###### Abstract

A general theory of electric charge is proposed. It is based on two
phenomenologies. Electric charge mutation and conservation law. Three charges
$\\{+,-,0\\}$ transformations physics succeeds. Quantum field theory underlies
corresponding creations and annihilation. A potential field’s quadruplet is
ruled. Microscopic electromagnetism is processed by four vectors bosons
intermediations.

The electromagnetism closure is accomplished. The quadruplet $A_{\mu
I}\equiv\\{A_{\mu},U_{\mu},V_{\mu}^{\pm}\\}$ completeness introduces the most
generic EM energy flux between electric charges. Charge mutation includes that
besides usual photon, EM phenomena is enlarged by massive and charged photons.
Charge conservation associates these four vector fields. Electric charge
symmetry, extends EM for an abelian symmetry $U_{Q}\equiv U(1)\times
SO(2)_{global}$. A new EM Lagrangian beyond Maxwell results. A symmetry
equation for electric charge is established through Noether theorem.

The electric charge transfer physics extends the EM phenomenon. Nonlinear
Electromagnetic fields modified electric charge symmetry, new EM regimes.
Potential fields become a physical entity producing conglomerates, collective
fields, mass, sources, charges, monopoles, forces. EM features ruled from an
extended electric charge abelian symmetry. Systemic, nonlinear, neutral,
spintronics, photonics, electroweak EM regimes are constituted.

Electric Charge transfer by Four Vector Bosons
R<EMAIL_ADDRESS>and L.S. Mendes
<EMAIL_ADDRESS>

1Aprendanet, Petrópolis, Brazil; Quarks, Petropolis, Brazil
2Aprendanet, Petrópolis, Brazil; CEFET-RJ, Petropolis, Brazil

##  1 Introduction

Electromagnetism is the manifestation of electric and magnetic fields. It is
based on electric charge and spin. Maxwell equations are its standard
description [1]. A qualitative difference appears to Newton’s laws. Their
features are that nature is not only constituted by matter, but also through
Faraday’s lines of force [2].

Nevertheless, although all its success, Maxwell equations contain limitations.
It is a linear theory, introduces the potential fields as a subsidiary,
polarization and magnetization fields by hand, passive light, others.
Different subjects as condensate matter, plasma, superconductivity,
astrophysics are registering such restrictions. There is a physics beyond
Maxwell to be encompassed. Something is expected to be done with contemporary
electromagnetism. Advances are welcomed since preserving the four Maxwell
postulates. Consider light invariance, set EM physics, electric and magnetic
fields in pairs, electric charge conservation. There are 52 effective non-
Maxwellians extensions, organized into four categories. They are 12 extensions
considering the Standard Model, 14 extensions motivated by a physics beyond
the Standard Model, 16 nonlinear extensions, and 10 due to dimensionality [3].
The quire is whether the $21^{th}$ century is in front of new electromagnetism
or just expecting extensions to Maxwell theory. The effort here will be to
enlarge the EM phenomena with a fundamental model.

Historically, EM development may be divided into three phases. The first one
based on charges and magnets was described by William Gilbert in 1600 [4]. The
second, in 1865 due to Maxwell, was based on charges and fields [5]. And by
third, it is hoped as the $21^{th}$ century development, a relationship
between fields and fields. Consider the EM energy flux just depending on
fields. A relationship between fields and matter beyond energy and mass.
Nonlinear electromagnetism introduces matter dematerialization through EM
fields.

The pos-Maxwell concerns to the microscopic understanding of electric charge.
It appeared at $20^{th}$ century through the elementary particle physics.
Electromagnetism is the theory describing the relationships between electric
and magnetic fields. Faraday and Maxwell studied at macroscopic level. The
development of elementary particle physics introduced a microscopic
electromagnetism. The electric charge being ported by particles with different
flavours and spins.

Since 1874 the Irish physicist George Stoney had been suspecting of an
electric charge atom, naming it electron [6]. In 1894 stipulated its charge
value nearly $e=0.5\cdot 10^{-19}C$. An assignment was realized with the
electron discovery by Thomson in 1897 and by Millikan measurement in 1913 of
magnitude $e=1.6\cdot 10^{-19}C$ in 1913 [7]. The microscopic electromagnetism
becomes realistic. Four charges feature followed. Electric charge
quantization, carried by particles with different flavors and spins, charges
exchanges $\Delta Q=0,\pm 1$, and spin as a magnet.

The $20^{th}$ century did the EM qualitative change from Maxwell macroscopic
to a microscopic behavior. The following stages were developed. First, in
1905, Einstein relativity introduced EM fields under reference systems [8]. In
1921 and 1923, Compton proposes the magnetic electron and incorporated the
photon as a particle, and in 1924, de Broglie introduced the wave-particle
dualism [9]. Third, in 1927, Dirac studies the Maxwell field under second
quantization [10]. In 1928, Dirac equation introduces relativistic quantum
mechanics for electron particle quantization [11]. And so, at the end of the
1920s and the beginning of the 1930s, the first QED phase was studied. It was
a sequence of works with relativistic fields equations by Jordan, Klein,
Wigner, Pauli, Heisenberg, Fermi, Fock, Podolski, Weisskopf, Stueckelberg
[12]. In the forties, the second QED phase was developed. First, proposed by
Schwinger with spin-1 mesotrons (1940) [13] and Dirac with Lorentz group
(1942) [14]. Second, after the war, through Tomonaga (1946-1949) [15], Feynman
(1948-1951) [16], Schwinger (1948-1954) [17], Dyson (1949) [18]. Third,
continued with applications in the 1950s by Landau and others [19-21].

The next stage was scalar electrodynamics by Salam in the fifties [22]. Vector
bosons electrodynamics by Salam, Lee, Yang, and others appeared in the sixties
[23-29]. Nevertheless, all these quantum fields theories did not surpass the
Maxwell boundaries. Although extending Maxwell equations, they were not able
to incorporate new EM regimes. For instance, weak interactions as part of
electromagnetism. And so, they were replaced by the Standard Model [30-32].

Parallelly, effective nonlinear electromagnetism has been developed since 1925
due to the photon-photon collision. It was introduced by Schapocshnikov,
Vavilov, Delbruck, Breit, Wheeler [33]. The $\gamma\gamma-$ scattering
motivated nonlinear extensions to Maxwell equations [34]. Diverse scenarios
based on effective NLED (nonlinear electrodynamics) were developed since Born-
Infeld and Euler-Heisenberg [35]. Actually, there are NLED types as ED ModMax,
ED Axionic, ED Logarithmic, and so on [36].

Nevertheless, between these EM periods, a phenomenological missing happened.
It was on the electric charge transference. Prospect the electric charge set
$\\{+,0,-\\}$ exchanges. Instead of considering the reaction $n\to
p+e^{-}+\tilde{\nu}_{e}$ through Fermi theory of nuclear beta decay, propose
the physics of electric charge transfer. Understand the neutron transmutation
into a proton with emission of an electron and neutron as an EM regime with a
coupling constant diverse from electric charge. Assuming that, charge
transference is an EM phenomenon, and not, a distinct $G_{F}$ weak
interaction.

The phenomenology of electric charge mutation was already a theme for 1923. A
time with an electron, proton, photon exchanging three charges. Later in the
1930s, the positron discovery [37], neutron decay [38], muon decay [39], pions
cosmic rays decays [40] were confirmed on charges transference. In 1938, by
occasion of the Warsaw conference ’New theories in Physics’, there were 14
particles in the physics variety. They were these particles
$e^{-}$,$e^{+}$,$p(p^{-})$,$\gamma$,$n(\tilde{n})$,$\mu^{-}$,$\mu^{+}$,$\pi^{+}$,$\pi^{0}$,$\pi^{-}$,$\nu(\tilde{\nu})$.
There was a rich particle’s spectrum confirming the charges set $\\{+,0,-\\}$
phenomenology.

At that time, two possibilities to do physics were available. Consider these
particles sharing charge mutation, or look for forces of unification. Although
different particles reactions coming from cosmic rays and accelerators were
showing on charges transference, prevailed the unification scheme. The
research line taken by Oscar Klein was unification in five dimensions [41].
The reductionist proposal of unifying four forces becomes the zeitgeist. Klein
wicked the name Theory of Everything, ToE [42]. The electric charge mutation
argument remained secondary. Although in 1940, Schwinger added mesotrons as
spin-1 particles, introducing the existence of two charged vector bosons plus
photon, the presence of four-vector bosons intermediating electric charges
just appeared in the fifties as belonging to weak interactions.

As time goes by, the view of electric charge mutation was being left. QED took
the place for charges exchanges. It was successfully formulated as the theory
of electrons and photons. The three charges of electromagnetism were
unexplored. Forgotten. The completeness of a whole quantum field theory under
charges creation and destruction carried by a set with four intermediate
vector particles is still expected. There is a missing EM energy to be
considered. An EM energy where each charge is transformed through four-vector
bosons $\\{A_{\mu},U_{\mu},V_{\mu}^{\pm}\\}$. A purpose containing $A_{\mu}$
as the usual photon, $U_{\mu}$ massive photon, and $V_{\mu}^{\pm}$ massive
vector photons.

Thus, without violating any basic Maxwell principle, an EM enlargement is
projected beyond Maxwell and QED. It is called four bosons electromagnetism
[43]. While at the macroscopic level, the Coulomb balance did not register on
the zero electric charge presence, at a microscopic level, its physicality
appears through bosons exchanges. The three electric charge transformations
are displayed on the zero charge EM energy. A result already detected at QED
with photons carrying a zero charge EM energy.

This work objective is to investigate a new EM performance under charge
transfer. Given the electric charge, mutation to search for new
electromagnetic edges. In section 2, one studies the four potential fields
association through an extended Abelian gauge symmetry $U_{Q}\equiv U(1)\times
SO(2)_{global}$ and the corresponding observable in a Lagrangian. Euler-
Lagrangian equations of motion are derived in section 3. Section 4 electric
charge physicality is investigated through Noether theorem and identified as
the symmetry equation. Section 5 introduces the electromagnetic whole
equations of motion. Bianchi’s identities are left for section 6. In section 7
invariants are considered. Nonlinear electromagnetic waves are derived in
section 8. In section 9, Lorentz’s force is extended to the field set. In a
conclusion, new perspectives on electromagnetism are considered. Based on
nonlinear fields, new electromagnetic regimes, and extended electric charge
behavior.

##  2 Four Bosons Lagrangian

The purpose here is a General Electric charge theory. Consider that electric
charge alongside the conservation law contains the mutation factor. Two
aspects are supporting such charge exchange physics. The diversity of
elementary particles dressing positive, negative, zero charges and the notion
of creation and annihilation from quantum field theory. Even so, history
shows, the physics of electric charge mutation was not assigned. Physics
developed QED just based on positive and negative charges plus the photon.
However, despite all the (g-2) QED success [44-46], the existence of an EM
completeness intermediate by four bosons is still a missing point to be
excavated.

There is an EM energy to be explored. The electric charge mutation and
conservation join together for developing the charges set $\\{+,0,-\\}$
physics carried by four bosons messengers
$\\{A_{\mu},U_{\mu},V_{\mu}^{\pm}\\}$. A relationship where the gauge
principle is extended through the introduction of a family of potential fields
in association with an extended Abelian symmetry. An approach has been
supported under several points of view [47]. It yields Maxwell being extended
by nonlinear Abelian gauge theory. The usual photon field $A_{\mu}$ appears
associated with a massive photon $U_{\mu}$ and two charged photons
$V_{\mu}^{\pm}$.

The electric charge closure is focused. A structure systematized by a
quadruplet. While charge mutation requires four intermediate fields,
conservation makes their association. Following Noether theorem [48], the
quadruplet should be connected through a common gauge parameter. It yields an
extended Abelian gauge model $U_{Q}\equiv U(1)\times SO(2)_{global}$ [43],

$\displaystyle A^{\prime}_{\mu}$ $\displaystyle=$ $\displaystyle
A_{\mu}+k_{1}\partial_{\mu}\alpha$ (1) $\displaystyle U^{\prime}_{\mu}$
$\displaystyle=$ $\displaystyle U_{\mu}+k_{2}\partial_{\mu}\alpha$ (2)
$\displaystyle V_{\mu}^{+^{\prime}}$ $\displaystyle=$ $\displaystyle
e^{iq\alpha}\left(V_{\mu}^{+}+k_{+}\partial_{\mu}\alpha\right)$ (3)
$\displaystyle V_{\mu}^{-^{\prime}}$ $\displaystyle=$ $\displaystyle
e^{-iq\alpha}\left(V_{\mu}^{-}+k_{-}\partial_{\mu}\alpha\right)$ (4)

Eqs. (1 \- 4) introduce the physical fields. They have been defined as the
poles of two points Green’s functions. We should remember that different
fields’ basis was previously constructed in order to obtain the above physical
basis [49]. They are consistent with Kamechu-Raifeartaigh-Salam prescriptions
[50]. Consequently, $k_{1},k_{2},k_{\pm}$ are terms related to parameters at
the field basis transformations matrix. They are depending on theory-free
coefficients associated with Lagrangian terms [51].

EM physics is extended to a quadruplet $A_{\mu
I}\equiv\\{A_{\mu},U_{\mu},V_{\mu}^{\pm}\\}$. Interconnected potential fields
share an Abelian symmetry. Observable is formed. Fields conglomerates,
granular and collective fields strengths are derived. Potential fields
conglomerates are fields linear combinations, as
$\mathbf{e}_{[IJ]}A_{\mu}^{J}$, where $\mathbf{e}_{[IJ]}$ is a parameter on
free coefficients. Granular and collective fields will be described below.
Their corresponding gauge invariance properties are studied in Appendix A.

For the antisymmetric sector, one gets the following granular fields,

$\displaystyle F_{\mu\nu}$
$\displaystyle=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},$ $\displaystyle
F^{\prime}_{\mu\nu}=F_{\mu\nu}$ (5) $\displaystyle U_{\mu\nu}$
$\displaystyle=\partial_{\mu}U_{\nu}-\partial_{\nu}U_{\mu},$ $\displaystyle
U^{\prime}_{\mu\nu}=U_{\mu\nu}$ (6) $\displaystyle V_{\mu\nu}^{\pm}$
$\displaystyle=\partial_{\mu}V_{\nu}^{\pm}-\partial_{\nu}V_{\mu}^{\pm},$
$\displaystyle V_{\mu\nu}^{\pm^{\prime}}=e^{\pm iq\alpha}V_{\mu\nu}^{\pm}$ (7)

The collective antisymmetric field strengths are

$\displaystyle\mathbf{e}^{[12]}_{[\mu\nu]}=\mathbf{e}_{[12]}\left(A_{\mu}U_{\nu}-A_{\nu}U_{\mu}\right)$
(8)
$\displaystyle\mathbf{e}^{[+-]}_{[\mu\nu]}=-i\mathbf{e}_{[34]}\left(V_{\mu}^{+}V_{\nu}^{-}-V_{\nu}^{+}V_{\mu}^{-}\right)$
(9)
$\displaystyle\mathbf{e}^{[+1]}_{[\mu\nu]}+\mathbf{e}^{[-1]}_{[\mu\nu]}=\frac{1}{\sqrt{2}}\big{[}\left(\mathbf{e}_{[13]}+i\mathbf{e}_{[14]}\right)\left(A_{\mu}V_{\nu}^{+}-A_{\nu}V_{\mu}^{+}\right)$
$\displaystyle+\left(\mathbf{e}_{[13]}-i\mathbf{e}_{[14]}\right)\left(A_{\mu}V_{\nu}^{-}-A_{\nu}V_{\mu}^{-}\right)\big{]}$
(10)
$\displaystyle\mathbf{e}^{[+2]}_{[\mu\nu]}+\mathbf{e}^{[-2]}_{[\mu\nu]}=\frac{1}{\sqrt{2}}\big{[}\left(\mathbf{e}_{[23]}+i\mathbf{e}_{[24]}\right)\left(U_{\mu}V_{\nu}^{+}-U_{\nu}V_{\mu}^{+}\right)+$
$\displaystyle\left(\mathbf{e}_{[23]}-i\mathbf{e}_{[24]}\right)\left(U_{\mu}V_{\nu}^{-}-U_{\nu}V_{\mu}^{-}\right)\big{]}$
(11)

For the symmetric sector, the granular fields strengths are

$\displaystyle S_{\mu\nu 1}$
$\displaystyle=\partial_{\mu}A_{\nu}+\partial_{\nu}A_{\mu},$ $\displaystyle
S_{\mu\nu 1}^{\prime}=S_{\mu\nu 1}$ (12) $\displaystyle S_{\mu\nu 2}$
$\displaystyle=\partial_{\mu}U_{\nu}+\partial_{\nu}U_{\mu},$ $\displaystyle
S_{\mu\nu 2}^{\prime}=S_{\mu\nu 2}$ (13) $\displaystyle S_{\mu\nu}^{\pm}$
$\displaystyle=\partial_{\mu}V_{\nu}^{\pm}+\partial_{\nu}V_{\mu}^{\pm},$
$\displaystyle S_{\mu\nu}^{\pm^{\prime}}=e^{\pm iq\alpha}S^{\pm}_{\mu\nu}$
(14)

and

$\displaystyle S_{\alpha 1}^{\alpha}$
$\displaystyle=2\partial_{\alpha}A^{\alpha},$ $\displaystyle S_{\alpha
1}^{\alpha^{\prime}}=S_{\alpha 1}^{\alpha}$ (15) $\displaystyle S_{\alpha
2}^{\alpha}$ $\displaystyle=\partial_{\alpha}U^{\alpha},$ $\displaystyle
S_{\alpha 2}^{\alpha^{\prime}}=S_{\alpha 2}^{\alpha}$ (16) $\displaystyle
S_{\alpha}^{\alpha\pm}$ $\displaystyle=\partial_{\alpha}V^{\alpha\pm},$
$\displaystyle S_{\alpha}^{\alpha\pm^{\prime}}=e^{\pm
iq\alpha}S_{\alpha}^{\alpha\pm}$ (17)

