\chi_{\mu}=m\varphi\,.$ (1.8b)

However, taking the spatial sector of eq. (1.8a) with $\mu=i$, the constraints
$\chi_{i}=\frac{1}{m}\Pi_{i}\varphi$ is manifested, remaining only $\chi_{0}$
and $\varphi$ as dynamical fields. Working on the temporal sector of eqs.
(1.8), is possible to establish the relations

$\displaystyle i\partial_{t}\varphi=m\chi_{0}+(q\phi-gB)\varphi\,,$ (1.9a)
$\displaystyle i\partial_{t}\chi_{0}=m\varphi+(q\phi-
gB)\chi_{0}+\frac{\vec{\Pi}^{2}}{m}\varphi\,,$ (1.9b)

and recast them in a Schrödinger-like equation, $i\partial_{t}\rho=H_{0}\rho$,
with a two-component field

$\rho=\left(\begin{array}[]{c}\rho_{a}\\\
\rho_{b}\end{array}\right)=\frac{1}{2}\left(\begin{array}[]{c}\varphi+\chi_{0}\\\
\varphi-\chi_{0}\end{array}\right)\,.$ (1.10)

These algebraic manipulation described above allow to set the Hamiltonian

$H_{0}=\frac{\vec{\Pi}^{2}}{2m}\left(\mathcal{R}+\mathcal{N}\right)+m\mathcal{N}+(q\phi-
gB)\mathbb{I}\,,$ (1.11)

where $\mathbb{I}$ denotes the identity $2\times 2$, $\mathcal{R}=i\sigma_{y}$
and $\mathcal{N}=\sigma_{z}$, with $\sigma_{y}$ and $\sigma_{z}$ being the
Pauli matrices.

Recovering the formulation presented in Appendix A, more specifically in eq.
(A.6), $H_{0}=\beta m+\mathcal{E}+\mathcal{O}$, the operators $\mathcal{E}$
and $\mathcal{O}$ are

$\mathcal{O}=\frac{\vec{\Pi}^{2}}{2m}\mathcal{R}\,\;,\,\;\mathcal{E}=\frac{\vec{\Pi}^{2}}{2m}\mathcal{N}+(q\phi-
gB)\mathbb{I}\,,$ (1.12)

noticing that, in three spacetime dimensions, $\beta\equiv\mathcal{N}$. Thus,
the diagonalization process culminates in the expression (A.11) which
deliveries a diagonalized Hamiltonian $H_{FW,0}$ up to order $O(1/m^{3})$,

$\displaystyle H_{FW,0}$ $\displaystyle\approx$ $\displaystyle
m\mathcal{N}+\mathcal{E}+\mathcal{N}\frac{\mathcal{O}^{2}}{2m}-\frac{1}{8m^{2}}[\mathcal{O},\,[\mathcal{O},\mathcal{E}]+i\dot{\mathcal{O}}]-\mathcal{N}\frac{\mathcal{O}^{4}}{8m^{3}}\,,$
(1.13)

where the leading terms of the positive energy solution are

$\displaystyle H_{FW,0}$ $\displaystyle\approx$ $\displaystyle
m+\frac{(\vec{p}-q\vec{A}+g\widetilde{\vec{E}}\,)^{2}}{2m}+q\phi-
gB-\frac{(\vec{p}-q\vec{A}+g\widetilde{\vec{E}}\,)^{4}}{8m^{3}}\,.$ (1.14)

This Hamiltonian contains the familiar non-relativistic kinetic term composed
by the canonical momentum $\vec{\Pi}=\vec{p}-q\vec{A}+g\widetilde{\vec{E}}$.
It is worthy to signalize this canonical momentum is capable of generating the
Aharonov-Casher and Aharonov-Bohm phases [131, 155, 144]. The electric
potential term $q\phi$ is also present. In the sequence, the anomalous
magnetic dipole interaction $gB$ and the last term contributes to the
relativistic mass correction. In a heuristic way, this mass correction emerges
from the Taylor expansion of the relativistic kinetic energy
$\sqrt{\vec{\Pi}^{2}+m^{2}}$. Finally, it is highlighted that there is no term
proportional to $O(1/m^{2})$, which is a particularity of the spin-0 system.

The fermionic case is developed in the next subsection, following the same
steps adopt for the spin-0 system, with the presentation of the Lagrangian,
structuring of the Hamiltonian and the calculation of the Foldy-Wouthuysen
transformation. Furthermore, the resulted Hamiltonian in eq. (1.14) is
obtained again in the next subsection. It occurs, as will be demonstrated, due
to the convergence of a fermionic system into a scalar one, when the former is
exposed to a Maxwell-Chern-Simons external electromagnetic field in a
particular settlement.

####  1.3.2 Fermionic Field

At this stage, the fermionic field system is investigated. The presence of
spin, being a quantum number, widely enriches the physical content,
aggregating the group representation manifested in the Pauli matrices, new
degrees of freedom and the dynamic associated to it, also the spin-statistics
theory and spin peculiar properties as its unpredictability presented in
Stern-Gerlach experiment (see, for instance, [156]), or its states rising in
discrete values and so on.

Taking the massive fermionic field non-minimally coupled to an external field,
represented through the Dirac Lagrangian,

$\mathcal{L}_{D}=i\bar{\psi}\gamma^{\mu}\mathcal{\mathfrak{D}}_{\mu}\psi-m\bar{\psi}\psi\,,$
(1.15)

or in an unfolded expression resourcing to the minimal covariant derivative
(1.1),

$\mathcal{L}_{D}=i\bar{\psi}\gamma^{\mu}D_{\mu}\psi-m\bar{\psi}\psi-g\,j^{\mu}\widetilde{F}_{\mu}\,,$
(1.16)

with $j^{\mu}=\bar{\psi}\gamma^{\mu}\psi$ and
$\bar{\psi}=\psi^{\dagger}\gamma^{0}$. It is convenient to introduce some
definitions. The Dirac representation conducts the three dimensional spacetime
matrix algebra, which is defined in terms of the Pauli matrices:
$\gamma^{0}=\sigma_{z}$, $\gamma^{1}=i\sigma_{x}$ and
$\gamma^{2}=i\sigma_{y}$, respects the Clifford algebra
$\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=2\eta^{\mu\nu}\mathbb{I}\,$ and
satisfies the identity

$\gamma^{\mu}\gamma^{\nu}=\eta^{\mu\nu}\mathbb{I}-i\epsilon^{\mu\nu\kappa}\gamma_{\kappa}\,.$
(1.17)

The traditional Dirac equation is written in function of
$\mathcal{\mathfrak{D}}_{\mu}$,

$\left(i\gamma^{\mu}\mathcal{\mathfrak{D}}_{\mu}-m\right)\psi=0\,,$ (1.18)

and defining $\beta\equiv\gamma^{0}$ and
$\vec{\alpha}\equiv\beta\vec{\gamma}$, one is able to rearrange eq. (1.18)
into the form $i\partial_{t}\psi=H_{D}\,\psi$ and to determine the Dirac
Hamiltonian

$H_{D}=m\beta+\vec{\alpha}\cdot\vec{\Pi}+\left(q\phi-gB\right)\mathbb{I}\,.$
(1.19)

Since the Hamiltonian is found, one is in condition of beginning the FW
procedure. Then, the terms of $H_{D}$ are grouped in the operators
$\mathcal{E}$ and $\mathcal{O}$

$H_{D}=m\beta+\mathcal{E}+\mathcal{O}\,,$ (1.20)

$\mathcal{O}=\vec{\alpha}\cdot\vec{\Pi}\,\;,\,\;\mathcal{E}=\left(q\phi-
gB\right)\mathbb{I}\,,$ (1.21)

remembering the commutation and anti-commutation relationships pointed out in
Appendix A, $\mathcal{E}\beta=\beta\mathcal{E}$ and
$\mathcal{O}\beta=-\beta\mathcal{O}$, respectively. Proceeding to the last
stage of the diagonalization of $H_{D}$ – eq. (1.19) –, the odd and even
operators specified in (1.21) are substituted in the diagonalized Hamiltonian
$H_{FW}$ – eq. (A.11) –

$\displaystyle H_{FW,D}$ $\displaystyle\approx$ $\displaystyle
m\beta+\mathcal{E}+\beta\frac{\mathcal{O}^{2}}{2m}-\frac{1}{8m^{2}}[\mathcal{O},\,[\mathcal{O},\mathcal{E}]+i\dot{\mathcal{O}}]-\beta\frac{\mathcal{O}^{4}}{8m^{3}}\,.$
(1.22)

At this point is convenient to observe that the structures of the diagonalized
Hamiltonians in three spacetime dimensions for the scalar eq. (1.13) and
fermionic eq. (1.22) cases are the same (recovering that
$\beta=\mathcal{N}=\sigma_{z}$), where both derived from eq. (A.11). The
leading terms for positive energy solutions of the Hamiltonian resulted from
the substitutions are

$\displaystyle H_{FW,D}$ $\displaystyle\approx$ $\displaystyle
m+\frac{(\vec{p}-q\vec{A}+g\widetilde{\vec{E}}\,)^{2}}{2m}+q\phi-
gB-\frac{q}{2m}B-\frac{g}{2m}\vec{\nabla}\cdot\vec{E}$ (1.23) $\displaystyle-$
$\displaystyle\frac{iq}{8m^{2}}\,\vec{\nabla}\times\vec{\mathbb{E}}\,-\frac{q}{4m^{2}}\,\vec{\mathbb{E}}\times(\,\vec{p}-q\vec{A}+g\widetilde{\vec{E}}\,)\,-\frac{q}{8m^{2}}\,\vec{\nabla}\cdot\vec{\mathbb{E}}\,,$

with

$\vec{\mathbb{E}}\equiv\vec{E}+\frac{g}{q}(\,\vec{\nabla}B+\partial_{t}\widetilde{\vec{E}}\,)\,.$
(1.24)

Immediately at the first glance is verified the effects of the presence of the
spin generating a Hamiltonian with a physical content more robust than the
spin-0 scenario in $H_{FW,0}$ (eq. (1.14)). At the first line of $H_{FW,D}$
are the nonrelativistic contributions, then, comparing them with the ones in
$H_{FW,0}$ is patent the existence of two new terms in eq. (1.23). The first
from left to right is the usual Pauli term $\,-\frac{q}{2m}B\,$ representing
the dipole magnetic moment of spin, in which the spin projection
$S_{z}=\sigma_{z}/2$ is not evident, and the next is a Darwin-type term
$\,-\frac{g}{2m}\vec{\nabla}\cdot\vec{E}\,$ coming from the non-minimal
coupling [157]. Moving to the relativistic contributions sector at the second
line of $H_{FW,D}$ and starting from left to right, one comes across the spin-
orbit term with the non-minimal corrections. At last, a Darwin term concerning
to a correction on the electrostatic energy $q\phi$ due to the electron
fluctuation position (for detailing the physical origin of the last two terms,
see ref. [158]). Whether $g$ is taken trivial, $g=0$, hence
$\,(\vec{\mathbb{E}}\rightarrow\vec{E})\,$, the spin-orbit components and the
Darwin term assume the familiar interaction form of the four dimensions
system. The results of $H_{FW,D}$ are restricted to order $O(1/m^{2})$, in a
way that the relativistic mass correction to the kinetic term appeared in the
spin-0 system, $-(\vec{p}-q\vec{A}+g\widetilde{\vec{E}}\,)^{4}/8m^{3}\,$, is
unrevealed in spin-1/2 Hamiltonian eq. (1.23), once it is a contribution
proceeding from the $\mathcal{O}^{4}$ term (of order $O(1/m^{3}))$ in eq.
(1.22). In the coming section 1.4, a dimensional reduction is performed on a
FW transformed Hamiltonian for a massive fermionic system non-minimally
coupled in four spacetime dimensions and the resulting interactions are
compared to the ones of $H_{FW,D}$ in eq. (1.23).