The collective symmetric fields strengths are

$\displaystyle\mathbf{e}^{(11)}_{(\mu\nu)}=\mathbf{e}_{(11)}A_{\mu}A_{\nu}$
(18)
$\displaystyle\mathbf{e}^{(22)}_{(\mu\nu)}=\mathbf{e}_{(22)}U_{\mu}U_{\nu}$
(19)
$\displaystyle\mathbf{e}^{(12)}_{(\mu\nu)}=\mathbf{e}_{(12)}\left(A_{\mu}U_{\nu}+A_{\nu}U_{\mu}\right)$
(20)
$\displaystyle\mathbf{e}^{(+-)}_{(\mu\nu)}=\frac{1}{2}\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)\left(V_{\mu}^{+}V_{\nu}^{-}+V_{\nu}^{+}V_{\mu}^{-}\right)$
(21)
$\displaystyle\mathbf{e}^{(+1)}_{(\mu\nu)}+\mathbf{e}^{(-1)}_{(\mu\nu)}=\frac{1}{\sqrt{2}}\big{[}\left(\mathbf{e}_{(13)}+i\mathbf{e}_{(14)}\right)\left(A_{\mu}V_{\nu}^{+}+A_{\nu}V_{\mu}^{+}\right)+$
$\displaystyle+\left(\mathbf{e}_{(13)}-i\mathbf{e}_{(14)}\right)\left(A_{\mu}V_{\nu}^{-}+A_{\nu}V_{\mu}^{-}\right)\big{]}$
(22)
$\displaystyle\mathbf{e}^{(+2)}_{(\mu\nu)}+\mathbf{e}^{(-2)}_{(\mu\nu)}=\frac{1}{\sqrt{2}}\big{[}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)\left(U_{\mu}V_{\nu}^{+}+U_{\nu}V_{\mu}^{+}\right)+$
$\displaystyle\left(\mathbf{e}_{(23)}-i\mathbf{e}_{(24)}\right)\left(U_{\mu}V_{\nu}^{-}+U_{\nu}V_{\mu}^{-}\right)\big{]}$
(23)
$\displaystyle\mathbf{e}^{(++)}_{(\mu\nu)}+\mathbf{e}^{(--)}_{(\mu\nu)}=\frac{1}{2}\left(\mathbf{e}_{(33)}-\mathbf{e}_{(44)}\right)\left(V_{\mu}^{+}V_{\nu}^{+}+V_{\mu}^{-}V_{\nu}^{-}\right)$
(24)
$\displaystyle\mathbf{e}^{(++34)(\mu\nu)}+\mathbf{e}^{(--34)(\mu\nu)}=i\mathbf{e}_{(34)}V_{\mu}^{+}V_{\nu}^{+}-i\mathbf{e}_{(34)}V_{\mu}^{-}V_{\nu}^{-}$
(25)

Under the gauge transformations cited in Appendix A. Similarly for scalar
symmetric collective terms, eqs. (161-166) .

The field strengths are written at eqs. (5 \- 25) are real observable which
gauge invariance is studied in Appendix A. Notice that
$\mathbf{e}_{[12]},\mathbf{e}_{[13]},\mathbf{e}_{(11)}$ and so on are just
parameters. They are depending on the free coefficients generated by the
model. Consequently, they can take any value without violating the gauge
symmetry. This property will support the correspondent gauge invariance.

Thus, an Abelian formulation joining distinct granular and collective fields
strengths is obtained. While non-abelian models architect them together as
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+g[A_{\mu},A_{\nu}]$,
eqs. (1-4) split in two separated gauge invariance terms. Two distinct
physicality are derived to be understood dynamically.

A new EM energy appears. The following Lagrangian is organized in terms of the
above observable and free parameters

$L=L_{A}+L_{S}+L_{M}+L_{GF}$ (26)

The antisymmetric sector is

$L_{A}=L_{A}^{K}+L_{A}^{3}+L_{A}^{4}$ (27)

where

$L_{A}^{K}=a_{1}F_{\mu\nu}F^{\mu\nu}+a_{2}U_{\mu\nu}U^{\mu\nu}+a_{3}V_{\mu\nu}^{+}V^{\mu\nu-}$
(28)

and

$\displaystyle
L_{A}^{3}=\left(b_{1}F_{\mu\nu}+b_{2}U_{\mu\nu}\right)\left(\mathbf{e}^{[12][\mu\nu]}+\mathbf{e}^{[+-][\mu\nu]}\right)+$
$\displaystyle+b_{3}V_{\mu\nu}^{+}\left(\mathbf{e}^{[-1][\mu\nu]}+e^{[-2][\mu\nu]}\right)+b_{3}V_{\mu\nu}^{-}\left(\mathbf{e}^{[+1][\mu\nu]}+\mathbf{e}^{[+2][\mu\nu]}\right)$
(29)

and

$L_{A}^{4}=\left(\mathbf{e}^{[12][\mu\nu]}+\mathbf{e}^{[+-][\mu\nu]}\right)^{2}+\left(\mathbf{e}^{[-1][\mu\nu]}+e^{[-2][\mu\nu]}\right)\left(\mathbf{e}^{[+1][\mu\nu]}+\mathbf{e}^{[+2][\mu\nu]}\right)$
(30)

The symmetric sector is

$L_{S}=L_{S}^{K}+L_{S}^{3}+L_{S}^{4}$ (31)

where

$\displaystyle L_{S}^{K}=\beta_{1}S_{\mu\nu 1}\ S^{\mu\nu
1}+\beta_{2}S_{\mu\nu 2}S^{\mu\nu
2}+\beta_{+}\beta_{-}S_{\mu\nu}^{+}S^{\mu\nu-}+$
$\displaystyle+\big{(}\rho_{1}g^{\mu\nu}S_{\alpha}^{\alpha
1}+\rho_{2}g^{\mu\nu}S_{\alpha}^{\alpha}\big{)}^{2}+16\rho_{+}\rho_{-}S_{\alpha}^{\alpha+}S_{\beta}^{\beta-}+\rho_{+}\beta_{-}g^{\mu\nu}S_{\alpha}^{\alpha+}S_{\mu\nu}^{-}$
(32) $\displaystyle\left(\rho_{1}g^{\mu\nu}S_{\alpha
1}^{\alpha}+\rho_{2}g^{\mu\nu}S_{\alpha
2}^{\alpha}\right)\left(\beta_{1}S_{\mu\nu 1}+\beta_{2}S_{\mu\nu
2}\right)+\rho_{-}\beta_{+}g^{\mu\nu}S_{\alpha}^{\alpha-}S_{\mu\nu}^{+}$

and

$\displaystyle L_{S}^{3}=\left(\beta_{1}S_{\mu\nu 1}+\beta_{2}S_{\mu\nu
2}+\rho_{1}g^{\mu\nu}S_{\alpha 1}^{\alpha}+\rho_{2}g^{\mu}\nu S_{\alpha
2}^{\alpha}\right)\big{(}\mathbf{e}^{(11)(\mu\nu)}$
$\displaystyle++\mathbf{e}^{(22)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}+g^{\mu\nu}\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}$
$\displaystyle+\mathbf{e}^{(+-)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha}\big{)}+\left(\beta_{+}S_{\mu\nu}^{+}+\rho_{+}g^{\mu\nu}S_{\alpha}^{\alpha+}\right)\big{(}\mathbf{e}^{(-1)(\mu\nu)}$
(33)
$\displaystyle+g^{\mu\nu}\mathbf{e}^{(-1)\alpha}_{\alpha}+\mathbf{e}^{(-2)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(-2)\alpha}_{\alpha}\big{)}+\left(\beta_{-}S_{\mu\nu}^{-}+\rho_{-}g^{\mu\nu}S_{\alpha}^{\alpha-}\right)$
$\displaystyle\big{(}\mathbf{e}^{(+1)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(+1)\alpha}_{\alpha}+\mathbf{e}^{(+2)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(+2)\alpha}_{\alpha}\big{)}$

and

$\displaystyle
L_{S}^{4}=\big{(}\mathbf{e}^{(11)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(22)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)(\mu\nu)}+$
$\displaystyle+g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(+-)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(+-)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha}\big{)}^{2}+$
$\displaystyle\left(\mathbf{e}^{(-1)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(-1)\alpha}_{\alpha}+\mathbf{e}^{(-2)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(-2)\alpha}_{\alpha}\right)\big{(}\mathbf{e}^{(+1)(\mu\nu)}$
(34)
$\displaystyle+g^{\mu\nu}\mathbf{e}^{(+1)\alpha}_{\alpha}+\mathbf{e}^{(+2)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(+2)\alpha}_{\alpha}\big{)}+\mathbf{e}_{(\mu\nu)}^{(++)}\mathbf{e}^{(--)(\mu\nu)}+$
$\displaystyle+16\mathbf{e}_{(\mu\nu)}^{(++)}\mathbf{e}^{(--)(\mu\nu)}+\mathbf{e}_{(\mu\nu)}^{(++34)}e^{(--34)(\mu\nu)}+16\mathbf{e}_{(\mu\nu)}^{(++34)}\mathbf{e}^{(--34)(\mu\nu)}$

$L^{3}$ and $L^{4}$ develop an Abelian nonlinear EM. A Maxwell extension with
a new perspective for Faraday lines of force. Stipulated by electric charge
phenomenology, Lagrangian eq. (5) penetrates in the third EM historical stage.
A self-interacting structure was diverse from Yang-Mills.

The mass sector is

$L_{M}=\mathbf{m}^{2}_{U}U_{\mu}U^{\mu}+\mathbf{m}^{2}_{V}V_{\mu}^{+}V^{\mu-}$
(35)

where eq. (35) introduces a mass term without requiring the spontaneously
breaking symmetry [52]. Notice that $\mathbf{m}^{2}_{U}$ and
$\mathbf{m}^{2}_{V}$ are just mass parameters.

The gauge fixing term is [49]

$\displaystyle L_{GF}=\frac{1}{4}\xi_{(11)}S_{\alpha 1}^{\alpha}S_{\beta
1}^{\beta}+\frac{1}{4}\xi_{(22)}S_{\alpha 2}^{\alpha}S_{\beta
2}^{\beta}+\frac{1}{2}\xi_{(12)}S_{\alpha 1}^{\alpha}S_{\beta 2}^{\beta}$
$\displaystyle+\frac{1}{4}\left(\xi_{(33)}+\xi_{(44)}\right)S_{\alpha}^{\alpha+}S_{\beta}^{\beta-}$
(36)

Notice that there is only one gauge fixing relating the four potential fields.
It is due to the presence of just one gauge parameter. However, it provides
five arbitrary parameters
$\xi_{(11)},\xi_{(12)},\xi_{(22)},\xi_{(33)},\xi_{(44)}$ which can assume any
value. This means that, differently from the usual case, more than one
constraint relationship can be taken.

The four bosons Lagrangian generalizes previous results. In 1962, Lee and Yang
introduced the longitudinal bosons propagation and the term
$F_{\mu\nu}W^{\mu}W^{\nu*}$ in order to correct the dipole moment magnet for
charged bosons [25]. In 1963, Salam introduced the term
$\lambda\left(W_{\mu}W^{\mu*}\right)^{2}$ to ensure renormalization by power
counting [23-24]. These terms are included in $L_{A}^{3}$ plus other magnetic
moment terms, and $L_{S}^{4}$, respectively. Self-interacting photons are
included at $L_{S}^{3}$ and $L_{S}^{4}$. Yet, differently from Maxwell, the
model contains spin-1 and spin-0 sectors. There is a more generic EM to be
explored.

##  3 Euler-Lagrange equations

Physics requires equations. While in the past, homothety introduced symmetry,
modern physics does with symmetry [53]. Trigonometry generated functions like
sin and cos, eqs. (1-4) introduces observable. Both provide variables to be
analyzed through equations. Trigonometry establish $sin^{2}(x)+cos^{2}(x)=1$.
The corresponding minimal action principle produces Euler-Lagrange equations.
Consequently, similarly to trigonometry, a mathematical structure based on
fields set $\\{A_{\mu},U_{\mu},V_{\mu}^{\pm}\\}$ is generated. A nonlinear
system of hyperbolic differential equations is developed. The following four
whole relativistic equations are derived.

For photon-$A_{\mu}$,

$\displaystyle\partial_{\nu}\big{\\{}4a_{1}F^{\nu\mu}+4\beta_{1}\ S^{\nu\mu
1}+4\big{(}\rho_{1}\
\beta_{1}+11\rho_{1}+\frac{1}{2}\xi_{(11)}\big{)}g^{\nu\mu}S_{\alpha
1}^{\alpha}$
$\displaystyle+2\big{(}\rho_{1}\beta_{2}+\rho_{2}\beta_{2}+22\rho_{1}\rho_{2}+\frac{1}{2}\xi_{(12)}\big{)}g^{\nu\mu}S_{\alpha
2}^{\alpha}+2b_{1}\big{(}\mathbf{e}^{[12][\nu\mu]}$
$\displaystyle+\mathbf{e}^{[+-][\nu\mu]}\big{)}+34\rho_{1}g^{\nu\mu}\big{(}\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(+-)\alpha}_{\alpha}\big{)}$
$\displaystyle+2\beta_{1}\big{(}\mathbf{e}^{(11)(\nu\mu)}+g^{\nu\mu}\mathbf{e}^{(11)\alpha}_{\alpha}+g^{\nu\mu}\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)(\nu\mu)}$
(37)
$\displaystyle+g^{\nu\mu}\mathbf{e}^{(12)\alpha}_{\alpha}+g^{\nu\mu}\mathbf{e}^{(+-)\alpha}_{\alpha}+\mathbf{e}^{(+-)(\nu\mu)}\big{)}\big{\\}}=J_{A}^{\mu}$

where

$\displaystyle
J_{A}^{\mu}=2\mathbf{e}_{[12]}\left(b_{1}F^{\mu\nu}+b_{2}U^{\mu\nu}\right)U_{\nu}+\sqrt{2}b_{3}\left(\mathbf{e}_{[13]}-i\mathbf{e}_{[14]}\right)V^{\mu\nu+}V_{\nu}^{-}+\sqrt{2}b_{3}\big{(}\mathbf{e}_{[13]}$
$\displaystyle+i\mathbf{e}_{[14]}\big{)}V^{\mu\nu-}V_{\nu}^{+}+2\mathbf{e}_{(11)}\left(\beta_{1}S_{\mu\nu
1}+\beta_{2}S_{\mu\nu 2}+\rho_{1}g^{\mu\nu}S_{\alpha
1}^{\alpha}+\rho_{2}g^{\mu\nu}S_{\alpha
2}^{\alpha}\right)A_{\nu}+2\mathbf{e}_{(12)}\big{(}\beta_{1}S_{\mu\nu 1}$
$\displaystyle+\beta_{2}S_{\mu\nu 2}+\rho_{1}g^{\mu\nu}S_{\alpha
1}^{\alpha}+\rho_{2}g^{\mu\nu}S_{\alpha
2}^{\alpha}\big{)}U_{\nu}+2e_{(12)}\left[\left(\beta_{1}+16\rho_{1}\right)S_{\alpha
1}^{\alpha}+\left(\beta_{2}+16\rho_{2}\right)S_{\alpha
2}^{\alpha}\right]U^{\mu}$
$\displaystyle+\sqrt{2}\left(\mathbf{e}_{13}-i\mathbf{e}_{(14)}\right)\big{[}\beta_{+}S^{\mu\nu+}+\big{(}16\beta_{+}+17\rho_{+}\big{)}g^{\mu\nu}S_{\alpha}^{\alpha+}\big{]}V^{-}_{\nu}++\sqrt{2}\left(\mathbf{e}_{(13)}+i\mathbf{e}_{(14)}\right)$
$\displaystyle[\beta_{-}S^{\mu\nu-}+(16\beta_{-}+17\rho_{-})g^{\mu\nu}S_{\alpha}^{\alpha-}]V^{+}_{\nu}+4\mathbf{e}_{[12]}\left(\mathbf{e}^{[12][\mu\nu]}+\mathbf{e}^{[+-][\mu\nu]}\right)U_{\nu}+\sqrt{2}\big{(}\mathbf{e}_{[13]}$
$\displaystyle-i\mathbf{e}_{[14]}\big{)}\left(\mathbf{e}^{[+1][\mu\nu]}+\mathbf{e}^{[+2][\mu\nu]}\right)V_{\nu}^{-}+\sqrt{2}\big{(}\mathbf{e}_{[13]}+i\mathbf{e}_{[14]}\big{)}\left(\mathbf{e}^{[-1][\mu\nu]}+\mathbf{e}^{[-2][\mu\nu]}\right)V_{\nu}^{+}$
$\displaystyle+4\mathbf{e}_{(11)}(\mathbf{e}^{(11)(\mu\nu)}+2g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(22)(\mu\nu)}+\mathbf{e}^{(+-)(\mu\nu)}+18g^{\mu\nu}\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(22)(\mu\nu)}+$
(38) $\displaystyle
18g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}+17g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha})A_{\nu}+4\mathbf{e}_{(12)}(\mathbf{e}^{(11)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}+g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}+18\mathbf{e}^{(22)(\mu\nu)})U_{\nu}$
$\displaystyle\sqrt{2}\left(\mathbf{e}_{(13)}+i\mathbf{e}_{(14)}\right)(\mathbf{e}^{(-1)(\mu\nu)}+18g^{\mu\nu}\mathbf{e}^{(-1)\alpha}_{\alpha})V_{\nu}^{+}+\sqrt{2}\left(\mathbf{e}_{(13)}-i\mathbf{e}_{(14)}\right)(\mathbf{e}^{(+1)(\mu\nu)}$
$\displaystyle+18g^{\mu\nu}\mathbf{e}^{(+1)\alpha}_{\alpha})V_{\nu}^{-}+\sqrt{2}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)(\mathbf{e}^{(-2)(\mu\nu)}+18g^{\mu\nu}\mathbf{e}^{(-2)\alpha}_{\alpha})V_{\nu}^{+}$
$\displaystyle+\sqrt{2}\left(\mathbf{e}_{(23)}-i\mathbf{e}_{(24)}\right)(\mathbf{e}^{(+2)(\mu\nu)}+18g^{\mu\nu}\mathbf{e}^{(+2)\alpha}_{\alpha})V_{\nu}^{-}$

For massive photon-$U_{\mu}$,

$\displaystyle\partial_{\nu}\\{4a_{2}U^{\nu\mu}+4\beta_{2}S^{\nu\mu
2}+4\big{(}\rho_{2}\beta_{2}+11\rho_{2}+1/2\xi_{(22)}\big{)}S_{\alpha}^{\alpha
2}+$
$\displaystyle+2\big{(}\rho_{2}\beta_{1}+\rho_{1}\beta_{2}+22\rho_{1}\rho_{2}+1/2\xi_{(12)}\big{)}S_{\alpha}^{\alpha
1}+2b_{2}\big{(}\mathbf{e}^{[12][\nu\mu]}$
$\displaystyle+\mathbf{e}^{[+-][\nu\mu]}\big{)}+34\rho_{2}+g^{\nu\mu}\big{(}\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(+-)\alpha}_{\alpha}\big{)}$
(39) $\displaystyle
2\beta_{2}(\mathbf{e}^{(11)(\nu\mu)}+g^{\nu\mu}\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(22)(\nu\mu)}+g^{\nu\mu}\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)(\nu\mu)}$
$\displaystyle+g^{\nu\mu}\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(+-)(\nu\mu)}+g^{\nu\mu}\mathbf{e}^{(+-)\alpha})\\}-2\mathbf{m}_{U}U^{\mu}=J_{U}^{\mu}$

where

$\displaystyle
J_{U}^{\mu}=2\mathbf{e}_{[12]}\left(b_{1}F^{\mu\nu}+b_{2}U^{\mu\nu}\right)A_{\nu}+\sqrt{2}b_{3}\left(\mathbf{e}_{[23]}-i\mathbf{e}_{[24]}\right)V^{\mu\nu+}V_{\nu}^{-}$
$\displaystyle+\sqrt{2}b_{3}\big{(}\mathbf{e}_{[23]}+i\mathbf{e}_{[24]}\big{)}V^{\mu\nu-}V_{\nu}^{+}+2\mathbf{e}_{(22)}\big{[}(\beta_{2}+17\rho_{2})g^{\mu\nu}S^{\alpha}_{\alpha
2}+$ $\displaystyle+(\beta_{1}+17\rho_{1})g^{\mu\nu}S^{\alpha}_{\alpha
1}+\beta_{1}S^{\mu\nu}_{1}+\beta_{2}S^{\mu\nu}_{2}\big{]}U_{\nu}+2\mathbf{e}_{(12)}[\beta_{1}S^{\mu\nu}_{1}$
$\displaystyle+\beta_{2}S^{\mu\nu}_{2}+(\beta_{1}+17\rho_{1})g^{\mu\nu}S^{\alpha}_{\alpha
1}(\beta_{2}+17\rho_{2})g^{\mu\nu}S^{\alpha}_{\alpha 2}]A_{\nu}+$
$\displaystyle+4\mathbf{e}_{[12]}\mathbf{e}^{[12][\mu\nu]}A_{\nu}+\sqrt{2}\left(\mathbf{e}_{(23)}-i\mathbf{e}_{(24)}\right)[\beta_{+}S^{\alpha+}_{\alpha}+$
$\displaystyle+(\beta_{+}+17\rho_{+})g^{\mu\nu}S^{\alpha+}_{\alpha}]V_{\nu}^{-}+\sqrt{2}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)[\beta_{-}S^{\alpha-}_{\alpha}$
$\displaystyle+(\beta_{-}+17\rho_{-})g^{\mu\nu}S^{\alpha-}_{\alpha}]V_{\nu}^{+}+4\mathbf{e}_{[12]}\mathbf{e}^{[+-][\mu\nu]}A_{\nu}+$
$\displaystyle\sqrt{2}\left(\mathbf{e}_{[23]}-i\mathbf{e}_{[24]}\right)(\mathbf{e}^{[+1][\mu\nu]}+\mathbf{e}^{[+2][\mu\nu]})V_{\nu}^{-}+$
$\displaystyle\sqrt{2}\left(\mathbf{e}_{[23]}+i\mathbf{e}_{[24]}\right)(\mathbf{e}^{[-1][\mu\nu]}+\mathbf{e}^{[-2][\mu\nu]})V_{\nu}^{+}$
$\displaystyle+4\mathbf{e}_{(22)}(2\mathbf{e}^{(22)(\mu\nu)}+2g^{\mu\nu}\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(11)(\mu\nu)}+\mathbf{e}^{(+-)(\mu\nu)}$
(40)
$\displaystyle+18g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}+18g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}+17g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha})U_{\nu}$
$\displaystyle+4\mathbf{e}_{(12)}(+18g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}+18g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha}+\mathbf{e}^{(22)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}$
$\displaystyle+2\mathbf{e}^{(11)(\mu\nu)})A_{\nu}+\sqrt{2}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)(\mathbf{e}^{(-1)(\mu\nu)}$
$\displaystyle+18g^{\mu\nu}\mathbf{e}^{(-1)\alpha}_{\alpha})V_{\nu}^{+}+\sqrt{2}\left(\mathbf{e}_{(23)}-i\mathbf{e}_{(24)}\right)(\mathbf{e}^{(+1)}+18g^{\mu\nu}\mathbf{e}^{(+1)\alpha}_{\alpha})V_{\nu}^{-}+$
$\displaystyle+\sqrt{2}\left(\mathbf{e}_{(23)}-i\mathbf{e}_{(24)}\right)(\mathbf{e}^{(+2)(\mu\nu)}+18g^{\mu\nu}\mathbf{e}^{(+2)\alpha}_{\alpha})V_{\nu}^{-}+$
$\displaystyle+18\sqrt{2}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)(\mathbf{e}^{(-2)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(-2)\alpha}_{\alpha})V_{\nu}^{+}$

For charged photon-fields,

$\displaystyle\partial_{\nu}\\{2a_{3}V^{\mu\nu-}+2\beta_{+}\beta_{-}S^{\nu\mu-}+2\big{(}16\rho_{+}\rho_{-}+\rho_{+}\beta_{-}+\rho_{-}\beta_{+}$
$\displaystyle
1+1/4\left(\xi_{33}+\xi_{44}\right)\big{)}g^{\nu\mu}S_{\alpha}^{\alpha-}+2b_{3}\left(\mathbf{e}^{[-1][\nu\mu]}+\mathbf{e}^{[-2][\nu\mu]}\right)+$
$\displaystyle+\beta_{+}\left(\mathbf{e}^{(-1)(\nu\mu)}+g^{\nu\mu}\mathbf{e}^{(-1)\alpha}_{\alpha}+\mathbf{e}^{(-2)(\nu\mu)}+g^{\nu\mu}\mathbf{e}^{(-2)\alpha}_{\alpha}\right)+$
(41)
$\displaystyle+34\rho_{+}g^{\nu\mu}\left(\mathbf{e}^{(-1)\alpha}_{\alpha}+\mathbf{e}^{(-2)\alpha}_{\alpha}\right)\\}-\mathbf{m}_{V}V^{\mu}=J_{V}^{\mu-}$

where

$\displaystyle
J_{V}^{\mu-}=-2i\mathbf{e}_{[34]}\left(b_{1}F^{\mu\nu}+b_{2}U^{\mu\nu}\right)V_{\nu}^{-}+\sqrt{2}\left(\mathbf{e}_{[13]}+i\mathbf{e}_{[14]}\right)V^{\mu\nu-}A_{\nu}+\big{(}\mathbf{e}_{(33)}+$
$\displaystyle+\mathbf{e}_{(44)}\big{)}[\beta_{1}S^{\mu\nu}_{1}+\beta_{2}S^{\mu\nu}_{2}+(\beta_{1}+17\rho_{1})g^{\mu\nu}S^{\alpha}_{\alpha
1}(\beta_{1}+17\rho_{2})g^{\mu\nu}S^{\alpha}_{\alpha
2}]V_{\nu}^{-}+\sqrt{2}\big{(}\mathbf{e}_{(13)}+$ $\displaystyle
i\mathbf{e}_{(14)}\big{)}(\beta_{-}S^{\mu\nu-}+\beta_{-}g^{\mu\nu}S^{\alpha-}_{\alpha}+17\rho_{-}g^{\mu\nu}S^{\alpha-}_{\alpha})A_{\nu}+\sqrt{2}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)(\beta_{-}S^{\mu\nu-}+$
$\displaystyle+\beta_{-}g^{\mu\nu}S^{\alpha-}_{\alpha}+17\rho_{-}g^{\mu\nu}S^{\alpha-}_{\alpha})U_{\nu}-4i\mathbf{e}_{[34]}(\mathbf{e}^{[+-][\mu\nu]}+\mathbf{e}^{[12][\mu\nu]})V_{\nu}^{-}+\sqrt{2}\big{(}\mathbf{e}_{[13]}+$
$\displaystyle+i\mathbf{e}_{[14]}\big{)}(\mathbf{e}^{[-1][\mu\nu]}+\mathbf{e}^{[-2][\mu\nu]})A_{\nu}+\sqrt{2}\left(\mathbf{e}_{[13]}+i\mathbf{e}_{[14]}\right)(\mathbf{e}^{[-1][\mu\nu]}+\mathbf{e}^{[-2][\mu\nu]})U_{\nu}+$
$\displaystyle+2\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)(\mathbf{e}^{(11)(\mu\nu)}+g^{\mu\nu}+\mathbf{e}^{(22)(\mu\nu)}+\mathbf{e}^{(12)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(+-)(\mu\nu)}+$
(42)
$\displaystyle+g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha}+2g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}+2g^{\mu\nu}\mathbf{e}^{(22)\alpha}_{\alpha})V_{\nu}^{-}+\sqrt{2}\left(\mathbf{e}_{(13)}+i\mathbf{e}_{(14)}\right)\big{(}\mathbf{e}^{(-1)(\mu\nu)}+$
$\displaystyle+\mathbf{e}^{(-2)(\mu\nu)}+17g^{\mu\nu}\mathbf{e}^{(-1)\alpha}_{\alpha}+17g^{\mu\nu}\mathbf{e}^{(-2)\alpha}_{\alpha}\big{)}A_{\nu}+\sqrt{2}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)\big{(}\mathbf{e}^{(-1)(\mu\nu)}+$
$\displaystyle+\mathbf{e}^{(-2)(\mu\nu)}+17g^{\mu\nu}\mathbf{e}^{(-1)\alpha}_{\alpha}+17g^{\mu\nu}\mathbf{e}^{(-2)\alpha}_{\alpha}\big{)}U_{\nu}+\left(\mathbf{e}_{(33)}-\mathbf{e}_{(44)}\right)(\mathbf{e}^{(--)(\mu\nu)}+$
$\displaystyle+16g^{\mu\nu}\mathbf{e}^{(--)\alpha}_{\alpha})V_{\nu}^{+}+2i\mathbf{e}_{(34)}(\mathbf{e}^{(--34)(\mu\nu)}+16\mathbf{e}^{(--34)\alpha}_{\alpha})V_{\nu}^{+}$

and

$\displaystyle\partial_{\nu}\\{2a_{3}V^{\mu\nu+}+2\beta_{+}\beta_{-}S^{\nu\mu+}+2\big{(}16\rho_{+}\rho_{-}+\rho_{+}\beta_{-}+\rho_{-}\beta_{+}$
$\displaystyle+1/4\left(\xi_{33}+\xi_{44}\right)\big{)}g^{\nu\mu}S_{\alpha}^{\alpha+}+2b_{3}\left(\mathbf{e}^{[+1][\mu\nu]}+\mathbf{e}^{[+2][\mu\nu]}\right)+$
$\displaystyle+\beta_{-}\left(\mathbf{e}^{(+1)(\nu\mu)}+g^{\nu\mu}\mathbf{e}_{\alpha}^{(+1)\alpha}+\mathbf{e}^{(+2)(\nu\mu)}+g^{\nu\mu}\mathbf{e}_{\alpha}^{(+2)\alpha}\right)+$
(43)
$\displaystyle+34\rho_{-}g^{\nu\mu}\left(\mathbf{e}^{(+1)\alpha}_{\alpha}+\mathbf{e}^{(+2)\alpha}_{\alpha}\right)\\}=J_{V}^{\mu+}$
$\displaystyle
J_{V}^{\mu+}=-2i\mathbf{e}_{[34]}\left(b_{1}F^{\mu\nu}+b_{2}U^{\mu\nu}\right)V_{\nu}^{+}+\sqrt{2}\left(\mathbf{e}_{[13]}-i\mathbf{e}_{[14]}\right)V^{\mu\nu+}A_{\nu}$
$\displaystyle+\big{(}\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\big{)}[\beta_{1}S^{\mu\nu}_{1}+\beta_{2}S^{\mu\nu}_{2}+(\beta_{1}+17\rho_{1})g^{\mu\nu}S^{\alpha}_{\alpha
1}(\beta_{1}+17\rho_{2})g^{\mu\nu}S^{\alpha}_{\alpha 2}]V_{\nu}^{+}+$
$\displaystyle+\sqrt{2}\left(\mathbf{e}_{(13)}-i\mathbf{e}_{(14)}\right)(\beta_{+}S^{\mu\nu+}+\beta_{+}g^{\mu\nu}S^{\alpha+}_{\alpha}+17\rho_{+}g^{\mu\nu}S^{\alpha+}_{\alpha})A_{\nu}+\sqrt{2}\big{(}\mathbf{e}_{(23)}$
$\displaystyle-i\mathbf{e}_{(24)}\big{)}(\beta_{+}S^{\mu\nu+}+\beta_{+}g^{\mu\nu}S^{\alpha+}_{\alpha}+17\rho_{+}g^{\mu\nu}S^{\alpha+}_{\alpha})U_{\nu}-4i\mathbf{e}_{[34]}(\mathbf{e}^{[+-][\mu\nu]}+\mathbf{e}^{[12][\mu\nu]})V_{\nu}^{+}$
$\displaystyle+\sqrt{2}\left(\mathbf{e}_{[13]}-i\mathbf{e}_{[14]}\right)(\mathbf{e}^{[+1][\mu\nu]}+\mathbf{e}^{[+2][\mu\nu]})A_{\nu}+\sqrt{2}\left(\mathbf{e}_{[13]}-i\mathbf{e}_{[14]}\right)(\mathbf{e}^{[+1][\mu\nu]}+\mathbf{e}^{[+2][\mu\nu]})U_{\nu}$
$\displaystyle+2\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)(\mathbf{e}^{(11)(\mu\nu)}+g^{\mu\nu}+\mathbf{e}^{(22)(\mu\nu)}+g^{\mu\nu}+\mathbf{e}^{(12)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(+-)(\mu\nu)}$
(44)
$\displaystyle+g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha}+2g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}+2g^{\mu\nu}\mathbf{e}^{(22)\alpha}_{\alpha})V_{\nu}^{+}+\sqrt{2}\left(\mathbf{e}_{(13)}-i\mathbf{e}_{(14)}\right)\big{(}\mathbf{e}^{(+1)(\mu\nu)}+\mathbf{e}^{(+2)(\mu\nu)}$
$\displaystyle+17g^{\mu\nu}\mathbf{e}^{(+1)\alpha}_{\alpha}+17g^{\mu\nu}\mathbf{e}^{(+2)\alpha}_{\alpha}\big{)}A_{\nu}+\sqrt{2}\left(\mathbf{e}_{(23)}-i\mathbf{e}_{(24)}\right)\big{(}\mathbf{e}^{(+1)(\mu\nu)}+17g^{\mu\nu}\mathbf{e}^{(+1)\alpha}_{\alpha}+$
$\displaystyle++\mathbf{e}^{(+2)(\mu\nu)}+17g^{\mu\nu}\mathbf{e}^{(+2)\alpha}_{\alpha}\big{)}U_{\nu}+\left(\mathbf{e}_{(33)}-\mathbf{e}_{(44)}\right)(\mathbf{e}^{(++)(\mu\nu)}+16g^{\mu\nu}\mathbf{e}^{(++)\alpha}_{\alpha})V_{\nu}^{-}$
$\displaystyle++2i\mathbf{e}_{(34)}(\mathbf{e}^{(++34)(\mu\nu)}+16\mathbf{e}^{(--34)\alpha}_{\alpha})V_{\nu}^{-}$

A whole EM physics is established. Maxwell equations are extended to a field’s
strengths network. An associated and nonlinear dynamics are generated in terms
of observables and free coefficients. A dualism appears. A complementarity
between individual-collective fields strengths. Granular and collective
fields’ strengths propagate together in space-time and work as their own
sources by coupling with potential fields. The coupling between fields
strengths and potential fields is through adimensional free coefficients.
Sources beyond electric charge.