So far, the whole discussion works with no specification of the
Electrodynamics which dictates the external electromagnetic field behaviour.
For a more refined and profound examination, probing subtle and particular
information about the intrinsic physics interactions, is specified the
external Electrodynamics, adopting Maxwell-Chern-Simons theory through the
Lagrangian

$\displaystyle\mathcal{L}_{MCS}=-\frac{1}{4}F_{\mu\nu}^{2}+\frac{\lambda}{2}\epsilon^{\mu\alpha\beta}A_{\mu}\partial_{\alpha}A_{\beta}-J^{\mu}A_{\mu}\,,$
(1.25)

where $J^{\mu}=(\rho,\vec{J})$ is an unspecified current and $\lambda$ denotes
the Chern-Simons parameter. MCS theory, in a context of topologically massive
gauge theory was extensively studied by Deser, Templeton and Jackiw in ref.
[159]. The equations of motion from $\mathcal{L}_{MCS}$ are

$\displaystyle\vec{\nabla}\times\vec{E}=-\partial_{t}B\,,$ (1.26a)
$\displaystyle\vec{\nabla}\cdot\vec{E}-\lambda B=\rho\,,$ (1.26b)
$\displaystyle\widetilde{\vec{\nabla}}B-\lambda\widetilde{\vec{E}}=\vec{J}+\partial_{t}\vec{E}\,.$
(1.26c)

An appropriated manipulation on eq.(1.26c) allows to rewrite it as
$\,\vec{\nabla}B+\partial_{t}\vec{\widetilde{E}}=\lambda\vec{E}-\vec{\widetilde{J}}\,$,
which is in a suitable form to be substituted in eq. (1.24), altering it to

$\vec{\mathbb{E}}=\left(1+\frac{\lambda
g}{q}\right)\vec{E}-\frac{g}{q}\,\widetilde{\vec{J}}\,.$ (1.27)

This new configuration to $\vec{\mathbb{E}}$ suggests some inspections for
particular values of the constants and current contented in the equation.
Therefore, initially rescuing the Maxwell theory in vacuum, $\lambda=0$ and
$\widetilde{\vec{J}}=\vec{0}$, what leads to
$\vec{\mathbb{E}}\rightarrow\vec{E}$ and, consequently, is verified that the
only relativistic correction by non-minimal coupling remained in eq. (1.23) is
an electric density energy $\frac{q}{4m^{2}}g\vec{E}^{\,\,2}$. Within the
approximations practiced in eq. (1.23), the non-minimal relativistic
interactions are relevant only in the presence of matter, where
$\vec{J}\neq\vec{0}$. Zeldovich anapole moment in four spacetime dimensions
also presents a close behaviour, vanishing in absence of matter [160]. A
second inspection occurs setting $\widetilde{\vec{J}}=\vec{0}$ and $\rho=0$,
giving the Maxwell-Chern-Simons theory in vacuum. An effective form for the
electric field appears
$\vec{\mathbb{E}}\rightarrow\vec{E}_{\textrm{eff}}=\left(1+\frac{\lambda
g}{q}\right)\vec{E}$ and the same is applicable to the magnetic field
$B_{\textrm{eff}}=\left(1+\frac{\lambda g}{q}\right)B$ just managing the Pauli
and Darwin-type terms, situated at the non-relativistic contributions sector
of $H_{FW,D}$, with the modified Gauss equation
$\vec{\nabla}\cdot\vec{E}-\lambda B=0$ (eq. (1.26b)). These effective forms
for the electric and magnetic fields depend on the value of the coupling
constant $g$, what permits a critical value to it, namely $g_{c}=-q/\lambda$.
In the critical condition, the relativistic corrections for $H_{FW,D}$ are
null and the Pauli and Darwin-type terms cancel each other, remaining

$H_{c}\approx
m+\frac{(\vec{p}-q\vec{A}+g_{c}\widetilde{\vec{E}}\,)^{2}}{2m}+q\phi-g_{c}B\,$
(1.28)

that is the convergence of the spin-1/2 Hamiltonian to the spin-0 one (eq.
(1.14)) up to order $O(1/m^{2})$. The critical situation appears in different
physical contexts and is demanded to achieve specific effects, for example as
a special condition to acquire first order vortex solutions [161] or
disappearance of one-loop quantum corrections to the photon mass [162].

###  1.4 Dimensional Reduction Analysis

At this stage is brought up a philosophical reflection about the
interpretation and connection among physical systems in different
dimensionalities. Upon this point, by construction, systems in low dimensions
were studied, thus, they carry implicitly the conception of a planar geometry
as representative of the Nature. A second proposal is to start from higher
dimensions, so as to describe a system as a whole and containing the aimed
physical environment. Then, executing a dimensional reduction, it creates a
specie of delimited view on the original system, establishes particular
physical conditions to it, approximates the higher to the low dimensional
environment and brings physical information from the original one. Therefore,
it seems worthy to inspect what impact a physical system originally designed
in higher dimensions suffer under a dimensional reduction and, also, what
information is aggregated or, even, lost with the procedure. Witten, Qi and
Zhang demonstrated theoretical advantages of structuring in higher dimensions
topological superconductors [163], while in ref. [164] the authors presented
an emergence of a dark sector through a dimensional reduction from a five
dimensional Electrodynamics coupled to 3-form gauge field to four dimensional.
An investigation over the transformation of the spin properties under
compactification of a spatial dimension is done in ref. [165] and Helayël-Neto
and Ospedal calculated new effects for interparticle potentials initially in
five dimensions and, then, reduced to four [166]. Analysis of effects of
boundary conditions on fermionic and bosonic dimensionally reduced models are
developed by Cavalcanti et al [167]. The listed papers are some examples of
how the dimension reduction, and its philosophy inherent, is a potential theme
to reflect and debate. In the current research, is consider a fermionic field
non-minimally coupled to an external electromagnetic field in four dimensions,
which Hamiltonian is diagonalized by FW prescription and its result,
(non)relativistic interactions, are submit to a dimensional reduction
rendering a three spacetime dimensions physical system. The reduced
Hamiltonian is analytically compared to the one originally construct in three
dimensions (1.23). Henceforth, the indexes nomenclature is
$\hat{\mu},\hat{\nu}=(0,1,2,3)$ and the Minkowski metric
$\eta_{\hat{\mu}\hat{\nu}}=\textrm{diag}(+,-,-,-)$ for four dimensional
spacetime and the three dimensional space vectors are in bold, e.g., the gauge
potential is denoted as $A^{\hat{\mu}}=(\phi,\vec{\bm{A}})$.

The described fermionic Lagrangian is the Dirac one with the minimal covariant
derivative and the Pauli interaction as an extra term,

$\mathcal{L}_{DP}=i\bar{\psi}\gamma^{\hat{\mu}}D_{\hat{\mu}}\psi-m\bar{\psi}\psi+\frac{f}{2}\,\bar{\psi}\sigma^{\hat{\mu}\hat{\nu}}\psi\,F_{\hat{\mu}\hat{\nu}}\,$
(1.29)

with $D_{\hat{\mu}}=\partial_{\hat{\mu}}+iqA_{\hat{\mu}}$ and
$\sigma^{\hat{\mu}\hat{\nu}}=\frac{i}{2}\left[\gamma^{\hat{\mu}},\gamma^{\hat{\nu}}\right]$.
The non-minimal coupling constant $f$, as occurred in the low dimensional
cases, has negative mass dimension $[f]=-1$, signalizing that the theory
should be treated as effective. The Pauli interaction, recently investigated
in the context of spin Hall effect [168], is passive of being connected to the
non-minimal coupling in three dimensions $j^{\mu}\widetilde{F}_{\mu}$. The
resulting manipulation of the previous coupling and of the eq. (1.17) is
susceptible to be written as
$\frac{1}{2}\,\bar{\psi}\sigma^{\mu\nu}\psi\,F_{\mu\nu}$, representing, in
three spacetime dimension, the analogous to the Pauli term. In ref. [144], the
authors worked on some comparisons in the non-relativistic limit involving
these couplings, and now, is intended to extended the analyses to the
relativistic interactions, connecting them though dimensional reduction.

The equation of motion of the Lagrangian (1.29) is

$\left(i\gamma^{\hat{\mu}}D_{\hat{\mu}}-m+\frac{f}{2}\,\sigma^{\hat{\mu}\hat{\nu}}F_{\hat{\mu}\hat{\nu}}\right)\psi=0\,,$
(1.30)

where the gamma matrices are in the Dirac representation

$\gamma^{0}=\left(\begin{array}[]{cc}\mathbb{I}&0\\\
0&-\mathbb{I}\end{array}\right)\,\;,\,\;\gamma^{i}=\left(\begin{array}[]{cc}0&\sigma_{i}\\\
-\sigma_{i}&0\end{array}\right)\,.$ (1.31)

Reorganizing the equation of motion (1.30) and managing some algebra, is
obtained the Dirac-Pauli Hamiltonian

$\displaystyle H_{DP}$ $\displaystyle=$ $\displaystyle\beta
m+\vec{\bm{\alpha}}\cdot\vec{\bm{\pi}}+q\phi\,\bm{I}-if\,\beta\,\vec{\bm{\alpha}}\cdot\vec{\bm{E}}+f\,\beta\,\vec{\bm{\Sigma}}\cdot\vec{\bm{B}}\,,$
(1.32)

in which $\bm{I}$ stands for the identity $4\times 4\,$,
$\vec{\bm{\pi}}\equiv\vec{\bm{p}}-q\vec{\bm{A}}$, $\beta\equiv\gamma^{0}$,
$\vec{\alpha}\equiv\beta\vec{\gamma}$ and $\vec{\bm{\Sigma}}$ corresponds to
the spin matrix

$\vec{\bm{\Sigma}}=\left(\begin{array}[]{cc}\vec{\sigma}&0\\\
0&\vec{\sigma}\end{array}\right)\,.$ (1.33)

Exactly as proceeded in lower dimensional systems, the resulting Hamiltonian,
in this case, the Dirac-Pauli one in eq. (1.32), is set in terms of odd
$(\mathcal{O})$ and even $(\mathcal{E})$ operators ($H=\beta
m+\mathcal{E}+\mathcal{O}$), which are defined as

$\displaystyle\mathcal{O}$ $\displaystyle=$
$\displaystyle\vec{\bm{\alpha}}\cdot\vec{\bm{\pi}}-if\,\beta\,\vec{\bm{\alpha}}\cdot\vec{\bm{E}}\,\;\,,\,\;\,\mathcal{E}=q\phi\,\bm{I}+f\,\beta\,\vec{\bm{\Sigma}}\cdot\vec{\bm{B}}\,.$
(1.34)

Substituting $\mathcal{O}$ and $\mathcal{E}$ in eq. (1.22) and considering the
four dimensional gamma matrices eq. (1.31), one is prompted to the
diagonalized Hamiltonian (for positive energy solutions)

$\displaystyle H_{FW,DP}$ $\displaystyle\approx
m\,\mathbb{I}+\frac{1}{2m}\,\left(\vec{\bm{\pi}}\,\mathbb{I}-f\vec{\bm{E}}\times\vec{\sigma}\right)^{2}+q\phi\,\mathbb{I}+\left(f-\frac{q}{2m}\right)\vec{\sigma}\cdot\vec{\bm{B}}-\frac{f^{2}}{2m}\vec{\bm{E}}^{2}\,\mathbb{I}$
(1.35)
$\displaystyle+\left(\frac{f}{2m}-\frac{q}{8m^{2}}\right)\vec{\nabla}\cdot\vec{\bm{E}}\,\mathbb{I}-i\,\frac{q}{8m^{2}}\,\vec{\sigma}\cdot\left(\vec{\nabla}\times\vec{\bm{E}}\right)-\,\frac{q}{4m^{2}}\,\vec{\sigma}\cdot\left(\vec{\bm{E}}\times\vec{\bm{\pi}}\right)$
$\displaystyle-\frac{qf}{4m^{2}}\,\vec{\bm{E}}^{2}\,\mathbb{I}-i\,\frac{f}{8m^{2}}\,\left[\vec{\nabla}\cdot\left(\partial_{t}\vec{\bm{E}}\right)\right]\,\mathbb{I}+\frac{f}{8m^{2}}\,\vec{\sigma}\cdot\left(\vec{\nabla}\times\partial_{t}\vec{\bm{E}}\right)$
$\displaystyle+i\,\frac{f}{4m^{2}}\,\left(\vec{\nabla}\cdot\vec{\bm{B}}\right)\vec{\sigma}\cdot\vec{\bm{\pi}}+\frac{f}{4m^{2}}\,\mathbb{I}\left[\partial_{t}\vec{\bm{E}}-\vec{\nabla}\times\vec{\bm{B}}\right]\cdot\vec{\bm{\pi}}$
$\displaystyle+\frac{f}{8m^{2}}\,\left[\nabla^{2}\left(\vec{\sigma}\cdot\vec{\bm{B}}\right)\right]+i\,\frac{f}{4m^{2}}\,\left[\left(\vec{\sigma}\cdot\vec{\nabla}\right)\vec{\bm{B}}\right]\cdot\vec{\bm{\pi}}-\frac{f}{2m^{2}}\,\vec{\bm{B}}\cdot\Bigl{[}\Bigl{(}\vec{\sigma}\cdot\vec{\bm{\pi}}\Bigr{)}\vec{\bm{\pi}}\Bigr{]}$
$\displaystyle+\frac{f^{2}}{4m^{2}}\,\vec{\sigma}\cdot\left(\vec{\bm{E}}\times\partial_{t}\vec{\bm{E}}\right)-\frac{f^{2}}{4m^{2}}\,\left[\vec{\sigma}\cdot\vec{\nabla}\Bigl{(}\vec{\bm{E}}\cdot\vec{\bm{B}}\Bigr{)}\right]+\frac{f^{2}}{4m^{2}}\,\left[\vec{\bm{E}}\cdot\vec{\nabla}\Bigl{(}\vec{\sigma}\cdot\vec{\bm{B}}\Bigr{)}\right]$
$\displaystyle-\frac{f^{2}}{4m^{2}}\,\vec{\bm{E}}\cdot\left[\Bigl{(}\vec{\sigma}\cdot\vec{\nabla}\Bigr{)}\vec{\bm{B}}\right]-\frac{f^{2}}{4m^{2}}\,\left[\vec{\bm{B}}\cdot\vec{\nabla}\Bigl{(}\vec{\sigma}\cdot\vec{\bm{E}}\Bigr{)}\right]$
$\displaystyle-\frac{f^{3}}{2m^{2}}\,\Bigl{(}\vec{\bm{E}}\cdot\vec{\bm{B}}\Bigr{)}\Bigl{(}\vec{\sigma}\cdot\vec{\bm{E}}\Bigr{)}.$

This Hamiltonian with terms until order $O(1/m^{2})$ contains new interactions
and properties which demand some notes. Most of the (non)relativistic
interactions are fruit of the non-minimal coupling, hence, a familiar
expression is recovered for $f=0$ [157] with the habitual spin-dependent
interactions $\,\vec{\sigma}\cdot\vec{\bm{B}}$,
$\,\vec{\sigma}\cdot(\vec{\nabla}\times\vec{\bm{E}})\,$ and
$\,\vec{\sigma}\cdot(\vec{\bm{E}}\times\vec{\bm{\pi}})\,$, noticing that
$\vec{S}=\vec{\sigma}/2$. At the non-relativistic sector are the previewed
anomalous magnetic moment $\,f\vec{\sigma}\cdot\vec{\bm{B}}$ and the
correction to Darwin-type term $\,\frac{f}{2m}\vec{\nabla}\cdot\vec{\bm{E}}$.
Along the Hamiltonian, at the fourth line, the expressions
$\,\vec{\nabla}\cdot\vec{\bm{B}}\,$ and
$\,\partial_{t}\vec{\bm{E}}-\vec{\nabla}\times\vec{\bm{B}}\,$ are maintained
once the Electrodynamics is unspecified. In case of Maxwell theory in vacuum
describing the external electromagnetic field, both contributions are
identically null. In counterpart, $\,\vec{\nabla}\cdot\vec{\bm{B}}\,$ may
perform contribution in the presence of magnetic monopole, for example, as
suggested in ref. [169] for spin ice configuration. In parallel, extensions of
Maxwell Electrodynamics theories such as non-linear ones [170, 171], model
contemplating Lorentz symmetry violation [172], manifesting quantum gravity
effects [173, 174], are subjected to manifest non-trivial contributions of
Ampère-Maxwell equation, even in vacuum. The diagonalized Dirac-Pauli
Hamiltonian is compound by new spin-dependent interactions of magnitude
$f^{2}/m^{2}$ and $f^{3}/m^{2}$, involving magnetic and electric fields and
preserving parity symmetry. A last pointing is the observation that Chen and
Chiou [175], in regime of weak, homogeneous and static field, worked on the
high-order contributions of the FW transformation for the Dirac-Pauli
Hamiltonian and obtained some interactions similar to $H_{FW,DP}$.

Following the script, the next step is to reduce the dimensionality of the
system in eq. (1.35), generating a three spacetime dimension Hamiltonian (for
a detailed historical review of reduction dimension approach, consult ref.
[4]). The methodological approach adopted is similar to the Scherk-Schwarz
reduction [176], taken in a simple formulation. The core idea is to assume
that the fields are independent of the reduced spatial coordinate – "$z$" in
the present text –, what leads to $\partial_{z}(\textrm{any field})=0$. Facing
this condition, the integration of the action over the reduced coordinate
(normally taken compact) renders a length dimension factor. This factor is
absorbed by the coupling constants and fields, what assures the suitable mass
dimension in the three-dimensional spacetime. In this configuration, the gauge
vector potential $\,\vec{\bm{A}}\,$ is decomposed in a planar vector
$\,\vec{A}\,$ and a new scalar field $\,\varphi$, thus
$\,\vec{\bm{A}}=(\vec{A},\varphi)$. The calculus is conducted assuming
$\,\varphi=0$ (which is the simplest choice based on the purposes of this
Thesis), producing $\,\vec{\bm{B}}\rightarrow(0,0,B_{z})\,$,
$\,\vec{\bm{E}}\rightarrow(\vec{E},0)\,$ and $\,\vec{\bm{\pi}}_{z}\rightarrow
0\,$. Then, inserting these formulations for the electric and magnetic fields
and for the canonical moment in Dirac-Pauli Hamiltonian eq. (1.35) and
manipulating it algebraically, one finds the reduced Hamiltonian

$\displaystyle H_{\textrm{red}}$ $\displaystyle\approx
m\,\mathbb{I}+\frac{1}{2m}\,\left[\left(\vec{p}-q\vec{A}\right)\,\mathbb{I}-f\,\widetilde{\vec{E}}\,\sigma_{z}\right]^{2}+q\phi\,\mathbb{I}$
(1.36)
$\displaystyle+\left(f-\frac{q}{2m}\right)\,B_{z}\,\sigma_{z}+\frac{f}{2m}\,\vec{\nabla}\cdot\vec{E}\,\mathbb{I}$
$\displaystyle-i\,\frac{q}{8m^{2}}\,\sigma_{z}\,\vec{\nabla}\times\left[\vec{E}\,\mathbb{I}-\frac{f}{q}\,\partial_{t}\widetilde{\vec{E}}\,\sigma_{z}\right]$
$\displaystyle-\frac{q}{8m^{2}}\,\vec{\nabla}\cdot\left[\vec{E}\,\mathbb{I}-\frac{f}{q}\,\left(\vec{\nabla}B_{z}+\partial_{t}\widetilde{\vec{E}}\right)\,\sigma_{z}\right]$
$\displaystyle-\frac{q}{4m^{2}}\,\sigma_{z}\left[\vec{E}\,\mathbb{I}-\frac{f}{q}\,\left(\vec{\nabla}B_{z}+\partial_{t}\widetilde{\vec{E}}\right)\,\sigma_{z}\right]\times\left[\left(\vec{p}-q\vec{A}\right)\,\mathbb{I}-f\,\widetilde{\vec{E}}\,\sigma_{z}\right]\,,$

where is recognized $\epsilon_{3\,ij}=\epsilon_{ij}$ to recover the dual
electric field $\widetilde{\vec{E}}_{i}=\epsilon_{ij}\vec{E}_{j}$.

The eq. (1.36) precisely recovers the Hamiltonian set originally in three
spacetime dimensions in eq. (1.23). This verification is facilitated
resourcing to the associations among the magnetic fields $\,B_{z}\rightarrow
B\,$ and the non-minimal couplings $\,f\rightarrow-g\,$ and projecting the
result on the upper state ($\psi_{1}$) of the two-components spinor
$\,\psi=(\psi_{1},\psi_{2})^{T}\,$. In the end is demonstrated that the non-
minimal interactions in eq. (1.32), after a particular dimensional reduction,
become the ones in (1.23), suggesting an existence of connection through them.

Turn the attention to the matrix structure of both systems, one observes that
the spin arrangement in $H_{red}$ (eq. (1.36)) is evident
($S_{z}=\sigma_{z}/2$) and hidden in $H_{FW,D}$ (eq. (1.23)). $H_{red}$ is
generated from the Hamiltonian $H_{FW,DP}$ (eq. (1.32)) conceived in four
spacetime dimensions, based on gamma matrices $4\times 4$ and four-components
spinors. Considering only the positive energy solutions of $H_{FW,DP}$, set
this Hamiltonian in Pauli matrices and two-components spinors. On the other
hand, the three-dimensional $H_{FW,D}$ is structured in Pauli matrices and
two-components spinors, in a way that, projecting its positive energy
solutions results in a subspace $1\times 1$ and one-component spinor, hiding
the spin representation.

It is important to emphasize that the external electromagnetic field is
determinant to achieve the connection of the Hamiltonians through the
dimensional reduction. For example, taking the Maxwell Electrodynamics, the
Hamiltonians $H_{red}$ and $H_{FW,D}$ converge to the same expression.
However, remembering the Maxwell-Chern-Simons Electrodynamics in eq. (1.25),
the Levi-Civita symbol $\epsilon^{\mu\alpha\beta}$ is peculiar to three
spacetime dimensions, whereas an extension of four-dimensional Maxwell
Electrodynamics, when submitted to a dimensional reduction, results in a
theory with non-trivial equations of motion.

###  1.5 Partial Conclusion

This work is dedicated to effective low-dimensional systems carrying non-
minimal electromagnetic couplings expanded through the Foldy-Wouthuysen
transformation, revealing (non)relativistic interactions in the Hamiltonian
formalism. The procedure of expanding is applied directly on systems genuinely
constructed in the low dimension scenario or in reduced ones originated in
higher dimensions.

In the Klein-Gordon Lagrangian with the non-minimal coupling in three
spacetime dimensions, the FW Transformation until $\mathcal{O}(1/m^{3})$
presented the usual anomalous magnetic dipole interaction and the relativistic
kinetic term contributing to corrections on mass.

The three-dimensional fermionic system non-minimally coupled to
electromagnetic field, after the transformation up to $\mathcal{O}(1/m^{2})$,
unveiled that relativistic interactions are relevant only in presence of
matter $(\vec{J}\neq\vec{0})$, with the Maxwell Electrodynamics describing the
external electromagnetic field. Exchanging the Maxwell Electrodynamics by the
Maxwell-Chern-Simons one, results in non-trivial contributions even in vacuum.
Besides, in vacuum configuration, the effective electric field (eq. (1.27))
vanishes when pushed to a particular critical coupling constant
($g_{c}=-q/\lambda)$, converging Hamiltonians from bosonic and fermionic
systems to the same one. Taking the Hamiltonian generated from four
dimensional fermionic Lagrangian containing non-minimal Pauli coupling, and
reducing it dimensionality through a method similar to Scherk-Schwarz
procedure, one recovers the Hamiltonian obtained for the low-dimensional
fermionic system. This result should be taken as a particular achievement and
not be generalized, once this is not expected for systems with extension of
Maxwell Electrodynamics. This opens possibilities of further investigations,
in which are contemplated the influence of the different dimensional reduction
methods in overall results. The fermionic higher dimension is also passive of
being analysed in detail in the future due to its relativistic interactions
manifested from the FW transformation (eq. (1.35)).