Including the constraints equations,

$\partial_{\nu}S^{\nu\mu I}=\partial_{\nu}F^{\nu\mu
I}+g^{\nu\mu}\partial_{\mu}S^{\alpha I}_{\alpha}$ (45)

and

$\partial_{\nu}\mathbf{e}^{(\nu\mu)}=\frac{1}{2}\mathbf{e}_{(IJ)}S^{\alpha
I}_{\alpha}A^{\mu J}+\mathbf{e}_{(IJ)}S^{\nu\mu
I}A_{\mu}^{J}-\frac{1}{2}\partial_{\nu}g^{\nu\mu}\mathbf{e}^{\alpha}_{\alpha}$
(46)

one suppresses the field’s strengths, by mixing spin-1 and spin-0 together.

Next development is to split the above dynamics at spin-1 and spin-0 sectors.
Apply the transversal and longitudinal operators,
$\theta_{\mu\nu}=g_{\mu\nu}-\frac{\partial_{\mu}\partial_{\nu}}{\Box}$ and
$\omega_{\mu\nu}=\frac{\partial_{\mu}\partial_{\nu}}{\Box}$. Physically, such
spin-1 and spin-0 dynamics separation is valid, due to the fact that, although
non-local operators are applied, covariance is maintained. Lorentz’s
transformation and the electric charge symmetry preserved.

For spin-1 sector, it yields

$\partial_{\nu}\\{4\left(a_{1}+\beta_{1}\right)F^{\nu\mu}+2b_{1}\left(\mathbf{e}^{[12][\nu\mu]}+\mathbf{e}^{[+-][\nu\mu]}\right)\\}=J^{\mu}_{AT},$
(47)
$\partial_{\nu}\\{4\left(a_{2}+\beta_{2}\right)U^{\nu\mu}+2b_{2}\left(\mathbf{e}^{[12][\nu\mu]}+\mathbf{e}^{[+-][\nu\mu]}\right)\\}-2\mathbf{m}_{U}^{2}U_{L}^{\mu}=J^{\mu}_{UT},$
(48)
$\partial_{\nu}\\{2\left(a_{3}+\beta_{+}\beta_{-}\right)V^{\nu\mu-}+2b_{3}\left(\mathbf{e}^{[-1][\nu\mu]}+\mathbf{e}^{[-2][\nu\mu]}\right)\\}-\mathbf{m}_{V}^{2}V_{T}^{\mu-}=J^{\mu-}_{VT},$
(49)
$\partial_{\nu}\\{2\left(a_{3}+\beta_{+}\beta_{-}\right)V^{\nu\mu+}+2b_{3}\left(\mathbf{e}^{[+1][\nu\mu]}+\mathbf{e}^{[+2][\nu\mu]}\right)\\}-\mathbf{m}_{V}^{2}V_{T}^{\mu+}=J^{\mu+}_{VT}.$
(50)

where

$J_{IT}^{\mu}=\theta^{\mu\nu}J_{\nu I}$ (51)

For spin-0 sector:

$\displaystyle\partial_{\nu}\\{4\left(\beta_{1}+\rho_{1}\beta_{1}+11\rho_{1}+\frac{1}{2}\xi_{(11)}\right)S_{\alpha}^{\alpha
1}+2\big{(}\rho_{1}\beta_{2}+$
$\displaystyle\rho_{2}\beta_{1}+22\rho_{1}\rho_{2}+1/2\xi_{(12)}\big{)}S_{\alpha}^{\alpha
2}+\big{(}34\rho_{1}+$ (52)
$\displaystyle-2\beta_{1}\big{)}g^{\nu\mu}\left(\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(+-)\alpha}_{\alpha}\right)\\}=J^{\mu}_{AL},$
$\displaystyle\partial_{\nu}\\{4\left(\beta_{2}+\rho_{2}\beta_{1}+11\rho_{2}+\frac{1}{2}\xi_{(22)}\right)S_{\alpha}^{\alpha
2}+2\big{(}\rho_{1}\beta_{2}+\rho_{2}\beta_{1}+$ $\displaystyle
22\rho_{1}\rho_{2}+\frac{1}{2}\xi_{(12)}\big{)}S_{\alpha}^{\alpha
1}+\left(34\rho_{2}-2\beta_{2}\right)g^{\nu\mu}\big{(}\mathbf{e}^{(11)\alpha}_{\alpha}$
$\displaystyle+\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(+-)\alpha}_{\alpha}\big{)}\\}-2\mathbf{m}_{U}^{2}U_{L}^{\mu}=J^{\mu}_{UL},$
$\partial_{\nu}\\{2\left(\beta_{+}\beta_{-}+16\rho_{+}\rho_{-}+\rho_{+}\beta_{-}+\rho_{-}\beta_{+}+\frac{1}{4}\left(\xi_{(33)}+\xi_{(44)}\right)\right)g^{\nu\mu}S_{\alpha}^{\alpha-}\\\
+\left(34\rho_{+}-2\beta_{+}\right)g^{\nu\mu}\left(\mathbf{e}^{(-1)\alpha}_{\alpha}+\mathbf{e}^{(-2)\alpha}_{\alpha}\right)\\}-\mathbf{m}_{V}^{2}V^{\mu-}_{L}=J^{\mu-}_{VL},$
(53)
$\partial_{\nu}\\{2\left(\beta_{+}\beta_{-}+16\rho_{+}\rho_{-}+\rho_{+}\beta_{-}+\rho_{-}\beta_{+}\frac{1}{4}\left(\xi_{(33)}+\xi_{(44)}\right)\right)g^{\nu\mu}S_{\alpha}^{\alpha+}\\\
+\left(34\rho_{-}-2\beta_{-}\right)g^{\nu\mu}\left(\mathbf{e}^{(+1)\alpha}_{\alpha}+\mathbf{e}^{(+2)\alpha}_{\alpha}\right)\\}-\mathbf{m}_{V}^{2}V^{\mu+}_{L}=J^{\mu+}_{VL}.$
(54)

where

$J_{IL}^{\mu}=\omega^{\mu\nu}J_{\nu I}$ (55)

Thus, although the Lorentz group representation $A_{\mu
I}\in(\frac{1}{2},\frac{1}{2})$ contains spin-0 and spin-1, dynamically, they
are distinct. A symmetry pluriform is expressed. Splitting at LHS equations,
it yields vector and a scalar kind of photons with different quanta. However,
at RHS, the mixing spin terms are preserved. Notice that the granular spin-0
propagation can be controlled by gauge fixing parameters.

sectionElectric charge symmetry

A generic electric charge physics based on two phenomenologies is being
studied. It englobes the mutation and conservation properties.
Macroscopically, Maxwell understood the electric charge conservation. A
microscopic interpretation is expected based on four intermediate fields.
Consider the electric charge set $\\{+,0,-\\}$ transformations under the
quadruplet $\\{A_{\mu},U_{\mu},V_{\mu}^{\pm}\\}$. Connect these fields under a
modified electric charge symmetry.

The electric charge nature is extended. At $18^{th}$ century, Benjamin
Franklin identified positive and negative electric charges and formulated the
electric charge conservation. The microscopic charge transfer phenomena
introduce a new feature for electric charge. It is the physical presence of a
neutral charge. Considering that electric charge is a generator of $U_{Q}(1)$
global gauge group, the corresponding quantum number is continuous. Therefore,
the electric charge value, $q=N_{q}e$, can take any value as an integer,
fractional, or even zero. A formulation enclosing a neutral electric charge.

There is physics beyond Maxwell’s electric charge properties. The
corresponding continuity equation and coupling constant be enlarged. The gauge
parameter is related to eqs. (1-4) develops the quadruplet association
$U_{Q}\equiv U(1)\times SO(2)_{global}$. This new symmetry produces a whole
EM, expressed through Lagrangian and Noether theorem. Three identities are
obtained from Lagrangian invariance,

$\alpha\partial_{\mu}J_{N}^{\mu}+\partial_{\nu}\alpha\\{\partial_{\mu}K^{\mu\nu}+J_{N}^{\nu}\\}+\partial_{\mu}\partial_{\nu}\alpha
K^{\mu\nu}=0$ (56)

The first one corresponds to the electric charge conservation,

$\partial_{\mu}J^{\mu}_{N}=0$ (57)

with

$J^{\mu}_{N}=iq\left\\{\left[V^{+}_{\nu}\frac{\partial
L}{\partial\left(\partial_{\nu}V_{\mu}^{+}\right)}\right]-\left[V^{-}_{\nu}\frac{\partial
L}{\partial\left(\partial_{\nu}V_{\mu}^{-}\right)}\right]\right\\}$ (58)

It gives,

$J_{N}^{\mu}=-iq\\{4a_{3}\left(V_{\nu}^{+}V^{\nu\mu-}-V_{\nu}^{-}V^{\nu\mu+}\right)+4\beta_{+}\beta_{-}\left(V_{\nu}^{+}S^{\nu\mu-}-V_{\nu}^{-}S^{\nu\mu+}\right)+\\\
+4\left(16\rho_{+}\rho_{-}+\rho_{+}\beta_{-}+\rho_{-}\beta_{+}\right)g^{\nu\mu}\left(V^{+}_{\nu}S_{\alpha}^{\alpha-}-V^{-}_{\nu}S_{\alpha}^{\alpha+}\right)+4[b_{3}V_{\nu}^{+}\left(\mathbf{e}^{[-1][\nu\mu]}+\mathbf{e}^{[-2][\nu\mu]}\right)+\\\
-b_{3}V_{\nu}^{-}\left(\mathbf{e}^{[+1][\nu\mu]}+\mathbf{e}^{[+2][\nu\mu]}\right)]+4\beta_{+}V_{\nu}^{+}\left(\mathbf{e}^{(-1)(\nu\mu)}+g^{\nu\mu}\mathbf{e}^{(-1)\alpha}_{\alpha}+\mathbf{e}^{(-2)(\nu\mu)}+g^{\nu\mu}\mathbf{e}^{(-2)\alpha}_{\alpha}\right)+\\\
-4\beta_{-}V_{\nu}^{-}\left(\mathbf{e}^{(+1)(\nu\mu)}+g^{\nu\mu}\mathbf{e}^{(+1)\alpha}_{\alpha}+\mathbf{e}^{(+2)(\nu\mu)}+g^{\nu\mu}\mathbf{e}^{(+2)\alpha}_{\alpha}\right)+17\rho_{+}V_{\nu}^{+}g^{\nu\mu}\left(\mathbf{e}^{(-1)\alpha}_{\alpha}+\mathbf{e}^{(-2)\alpha}_{\alpha}\right)+\\\
-17\rho_{-}V_{\nu}^{-}g^{\nu\mu}\left(\mathbf{e}^{(+1)\alpha}_{\alpha}+\mathbf{e}^{(+2)\alpha}_{\alpha}\right)\\}$
(59)

Eq. (59) reproduces the electric charge coupling between positive and negative
charges. The first novelty is electric charge coupling with observable beyond
Maxwell. It is also not restricted to charged fields. Chargeless fields are
included through collective fields, as term $\mathbf{e}^{[+1][\mu\nu]}$ shows.

The second Noether identity is the symmetry equation. It is identified as the
electric charge equation,

$\partial_{\nu}K^{\nu\mu}+J^{\mu}_{N}=0$ (60)

where

$K^{\nu\mu}=k_{1}\frac{\partial
L}{\partial\left(\partial_{\nu}A_{\mu}\right)}+k_{2}\frac{\partial
L}{\partial\left(\partial_{\nu}U_{\mu}\right)}+k_{+}\frac{\partial
L}{\partial\left(\partial_{\nu}V_{\mu}^{+}\right)}+k_{-}\frac{\partial
L}{\partial\left(\partial_{\nu}V_{\mu}^{-}\right)}$ (61)

which gives the following gauge-invariant equation,

$\displaystyle\partial_{\nu}\\{4k_{1}a_{1}F^{\nu\mu}+4k_{2}a_{2}U^{\nu\mu}+2k_{-}a_{3}V^{\nu\mu+}+2k_{+}a_{3}V^{\nu\mu-}+$
$\displaystyle+4(11k_{1}\rho_{1}+\frac{1}{2}k_{2}\rho_{2}\beta_{1}+11k_{2}\rho_{1}\rho_{2}+\frac{1}{4}k_{2}\xi_{(12)})g^{\nu\mu}S_{\alpha}^{\alpha
1}+$
$\displaystyle+4(11k_{2}\rho_{2}+\frac{1}{2}k_{1}\rho_{2}\beta_{1}+11k_{1}\rho_{1}\rho_{2}+\frac{1}{4}k_{1}\xi_{(12)})g^{\nu\mu}S^{\alpha}_{\alpha
2}$ (62)
$\displaystyle+2\left(b_{1}k_{1}+b_{2}k_{2}\right)\mathbf{e}^{[+-][\nu\mu]}\\}=-J_{N}^{\mu}.$

Eq.(3) establishes the electric charge flow through a quadruplet. It shows
electric charge as a source for propagating diverse granular and collective
fields strengths. The new behavior is produced on its flux features given by
conduction, transmission, conservation, interaction. Conducted by fields
quadruplet, transmitted through positive, negative, zero charges, conserved by
gauge symmetry, and interacting with potential fields coupling with fields
strengths.

Appendix B introduces a complementary consideration to eq. (61). Two distinct
electric charge equations are derived by applying the transversal and
longitudinal operators. Electric charge equation is separated into spin-1 and
spin-0 sectors.

For spin-1:

$\partial_{\nu}\\{4k_{1}a_{1}F^{\nu\mu}+4k_{2}a_{2}U^{\nu\mu}+2k_{+}a_{3}V^{\nu\mu+}+2k_{-}a_{3}V^{\nu\mu+}+2\left(b_{1}k_{1}+b_{2}k_{2}\right)\mathbf{e}^{[+-][\nu\mu]}=-J_{NT}^{\mu}$
(63)

For spin-0:

$\partial_{\nu}\\{2\left(22k_{1}\rho_{1}+k_{2}\rho_{2}\beta_{1}+\frac{1}{2}k_{2}\xi_{(12)}+22k_{2}\rho_{1}\rho_{2}\right)g^{\mu\nu}S^{\alpha}_{\alpha
1}+\\\
+2\left(k_{1}\rho_{1}\beta_{2}+22k_{1}\rho_{1}\rho_{2}+\frac{1}{2}\xi_{(12)}+22k_{2}\rho_{2}\right)g^{\nu\mu}S^{\alpha}_{\alpha
2}\\}=-J_{NL}^{\mu}$ (64)

where $J^{\mu}_{NT}=\theta^{\mu}_{\nu}J^{\nu}_{N}$ and
$J^{\mu}_{NL}=\omega^{\mu}_{\nu}J^{\nu}_{N}$

The third Noether identity is the constraint

$\partial_{\mu}\partial_{\nu}K^{(\nu\mu)}=0$ (65)

Considering that a symmetric tensor $K_{(\mu\nu)}$ can be separated as

$K_{(\mu\nu)}=\tilde{K}_{(\mu\nu)}+g_{\mu\nu}K_{\alpha}^{\alpha}$ (66)

where $\tilde{K}_{(\mu\nu)}$ is a tensor with trace zero, it gives

$\Box K_{\alpha}^{\alpha}+\partial^{\mu}\partial^{\nu}\tilde{K}_{(\mu\nu)}=0$
(67)

Eq. (67) shows that Noether constraint does not interfere with Lagrangian
physicality. No term at eq. (5) is suppressed by the constraint. The presence
of granular and collective fields strengths is preserved.

The quadruplet electric charge symmetry enlarges the usual electric charge
electromagnetic scenario. It replaces Maxwell to an $U(1)\times
SO(2)_{global}$ symmetry from where the potential fields are tied by electric
charge. The experimental conserved charge [54] relationship is reproduced
under four fields associations, as eq. (57) shows. However, interactions not
necessarily related to electric charge coupling appear. The quadruplet
mutation physics introduces that the electric charge is no more a universal
coupling constant. Consequently, the beta decay reaction may be reinterpreted
as a phenomenon belonging to electric charge transfer. The Fermi coupling
constant dimension is indicating the presence of a massive propagator, and so,
instead, is interpreted as weak interaction to consider as an extended EM set
intermediate by massive bosons.

##  4 Electromagnetic whole dynamics

Associative physic is being expressed. A quadruplet interconnected dynamics is
formed. Whole equations are constituted. A set of equations is derived
combining minimal action principle, identities equations, and electric charge
equation. A nonlinear EM whole dynamic is generated.