One more study branch is the Landau-Lifshitz-Gilbert (LLG) equation, observing
that the magnetic sector of eq. (1.35) is capable of originating the
corresponding extension to the LLG equation and its non-minimal relativistic
interactions may result in torque or damping contributions to magnetization.
Relativistic interactions in magnetization dynamics are matter of research
(see refs. [177, 178] and references therein) and may motivate further works.

##  Chapter 2 Beyond Monopole-Monopole Gravitational Interactions

###  2.1 Introduction

One of the front line researches is the attempting to conciliate the General
Relativity and the Quantum Field Theory, fusing Riemannian geometry and
particle interaction physics. To elaborate a quantum gravity theory
conciliating unitarity and renormalizability (covering from infrared to
ultraviolet regime) stands as incompatible, until nowadays. On the other hand,
consistent unitary theories respecting a cutoff scale have been formulated as
effective field theories [179] and gained relevance in the last forty years.

The gravitation theories containing terms with higher order derivatives back
to around the begin of twenties with Weyl [180] and Eddington [181]. Utiyama
and De-Witt [182] pointed the necessity of inclusion of high-order curvature
terms in Einstein-Hilbert action to permit a renormalizable theory at one-
loop. Such theories were promoted when Stelle [183] (see also ref. [184])
demonstrated they are renormalizable in purely gravitational scenario or when
coupled to a neutral massive scalar field, although their non-unitarity
formulation, manifesting a complex pole in the modified graviton propagator
[185]. This aspect was reinforced by Johnston [186], observing that the
appearance of ghost pole is independent of the chosen gauge. In parallel, it
was verified that, whether taken the General Relativity as a perturbative
theory, this theory was non-renormalizable at one-loop condition coupled to
neutral scalar particle [187]. However, the paper kept opened the possibility
of the theory be renormalizable in a free particle scenario at two-loops. In
the middle of eighties, it was refuted by Marc Goroff and Augusto Sagnotti
[188], proving the non-renormalizability of the Einstein gravity at two-loops.

As the challenges around General Relativity renormalization and its
harmonization with Quantum Field Theory have raised, modified gravitation
theories containing high-order curvature terms have been unfolding in several
models. Among them, one finds theories that switch the Ricci curvature scalar
$R$ by a general function ${\it f(R)}$ in the Einstein-Hilbert action (see the
review [189]), massive three spacetime dimension and unitary at tree-level
[190] (for a review, see ref. [191]), non-perturbative methods considering
$4-\epsilon$ dimensionality systems [192], approaches proposing the re-scaling
of a mass parameter associated to the Einstein-Hilbert action, which leads the
system to a strongly interacting regime type [193], non-local $D$-dimensional
gravity [194] (see also ref. [195]), a Yang-Mills gauge theory assisting a
weakly coupled quadratic gravity, in a way to provide a ultraviolet completion
[196], super-renormalizable theory with complex poles [197] and propositions
involving Lee-Wick prescription to construct an unitary S-matrix [198].

In effective field theories and some other methods, the calculation of
gravitational quantum corrections for inter-particle potentials have been
received attention ( e.g. refs. [199, 200, 201, 202, 203, 204, 205]) since
pointed by Donoghue in ref. [179] that these corrections for Newtonian
potential, derived from quantum gravity model based in Quantum Field Theory,
compose an efficient evaluation of consistency of the model. The research on
inter-particles potentials reaches beyond the traditional monopole-monopole
interaction, delivering also velocity- and spin-dependent corrections. The
authors in [206] obtained spin contributions to a graviton exchanging between
particles of varied spin representations, Khriplovich and Kirilin [207]
calculated "spin" (in the particular case, they defined as the internal
angular momenta of rotating compound body) and velocity corrections to the
Newtonian potential for scalar particles and, latter, Kirilin compared these
results with spinorial system and calculated spin-orbit and spin-spin quantum
corrections [208]. A Dirac-Einstein system forming an effective field theory
is studied in ref. [209], where the leading quantum corrections to
gravitational coupling of a charged massive particle with spin-1/2 are
achieved. Ross and Holstein [210, 211] analysed the non-relativistic
gravitational scattering amplitudes for different spin combinations, obtaining
classical and quantum corrections in spin-orbit and spin-spin sectors. They
extended the similar investigation for amplitudes of electromagnetic-
gravitational scattering systems [212]. For theoretical and experimental
reviews about the spin role in gravity, see refs. [213, 214].

The chapters 2 and 3 are part of the fruitful collaboration with Gustavo P. de
Brito, Judismar T. Guaitolini Jr., Leonardo P.R. Ospedal and Kim P.B. Veiga,
which resulted in the paper [2]. They are organized in a total of seven
sections, so that, four are in the chapter 2 and three in the third one. The
former is divided in the section 2.2, containing some definitions and main
structural tensors of the linearized gravity; followed by sec. 2.3 with a very
short oversight about effective action definition and formulation. The section
2.4 describes the methodological approach and formalism that supported the
calculation of the gravitational potentials and in the sec. 2.4 are obtained
the inter-particles non-relativistic potentials for two scalars and two
spinors exchanging one graviton. The next chapter is composed by the sec. 3.1,
dedicated to detailing and comparison of the non-relativistic potentials
sector by sector (velocity-velocity, orbit-spin and spin-spin) and in static
limit; the subsequent section 3.2 contains a comparative analysis among the
obtained potentials and the results of ref. [3], where the authors, in a
modified (effective) electrodynamics, calculated inter-particles potentials
for spin-0, -1/2 and -1, achieving velocity and spin contributions. Then, the
results are submitted to a praxis test in section 3.3, where is chosen a
particular effective quantum gravity model for low-energy condition (based on
Casual Dynamics Triangulation) [5] to support the definition of the form
factors and, thereby, to calculate the particular inter-particle potentials.
The last section (3.4) is dedicated for final comments and conclusions.

###  2.2 Linearized Gravity

The starting point is to set the building blocks of the linearized gravity. In
this way, it is worked in a Riemannian four dimensional spacetime, where an
approximation of a local flat spacetime is regarded, keeping the coordinate
invariance. With this approximation is possible to work with the Riemannian
metric ($g_{\mu\nu}$) equal to the Minkowski one
($\eta_{\mu\nu}=\text{diag}(1,-1,-1,-1)$ \- convention adopted) summed to a
smooth perturbation designated by the symmetric rank two tensor $h_{\mu\nu}$.
Making use of the familiar choice

$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu},$ (2.1)

with $\kappa$ being the Einstein constant ($\kappa=\sqrt{32\pi\text{G}}$ , G
denotes the Newtonian constant) and also considering $|\kappa h_{\mu\nu}|\ll
1$. It is demanded $g_{\mu\alpha}g^{\alpha\nu}=\delta_{\mu}^{\,\,\,\nu}$, what
results in

$g^{\mu\nu}=\eta^{\mu\nu}-\kappa
h^{\mu\nu}+\kappa^{2}h^{\mu\alpha}{h_{\alpha}}^{\nu}+...,$ (2.2)

still in time, it is defined

$\sqrt{-\text{det}(g_{\mu\nu})}\equiv\sqrt{-g}=1+\frac{1}{2}\kappa
h+O(h^{2}),$ (2.3)

in which $h\equiv{h_{\mu}}^{\mu}$.

The next step is the linearization of the main geometric elements of the
General Relativity. Thus, the Riemann-Christoffel connection $\Gamma$
($\Gamma^{\alpha}_{\mu\nu}=\frac{1}{2}g^{\alpha\rho}(\partial_{\mu}g_{\nu\rho}+\partial_{\nu}g_{\mu\rho}-\partial_{\rho}g_{\mu\nu})$),
after linearization, assumes the form

$\Gamma^{\alpha}_{\mu\nu}=\frac{1}{2}\kappa(\partial_{\mu}{h_{\nu}}^{\alpha}+\partial_{\nu}{h_{\mu}}^{\alpha}-\partial^{\alpha}h_{\mu\nu})+O(\kappa^{2}),$
(2.4)

in the same way, the Riemann-Christoffel tensor
(${R^{\alpha}}_{\mu\nu\beta}=\partial_{\nu}\Gamma^{\alpha}_{\mu\beta}-\partial_{\beta}\Gamma^{\alpha}_{\mu\nu}+\Gamma^{\alpha}_{\rho\nu}\Gamma^{\rho}_{\mu\beta}-\Gamma^{\alpha}_{\rho\beta}\Gamma^{\rho}_{\mu\nu}$),
the Ricci tensor ($R_{\mu\nu}={R^{\alpha}}_{\mu\nu\alpha}$) and the curvature
scalar ($R=g^{\mu\nu}R_{\mu\nu}$) are linearized, respectively,

${R^{\alpha}}_{\mu\nu\beta}=\frac{1}{2}\kappa(\partial_{\mu}\partial_{\nu}{h_{\beta}}^{\alpha}+\partial^{\alpha}\partial_{\beta}h_{\mu\nu}-\partial_{\mu}\partial_{\beta}{h_{\nu}}^{\alpha}-\partial_{\nu}\partial^{\alpha}{h_{\mu}}^{\beta})+O(\kappa^{2}),$
(2.5)

$R_{\mu\nu}=\frac{1}{2}\kappa(\Box
h_{\mu\nu}+\partial_{\mu}\partial_{\nu}h-\partial_{\rho}\partial_{\mu}{h_{\nu}}^{\rho}-\partial_{\rho}\partial_{\nu}{h_{\mu}}^{\rho})+O(\kappa^{2}),$
(2.6)

$R=\kappa(\Box h-\partial_{\mu}\partial_{\nu}h^{\mu\nu})+O(\kappa^{2}),$ (2.7)

using the symbol $\Box=\partial_{\mu}\partial^{\mu}$ to represent the
d‘Alembertian differential operator.

Those are the main elements of the Riemannian geometry and their linearized
form and the tensor $h_{\mu\nu}$, in this specific context, assumes the role
of the graviton as a spin-2 field [215, 216, 217].

###  2.3 The Effective Action: A Brief Review

The n-point Green‘s functions have a central role in the Quantum Field Theory,
once they structure the S-Matrix elements. They are the expected vacuum value
of the chronological ordered product of the field operators

${G_{n}}=\langle 0|\textit{T}\Phi_{1}\Phi_{2}...\Phi_{n}|0\rangle.$ (2.8)

In the functional approach, one extracts the Green‘s function $G_{n}(x)$
(where $x\equiv x_{1},x_{2},...,x_{n}$) of a system from the differentiation
of its generating functional $Z[J]$ by an external source $J(x)$

$i^{n}\,G(x_{1},x_{2},...,x_{n})=\dfrac{\delta^{n}{Z[J]}}{\delta{J(x_{1})}\delta{J(x_{2})}...\delta{J(x_{n})}}\,\bigg{|}_{J=0}.$
(2.9)

In a similar way, the connected Green‘s function ${G^{c}}(x)$ \- "connected"
refers to the Feynman diagrams that are impossible to separate the parts which
are not directly joined by a line - is expressed by

$i^{n}\,G^{c}_{n}(x_{1},x_{2},...,x_{n})={\dfrac{\delta^{n}{W[J]}}{\delta{J(x_{1})}\delta{J(x_{2})}...\delta{J(x_{n})}}}\,\bigg{|}_{J=0},$
(2.10)

with $W[J]$ representing the generating functional of $G^{c}_{n}(x)$. The
generating functional in eqs. (2.9) and (2.10) are related by
$Z[J]=\exp({iW[J]})$.

Now, defining the so called effective action as a functional Legendre
transformation (for more details, see ref. [218])

$\Gamma[\Phi]=W[J]+\int d^{4}x\,\Phi(x)\,J(x),$ (2.11)

with $\Phi(x)$ representing a function of infinity class decreasing fast and
$\Gamma[\Phi]$ being the generating functional of the n-point vertex function
$\Gamma_{n}(x)$, $n\geq 2$,

$\Gamma_{n}(x)=\dfrac{\delta^{n}\Gamma[\Phi]}{\delta{\Phi(x_{1})}\delta{\Phi(x_{2})}...\delta{\Phi(x_{n})}}\,\bigg{|}_{J=0},$
(2.12)

noting that $\Gamma_{n}(x)$ involves strictly the n-point connected Feynman
diagrams that are one particle irreducible - means the diagram that is
impossible, by breaking one of its line, to transform it in two - and have no
external lines (named "amputated").

After this superficial and brief presentation of the functional effective
action concept, which stands as the protagonist of this study, the focus is
moved toward the description of the methodological approach adopted for
calculation of the non-relativistic potentials.

###  2.4 Non-relativistic Potentials

#### Methodological Approach and Mathematical Formalism

It seems convenient firstly to introduce the fundamental equations,
conventions and methodology chosen to calculate the potentials. They are
achieved through the Fourier transform of the Born approximation to first
order in $V(r)$ (see, for example, ref.[219])

$V(r)=-\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}\mathcal{M}_{{}_{\textmd{NR}}}(\vec{q})\,e^{i\vec{q}\cdot\vec{r}}\,,$
(2.13)

where $\mathcal{M}_{{}_{\textmd{NR}}}(\vec{q})$ denotes the scattering
amplitude in the non-relativistic limit. The former amplitude can be obtained
from the adequate normalized relativistic scattering amplitude
($\mathcal{M}(\vec{q})$)

$\mathcal{M}_{{}_{\textmd{NR}}}(\vec{q})={\lim}_{{}_{\textmd{NR}}}\prod_{i=1,2}(2E_{i})^{-1/2}\prod_{j=1,2}(2E^{\prime}_{j})^{-1/2}\,\mathcal{M}(\vec{q})\,,$
(2.14)

noting that the normalization factor is composed by the product of the
scattered particles energies $E_{i}$ and $E^{\prime}_{j}$, which are resulted
of the Taylor expansion around the rest mass limit of the in and out going
particles, respectively.

An important remark on the physics approach methodology is the fact that the
endeavour is destined exclusively to calculations of the graviton propagator
quantum corrections. Therefore, the vertices are carried in their tree-level
processes. This structure is pictorially expressed by the Feynman diagram in
the fig. 2.1, in which the centered circle represents the quantum corrections
to the graviton propagator. Then, the relativistic scattering amplitude is
expressed by

$\displaystyle i\mathcal{M}=i\,T^{\mu\nu}(p_{1},p_{1}^{\prime})\,\langle
h_{\mu\nu}(-q)h_{\alpha\beta}(q)\rangle\,i\,T^{\alpha\beta}(p_{2},p_{2}^{\prime})\,,$
(2.15)

being $T^{\mu\nu}$ (and $T^{\alpha\beta}$) the tree-level energy-momentum
tensors of the scattered particles 1 and 2 (see fig. 2.1) and $\langle
h_{\mu\nu}(-q)h_{\alpha\beta}(q)\rangle$ the graviton propagator (accompanied
by their quantum corrections).

(2)$p_{2}$(1)$p_{1}$($2^{\prime}$)$p_{2}^{\prime}$($1^{\prime}$)$p_{1}^{\prime}$$q$$q$

Figure 2.1: Diagrammatic representation of the 3-momentum convention and
approximation adopted for the calculation of the scattering amplitude and
potential.

The opted 3-momentum conventions to the scattering amplitudes and potentials
calculations are represented in the fig. 2.1. The algebra is developed in the
center-of-mass (CM) reference frame, what allows to assume the following
3-momentum convention

$\vec{p}_{1}=-\vec{p}_{2}=\vec{p}-\frac{\vec{q}}{2}\,,\qquad\vec{p^{\prime}}_{1}=-\vec{p^{\prime}}_{2}=\vec{p}+\frac{\vec{q}}{2}\,,$
(2.16)

in which is made use of the 3-momentum arithmetic average $\vec{p}$, obtained
from each pair $\vec{p}_{i}$ and $\vec{p^{\prime}}_{i}$, and the momentum
transfer $\vec{q}$. For the analyses of the system dynamics is considered an
elastic scattering behaviour, what means the conservation of the system energy
$E_{final}=E_{initial}$. Expanding the energies around $\vec{p}_{i}=\vec{0}$
one has $E_{i}\approx m_{i}+\frac{{\vec{p}_{i}}^{2}}{2m_{i}}$, hence, applying
the mentioned conservation of the energy system and resourcing to eqs. (2.16),
the condition $\,\vec{q}\cdot\vec{p}=0\,$ is acquired. Due to the former
condition, new relations among the energies come out $E_{1}=E^{\prime}_{1}$
and $E_{2}=E^{\prime}_{2}$.

At this point, it is introduced the functional effective action formalism. The
generating functional - conventionally named effective action \- has the
general form $\Gamma=\bar{\Gamma}+\hat{\Gamma}$, in which, $\bar{\Gamma}$
expresses the sector containing diffeomorphism transformations invariance
($\delta_{\text{diff}}\bar{\Gamma}=0$) and $\hat{\Gamma}$ the sector that does
not dispose of this property ($\delta_{\text{diff}}\hat{\Gamma}\neq 0$). In
terms of covariant gauge theory, the diffeomorphism transformations are
interpreted as the gauge symmetry transformations. In this context of Quantum
Gravity Field Theory scenario, with the field $h_{\mu\nu}$ being a small
perturbation associated to the metric $g_{\mu\nu}$, the effective action
assumes the form

$\displaystyle\Gamma[h;g]=\bar{\Gamma}[g]+\hat{\Gamma}[h]\,.$ (2.17)

The eq. (2.17) is composed by a gauge invariant term ($\bar{\Gamma}[g]$) for
gauge transformations on $g_{\mu\nu}$ and by a non-invariant term
($\hat{\Gamma}[h]$), representing the gauge-fixing sector
$\delta_{h_{\mu\nu}}\hat{\Gamma}\neq 0$. The portion $\bar{\Gamma}[g]$
represents

$\bar{\Gamma}[g]=\frac{2}{\kappa^{2}}\int
d^{4}x\,\sqrt{-g}\left(-2\Lambda-R-\frac{1}{3}RF(\Box)R+C_{\mu\nu\alpha\beta}W(\Box)C^{\mu\nu\alpha\beta}\right)+\mathcal{O}(\mathcal{R}^{3})\,,$
(2.18)

with $\Lambda$ standing for the cosmological constant, $R$ the scalar of
curvature, $C_{\mu\nu\alpha\beta}$ the Weyl tensor

$\displaystyle C_{\mu\nu\alpha\beta}$ $\displaystyle=$ $\displaystyle
R_{\mu\nu\alpha\beta}+\frac{1}{2}(g_{\mu\beta}R_{\alpha\nu}+g_{\nu\alpha}R_{\beta\mu}-g_{\mu\alpha}R_{\beta\nu}-g_{\nu\beta}R_{\alpha\mu})$
(2.19) $\displaystyle+$
$\displaystyle\frac{1}{6}(g_{\mu\alpha}g_{\beta\nu}-g_{\mu\beta}g_{\alpha\nu})R,$

where $R_{\mu\nu\alpha\beta}$ and $R_{\mu\nu}$ are the Riemann and Ricci
tensors. Each one of the mentioned tensors, including the constant
$\sqrt{-g}$, will be linearized following the equations (2.7), (2.5), (2.6)
and (2.3) respectively. $F(\Box)$ and $W(\Box)$ denote the form factors as
functions of the covariant d‘Alembertian operator and
$\mathcal{O}(\mathcal{R}^{3})$ includes terms contributions involving
curvature invariant compositions of order higher than 2. The action (2.18) in
a region around Minkowskian four dimensional background scenario, represents
the more general torsion-free and parity invariant form that holds unitarity
(ghost-free) [220, 221, 222]. $\bar{\Gamma}[g]$ contains explicitly the
Einstein-Hilbert action term ($\Gamma_{\text{EH}}\sim\int d^{4}x\,\sqrt{-g}R$)
and two others geometrical ones quadratic in curvature. They are associated to
form factors, which, in this approach, once expanded the Green‘s functions,
the quantum corrections of the interactions potentials are calculated in
function of them. The effective action piece displayed above can be faced as a
generalization of the van Nieuwenhuizen proposal [223], where $F(\Box)$ and
$W(\Box)$ could assume the form of non-local arrangements [224], as adopted,
for instance, by Moffat, Barvinsky and Modesto [225, 226, 227], until the form
of local polynomials functions (some references are available in Giacchini and
Netto [228] and Accioly et al [222]).

Now, turning the attention to the expression $\hat{\Gamma}[h]$,

$\displaystyle\hat{\Gamma}[h]=\frac{1}{2\alpha}\int
d^{4}x\sqrt{-{g}}\,{\eta}^{\mu\nu}F_{\mu}[h]F_{\nu}[h]\,,$ (2.20)

where $\alpha$ is the gauge parameter and
$F_{\mu}[h]=\bar{\nabla}^{\nu}h_{\mu\nu}-\frac{1}{2}\bar{\nabla}_{\mu}h$, with
$\bar{\nabla}_{\mu}$ indicating the covariant derivative
$\bar{\nabla}_{\mu}\equiv\partial_{\mu}+\bar{\Gamma}_{\mu}$, with
$\bar{\Gamma}_{\mu}$ representing an appropriate connection. Basically, it
exhibits the same appearance of the usual gauge-fixing term in the classical
action. However, it is worked on the mass shell, what results in trivial Ward
identities and makes the calculation gauge-independent.

From the eq. (2.17) is calculated the Green‘s functions for the graviton
propagator, which are the ones described in Sec (2.3) - diagrammatically
represented by connected, amputated and one particle irreducible Feynman
diagrams. This is accomplished taking the vertex function (2.12) and
calculating the inverse of its two point function ${\delta^{2}\Gamma}/{{\delta
h}^{2}}\,\big{|}_{h=0}$. In this way, the graviton propagator reads (in
chapter 2 of ref. [218], there is the detailing of this approaching)

$\displaystyle\langle
h_{\mu\nu}(-q)h_{\alpha\beta}(q)\rangle=\frac{i}{q^{2}}\Bigg{[}\frac{1}{Q_{2}(q^{2})}\mathcal{P}_{\mu\nu\alpha\beta}^{(2)}-\frac{1}{2Q_{0}(q^{2})}\mathcal{P}_{\mu\nu\alpha\beta}^{(0)}\Bigg{]}\,+\,i\Delta_{\mu\nu\alpha\beta}(q)\,,$
(2.21)

where $Q_{0}$ and $Q_{2}$ are defined as

$\displaystyle Q_{2}(q^{2})=1+\frac{2\Lambda}{q^{2}}+2q^{2}\,W(-q^{2})\,,$
(2.22a)

$\displaystyle Q_{0}(q^{2})=1+\frac{2\Lambda}{q^{2}}+2q^{2}\,F(-q^{2})\,,$
(2.22b)

and the tensors $\mathcal{P}_{\mu\nu\alpha\beta}^{(2)}$ and
$\mathcal{P}_{\mu\nu\alpha\beta}^{(0)}$ denote

$\displaystyle\mathcal{P}_{\mu\nu\alpha\beta}^{(2)}=\frac{1}{2}(\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha})-\frac{1}{3}\eta_{\mu\nu}\eta_{\alpha\beta}\,,$
(2.23a)

$\displaystyle\mathcal{P}_{\mu\nu\alpha\beta}^{(0)}=\frac{1}{3}\eta_{\mu\nu}\eta_{\alpha\beta}\,.$
(2.23b)

Additionally, the structure $\Delta_{\mu\nu\alpha\beta}$ encloses the terms
that contracted to the energy-momentum tensors of the scattered particles and
vanish. It seems relevant to point out the disconnection feature manifested by
the propagator (2.21) through the "$Q$" factors (eqs. (2.22a) and (2.22b)).
The form factor $W(-q^{2})$ is restricted to $Q_{2}$ and, likely, $F(-q^{2})$
is to the factor $Q_{0}$, what evidences their exclusive contribution by the
tensor $\mathcal{P}_{\mu\nu\alpha\beta}^{(2)}$, in the case of $W$, and by
$\mathcal{P}_{\mu\nu\alpha\beta}^{(0)}$ to $F$.