Separating the spin-1 and spin-0 sectors, one gets For $A_{\mu T}$,

$\partial_{\nu}\\{4\left(a_{1}+\beta_{1}+k_{1}a_{1}\right)F^{\nu\mu}+2b_{1}[\mathbf{e}^{[12][\nu\mu]}+(1+k_{1})\mathbf{e}^{[+-][\nu\mu]}]\\}=M^{A\mu}_{IT}+J^{\mu}_{AT}-J^{\mu}_{NT}-k_{2}J^{\mu}_{UT}$
(68)

where the interactive mass term is

$M_{IT}^{A\mu}=-2k_{2}\mathbf{m}_{U}^{2}U^{\mu}-k_{-}\mathbf{m}_{V}^{2}V^{\mu+}-k_{+}\mathbf{m}_{V}^{2}V^{\mu-}$
(69)

Eq.(68) extends the usual massless photon equation. Eq.(69) shows that
although the photon is massless, it can have a penetration length.
$M_{IT}^{A\mu}$ works as a source to superconductivity [55].

For $U_{\mu T}$,

$\displaystyle\partial_{\nu}\\{4\left(a_{2}+\beta_{2}+k_{2}a_{2}\right)U^{\nu\mu}+2b_{2}[\mathbf{e}^{[12][\nu\mu]}+(1+k_{2})\mathbf{e}^{[+-][\nu\mu]}]\\}-2\mathbf{m}_{U}^{2}U_{T}^{\mu}=M^{A\mu}_{IT}+J^{\mu}_{UT}$
(70) $\displaystyle-J^{\mu}_{NT}-k_{1}J^{\mu}_{AT}$ (71)

with the interactive mass term

$M^{U\mu}_{I}=-k_{-}\mathbf{m}_{V}^{2}V^{\mu+}-k_{+}\mathbf{m}_{V}^{2}V^{\mu-}$
(72)

Eq. (71) introduces a photon with mass [56] and eq. (72) its correspondent
mass source

For $V^{\pm}_{\mu T}$,

$\partial_{\nu}\\{2\left(a_{3}+\beta_{+}\beta_{-}\right)V^{\nu\mu-}+2b_{3}\left(\mathbf{e}^{[-1][\nu\mu]}+\mathbf{e}^{[-2][\nu\mu]}\right)\\}-\mathbf{m}_{V}^{2}V_{T}^{\mu-}=J^{\mu-}_{VT}$
(73)

and

$\partial_{\nu}\\{2\left(a_{3}+\beta_{+}\beta_{-}\right)V^{\nu\mu+}+2b_{3}\left(\mathbf{e}^{[+1][\nu\mu]}+\mathbf{e}^{[+2][\nu\mu]}\right)\\}-\mathbf{m}_{V}^{2}V_{T}^{\mu+}=J^{\mu+}_{VT}$
(74)

Eqs. (73-74) are including charged vector bosons [13].

The longitudinal equations are

For $A_{\mu L}$,

$\partial_{\nu}\\{s_{11}g^{\nu\mu}S^{\alpha}_{\alpha
1}+c_{11}g^{\nu\mu}\left(\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(+-)\alpha}_{\alpha}\right)\\}=J^{\mu}_{AL}-J^{\mu}_{NL}+t_{11}\mathbf{m}^{2}_{U}U^{\mu}_{L}+\frac{1}{2}t_{11}J^{\mu}_{UL}$
(75)

For $U_{\mu L}$,

$\partial_{\nu}\\{s_{22}g^{\nu\mu}S^{\alpha}_{\alpha
2}+c_{22}g^{\nu\mu}\left(\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(+-)\alpha}_{\alpha}\right)\\}-2\mathbf{m}^{2}_{U}U^{\mu}_{L}=J^{\mu}_{AL}-J^{\mu}_{NL}+t_{22}J^{\mu}_{UL}$
(76)

For $V^{\pm}_{\mu L}$,

$\partial_{\nu}\\{2\left(\beta_{+}\beta_{-}+16\rho_{+}\rho_{-}+\rho_{+}\beta_{-}+\rho_{-}\beta_{+}\right)g^{\nu\mu}S_{\alpha}^{\alpha-}+\frac{1}{2}\left(\xi_{(33)}+\xi_{(44)}\right)g^{\nu\mu}S_{\alpha}^{\alpha-}+\\\
+\left(34\rho_{+}-2\beta_{+}\right)g^{\nu\mu}\left(\mathbf{e}^{(-1)\alpha}_{\alpha}+\mathbf{e}^{(-2)\alpha}_{\alpha}\right)\\}-\mathbf{m}_{V}^{2}V^{\mu-}_{L}=J^{\mu-}_{VL},$
(77)

and

$\partial_{\nu}\\{2\left(\beta_{+}\beta_{-}+16\rho_{+}\rho_{-}+\rho_{+}\beta_{-}+\rho_{-}\beta_{+}+\frac{1}{2}\xi_{(33)}+\frac{1}{2}\xi_{(44)}\right)g^{\nu\mu}S_{\alpha}^{\alpha+}\\\
+\left(34\rho_{-}-2\beta_{-}\right)g^{\nu\mu}\left(\mathbf{e}^{(+1)\alpha}_{\alpha}+\mathbf{e}^{(+2)\alpha}_{\alpha}\right)\\}-\mathbf{m}_{V}^{2}V^{\mu+}_{L}=J^{\mu+}_{VL}.$
(78)

Three kinds of EM observables appear. They are potential fields conglomerates,
granular and collective fields strengths. A whole dynamic is developed. Fields
strengths are derived under individual-collective dynamical correspondence at
space-time and act as their own sources by coupling with potential fields. The
spin-1 and spin-0 sectors embedded at Minkowski fields $A_{\mu
I}\equiv(\phi_{I},\vec{A}_{I})$ are separated by non-local operators
preserving covariance. A diverse dynamic for spin-1 and spin-0 sectors is
proposed. Similarly, in supersymmetry, one gets correlated spin-1 and spin-0
particles with different masses. They are physically viable but with a dynamic
distinct for each spin- sector. The above equations are also expressing the
four distinct scalar photons inserted at the Lorentz group.

A systemic relativistic equation is expressed. It provides a new EM energy
manifestation. New aspects on a mass, electric charge, and couplings are
derived. Besides mass without Higgs and couplings between electric and
magnetic fields without electric charge, the quadruplet equations of motion
express three kinds of fields associations. They are collective fields,
London, and conglomerates terms. The last two are written respectively as
$L=A_{\mu}A^{\mu}$ and $M_{IJ}=A_{\mu I}A^{\mu}_{J}$. They introduce the
following terms at equations

$l_{I}^{\mu}\equiv gLA_{I}^{\mu},\ L\equiv A_{\nu J}A^{\nu}_{J}$ (79)

and

$c_{I}^{\mu}\equiv g_{JK}M_{JK}A_{I}^{\mu},\ M_{JK}\equiv A_{\nu
J}A^{\nu}_{K},$ (80)

The current term is

$j_{I}^{\mu}\equiv f_{I}X^{\mu\nu}_{J}A_{\nu J}$ (81)

where $A_{\mu I}$ is any field belonging to the quadruplet $A_{\mu
I}=\\{A_{\mu},U_{\mu},V^{\pm}_{\mu}\\}$ and $X^{\mu\nu}_{I}$ any granular or
collective field strength. For the particular case where eq.(81) is expressed
as

$j^{\mu
s}_{I}=f_{J}\partial_{\nu}\left(A_{I}^{\nu}A^{\mu}_{J}-A_{J}^{\nu}A^{\mu}_{I}\right)$
(82)

it can be rewritten in terms of the spin operator.

$j^{\mu
s}_{I}=f_{I}i\partial_{\nu}\left(A_{J}^{\alpha}\left(\Sigma^{\nu\mu}\right)_{\alpha}A^{\beta}_{J}\right)$
(83)

where
$\left(\Sigma_{\alpha\beta}\right)^{\mu}_{\nu}=-i\left(\delta^{\mu}_{\alpha}\eta_{\alpha\beta}-\eta_{\alpha\nu}\delta^{\mu}_{\rho}\right)$.

Maxwell’s limitations are surrounded. The quadruplet dynamics crosses
consistently the Maxwell frontier. A nonlinear EM appears. Potential fields
are expressed explicitly in the equations. Granular and collective fields
strengths are developed. The collective fields’ strengths are identified to
polarization and magnetization vectors [57], Eq. (79) to London
superconductivity term [58], eq. (80) to Anderson plasmon [59]. Photon-photon
interaction is revived [60]. Eq. (83) incorporates spin presence [61].

Electric charge properties are derived. Historically, physics development has
been showing diverse performances on electric charge. The renormalization
group equation introduced the value of structure constant depending on
momentum, the standard model modulates its value, quarks introduce a
fractional value, Gell-Mann-Nishijima on quantum numbers [62]. A further
feature is electric charge mutation. Writing a generic current, one gets

$\displaystyle J^{\mu}\sim
2b_{1}\mathbf{e}_{[12]}F^{\mu\nu}A_{\nu}+4\mathbf{e}_{[12]}\mathbf{e}^{[+-][\mu\nu]}A_{\nu}+\left(\mathbf{e}_{[13]}-i\mathbf{e}_{[14]}+qa_{3}\right)V^{\mu\nu+}V_{\nu-}$
$\displaystyle+\left(\mathbf{e}_{[13]}-i\mathbf{e}_{[14]}+q\right)\mathbf{e}^{[+1][\mu\nu]}V_{\nu}^{+}$
(84)

where eq.(84) shows diverse types of EM coupling beyond electric charge.
Showing that EM interactions are not restricted to electric charge.
Physically, the EM flux comes first, and it includes neutral couplings.

Thus, an extended EM is proposed. Given the above features, the whole
quadruplet dynamics are derived. It gives,

For photon field:

$A_{\mu}$ spin-1:

$\displaystyle\partial_{\nu}\\{4\left(a_{1}+\beta_{1}+k_{1}a_{1}\right)F^{\nu\mu}+2b_{1}[\mathbf{e}^{[12][\nu\mu]}+(1+k_{1})\mathbf{e}^{[+-][\nu\mu]}]\\}+$
$\displaystyle+l_{AT}^{\mu}+c_{AT}^{\mu}=M^{A\mu}_{IT}+j^{\mu}_{AT}-J^{\mu}_{NT}-k_{2}J^{\mu}_{UT}$
(85)

$A_{\mu}$ spin-0:

$\displaystyle\partial_{\nu}\\{s_{11}g^{\nu\mu}S^{\alpha}_{\alpha
1}+c_{11}g^{\nu\mu}\left(\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(+-)\alpha}_{\alpha}\right)\\}$
$\displaystyle+l_{AL}^{\mu}+c_{AL}^{\mu}=j^{\mu}_{AL}-J^{\mu}_{NL}+t_{11}\mathbf{m}_{U}U^{\mu}_{L}+\frac{1}{2}t_{11}J^{\mu}_{UL}$
(86)

where

$\displaystyle
l_{A}^{\mu}=-A_{\nu}\big{\\{}4\mathbf{e}_{(11)}\big{(}\mathbf{e}^{(+-)(\mu\nu)}+2g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}+g^{\mu\nu}\mathbf{e}^{(22)\alpha}_{\alpha}+2g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}+17g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}\big{)}$
$\displaystyle+4\mathbf{e}_{(12)}\mathbf{e}^{(11)(\mu\nu)}\big{\\}}-U_{\nu}\\{4\mathbf{e}_{[12]}\mathbf{e}^{[12][\mu\nu]}+4\mathbf{e}_{(12)}(17g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}+2\mathbf{e}^{(22)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}$
(87) $\displaystyle+18g^{\mu\nu}\mathbf{e}^{(22)\alpha}_{\alpha})\\},$
$\displaystyle
c^{\mu}_{A}=-A_{\nu}\\{4\mathbf{e}_{(11)}(\mathbf{e}^{(+-)(\mu\nu)}+17g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha})\\}-U_{\nu}\\{4\mathbf{e}_{[12]}\mathbf{e}^{[+-][\mu\nu]}+72\mathbf{e}_{(12)}g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha}\\}+$
$\displaystyle-
V_{\nu}^{-}\\{\sqrt{2}\left(\mathbf{e}_{[13]}-i\mathbf{e}_{[14]}\right)(\mathbf{e}^{[+1][\mu\nu]}+\mathbf{e}^{[+2][\mu\nu]})+\sqrt{2}\left(\mathbf{e}_{(13)}-i\mathbf{e}_{(14)}\right)(\mathbf{e}^{(+1)(\mu\nu)}+$
$\displaystyle+18g^{\mu\nu}\mathbf{e}^{(+1)\alpha}_{\alpha}+\mathbf{e}^{(+2)(\mu\nu)}+18g^{\mu\nu}\mathbf{e}^{(+2)\alpha}_{\alpha})\\}-V_{\nu}^{+}\\{\sqrt{2}\left(\mathbf{e}_{[13]}+i\mathbf{e}_{[14]}\right)(\mathbf{e}^{[-1][\mu\nu]}+$
(88)
$\displaystyle+\mathbf{e}^{[-2][\mu\nu]})+\sqrt{2}\left(\mathbf{e}_{(13)}+i\mathbf{e}_{(14)}\right)(\mathbf{e}^{(-1)(\mu\nu)}+18g^{\mu\nu}\mathbf{e}^{(-1)\alpha}_{\alpha}+\mathbf{e}^{(-2)(\mu\nu)}+18g^{\mu\nu}\mathbf{e}^{(-2)\alpha}_{\alpha}\\}$

The current term is

$\displaystyle
j^{\mu}_{A}=A_{\nu}\\{2\mathbf{e}_{(11)}\big{[}(\frac{1}{2}\beta_{1}+17\rho_{1})g^{\mu\nu}S^{\alpha}_{\alpha
1}+\left(\beta_{2}+17\rho_{2}\right)g^{\mu\nu}S^{\alpha}_{\alpha
2}+\beta_{1}S^{\mu\nu}_{1}\big{]}-\beta_{1}\mathbf{e}_{(12)}g^{\mu\nu}S^{\alpha}_{\alpha
2}$
$\displaystyle+\beta_{1}\mathbf{e}_{(22)}S^{(\mu\nu)}_{2}\\}+U_{\nu}\\{2\mathbf{e}_{[12]}\left(b_{1}F^{\mu\nu}+b_{2}U^{\mu\nu}\right)+2\mathbf{e}_{(12)}\big{[}\beta_{2}S^{\mu\nu}_{2}+\left(\beta_{1}+17\rho_{1}\right)g^{\mu\nu}S^{\alpha}_{\alpha
1}$
$\displaystyle+\left(\beta_{2}+17\rho_{2}\right)g^{\mu\nu}S^{\alpha}_{\alpha
2}\big{]}-\beta_{1}\mathbf{e}_{(22)}g^{\mu\nu}S^{\alpha}_{\alpha
2}-\beta_{1}\mathbf{e}_{(22)}S^{\mu\nu}_{2}\\}+V_{\nu}^{+}\big{\\{}\sqrt{2}\left(\mathbf{e}_{[13]}+i\mathbf{e}_{[14]}\right)V^{\mu\nu-}$
$\displaystyle+\sqrt{2}\left(\mathbf{e}_{(13)}+i\mathbf{e}_{(14)}\right)\left(\beta_{-}S^{\mu\nu-}+17\rho_{-}g^{\mu\nu}S^{\alpha-}_{\alpha}+\right)-\beta_{1}\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)\big{(}g^{\mu\nu}S_{\alpha}^{\alpha-}+$
(89)
$\displaystyle+2S^{\mu\nu-}\big{)}\big{\\}}+V_{\nu}^{-}\\{\sqrt{2}\left(\mathbf{e}_{[13]}-i\mathbf{e}_{[14]}\right)V^{\mu\nu+}-\beta_{1}\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)\left(g^{\mu\nu}S_{\alpha}^{\alpha+}+2S^{\mu\nu+}\right)$
$\displaystyle+\sqrt{2}\left(\mathbf{e}_{(13)}-i\mathbf{e}_{(14)}\right)\left(\beta_{+}S^{\mu\nu+}+17\rho_{+}g^{\mu\nu}S^{\alpha+}_{\alpha}+\beta_{+}g^{\mu\nu}S^{\alpha+}_{\alpha}\right)\\}$

For massive photon field:

$U_{\mu}$ spin-1:

$\displaystyle\partial_{\nu}\left\\{4\left(a_{2}+\beta_{2}+a_{2}k_{2}\right)U^{\nu\mu}+2b_{2}[\mathbf{e}^{[12][\nu\mu]}+(1+k_{2})\mathbf{e}^{[+-][\nu\mu]}]+2b_{2}k_{2}\mathbf{e}^{[+-][\nu\mu]}\right\\}+$
$\displaystyle-2\mathbf{m}_{U}^{2}U^{\mu}_{T}+l^{\mu}_{UT}+c^{\mu}_{U}=M^{U\mu}_{IT}+j^{\mu}_{UT}-k_{1}J^{\mu}_{AT}-J^{\mu}_{NT}$
(90)

$U_{\mu}$ spin-0:

$\partial_{\nu}\left\\{4s_{22}g^{\mu\nu}S^{\alpha}_{\alpha
2}+c_{22}g^{\mu\nu}\left(\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(11)\alpha}_{\alpha}\right)\right\\}-2\mathbf{m}_{U}^{2}U^{\mu}_{L}+l^{\mu}_{UL}+c^{\mu}_{UL}=\\\
=j^{\mu}_{UL}-k_{1}J^{\mu}_{AL}-J^{\mu}_{NL}$ (91)

where

$\displaystyle
l^{\mu}_{U}=-A_{\nu}\\{4\mathbf{e}_{[12]}\mathbf{e}^{[12][\mu\nu]}+4\mathbf{e}_{(12)}\mathbf{e}^{(22)(\mu\nu)}+8\mathbf{e}_{(12)}\mathbf{e}^{(12)(\mu\nu)}$
$\displaystyle+72\mathbf{e}_{(12)}g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}+4\mathbf{e}_{(12)}g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}+68\mathbf{e}_{(12)}g^{\mu\nu}\mathbf{e}^{(22)\alpha}_{\alpha}\\}$
$\displaystyle-
U_{\nu}\\{4\mathbf{e}_{(22)}\mathbf{e}^{(22)(\mu\nu)}+8\mathbf{e}_{(22)}g^{\mu\nu}\mathbf{e}^{(22)\alpha}_{\alpha}+4\mathbf{e}_{(22)}\mathbf{e}^{(11)(\mu\nu)}$
(92)
$\displaystyle+\mathbf{e}_{(22)}g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}+4\mathbf{e}_{(12)}\mathbf{e}^{(12)(\mu\nu)}+72\mathbf{e}_{(22)}g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}\\},$
$\displaystyle
c^{\mu}_{U}=-A_{\nu}\\{4\mathbf{e}_{[12]}\mathbf{e}^{[+-][\mu\nu]}+72\mathbf{e}_{(12)}g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha}\\}-U_{\nu}\\{4\mathbf{e}_{(22)}\mathbf{e}^{(+-)\alpha}_{\alpha}$
$\displaystyle+4\mathbf{e}_{(22)}\mathbf{e}^{(+-)(\mu\nu)}+68\mathbf{e}_{(22)}g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha}\\}+V_{\nu}^{-}\\{\sqrt{2}\left(\mathbf{e}_{[23]}-i\mathbf{e}_{[24]}\right)\big{(}\mathbf{e}^{[+2][\mu\nu]}\mathbf{e}^{[+1][\mu\nu]}\big{)}$
$\displaystyle\sqrt{2}\left(\mathbf{e}_{(23)}-i\mathbf{e}_{(24)}\right)\big{(}\mathbf{e}^{(+1)(\mu\nu)}+\mathbf{e}^{(+2)(\mu\nu)}+18g^{\mu\nu}\mathbf{e}^{(+1)\alpha}_{\alpha}+18g^{\mu\nu}\mathbf{e}^{(+2)\alpha}_{\alpha}\big{)}\\}$
$\displaystyle
V_{\nu}^{+}\\{\sqrt{2}\left(\mathbf{e}_{[23]}+i\mathbf{e}_{[24]}\right)\mathbf{e}^{[-2][\mu\nu]}+\sqrt{2}\left(\mathbf{e}_{[23]}+i\mathbf{e}_{[24]}\right)\mathbf{e}^{[-1][\mu\nu]}$
(93)
$\displaystyle+\sqrt{2}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)\mathbf{e}^{(-1)(\mu\mu)}+18\sqrt{2}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)g^{\mu\nu}\mathbf{e}^{(-1)\alpha}_{\alpha}$
$\displaystyle+\sqrt{2}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)\mathbf{e}^{(-2)(\mu\nu)}+18\sqrt{2}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)\mathbf{e}^{(-2)\alpha}_{\alpha}\\}$

The current term is

$\displaystyle
j^{\mu}_{U}=A_{\nu}\\{\mathbf{e}_{[12]}\left(b_{1}F^{\mu\nu}+b_{2}U^{\mu\nu}\right)+2\mathbf{e}_{(12)}[(\beta_{1}+17\rho_{1})g^{\mu\nu}S^{\alpha}_{\alpha
1}$
$\displaystyle+\left(1/2\beta_{2}+17\rho_{2}\right)g^{\mu\nu}S^{\alpha}_{\alpha}+\beta_{1}S^{\mu\nu}_{1}\big{]}-\beta_{2}\mathbf{e}_{(12)}g^{\mu\nu}S^{\alpha}_{\alpha
1}-2\beta_{2}\mathbf{e}_{(11)}S^{\mu\nu}_{1}\\}+$
$\displaystyle+U_{\nu}\\{2\mathbf{e}_{(22)}[\beta_{1}S^{\mu\nu}_{1}+(\beta_{1}+17\rho_{1})g^{\mu\nu}S^{\alpha}_{\alpha
1}+\left(1/2\beta_{2}+17\rho_{2}\right)g^{\mu\nu}S^{\alpha 2}_{\alpha}]$
$\displaystyle-2\beta_{2}\mathbf{e}_{(12)}g^{\mu\nu}S^{\alpha}_{\alpha
1}+V_{\nu}^{-}\\{\sqrt{2}\left(\mathbf{e}_{[23]}-i\mathbf{e}_{[24]}\right)V^{\mu\nu+}+\sqrt{2}\big{(}\mathbf{e}_{(23)}+$
(94)
$\displaystyle-i\mathbf{e}_{(24)}\big{)}[\beta_{+}S^{\mu\nu+}+\left(\beta_{+}+17\rho_{+}\right)g^{\mu\nu}S^{\alpha+}_{\alpha}]-\beta_{2}\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)\big{(}g^{\mu\nu}S^{\alpha+}_{\alpha}$
$\displaystyle+2S^{\mu\nu+}\big{)}\\}+V_{\nu}^{+}\\{\sqrt{2}\left(\mathbf{e}_{[23]}+i\mathbf{e}_{[24]}\right)V^{\mu\nu-}+\sqrt{2}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)$
$\displaystyle\left(\beta_{-}+17\rho_{-}\right)g^{\mu\nu}S^{\alpha-}_{\alpha}-\beta_{2}\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)\left(g^{\mu\nu}S^{\alpha-}_{\alpha}+2S^{\mu\nu-}\right)\\}$

Considering positive charged photon,

$V^{+}_{\mu}$ spin-1:

$\displaystyle\partial_{\nu}\big{\\{}2\left(a_{3}+\beta_{+}\beta_{-}\right)V^{\nu\mu-}+2b_{3}\left(\mathbf{e}^{[-1][\nu\mu]}+\mathbf{e}^{[-2][\nu\mu]}\right)\big{\\}}+$
$\displaystyle-\mathbf{m}_{V}^{2}V^{\mu-}_{T}+l_{VT}^{\mu-}+c_{VT}^{\mu-}=j^{\mu-}_{VT}$
(95)

$V^{+}_{\mu}$ spin-0:

$\partial_{\nu}\\{4s_{-}g^{\mu\nu}S^{\alpha-}_{\alpha}+c_{-}g^{\mu\nu}\left(\mathbf{e}^{(-1)\alpha}_{\alpha}+\mathbf{e}^{(-2)\alpha}_{\alpha}\right)\\}-\mathbf{m}_{V}^{2}V^{\mu-}_{L}+l^{\mu-}_{VL}+c^{\mu-}_{VL}=j^{\mu-}_{VL}$
(96)

where

$\displaystyle
l^{\mu-}_{V}=-V_{\nu}^{-}\big{\\{}\big{[}34\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)g^{\mu\nu}\left(\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(22)\alpha}_{\alpha}\right)\big{]}\big{\\}}$
$\displaystyle-
V_{\nu}^{+}\big{\\{}16\left(\mathbf{e}_{(33)}-\mathbf{e}_{(44)}\right)g^{\mu\nu}\mathbf{e}^{(--)\alpha}_{\alpha}++32i\mathbf{e}_{(34)}\mathbf{e}_{(34)}\mathbf{e}^{(--34)\alpha}_{\alpha}\big{\\}}$
(97) $\displaystyle
c^{\mu-}_{V}=-A_{\nu}\\{\sqrt{2}\left(\mathbf{e}_{[13]}+i\mathbf{e}_{[14]}\right)\big{(}\mathbf{e}^{[-1][\mu\nu]}+\mathbf{e}^{[-2][\mu\nu]}\big{)}+$
$\displaystyle\sqrt{2}\left(\mathbf{e}_{(13)}+i\mathbf{e}_{(14)}\right)(\mathbf{e}^{(-1)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(-1)\alpha}_{\alpha}+\mathbf{e}^{(-2)(\mu\nu)}$
$\displaystyle+g^{\mu\nu}\mathbf{e}^{(-2)\alpha}_{\alpha})+16\sqrt{2}\left(\mathbf{e}_{(13)}+i\mathbf{e}_{(14)}\right)g^{\mu\nu}\big{(}\mathbf{e}^{(-1)\alpha}_{\alpha}$
$\displaystyle+\mathbf{e}^{(-2)\alpha}_{\alpha}\big{)}\\}-U_{\nu}\\{\sqrt{2}\left(\mathbf{e}_{[23]}+i\mathbf{e}_{[24]}\right)\big{(}\mathbf{e}^{[-1][\mu\nu]}+\mathbf{e}^{[-2][\mu\nu]}\big{)}$
$\displaystyle+\sqrt{2}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)(\mathbf{e}^{(-1)(\mu\nu)}+\mathbf{e}^{(-2)\alpha}_{\alpha}+g^{\mu\nu}\mathbf{e}^{(-1)\alpha}_{\alpha}$
(98)
$\displaystyle+\mathbf{e}^{(-2)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(-2)\alpha}_{\alpha})+16g^{\mu\nu}\mathbf{e}^{(-1)\alpha}_{\alpha}\\}$
$\displaystyle-
V_{\nu}^{-}\\{4i\mathbf{e}_{[34]}\mathbf{e}^{[+-][\mu\nu]}+4i\mathbf{e}_{[34]}\mathbf{e}^{[12][\mu\nu]}+2\big{(}\mathbf{e}^{(11)(\mu\nu)}+\mathbf{e}^{(-2)(\mu\nu)}$
$\displaystyle+g^{\mu\nu}\mathbf{e}^{(-2)\alpha}_{\alpha})+16g^{\mu\nu}\mathbf{e}^{(-1)\alpha}_{\alpha}\\}+2\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)\mathbf{e}^{(+-)(\mu\nu)}+$
$\displaystyle+2\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha}\\}+V_{\nu}^{+}\\{\left(\mathbf{e}_{(33)}-\mathbf{e}_{(44)}\right)\mathbf{e}^{(--)(\alpha)}_{\alpha}$
$\displaystyle+2i\mathbf{e}_{(34)}\mathbf{e}^{(--34)(\mu\nu)}\\}$

The current term is

$\displaystyle
j_{V}^{\mu-}=V_{\nu}^{-}\big{\\{}-2\mathbf{e}_{[34]}\left(b_{1}F^{\mu\nu}+b_{2}U^{\mu\nu}\right)+\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)[\beta_{1}S^{\mu\nu}_{1}$
$\displaystyle+\left(\beta_{1}+17\rho_{1}\right)g^{\mu\nu}S^{\alpha}_{\alpha
1}+\beta_{2}S^{\mu\nu}_{2}+\left(\beta_{2}+17\rho_{2}\right)g^{\mu\nu}S^{\alpha}_{\alpha
2}]$
$\displaystyle-\beta_{+}\big{(}\mathbf{e}_{(13)}-i\mathbf{e}_{(14)}\big{)}\big{(}\frac{1}{\sqrt{2}}g^{\mu\nu}S^{\alpha}_{\alpha
1}+\sqrt{2}S^{\mu\nu}_{1}\big{)}+$
$\displaystyle-\beta_{+}\left(\mathbf{e}_{(23)}-i\mathbf{e}_{(24)}\right)\big{(}\frac{1}{\sqrt{2}}g^{\mu\nu}S^{\alpha}_{\alpha
2}+\sqrt{2}S^{\mu\nu}_{2}\big{)}\big{\\}}$
$\displaystyle+A_{\nu}\\{\sqrt{2}\left(\mathbf{e}_{[13]}+i\mathbf{e}_{[14]}\right)V^{\mu\nu-}+\sqrt{2}\left(\mathbf{e}_{(13)}+i\mathbf{e}_{(14)}\right)[\beta_{-}S^{\mu\nu-}$
(99)
$\displaystyle+\left(\beta_{-}+17\rho_{-}\right)g^{\mu\nu}S^{\alpha-}_{\alpha}]-\beta_{+}\big{(}\mathbf{e}_{(13)}-i\mathbf{e}_{(14)}\big{)}\big{(}\frac{1}{\sqrt{2}}g^{\mu\nu}S^{\alpha-}_{\alpha}$
$\displaystyle+\sqrt{2}S^{\mu\nu-}\big{)}\\}+U_{\nu}\big{\\{}\sqrt{2}\left(\mathbf{e}_{[23]}+i\mathbf{e}_{[24]}\right)V^{\mu\nu-}+$
$\displaystyle+\sqrt{2}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)\big{[}\beta_{-}S^{\mu\nu-}+\left(\beta_{-}+17\rho_{-}\right)g^{\mu\nu}S^{\alpha-}_{\alpha}\big{]}$
$\displaystyle-\beta_{+}\left(\mathbf{e}_{(23)}-i\mathbf{e}_{(24)}\right)\big{(}1/\sqrt{2}g^{\mu\nu}S^{\alpha-}_{\alpha}+\sqrt{2}S^{\mu\nu-}\big{)}\big{\\}}$

Considering negative charged photon,

$V^{-}_{\mu}$ spin-1:

$\displaystyle\partial_{\nu}\\{2\left(a_{3}+\beta_{+}\beta_{-}\right)V^{\nu\mu+}+2b_{3}\left(\mathbf{e}^{[+1][\nu\mu]}+\mathbf{e}^{[+2][\nu\mu]}\right)\\}$
$\displaystyle-\mathbf{m}_{V}^{2}V^{\mu+}_{T}+l_{VT}^{\mu+}+c_{VT}^{\mu+}=j^{\mu+}_{VT}$
(100)

$V^{-}_{\mu}$ spin-0:

$\partial_{\nu}\\{4s_{+}g^{\mu\nu}S^{\alpha+}_{\alpha}+c_{+}g^{\mu\nu}\left(\mathbf{e}^{(+1)\alpha}_{\alpha}+\mathbf{e}^{(+2)\alpha}_{\alpha}\right)\\}-\mathbf{m}_{V}^{2}V^{\mu+}_{L}+l^{\mu+}_{VL}+c^{\mu+}_{VL}=j^{\mu+}_{VL}$
(101) $\displaystyle
l^{\mu+}_{V}=-V_{\nu}^{+}\\{\left[34\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)g^{\mu\nu}\left(\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(22)\alpha}_{\alpha}\right)\right]\\}$
$\displaystyle-
V_{\nu}^{-}\\{16\left(\mathbf{e}_{(33)}-\mathbf{e}_{(44)}\right)g^{\mu\nu}\mathbf{e}^{(++)\alpha}_{\alpha}++32i\mathbf{e}_{(34)}\mathbf{e}_{(34)}\mathbf{e}^{(++34)\alpha}_{\alpha}\\}$
(102) $\displaystyle
c^{\mu+}_{V}=-A_{\nu}\big{\\{}\sqrt{2}\left(\mathbf{e}_{[13]}-i\mathbf{e}_{[14]}\right)\big{(}\mathbf{e}^{[+1][\mu\nu]}+\mathbf{e}^{[+2][\mu\nu]}\big{)}$
$\displaystyle+\sqrt{2}\big{(}\mathbf{e}_{(13)}-i\mathbf{e}_{(14)}\big{)}\big{(}\mathbf{e}^{(+1)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(+1)\alpha}_{\alpha}+\mathbf{e}^{(+2)(\mu\nu)}$
$\displaystyle+g^{\mu\nu}\mathbf{e}^{(+2)\alpha}_{\alpha}+16g^{\mu\nu}\mathbf{e}^{(+1)\alpha}_{\alpha}+16g^{\mu\nu}\mathbf{e}^{(+2)\alpha}_{\alpha}\big{)}\big{\\}}$
$\displaystyle-
U_{\nu}\big{\\{}\sqrt{2}\left(\mathbf{e}_{[23]}-i\mathbf{e}_{[24]}\right)\big{(}\mathbf{e}^{[+1][\mu\nu]}+\mathbf{e}^{[+2][\mu\nu]}\big{)}+$
$\displaystyle+\sqrt{2}\left(\mathbf{e}_{(23)}-i\mathbf{e}_{(24)}\right)\big{(}\mathbf{e}^{(+1)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(+1)\alpha}_{\alpha}+\mathbf{e}^{(+2)(\mu\nu)}$
$\displaystyle+g^{\mu\nu}\mathbf{e}^{(+2)\alpha}_{\alpha}+16g^{\mu\nu}\mathbf{e}^{(+1)\alpha}_{\alpha}+16g^{\mu\nu}\mathbf{e}^{(+2)\alpha}_{\alpha}\big{)}\big{\\}}$
(103) $\displaystyle-
V_{\nu}^{+}\\{4i\mathbf{e}_{[34]}\mathbf{e}^{[+-][\mu\nu]}+4i\mathbf{e}_{[34]}\mathbf{e}^{[12][\mu\nu]}+2\big{(}\mathbf{e}^{(11)(\mu\nu)}$
$\displaystyle+\mathbf{e}^{(22)(\mu\nu)}+\mathbf{e}^{(12)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}\big{)}+2\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)\mathbf{e}^{(+-)(\mu\nu)}$
$\displaystyle+2\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha}\\}+$
$\displaystyle+V_{\nu}^{-}\\{\left(\mathbf{e}_{(33)}-\mathbf{e}_{(44)}\right)\mathbf{e}^{(++)(\alpha)}_{\alpha}+2i\mathbf{e}_{(34)}\mathbf{e}^{(++34)(\mu\nu)}\\}$

The current term is

$\displaystyle
j_{V}^{\mu+}=V_{\nu}^{+}\big{\\{}-2\mathbf{e}_{[34]}\left(b_{1}F^{\mu\nu}+b_{2}U^{\mu\nu}\right)+\left(\mathbf{e}_{(33)}+\mathbf{e}_{(44)}\right)\big{[}\left(\beta_{1}+17\rho_{1}\right)g^{\mu\nu}S^{\alpha}_{\alpha
1}$
$\displaystyle\beta_{1}S^{\mu\nu}_{1}+\beta_{2}S^{\mu\nu}_{2}+\left(\beta_{2}+17\rho_{2}\right)g^{\mu\nu}S^{\alpha}_{\alpha
2}\big{]}-\beta_{-}\left(\mathbf{e}_{(13)}+i\mathbf{e}_{(14)}\right)\big{(}1/\sqrt{2}g^{\mu\nu}S^{\alpha}_{\alpha
1}$
$\displaystyle+\sqrt{2}S^{\mu\nu}_{1}\big{)}-\beta_{-}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)\big{(}\frac{1}{\sqrt{2}}g^{\mu\nu}S^{\alpha}_{\alpha
2}+\sqrt{2}S^{\mu\nu}_{2}\big{)}\big{\\}}+A_{\nu}\\{\sqrt{2}\big{(}\mathbf{e}_{[13]}+$
(104)
$\displaystyle-i\mathbf{e}_{[14]}\big{)}V^{\mu\nu+}+\sqrt{2}\left(\mathbf{e}_{(13)}-i\mathbf{e}_{(14)}\right)[\beta_{+}S^{\mu\nu+}+\left(\beta_{+}+17\rho_{+}\right)g^{\mu\nu}S^{\alpha+}_{\alpha}]+$
$\displaystyle-\beta_{-}\left(\mathbf{e}_{(13)}+i\mathbf{e}_{(14)}\right)\big{(}1/\sqrt{2}g^{\mu\nu}S^{\alpha+}_{\alpha}+\sqrt{2}S^{\mu\nu+}\big{)}\\}+U_{\nu}\big{\\{}\sqrt{2}\big{(}\mathbf{e}_{[23]}+$
$\displaystyle-i\mathbf{e}_{[24]}\big{)}V^{\mu\nu+}+\sqrt{2}\left(\mathbf{e}_{(23)}-i\mathbf{e}_{(24)}\right)\big{[}\beta_{+}S^{\mu\nu+}+\left(\beta_{+}+17\rho_{+}\right)g^{\mu\nu}S^{\alpha+}_{\alpha}\big{]}$
$\displaystyle-\beta_{-}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)\big{(}\frac{1}{\sqrt{2}}g^{\mu\nu}S^{\alpha+}_{\alpha}+\sqrt{2}S^{\mu\nu+}\big{)}\big{\\}}$

A fields-fields EM is encountered. EM energy is developed with features beyond
$E=mc^{2}$ and electric charge interaction. New opportunities for electric-
magnetic fields pairs, mass, electric charge, spin are expressed. The EM pairs
develop relationships beyond Maxwell, for instance, collective EM pairs are
created. Fields are attached to mass as interconvertible. The energy-mass
relationship is expressed without Higgs. Fields converted to masses are
obtained through London and conglomerates terms. The interactive mass appears
to work as the source.

A whole interlaced dynamics is expressed. New properties from such fundamental
whole dynamics laws are expected. First, with NLED. Physics contains critical
electric and magnetic fields $E_{c}=\frac{m^{2}_{e}c^{3}}{eh}\sim 1.32\times
10^{18}V\cdot m^{-}1$, $B_{c}=\frac{m^{2}_{e}c^{2}}{e\hbar}\sim 4.41\times
10^{9}T$ which non-linearities have to be understood theoretically. Consider a
fundamental NLED instead of effective models. Second, a physics-based on
correlated fields is derived. Faraday lines are obtained for fields strengths.
Third, a duality similar Seilberg-Witten equations are expected [63], [49].
Fourth, it hypothesis different EM phases, as Mandelstam considered non-
abelian theories [64]. Fifth, whole equations introduce the evolution scope. A
quanta evolution physics where physical entities are submitted to fields of
environment, free will, and adaptation. The quadruplet fields family is
expressing a field’s environment. Quanta generated from fields set and with a
chance. The whole quanta with a change are determined from the free
coefficients at quadruplet fields set Lagrangian. Quanta free will appears
from circumstances given by free coefficients. Quanta adaptation appears as
physics depending on energy distribution, as the system has minimal energy.
Another type of adaptation should be considered from mathematics. Through
various possible fields, solutions are to be derived from a nonlinear system.

##  5 Bianchi Identities

The third type of equation, to analyze the system, is constituted by eqs. (1
-4) are the Bianchi identities. The following granular expressions are derived

$\partial_{\mu}F_{\nu\rho}+\partial_{\nu}F_{\rho\mu}+\partial_{\rho}F_{\mu\nu}=0$
(105)
$\partial_{\mu}U_{\nu\rho}+\partial_{\nu}U_{\rho\mu}+\partial_{\rho}U_{\mu\nu}=0$
(106)
$\partial_{\mu}V^{\pm}_{\nu\rho}+\partial_{\nu}V^{\pm}_{\rho\mu}+\partial_{\rho}V^{\pm}_{\mu\nu}=0$
(107)

They will be complemented by twelve more collective Bianchi identities
corresponding to each physical collective field. For the collective
antisymmetric tensor, one gets

$\displaystyle\partial_{\mu}\mathbf{e}^{[12]}_{[\nu\rho]}+\partial_{\nu}\mathbf{e}^{[12]}_{[\rho\mu]}+\partial_{\rho}\mathbf{e}^{[12]}_{[\mu\nu]}=\mathbf{e}_{[12]}\\{A_{\mu}U_{\nu\rho}+A_{\nu}U_{\rho\mu}$
$\displaystyle+A_{\rho}U_{\mu\nu}-U_{\mu}F_{\nu\rho}-U_{\nu}F_{\rho\mu}+U_{\rho}F_{\mu\nu}\\},$
(108)
$\displaystyle\partial_{\mu}\mathbf{e}^{[+-]}_{[\nu\rho]}+\partial_{\nu}\mathbf{e}^{[+-]}_{[\rho\mu]}+\partial_{\rho}\mathbf{e}^{[+-]}_{[\mu\nu]}=-i\mathbf{e}_{[34]}\\{V_{\mu}^{+}V^{-}_{\nu\rho}+V_{\nu}^{+}V^{-}_{\rho\mu}+$
(109)
$\displaystyle+V_{\rho}^{+}V^{-}_{\mu\nu}-V_{\mu}^{-}V^{+}_{\nu\rho}-V_{\nu}^{-}V^{+}_{\rho\mu}-V_{\rho}^{-}V^{+}_{\mu\nu}\\},$
$\displaystyle\partial_{\mu}\left(\mathbf{e}^{[+1]}_{[\nu\rho]}+\mathbf{e}^{[-1]}_{[\nu\rho]}\right)+\partial_{\nu}\left(\mathbf{e}^{[+1]}_{[\rho\mu]}+\mathbf{e}^{[-1]}_{[\rho\mu]}\right)+\partial_{\rho}\left(\mathbf{e}^{[+1]}_{[\mu\nu]}+\mathbf{e}^{[-1]}_{[\mu\nu]}\right)=$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\mathbf{e}_{[13]}+i\mathbf{e}_{[14]}\right)\\{A_{\mu}V^{+}_{\nu\rho}+A_{\nu}V^{+}_{\rho\mu}+A_{\rho}V^{+}_{\mu\nu}+V^{+}_{\mu}F_{\nu\rho}+$
$\displaystyle+V^{+}_{\nu}F_{\rho\mu}+V^{+}_{\mu}F_{\nu\rho}\\}+\frac{1}{\sqrt{2}}\left(\mathbf{e}_{[13]}+i\mathbf{e}_{[14]}\right)\\{A_{\mu}V^{-}_{\nu\rho}+A_{\nu}V^{-}_{\rho\mu}+$
(110) $\displaystyle
A_{\rho}V^{-}_{\mu\nu}+V^{-}_{\mu}F_{\nu\rho}+V^{-}_{\nu}F_{\rho\mu}+V^{-}_{\mu}F_{\nu\rho}\\},$

and

$\displaystyle\partial_{\mu}\left(\mathbf{e}^{[+2]}_{[\nu\rho]}+\mathbf{e}^{[-2]}_{[\nu\rho]}\right)+\partial_{\nu}\left(\mathbf{e}^{[+2]}_{[\rho\mu]}+\mathbf{e}^{[-2]}_{[\rho\mu]}\right)+\partial_{\rho}\left(\mathbf{e}^{[+2]}_{[\mu\nu]}+\mathbf{e}^{[-2]}_{[\mu\nu]}\right)=$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\mathbf{e}_{[23]}+i\mathbf{e}_{[24]}\right)\\{U_{\mu}V^{+}_{\nu\rho}+U_{\nu}V^{+}_{\rho\mu}+U_{\rho}V^{+}_{\mu\nu}+V^{+}_{\mu}U_{\nu\rho}$
(111)
$\displaystyle+V^{+}_{\nu}U_{\rho\mu}+V^{+}_{\mu}U_{\nu\rho}\\}+\frac{1}{\sqrt{2}}\left(\mathbf{e}_{[23]}+i\mathbf{e}_{[24]}\right)\\{U_{\mu}V^{-}_{\nu\rho}+U_{\nu}V^{-}_{\rho\mu}+$
$\displaystyle
U_{\rho}V^{-}_{\mu\nu}+V^{-}_{\mu}U_{\nu\rho}+V^{-}_{\nu}U_{\rho\mu}+V^{-}_{\mu}U_{\nu\rho}\\}.$

For collective symmetric tensor, it yields

$\partial_{\mu}\mathbf{e}^{(11)}_{(\nu\rho)}+\partial_{\nu}\mathbf{e}^{(11)}_{(\rho\mu)}+\partial_{\rho}\mathbf{e}^{(11)}_{(\mu\nu)}=\mathbf{e}_{(11)}\\{A_{\mu}S_{\nu\rho
1}+A_{\nu}S_{\rho\mu 1}+A_{\rho}S_{\mu\nu 1}\\},$ (112)
$\partial_{\mu}\mathbf{e}^{(22)}_{(\nu\rho)}+\partial_{\nu}\mathbf{e}^{(22)}_{(\rho\mu)}+\partial_{\rho}\mathbf{e}^{(22)}_{(\mu\nu)}=\mathbf{e}_{(22)}\\{U_{\mu}S_{\nu\rho
2}+U_{\nu}S_{\rho\mu 2}+U_{\rho}S_{\mu\nu 2}\\},$ (113)
$\partial_{\mu}\mathbf{e}^{(12)}_{(\nu\rho)}+\partial_{\nu}\mathbf{e}^{(12)}_{(\rho\mu)}+\partial_{\rho}\mathbf{e}^{(12)}_{(\mu\nu)}=\mathbf{e}_{(12)}\\{A_{\mu}S_{\nu\rho
2}+A_{\nu}S_{\rho\mu 2}+A_{\rho}S_{\mu\nu 2}+U_{\mu}S_{\nu\rho
1}+U_{\nu}S_{\rho\mu 1}+U_{\rho}S_{\mu\nu 1}\\},$ (114)
$\displaystyle\partial_{\mu}\mathbf{e}^{(+-)}_{(\nu\rho)}+\partial_{\nu}\mathbf{e}^{(+-)}_{(\rho\mu)}+\partial_{\rho}\mathbf{e}^{(+-)}_{(\mu\nu)}=\frac{1}{2}\left(\mathbf{e}_{(33)}+\mathbf{e}_{44}\right)\\{V^{-}_{\mu}S^{+}_{\nu\rho}+V^{-}_{\nu}S^{+}_{\rho\mu}$
$\displaystyle+V^{-}_{\rho}S^{+}_{\mu\nu}+V^{+}_{\mu}S^{-}_{\nu\rho}+V^{+}_{\nu}S^{-}_{\rho\mu}+V^{+}_{\rho}S^{-}_{\mu\nu}\\},$
(115)
$\displaystyle\partial_{\mu}\left(\mathbf{e}^{(+1)}_{(\nu\rho)}+\mathbf{e}^{(-1)}_{(\nu\rho)}\right)+\partial_{\nu}\left(\mathbf{e}^{(+1)}_{(\rho\mu)}+\mathbf{e}^{(-1)}_{(\rho\mu)}\right)+\partial_{\rho}\left(\mathbf{e}^{(+1)}_{(\mu\nu)}+\mathbf{e}^{(-1)}_{(\mu\nu)}\right)=$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\mathbf{e}_{(13)}+i\mathbf{e}_{(14)}\right)\\{A_{\mu}S^{+}_{\nu\rho}+A_{\nu}S^{+}_{\rho\mu}+A_{\rho}S^{+}_{\mu\nu}+V^{+}_{\mu}S_{\nu\rho
1}+$ $\displaystyle V^{+}_{\nu}S_{\rho\mu 1}+V^{+}_{\rho}S_{\mu\nu
1}\\}+\frac{1}{\sqrt{2}}\left(\mathbf{e}_{(13)}-i\mathbf{e}_{(14)}\right)\\{A_{\mu}S^{-}_{\nu\rho}+A_{\nu}S^{-}_{\rho\mu}$
(116) $\displaystyle+A_{\rho}S^{-}_{\mu\nu}+V^{-}_{\mu}S_{\nu\rho
1}+V^{-}_{\nu}S_{\rho\mu 1}+V^{-}_{\rho}S_{\mu\nu 1}\\},$
$\displaystyle\partial_{\mu}\left(\mathbf{e}^{(+2)}_{(\nu\rho)}+\mathbf{e}^{(-2)}_{(\nu\rho)}\right)+\partial_{\nu}\left(\mathbf{e}^{(+2)}_{(\rho\mu)}+\mathbf{e}^{(-2)}_{(\rho\mu)}\right)+\partial_{\rho}\left(\mathbf{e}^{(+2)}_{(\mu\nu)}+\mathbf{e}^{(-2)}_{(\mu\nu)}\right)=$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\mathbf{e}_{(23)}+i\mathbf{e}_{(24)}\right)\\{U_{\mu}S^{+}_{\nu\rho}+U_{\nu}S^{+}_{\rho\mu}+U_{\rho}S^{+}_{\mu\nu}+V^{+}_{\mu}S_{\nu\rho
2}+V^{+}_{\nu}S_{\rho\mu 2}$ $\displaystyle+V^{+}_{\rho}S_{\mu\nu
2}\\}+\frac{1}{\sqrt{2}}\left(\mathbf{e}_{(23)}-i\mathbf{e}_{(24)}\right)\\{U_{\mu}S^{-}_{\nu\rho}+U_{\nu}S^{-}_{\rho\mu}+U_{\rho}S^{-}_{\mu\nu}+$
(117) $\displaystyle+V^{-}_{\mu}S_{\nu\rho 2}+V^{-}_{\nu}S_{\rho\mu
2}++V^{-}_{\rho}S_{\mu\nu 2}\\},$
$\displaystyle\partial_{\mu}\left(\mathbf{e}^{(++)}_{(\nu\rho)}+\mathbf{e}^{(--)}_{(\nu\rho)}\right)+\partial_{\nu}\left(\mathbf{e}^{(++)}_{(\rho\mu)}+\mathbf{e}^{(--)}_{(\rho\mu)}\right)+\partial_{\rho}\left(\mathbf{e}^{(++)}_{(\mu\nu)}+\mathbf{e}^{(--)}_{(\mu\nu)}\right)=$
$\displaystyle=\frac{1}{2}\left(\mathbf{e}_{(33)}-\mathbf{e}_{(44)}\right)\\{V^{+}_{\mu}V^{+}_{\nu\rho}+V^{+}_{\nu}V^{+}_{\rho\mu}+V^{+}_{\rho}V^{+}_{\mu\nu}+$
(118) $\displaystyle
V^{-}_{\mu}V^{-}_{\nu\rho}+V^{-}_{\nu}V^{-}_{\rho\mu}+V^{-}_{\rho}V^{-}_{\mu\nu}\\},$

and

$\displaystyle\partial_{\mu}\left(\mathbf{e}^{(++34)}_{(\nu\rho)}+\mathbf{e}^{(--34)}_{(\nu\rho)}\right)+\partial_{\nu}\left(\mathbf{e}^{(++34)}_{(\rho\mu)}+\mathbf{e}^{(--34)}_{(\rho\mu)}\right)+$
$\displaystyle\partial_{\rho}\left(\mathbf{e}^{(++34)}_{(\mu\nu)}+\mathbf{e}^{(--34)}_{(\mu\nu)}\right)=i\mathbf{e}_{(34)}\\{V^{+}_{\mu}V^{+}_{\nu\rho}+V^{+}_{\nu}V^{+}_{\rho\mu}+$
(119) $\displaystyle
V^{+}_{\rho}V^{+}_{\mu\nu}-V^{-}_{\mu}V^{-}_{\nu\rho}-V^{-}_{\nu}V^{-}_{\rho\mu}-V^{-}_{\rho}V^{-}_{\mu\nu}\\}.$

The above equations are showing induced relationships derived from the
quadruplet correlations. New Faraday-Lenz-Neumann laws are developed [65]. It
produces (4+4) equations of motion and 15 Bianchi identities. Monopoles appear
related to fields associations. The collective antisymmetric Bianchi
identities, eqs. (109 \- 5) can be rewritten as

$\partial_{\mu}\tilde{e}^{\mu\nu}=j^{\nu}_{B},\ \partial_{\nu}j^{\nu}_{B}=0$
(120)

Eqs. (112-5) are also introducing sources.