In the next subsection is evolved the calculations of the scattering
amplitudes and potentials between two spin-0 scattered particles and two
spin-1/2.

#### Results

####  2.4.1 Spin-0 External Particles

The start point is the calculation of the gravitational potential in an
elastic scattering among massive spin-0 and spin-2 particles. The vertex is
taken in the tree level. To suppress quantum corrections for the vertex and
the external particles propagator is unfeasible when one tries to reach
results experimentally reliable. Therefore, it is decided to work with this
configuration as an approximation and a first stage for further calculations.
In this way, the scalar particles dynamic assumes the form

$\displaystyle\Gamma_{\textmd{scalar}}[\phi,g]=\int{d^{4}x\,\sqrt{-g}\Bigg{(}\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}m^{2}\phi^{2}}\Bigg{)}\,,$
(2.24)

observing that it derives from the classical Klein-Gordon Lagrangean density
submitted to the principle of minimal gravitational coupling
($d^{4}x\rightarrow{d^{4}x}\,\sqrt{-g}$. In this case, the covariant
derivative is identical to the regular derivative,
$\nabla_{\mu}\equiv\partial_{\mu}$). Then, the energy-momentum tensor resulted
from the interaction sector of eq. (2.24) is written in the momenta space,

$\displaystyle
T_{\mu\nu}(p,p^{\prime})=-\frac{\kappa}{2}\Big{[}\,p_{\mu}p^{\prime}_{\nu}+p_{\nu}p^{\prime}_{\mu}-\eta_{\mu\nu}\left(p\cdot
p^{\prime}-m^{2}\right)\Big{]}\,,$ (2.25)

noting that is considered terms up to first order in $\kappa$ and the "prime"
symbol on momenta follows the definition expressed in fig. 2.1, namely, the
particle 4-momentum after the scattering. It is worthy to remember that
$E_{1}=E^{\prime}_{1}$ and $E_{2}=E^{\prime}_{2}$ and the external particles
in the vertex are on-shell, ${p_{1}}^{2}={p^{\prime}_{1}}^{2}={m_{1}}^{2}$ and
${p_{2}}^{2}={p^{\prime}_{2}}^{2}={m_{2}}^{2}$. The algebraic manipulation of
eqs. (2.21) and (2.25) based on eq. (2.15) provides the relativistic
scattering amplitudes. Firstly, presenting the partial result

$\displaystyle
i\mathcal{M}^{(s=0)}=\frac{i}{q^{2}}\left[\frac{1}{3Q_{2}}\left(T_{1\,\,\mu}^{\mu}T_{2\,\,\beta}^{\beta}-\,3\,T_{1}^{\mu\nu}T_{2\,\mu\nu}\right)+\frac{1}{6Q_{0}}T_{1\,\,\mu}^{\mu}T_{2\,\,\beta}^{\beta}\right]\,,$
(2.26)

in which is adopted the notations $T^{\mu\nu}_{i}\equiv
T^{\mu\nu}(p_{i},p^{\prime}_{i})$ and $Q_{i}\equiv Q_{i}(q^{2})$. The
expression for the amplitude in terms of the 4-momentum and masses is

$\displaystyle\mathcal{M}^{(s=0)}$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}}{6q^{2}\,Q_{2}}\Big{[}2m_{1}^{2}m_{2}^{2}-3(p_{1}\cdot
p_{2})(p_{1}^{\prime}\cdot p_{2}^{\prime})-3(p_{1}\cdot
p_{2}^{\prime})(p_{1}^{\prime}\cdot p_{2})$ (2.27) $\displaystyle+$
$\displaystyle 2(p_{1}\cdot p_{1}^{\prime})(p_{2}\cdot
p_{2}^{\prime})-m_{1}^{2}\,p_{2}\cdot p_{2}^{\prime}-m_{2}^{2}\,p_{1}\cdot
p_{1}^{\prime}\Big{]}$ $\displaystyle+$
$\displaystyle\frac{\kappa^{2}}{6q^{2}\,Q_{0}}\Big{[}(p_{1}\cdot
p_{1}^{\prime})(p_{2}\cdot p_{2}^{\prime})-2m_{1}^{2}\,p_{2}\cdot
p_{2}^{\prime}$ $\displaystyle-$ $\displaystyle 2m_{2}^{2}\,p_{1}\cdot
p_{1}^{\prime}+4m_{1}^{2}m_{2}^{2}\Big{]}.$

Following the methodology prescribed along the subsection 2.4, it is opted by
the reference of center-of-mass and, hence, the 3-momentum transfer $\vec{q}$
and the average $\vec{p}$, both detailed in eq. (2.16). Thus, normalizing the
relativistic scattering amplitude according to eq. (2.14), one finds the non-
relativistic limit of the amplitude

$\displaystyle\mathcal{M}^{(s=0)}_{\textrm{NR}}$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}m_{1}m_{2}}{6\,Q_{2}\,\vec{q}^{\,2}}\,\left[1+\vec{p}^{\,2}\left(\frac{3}{m_{1}m_{2}}+\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)+\frac{\vec{q}^{\,2}}{8}\,\left(\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)+\mathcal{O}(3)\right]$
(2.28) $\displaystyle-$
$\displaystyle\frac{\kappa^{2}m_{1}m_{2}}{24\,Q_{0}\,\vec{q}^{\,2}}\,\left[1-\frac{\vec{p}^{\,2}}{2}\left(\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)-\frac{5\,\vec{q}^{\,2}}{8}\left(\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)+\mathcal{O}(3)\right]\,,$

where $\mathcal{O}(3)$ indicates terms higher than second order in
$|\vec{p}|/m_{1,2}$ and/or $|\vec{q}|/m_{1,2}$, which are neglected.

Then, once substituted the expression (2.28) in (2.13), the Fourier transform
of the non-relativistic scattering amplitude determines the non-relativistic
inter-particle (gravitational) potential

$\displaystyle V^{(s=0)}(r)$ $\displaystyle=$
$\displaystyle-\frac{\kappa^{2}m_{1}m_{2}}{6}\Bigg{[}I_{1}^{(2)}(r)+\vec{p}^{\,2}\left(\frac{3}{m_{1}m_{2}}+\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)I_{1}^{(2)}(r)$
(2.29) $\displaystyle+$
$\displaystyle\frac{1}{8}\left(\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)I_{0}^{(2)}(r)\Bigg{]}+\frac{\kappa^{2}m_{1}m_{2}}{24}\Bigg{[}I_{1}^{(0)}(r)$
$\displaystyle-$
$\displaystyle\frac{\vec{p}^{\,2}}{2}\left(\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)I_{1}^{(0)}(r)-\frac{5}{8}\left(\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)I_{0}^{(0)}(r)\Bigg{]},$

being the integrals $I_{n}^{(a)}(r)$ defined in Appendix B, eq. (B.1), with
$n=0,1$ and $a=0,2$. The potential $V^{(s=0)}(r)$ reveals a structure
compounded by terms beyond the monopole-monopole sector. This theme will be
debated in the section 3.1, in which both potentials results (scalar and
fermionic) are compared. In a section apart (sec. 3.2), the analyses are
extended to a comparison with the electromagnetic potential cases. The next
subsection is dedicated to the calculations of the fermionic non-relativistic
gravitational potential.

####  2.4.2 Spin-1/2 External Particles

In this subsection is established the non-relativistic gravitational potential
involving two fermionic particles. As practiced in the last subsection 2.4.1
for the scalar case, the fermionic particles are taken on-shell, calculated
quantum corrections for the graviton propagator and treated the fermion-
graviton vertices at the classical level, suppressing their quantum
corrections.

The start point is the Dirac Lagrangean density, described by the functional
action $\Gamma_{\textmd{ferm}}[\bar{\psi},\psi,g]$

$\displaystyle\Gamma_{\textmd{ferm}}[\bar{\psi},\psi,g]=\int{d^{4}x\,\sqrt{-g}\left[\frac{i}{2}(\bar{\psi}\,\gamma^{\mu}_{g}\,\nabla_{\mu}\psi-\nabla_{\mu}\bar{\psi}\,\gamma^{\mu}_{g}\,\psi)-m\bar{\psi}\psi\right]}\,,$
(2.30)

which is rewritten in the context of gravitational geometry scenario. It means
to submit it to the prescription of gravitational minimal coupling, although,
in this case, differently from what occurs in the scalar one (2.24), the
covariant derivative $\nabla_{\mu}$ carries the Lorentz connection
$\Gamma_{\mu}$ responsible for restoring the local Lorentz symmetry [229],
thus $\nabla_{\mu}\psi=\partial_{\mu}\psi+\Gamma_{\mu}\psi$ or
$\nabla_{\mu}\bar{\psi}=\partial_{\mu}\bar{\psi}-\bar{\psi}\Gamma_{\mu}$. One
comes across with the gamma matrices $\gamma^{\mu}_{g}$ that obey to the
Clifford‘s algebra
$\left\\{\gamma^{\mu}_{g},\gamma^{\nu}_{g}\right\\}=2\,g_{\mu\nu}$, the
symbols $\psi$, meaning the spin-1/2 field, and
$\bar{\psi}=\psi^{\dagger}\gamma^{0}$, where $\gamma^{0}$ belongs to the usual
gamma matrices in a flat background.

Once substituted eq. (2.3) in the action (2.30), one expands
$\Gamma_{\textmd{ferm}}[\bar{\psi},\psi,g]$ in terms of the fluctuation field
$h_{\mu\nu}$, up to its first order, and extracts the energy-momentum tensor
relative to the fermion-graviton tree-level vertex

$\displaystyle T_{\mu\nu}(p,p^{\prime})=$
$\displaystyle\,\frac{\kappa}{8}\Big{\\{}2\eta_{\mu\nu}\big{[}(p+p^{\prime})_{\alpha}\,\mathcal{J}^{\alpha}(p,p^{\prime})-2m\,\rho(p,p^{\prime})\big{]}$
$\displaystyle\,\,-(p+p^{\prime})_{\mu}\mathcal{J}_{\nu}(p,p^{\prime})-(p+p^{\prime})_{\nu}\mathcal{J}_{\mu}(p,p^{\prime})\Big{\\}}\,.$
(2.31)

The tensorial expression in eq. (2.31) has the bi-linear
$\rho(p,p^{\prime})=\bar{u}(p^{\prime})u(p)$ and
$\mathcal{J}^{\mu}(p,p^{\prime})=\bar{u}(p^{\prime})\gamma^{\mu}u(p)$, where
$u(p)$ denotes the free positive energy solution for the four-component spinor
and $\bar{u}(p)={u}^{\dagger}(p)\gamma^{0}$. In detail

$\displaystyle u(p)=\sqrt{E+m}\left(\begin{array}[]{c}\xi\\\
\frac{\vec{\sigma}\cdot\vec{p}}{E+m}\,\xi\end{array}\right),$ (2.34)

with $\xi$ representing the two component spinor eigenstates and
$\vec{\sigma}$ the Pauli matrices. In on-shell condition, the Dirac equation
reads $(\gamma^{\mu}\,p_{\mu}-m)\,u(p)=0$.

Repeating the same procedure as done for the scalar case and described in
section 2.4, specifically working on the eq. (2.15), one calculates the
relativistic scattering amplitude

$\displaystyle
i\mathcal{M}^{(s=1/2)}=\frac{i}{q^{2}}\left[\frac{1}{3Q_{2}}\left(T_{1\,\mu}^{\,\,\mu}T_{2\,\beta}^{\,\,\beta}-3\,T_{1}^{\,\mu\nu}T_{2\,\mu\nu}\right)+\frac{1}{6Q_{0}}T_{1\,\mu}^{\,\,\mu}T_{2\,\beta}^{\,\,\beta}\right]\,$
(2.35)

This equation has the same form of the scalar one (2.26) with its appropriated
energy-momentum tensors. After some algebraic manipulations and using a
compact notation $\rho_{j}\equiv\rho(p_{j},p_{j}^{\prime})$ and
$\mathcal{J}_{j}^{\mu}\equiv\mathcal{J}^{\mu}(p_{j},p_{j}^{\prime})$, noting
that the particles are recognized by index $j=1,2$, $\mathcal{M}^{(s=1/2)}$ is
obtained in the following result

$\displaystyle\mathcal{M}^{(s=1/2)}$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}}{q^{2}Q_{2}}\Bigg{\\{}\frac{1}{16}(p_{1}+p_{1}^{\prime})_{\mu}(p_{2}+p_{2}^{\prime})_{\nu}\mathcal{J}_{1}^{\mu}\mathcal{J}_{2}^{\nu}-\frac{m_{1}}{8}\rho_{1}(p_{2}+p_{2}^{\prime})_{\mu}\mathcal{J}_{2}^{\mu}$
(2.36) $\displaystyle-$
$\displaystyle\frac{m_{2}}{8}\rho_{2}(p_{1}+p_{1}^{\prime})_{\mu}\mathcal{J}_{1}^{\mu}-\frac{1}{32}(p_{1}+p^{\prime}_{1})^{\nu}(p_{2}+p^{\prime}_{2})_{\nu}\mathcal{J}_{1}^{\mu}\mathcal{J}_{2\mu}$
$\displaystyle-$
$\displaystyle\frac{1}{32}(p_{1}+p_{1}^{\prime})_{\mu}(p_{2}+p_{2}^{\prime})_{\nu}\mathcal{J}_{2}^{\mu}\mathcal{J}_{1}^{\nu}+\frac{m_{1}m_{2}}{3}\rho_{1}\rho_{2}\Bigg{\\}}$
$\displaystyle+$
$\displaystyle\frac{\kappa^{2}}{q^{2}Q_{0}}\,\Bigg{\\{}\frac{3}{32}(p_{1}+p_{1}^{\prime})_{\mu}(p_{2}+p_{2}^{\prime})_{\nu}\mathcal{J}_{1}^{\mu}\mathcal{J}_{2}^{\nu}+\frac{2m_{1}m_{2}}{3}\rho_{1}\rho_{2}$
$\displaystyle-$
$\displaystyle\frac{m_{1}}{4}\rho_{1}(p_{2}+p_{2}^{\prime})_{\mu}\mathcal{J}_{2}^{\mu}-\frac{m_{2}}{4}\rho_{2}(p_{1}+p_{1}^{\prime})_{\mu}\mathcal{J}_{1}^{\mu}\Bigg{\\}}\,,$

with $m_{1}$ and $m_{2}$ standing for the fermion masses. Then, moving toward
the non-relativistic amplitude, one transfers the physical system to the
center-of-mass referential writing the eq. (2.36) in function of the momentum
transfer $\vec{q}$ and the average $\vec{p}$ (see eq. (2.16)). The bi-linear
$\mathcal{J}_{j}^{\mu}$ and $\rho_{j}$ in non-relativistic approach are

$\displaystyle\rho_{j}|_{\textmd{NR}}=2\,m_{j}\bigg{[}1+\frac{1}{8m_{j}^{2}}\bigg{(}\vec{q}^{\,2}-4i(\vec{q}\times\vec{p}\,)\cdot\vec{S}_{j}\bigg{)}+\mathcal{O}(3)\bigg{]}\,,$
(2.37a)
$\displaystyle\mathcal{J}^{0}_{j}|_{\textmd{NR}}=2\,m_{j}\bigg{[}1+\frac{1}{2m_{j}^{2}}\bigg{(}\vec{p}^{\,2}+i(\vec{q}\times\vec{p}\,)\cdot\vec{S}_{j}\bigg{)}+\mathcal{O}(3)\bigg{]}\,,$
(2.37b)
$\displaystyle\vec{\mathcal{J}}_{j}|_{\textmd{NR}}=2\,\chi_{j}\bigg{[}\vec{p}-i(\vec{q}\times\vec{S}_{j})\bigg{]}\,.$
(2.37c)

The eqs. (2.37) are managed in order to present them in a compact form, hence,
$\vec{S}_{j}=\xi^{\prime\dagger}_{j}\vec{\sigma}\xi_{j}$, the factors
$\xi^{\prime\dagger}_{j}\xi_{j}$ are omitted, $\chi_{1}=1$, $\chi_{2}=-1$ and
terms in powers higher than quadratic order for $|\vec{p}|/m_{1,2}$ and/or
$|\vec{q}|/m_{1,2}$ are neglected (referenced as $\mathcal{O}(3)$),
highlighting that eq. (2.37c) is exact.

The non-relativistic elastic scattering amplitude is achieved after the
substitution of the eqs. (2.37) in eq. (2.36) and following the same procedure
commented in the scalar case and described in sec. 2.4, precisely by eq.
(2.14), resulting in

$\displaystyle\mathcal{M}^{(s=1/2)}_{\textrm{NR}}$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}m_{1}m_{2}}{6Q_{2}\,\vec{q}^{\,2}}\,\Bigg{\\{}1+\vec{p}^{\,2}\left(\frac{3}{m_{1}m_{2}}+\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)$
(2.38) $\displaystyle+$ $\displaystyle
i\left[\left(\frac{1}{m_{1}^{2}}+\frac{3}{2}\frac{1}{m_{1}m_{2}}\right)\vec{S}_{1}+\left(\frac{1}{m_{2}^{2}}+\frac{3}{2}\frac{1}{m_{1}m_{2}}\right)\vec{S}_{2}\right]\cdot\left(\vec{q}\times\vec{p}\,\right)$
$\displaystyle-$
$\displaystyle\frac{3}{4}\frac{\vec{q}^{\,2}}{m_{1}m_{2}}\vec{S}_{1}\cdot\vec{S}_{2}+\frac{3}{4}\frac{1}{m_{1}m_{2}}\left(\vec{q}\cdot\vec{S_{1}}\right)\left(\vec{q}\cdot\vec{S_{2}}\right)+\mathcal{O}(3)\Bigg{\\}}$
$\displaystyle-$
$\displaystyle\frac{\kappa^{2}m_{1}m_{2}}{24Q_{0}\,\vec{q}^{\,2}}\,\Bigg{\\{}1-\frac{\vec{p}^{\,2}}{2}\left(\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)$
$\displaystyle-$
$\displaystyle\frac{i}{2}\left[\frac{1}{m_{1}^{2}}\vec{S}_{1}+\frac{1}{m_{2}^{2}}\vec{S}_{2}\right]\cdot\left(\vec{q}\times\vec{p}\,\right)+\mathcal{O}(3)\Bigg{\\}}\,,$

and, therefore, the gravitational potential for two spin-1/2 particles
interacting via a graviton, resulted from the application of the first Born
approximation (2.13) on the amplitude (2.38), is

$\displaystyle
V^{(s=1/2)}(r)=-\frac{\kappa^{2}m_{1}m_{2}}{6}\,\Bigg{\\{}I^{(2)}_{1}(r)+\vec{p}^{\,2}\left(\frac{3}{m_{1}m_{2}}+\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)I^{(2)}_{1}(r)$
$\displaystyle\quad+\left[\left(\frac{1}{m_{1}^{2}}+\frac{3}{2}\frac{1}{m_{1}m_{2}}\right)\vec{S}_{1}+\left(\frac{1}{m_{2}^{2}}+\frac{3}{2}\frac{1}{m_{1}m_{2}}\right)\vec{S}_{2}\right]\cdot\frac{\vec{L}}{r}\frac{d}{dr}I^{(2)}_{1}(r)$
$\displaystyle\quad-\frac{3}{4}\frac{\vec{S}_{1}\cdot\vec{S}_{2}}{m_{1}m_{2}}\,I^{(2)}_{0}(r)+\frac{3}{4}\sum_{i,j=1}^{3}\frac{(\vec{S}_{1})_{i}\,(\vec{S}_{2})_{j}}{m_{1}m_{2}}\,I_{ij}^{(2)}(r)\Bigg{\\}}$
$\displaystyle\quad+\frac{\kappa^{2}m_{1}m_{2}}{24}\,\Bigg{\\{}I^{(0)}_{1}(r)-\frac{\vec{p}^{\,2}}{2}\left(\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)I_{1}^{(0)}(r)-\frac{1}{2}\left[\frac{\vec{S}_{1}}{m_{1}^{2}}+\frac{\vec{S}_{2}}{m_{2}^{2}}\right]\cdot\frac{\vec{L}}{r}\frac{d}{dr}I^{(0)}_{1}(r)\Bigg{\\}}\,,$
(2.39)

observing that $\vec{L}=\vec{r}\times\vec{p}\,\,$ denotes the orbital angular
momentum and the appearance of a derivative in the spin-orbit sector is due to
the manipulation of particulars Fourier transform and the adoption of
spherical coordinate to solve their angular components (for more details, see
eq. (B.3)). A last comment is about the anisotropic integral
$I_{ij}^{(2)}(r)\,\,$ which is defined in eq. (B.2).

The analyses finalizes with the richness of the gravitational potentials
results. The next chapter is dedicated to an exploration of their
particularities and a comparison among both expressions eqs. (2.29) and
(2.39).

##  Chapter 3 Spin and Velocity Corrections in Gravitational Potentials

###  3.1 Scalar and Fermionic Gravitational Potentials: Aspects and
Comparisons

Continuing the analysis of the gravitational potentials for external spin-0
and spin-1/2 particles calculated in the last two subsections 2.4.1 and 2.4.2,
each one of the potentials is segregated according to the sectors that
compound them. Beginning by the spin-0 scenario, one evidences its two
sectors:

$\bullet$ Monopole-monopole sector

$\displaystyle V_{\textmd{mon-mon}}^{(s=0)}(r)=$ $\displaystyle-$
$\displaystyle\frac{\kappa^{2}m_{1}m_{2}}{6}\Bigg{[}\bigg{(}I^{(2)}_{1}(r)-\frac{1}{4}I^{(0)}_{1}(r)\bigg{)}$
(3.1a) $\displaystyle+$
$\displaystyle\left(\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)\bigg{(}\frac{1}{8}I^{(2)}_{0}(r)+\frac{5}{32}I^{(0)}_{0}(r)\bigg{)}\Bigg{]}\,.$

$\bullet$ Velocity-velocity sector

$\displaystyle V_{\textrm{vel-vel}}^{(s=0)}(r)=$ $\displaystyle-$
$\displaystyle\frac{\kappa^{2}m_{1}m_{2}}{6}\,\vec{p}^{\,2}\Bigg{[}\frac{3}{m_{1}m_{2}}I^{(2)}_{1}(r)$
(3.1b) $\displaystyle+$
$\displaystyle\left(\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)\left(I^{(2)}_{1}(r)+\frac{1}{8}I^{(0)}_{1}(r)\right)\Bigg{]}\,.$

In the same way, is presented the spin-1/2 sectors:

$\bullet$ Monopole-monopole sector

$\displaystyle V_{\textmd{mon-
mon}}^{(s=1/2)}(r)=-\frac{\kappa^{2}m_{1}m_{2}}{6}\bigg{(}I^{(2)}_{1}(r)-\frac{1}{4}I^{(0)}_{1}(r)\bigg{)}\,.$
(3.2a)

$\bullet$ Velocity-velocity sector

$\displaystyle V_{\textrm{vel-vel}}^{(s=1/2)}(r)=$ $\displaystyle-$
$\displaystyle\frac{\kappa^{2}m_{1}m_{2}}{6}\,\vec{p}^{\,2}\Bigg{[}\frac{3}{m_{1}m_{2}}I^{(2)}_{1}(r)$
(3.2b) $\displaystyle+$
$\displaystyle\left(\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)\left(I^{(2)}_{1}(r)+\frac{1}{8}I^{(0)}_{1}(r)\right)\Bigg{]}\,.$

$\bullet$ Spin-orbit sector

$\displaystyle V_{\textrm{spin-orbit}}^{(s=1/2)}(r)=$
$\displaystyle-\frac{\kappa^{2}m_{1}m_{2}}{6}\,\Bigg{\\{}\Bigg{[}\Bigg{(}\frac{1}{m_{1}^{2}}+\frac{3}{2}\frac{1}{m_{1}m_{2}}\Bigg{)}\vec{S}_{1}$
(3.2c)
$\displaystyle+\left(\frac{1}{m_{2}^{2}}+\frac{3}{2}\frac{1}{m_{1}m_{2}}\right)\vec{S}_{2}\Bigg{]}\cdot\frac{\vec{L}}{r}\frac{d}{dr}I^{(2)}_{1}(r)$
$\displaystyle+\frac{1}{8}\bigg{(}\frac{1}{m_{1}^{2}}\vec{S}_{1}+\frac{1}{m_{2}^{2}}\vec{S}_{2}\bigg{)}\cdot\frac{\vec{L}}{r}\frac{d}{dr}I^{(0)}_{1}(r)\Bigg{\\}}\,.$

$\bullet$ Spin-spin sector

$\displaystyle V_{\textrm{spin-
spin}}^{(s=1/2)}(r)=-\frac{\kappa^{2}m_{1}m_{2}}{6}\bigg{[}\\!-\\!\frac{3}{4}\frac{\vec{S}_{1}\cdot\vec{S}_{2}}{m_{1}m_{2}}\,I^{(2)}_{0}(r)+\frac{3}{4}\sum_{i,j=1}^{3}\frac{(\vec{S}_{1})_{i}\,(\vec{S}_{2})_{j}}{m_{1}m_{2}}\,I_{ij}^{(2)}(r)\bigg{]}\,.$
(3.2d)

The difference of the potential expressions related to both spin cases are
evident and expected, since the spin-1/2 field is richer in terms of physical
content. Observing the monopole-monopole sector for both scenarios, one
verifies the universal potential term present in eqs. (3.1a) and (3.2a). At
the same time, it points out the extra term proportional to the Fourier
transform eq. (B.1) containing the form factors $W$ and $F$ in spin-0
potential (eqs. (2.22a) and (2.22b)). Once chosen a particular action which
defines $W$ and $F$, this term may be managed to recover the universal
potential due to its suppressed behaviour. This is exemplified for a specific
choice of form factors in the subsection 3.3. On the other hand, the velocity
dependence exists in the two cases under analysis and, comparing eqs. (3.1b)
and (3.2b), they have identical structure. The spin-1/2 particles potential
reveals two exclusive sectors: the spin-orbit and spin-spin manifestations
(eqs. (3.2c) and (3.2d), respectively). Taking a closer look at
$V_{\text{spin-orbit}}^{(s=1/2)}$, one realizes that both form factors are
present in the mentioned potential, while in the sector $V_{\textrm{spin-
spin}}^{(s=1/2)}$ there is participation only of $W$. An important note is the
fact that, even though the particularities of the scalar and fermionic fields
as the existence of an extra term in monopole-monopole sector for spin-0
particle and the exclusive contributions of spin sectors in spin-1/2 case, the
gravitational potential resulted from the imposition of the static limit
restrictions ($\vec{p}\rightarrow{\vec{0}}\,$ and
$\,m_{i}\rightarrow{\infty}$),

$\displaystyle\frac{1}{m_{1}m_{2}}V_{\textmd{stat.}}^{(s)}(r)=\lim_{\begin{subarray}{c}\vec{p}\to\vec{0}\\\
m_{i}\to\infty\end{subarray}}\,\frac{1}{m_{1}m_{2}}V^{(s)}(r),$ (3.3)

is equal for both type of particles,

$\displaystyle
V_{\textmd{stat.}}^{(s)}(r)=-\frac{\kappa^{2}m_{1}m_{2}}{6}\bigg{(}I^{(2)}_{1}(r)-\frac{1}{4}I^{(0)}_{1}(r)\bigg{)}\,,$
(3.4)

and recovers $V_{\textrm{mon-mon}}^{(s=1/2)}(r)$ (3.2a). Furthermore, the
classical Newtonian potential
$V_{\textmd{stat.}}^{(s)}(r)=-\frac{\kappa^{2}m_{1}m_{2}}{32\pi
r}=-\frac{Gm_{1}m_{2}}{r}$ is recovered when the form factors and the
cosmological constant tend to zero.

One last observation should be done, the obtained results for spin-0 are
passive of being extended to gravitational interaction of spin-0 particles in
arbitrary dimensions as worked in refs.[230] and [231] for modified theories
of gravity in monopole-monopole sector. Things are not so trivial for spin-1/2
particles. The spin arrangement assumes varied structure as the spacetime
dimension vary. For instance, see refs. [232] and [233] for electromagnetic
cases, where spacetime with odd dimension are considered and the rising of new
spin-dependent effects are discussed.

###  3.2 Comparison with Electromagnetic Potentials

At this stage, there is the opportunity of comparing the gravitational
potentials results achieved in the last section 3.1 with the electromagnetic
potentials calculated in ref. [3] for modified electrodynamics. In the
mentioned paper, the authors adopted a very similar physical system
configuration and approaching of the problem. The target was the
electromagnetic potentials in the non-relativistic limit derived from spin-0,
-1/2 and -1 particles interactions with the exchanging of one bosonic vector,
what is summarized by the Feynman diagram fig. 2.1. Turning the attention to
spin-0 and -1/2 particles interactions, the same path chosen in sec. 2.4 was
gone through by the authors in the last mentioned reference, in which the
first Born approximation (2.13), amplitude expressions (2.15) and (2.14),
center-of-mass reference frame, 3-momentum convention (2.16) and elastic
scattering behaviour were took in account identically. In addition, they
worked until the second order in 3-momentum
$\mathcal{O}\big{(}|\vec{p}\,|^{\,2}/{m^{2}}\big{)}$, so the spin and orbit
couplings became notables. The electrodynamics effective action worked on was

$\displaystyle\Gamma_{\textmd{EM}}[A]=-\frac{1}{4}\int
d^{4}x\,F_{\mu\nu}(1+H(\Box))F^{\mu\nu}-\frac{1}{2\alpha}\int
d^{4}x\,(\partial_{\mu}A^{\mu})^{2}+\mathcal{O}(F^{3}),$ (3.5)

the structure $H(\Box)$ equally stood for a form factor as function of
d’Alembertian operator. The additional integral was the gauge-fixing term.
Vertices and the modified photon propagator were calculated at the tree level.
The found resultant photon propagator was

$\displaystyle\langle
A_{\mu}(-q)A_{\nu}(q)\rangle=-\frac{i}{q^{2}}\frac{1}{(1+H(-q^{2}))}\eta_{\mu\nu}+i\Delta_{\mu\nu}(q)\,,$
(3.6)

which presented similar configuration to eq. (2.21), with $H(\Box)$ factoring
the longitudinal and transversal projections operators contented in the
Minkowskian metric $\eta_{\mu\nu}$ and $\Delta_{\mu\nu}(q)$ concentrating the
terms that contracted with the vector current and vanish. For spin-0 case the
3-vertex originated from an Abelian vector and scalar particles, once they
worked on-shell, was $V^{\mu}(p,p^{\prime})=-ie(p^{\prime\mu}+p^{\mu})$. In
parallel, for spin-1/2 situation the conserved vector current taking place of
the vertex was $J^{\mu}(p,p^{\prime})=e\bar{u}(p^{\prime})\gamma^{\mu}u(p)$,
being $u(p)$ the positive solutions for Dirac equation and $\bar{u}(p)$ its
conjugated (the same terminology and convention described in subsection 2.4.2)
and "$e$" the electric charge.

This brief description of the elegant and rich content paper points out the
proximity of physics scenarios and methodology applied among it and the
present work, constituting a powerful voluntary to offer a valuable comparison
between gravitation and electromagnetic modified theories settled in very
close physics arrangements. Segregating the electromagnetic potentials, one
has the monopole-monopole and velocity-velocity sectors for spin-0 particles
interaction:

$\bullet$ Monopole-monopole sector

$\displaystyle V_{\textmd{EM, mon-
mon}}^{(s=0)}(r)=e_{1}e_{2}I^{\textmd{EM}}_{1}(r)\,.$ (3.7a)

$\bullet$ Velocity-velocity sector

$\displaystyle V_{\textmd{EM, vel-
vel}}^{(s=0)}(r)=\frac{e_{1}e_{2}}{m_{1}m_{2}}\,\vec{p}^{\,2}I^{\textmd{EM}}_{1}(r)\,.$
(3.7b)

And monopole-monopole, velocity-velocity, spin-orbit and spin-spin for
spin-1/2 particles:

$\bullet$ Monopole-monopole sector

$\displaystyle V_{\textmd{EM, mon-
mon}}^{(s=1/2)}(r)=e_{1}e_{2}\left[I^{\textmd{EM}}_{1}(r)-\frac{1}{8}\left(\frac{1}{m_{1}^{2}}+\frac{1}{m_{2}^{2}}\right)I^{\textmd{EM}}_{0}(r)\right]\,.$
(3.8a)

$\bullet$ Velocity-velocity sector

$\displaystyle V_{\textmd{EM, vel-
vel}}^{(s=1/2)}(r)=\frac{e_{1}e_{2}}{m_{1}m_{2}}\,\vec{p}^{\,2}I^{\textmd{EM}}_{1}(r)\,.$
(3.8b)

$\bullet$ Spin-orbit sector

$\displaystyle V_{\textmd{EM, spin-
orbit}}^{(s=1/2)}(r)=e_{1}e_{2}\,\left[\left(\frac{1}{2m_{1}^{2}}+\frac{1}{m_{1}m_{2}}\right)\vec{S}_{1}+\left(\frac{1}{2m_{2}^{2}}+\frac{1}{m_{1}m_{2}}\right)\vec{S}_{2}\right]\cdot\frac{\vec{L}}{r}\frac{d}{dr}I_{1}^{\textmd{EM}}(r)\,.$
(3.8c)

$\bullet$ Spin-spin sector

$\displaystyle V_{\textrm{EM, spin-
spin}}^{(s=1/2)}(r)=e_{1}e_{2}\bigg{[}\\!-\\!\frac{\vec{S}_{1}\cdot\vec{S}_{2}}{m_{1}m_{2}}\,I_{0}^{\textmd{EM}}(r)+\sum_{i,j=1}^{3}\frac{(\vec{S}_{1})_{i}\,(\vec{S}_{2})_{j}}{m_{1}m_{2}}\,I_{ij}^{\textmd{EM}}(r)\bigg{]}\,.$
(3.8d)

The integrals $I_{n}^{\textmd{EM}}(r)$ and $I_{ij}^{\textmd{EM}}(r)$ assume
the forms described in eqs. (B.1) and (B.2) respectively, considering the
substitution of $Q_{a}(q^{2})$ by $1+H(-q^{2})$. Comparing the results for
gravitational and electromagnetic potentials, one realizes some
correspondences of the sectors acquired in both spin-0 and spin-1/2. In eqs.
(3.1a) and (3.7a) are observed the universal terms (Newtonian and Coulombian),
adequate to each interaction. For the electromagnetic situation, the integral
$I_{1}^{\textmd{EM}}(r)$ "agglutinates" the contributions from the
longitudinal (trivial) and transversal projections once is considered a
massless photon and no anisotropy, while, for gravitational case, their
projections appear separately through the integrals $I_{1}^{(2)}(r)$ and
$I_{1}^{(0)}(r)$. The velocity sector manifests the same behaviour as the
monopole-monopole one does, that is, contributions of $I_{1}(r)$. There is no
quantum corrections attributed to $I_{0}^{\textmd{EM}}(r)$ inside the
electromagnetic potential for spin-0 external particles. Dislocating towards
spin-1/2 particles, the electromagnetic and gravitational potentials in
monopole and velocity sectors perform a similar relation as obtained for
spin-0. The monopole-monopole sector (eq. (3.8a)) contains the
$I_{1}^{\textmd{EM}}(r)$ term and, differently of occurred for spin-0,
manifests a quantum correction with $I_{0}^{\textmd{EM}}(r)$, in contrast to
gravitational potential monopole sector that has its result (eq. (3.2a))
concentrated in the integrals $I_{1}$. Moving to the velocity-dependent part,
the expressions for both potentials are preserved in relation to spin-0 case.