Monopoles have a rich story in physics. The magnet slicing experiment was
first performed in 1269 by the French scholar Peregrinus [66]. In 1931 Dirac
introduced the magnetic charge [67]. Different monopoles scenarios follow as
monopoles as gauge particles [68], GUTs [69], Kaluza-Klein [70], Superstrings
[71], Quantum Gravity and Cosmology [72], Electroweak monopoles [73],
monopoles produced in intense magnetic fields [74], monopoles and photons
decay [75], monopoles condensated in superconductors’ [76], monopoles and dark
matters [77].

The four bosons EM introduces monopoles as potential fields conglomerates.
Similar to ’t Hooft and Polyakov monopoles based on Higgs fields [78].
Monopoles depending on fields contain opportunities at the condensed matter as
spin ice monopoles [79].

##  6 Invariants

The extended Yang-Mills functional is invariant by the gauge group action
[80]. Consequently, quadruplet invariants are derived. Lorentz’s scalars and
conserved charges are obtained. EM invariants are obtained under Lorentz
transformations [81]. Diverse scalars are build up through combinations
between fields strengths as
$F_{\mu\nu}^{2},F_{\mu\nu}U^{\mu\nu},F_{\mu\nu}\mathbf{e}^{\mu\nu}$ and so on.

Four types of fields conserved charges are provided. First, by considering the
(4+4) transverse and longitudinal whole equations.

For spin-1,

$\partial\cdot\left(\mathbf{m}^{2}A^{T}_{I}+l^{T}_{I}+c^{T}_{I}-j^{T}_{I}\right)=0$
(121)

and for spin-0,

$\Box\left(S^{\alpha}_{\alpha I}+\mathbf{e}_{\alpha
I}^{\alpha}\right)=\partial\cdot\left(j-M-l-c\right)$ (122)

A second conservation law comes from antisymmetric collective Bianchi
identities. From eq. (120), one gets

$\partial_{\mu}J^{\mu}_{B}=0\text{ where the source is
}J_{\mu}^{B}=\epsilon_{\mu\nu\rho\sigma}\mathbf{e}_{[IJ]}A_{I}^{\nu}A_{J}^{\rho\sigma}$
(123)

where eq. (123) is showing the presence of fields monopoles

Noether current expresses a third continuity equation. A relevant aspect is
that electric charge is associated with a group of fields not necessarily
charged. The fourth conservation law is given by the second type of global
conservation associated to the improved energy-momentum tensor,
$\partial_{\mu}\theta^{\mu\nu}=0$ [82]. The quadruplet EM flux develops
conservation laws between energy dimity, Poynting vector, and stress tensor.
They are expressed as

$\frac{\partial U}{\partial t}+\vec{\bigtriangledown}\cdot\vec{S}=0$ (124)
$\frac{\partial S_{i}}{\partial t}+\partial_{j}T_{ji}=0$ (125)

A physicality beyond matter appears. The quadruplet is expressing physics,
providing the meaning of the field’s set. Fields conservation laws based on
fields and fields relationships are obtained.

##  7 Nonlinear wave equations

Maxwell focused on charge conservation and distribution; the four bosons
electromagnetism introduces the charge transfer. A difference appears while
Hertz’s experiments in 1888 comprised Maxwell linear EM waves [83], a further
investigation must be done. Nonlinear wave equations have been explored [84].
New types of transverse and longitudinal waves are obtained.

For $A_{\mu T}$,

$\Box\\{4\left(a_{1}+\beta_{1}+a_{1}k_{1}\right)F^{\nu\mu}+2b_{1}[\mathbf{e}^{[12][\nu\mu]}+(1+k_{1})\mathbf{e}^{[+-][\nu\mu]}]\\}=\partial^{\nu}d^{\mu}_{AT}-\partial^{\mu}d^{\nu}_{AT}$
(126)

where

$d^{\mu}_{AT}=M^{A\mu}_{IT}+j^{\mu}_{AT}-J^{\mu}_{NT}-k_{2}J^{\mu}_{UT}-l_{AT}^{\mu}+c^{\mu}_{AT}$
(127)

For $A_{\mu L}$,

$\displaystyle\Box\\{s_{11}S^{\alpha}_{\alpha
1}+2\left(\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(+-)\alpha}_{\alpha}\right)\\}=$
$\displaystyle\partial\cdot\left(j_{AL}+t_{11}M^{A}_{gL}+\frac{1}{2}t_{11}J_{UL}-l_{AL}-c_{AL}\right)$
(128)

For $U_{\mu T}$,

$\displaystyle\Box\\{4\left(a_{2}+\beta_{2}+a_{2}k_{2}\right)U^{\nu\mu}+2b_{2}[\mathbf{e}^{[12][\nu\mu]}+(1+k_{2})\mathbf{e}^{[+-][\nu\mu]}]\\}=$
$\displaystyle=\partial^{\nu}d^{\mu}_{UT}-\partial^{\mu}d^{\nu}_{UT}$ (129)

where

$d^{\mu}_{UT}=M^{U\mu}_{gT}+j^{\mu}_{UT}-J^{\mu}_{NT}-k_{1}J^{\mu}_{AT}-M^{\mu}_{UT}-l_{AT}^{\mu}+c^{\mu}_{AT}$
(130)

For $U_{\mu L}$,

$\displaystyle\Box\\{s_{22}S^{\alpha}_{\alpha
2}+2\left(\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)\alpha}_{\alpha}+\mathbf{e}^{(+-)\alpha}_{\alpha}\right)\\}=$
$\displaystyle=\partial\cdot\left(M^{U}_{gL}+j_{UL}+t_{22}J_{AL}-M_{UL}-l_{UL}-c_{UL}\right)$
(131)

For $V^{-}_{\mu T}$,

$\Box\\{2\left(a_{3}+\beta_{+}\beta_{-}\right)V^{\nu\mu+}+2b_{3}\left(\mathbf{e}^{[+1][\nu\mu]}+\mathbf{e}^{[+2][\nu\mu]}\right)\\}=\partial^{\nu}d^{\mu+}_{VT}-\partial^{\mu}d^{\nu+}_{VT}$
(132)

where

$d^{\mu+}_{V}=j^{\mu+}_{VT}-M^{\mu+}_{VT}-l^{\mu+}_{VT}-c^{\mu+}_{VT}$ (133)

For $V^{-}_{\mu L}$,

$\Box\\{4s_{+}S^{\alpha+}_{\alpha}+c_{+}\left(\mathbf{e}^{(+1)\alpha}_{\alpha}+\mathbf{e}^{(+2)\alpha}_{\alpha}\right)\\}=\partial\cdot\left(j^{+}_{VL}-M^{+}_{VL}-l^{+}_{VL}-c^{+}_{VL}\right)$
(134)

For $V^{+}_{\mu T}$,

$\Box\\{2\left(a_{3}+\beta_{+}\beta_{-}\right)V^{\nu\mu-}+2b_{3}\left(\mathbf{e}^{[-1][\nu\mu]}+\mathbf{e}^{[-2][\nu\mu]}\right)\\}=\partial^{\nu}d^{\mu-}_{VT}-\partial^{\mu}d^{\nu-}_{VT}$
(135)

where

$d^{\mu-}_{V}=j^{\mu-}_{VT}-M^{\mu-}_{VT}-l^{\mu-}_{VT}-c^{\mu-}_{VT}$ (136)

For $V^{+}_{\mu L}$,

$\Box\\{4s_{-}S^{\alpha-}_{\alpha}+c_{-}\left(\mathbf{e}^{(-1)\alpha}_{\alpha}+\mathbf{e}^{(-2)\alpha}_{\alpha}\right)\\}=\partial\cdot\left(j^{-}_{VL}-M^{-}_{VL}-l^{-}_{VL}-c^{-}_{VL}\right)$
(137)

##  8 Whole Lorentz forces

Special relativity defines force through the equation
$\partial_{\mu}\theta^{\mu\nu}=f^{\nu}$ which means a change of energy and
momentum between two systems made of fields and external sources [85]. This
leads to us following Lagrangian density.

$L=L_{0}+A^{\mu I}j_{\mu I}$ (138)

where $j_{\mu I}$ means N-external currents, not necessarily corresponding to
electric charge. The generic whole fields equation becomes

$\partial_{\mu}\left(F^{\nu\mu}_{I}+\mathbf{e}_{I}^{[\nu\mu]}+g^{\nu\mu}S_{\alpha
I}^{\alpha}+\mathbf{e}_{I}^{(\nu\mu)}\right)+\frac{1}{2}\mathbf{m}^{2}_{I}A^{\mu}_{I}=J^{\mu}_{I}(A)+j^{\mu}_{I}$
(139)

It yields the fundamental field’s equation [49]

$f^{\nu}=f^{\nu}_{L}+f^{\nu}_{M}+f^{\nu}_{E}$ (140)

where

$\displaystyle f^{\nu}_{L}=4F^{\nu\mu}_{I}j^{I}_{\mu},$ $\displaystyle
f^{\nu}_{M}=-2\mathbf{m}^{2}_{I}A^{\mu}_{I}(\partial^{\mu}A^{I}_{\mu}),$
$\displaystyle
f^{\nu}_{E}=4A^{\nu}_{I}\left(\partial^{\mu}J_{\mu}^{I}\left(A\right)\right)$
(141)

where $A_{\mu I}\equiv\\{A_{\mu},U_{\mu},V_{\mu}^{\pm}\\}$ and $J_{\mu I}$ the
corresponding fields current. $f^{\nu}_{L}$ means the extended Lorentz force,
$f^{\nu}_{M}$ a mass force and $f^{\nu}_{E}$ a fields environmental force.

##  9 Conclusion

EM under charge transference is studied. A new border is set up. Three
historical moments retreat the EM development. The third EM period is crossed,
it is between fields and fields. A new physics for Faraday lines of force. The
meaning of electric and magnetic fields is expanded. New terms, are provided
EM fields working as own sources, fields and matter.

A four bosons electromagnetism appears. The charges set $\\{+,0,-\\}$
intermediates a quadruplet $\\{A_{\mu},U_{\mu},V^{\pm}_{\mu}\\}$. A whole
abelian quantum field theory is derived under electric charge symmetry. New
properties for the EM flow are determined. Physicalities from nonlinear
abelian fields, new EM regimes, and electric charge symmetry.

A physics beyond Maxwell is opened. The charge transfer phenomenology is
explored. Nonlinear Faraday lines of force appear. Potential fields are
empowered by their own charges. The four bosons EM moves electric charge
symmetry from $U(1)$ to $U(1)\times SO(2)_{global}$. A neutral EM energy
appears. The third nature for electric charge is included. Zero electric
charges is introduced as an abelian quantum number.

Nevertheless, a model should first be tested on how far it covers old
limitations before proposing new properties. Four consistencies are observed
from the electromagnetic whole equations. First, a nonlinear EM is obtained at
a fundamental level. Second, potential fields are coupled with granular and
collective fields’ strengths instead of subsidiaries. As in quantum mechanics,
appear explicitly at equations. A result supporting the 1845 Faraday effect in
1845 between light and magnetism [2] and Aharanov-Bohm and Aharanov-Casher
experiments [86]. Third, the polarization and magnetization vectors are
derived from the first principles [57]. Identified in the equations of motion
with the collective fields strengths. Fourth, light is no more a passive
consequence of electric charge oscillations. The photon equation of motion,
eq. (68), is showing a photon field with its charge depending on fields and
with self-interacting photons at tree level.

Based on such consistencies, new EM features are expressed. EM phenomena are
accomplished by four photons. Nonlinear whole relativistic equations with new
relationships between electric and magnetic. Granular and collective fields’
strengths share a common dynamic. Although under the same Lorentz symmetry,
the spin-1 and spin-0 sectors are decoupled. Two spin dynamics succeed. Each
sector contains a proper physics. Their quanta differ on spin, masses,
charges, coupling constants.

Thus, three observances are supporting this search for a new EM energy. Two
electric charge phenomenology, four surpassed Maxwell limitations, and a
renormalizable theory [87]. These achievements bring confidence to the model
being investigated. Explore their fields, energy, and electric charge new
significances.

The first meaning is on physical features depending just on fields. A deeper
relationship between energy and matter is performed. Beyond $E=mc^{2}$, fields
and matter physics are developed. London and conglomerate expressions at eqs.
(79) and (80) are introducing a relationship between fields and mass. Fields
become physical entities with their own character. Phenomena as
superconductivity [55], dark mass [88], dark energy [89] may have consequences
for fields and fields of physics. Analysing the energy-momentum tensor, the
quadruplet model also includes fields dependences on energy, momentum, stress
tensor [82].

The second meaning is that six new interlaced EM regimes are produced. They
are systemic, nonlinear, neutral, spintronics, photonics, electroweak. First,
a field’s set of physics is constituted. While Maxwell related just the pair
$\\{\vec{\mathbf{E}},\vec{\mathbf{B}}\\}$ under four equations, eqs. (1-4)
expands to a system expressed by a set of potential fields associations [43].
A whole EM is determined under a set causality. The quadruplet integrates
twenty-three equations. Their variables and parameters are expressed in the
function of the whole. The whole Lorentz force is derived.

The second regime is nonlinearity. There are sixteen nonlinear EM models in
literature [36]. Most of them are effective theories derived from Born-Infeld
and Euler-Heisenberg in the 1930s. Diversely, eqs. (1-4) propose a fundamental
model. Instead of powering, as $\vec{\mathbf{E}}^{2}$, introduce as sources
interactions between fields strengths and potentials, as
$\vec{\mathbf{E}}\cdot\vec{A}$, with adimensional coupling constants. A
renormalizable and unitary nonlinear EM is produced.

A neutral EM follows. EM energy is more primitive than electric charge. The
interconnected charges set $\\{+,0,-\\}$ introduces the neutral charge
physicality. An EM energy with a coupling beyond Maxwell-QED. Potential
fields, electric and magnetic fields developing a physics not restricted to
the fine structure constant. The corresponding equations of motion, energy-
momentum tensor, and forces are showing sectors with couplings diverse from
electric charge. The zero charge from passive becomes active.

The fourth EM regime is on spintronics. Originally, spin was introduced
heuristically, through the magnetic moment interaction
$\vec{\mu}\cdot\vec{\mathbf{B}}$ [90]. The four bosons model introduces spin
from first principles as Lie Algebra valued [61]. Defining the potential field
$A_{\mu,\kappa\lambda}$ where the first indices mean the space-time symmetry,
$x^{\prime}_{\mu}=\Lambda_{\mu}^{\nu}x_{\nu}$, and others the $SO(1,3)$
rotation,
$A^{\prime}_{\mu}=\left(e^{i\omega_{\alpha\beta}\Sigma^{\alpha\beta}}\right)_{\mu}^{\nu}A_{\nu}$,
one incorporates as ab initio the spin generator
$\left(\Sigma_{\alpha\beta}\right)_{\mu}^{\nu}$. Similarly to Yang-Mills,
$A_{\mu}\equiv A_{\mu a}t_{a}$, the quadruplet fields are introduced as spin
algebra value. Consequently, the electric and magnetic dipole couplings are
derived from first principles through spin fields interactions, as
$gF^{\alpha\beta}\left(\Sigma_{\alpha\beta}\right)^{\nu}_{\mu}A_{\nu}$.

A fifth EM regime is photonics. Consider the photon as originally. Diversely
from Big-Bang theory and Maxwell introduce a primordial light. Big-Bang
derives the photon after the second phase transition $SU_{C}(3)\times
SU_{L}(2)\times U_{Y}(1)\stackrel{{\scriptstyle
SSB}}{{\rightarrow}}SU_{C}(3)\times U_{em}(1)$ at first $10^{-10}s$ due to the
Higgs presence at $10^{15}K$ temperature, $250GeV$ energy, $10^{19}T$ magnetic
field [91]. Consequently, the Standard Model photon appears after spontaneous
breaking symmetry, as $U_{em}(1)_{photon}=cos\theta_{w}\cdot
U_{Y}(1)+sin\theta_{w}I_{3}$. Concurrently, Maxwell photon is a radiation from
oscillating electric charge. It does not produce even its own electromagnetic
fields.

The photon should be considered as cause and not effect. An original photon
containing light invariance, ubiquitous lux, and self-interacting photons is
expected from a new EM model. Consider the light emulation without any
premise. Eq. (68) is introducing a physics with self-interacting photons not
depending on electric charge. A nonlinearity to create matter from pure
energy. A possibility to smash two photons and create particles.

The photonics turning point is on photon interaction. A discussion returning
to 1934 with Delbruck and Breit-Wheeler [60]. In 1950 the $\gamma\gamma$
scattering was studied by Karplus and Neuman through QED [92]. Complemented in
the decade by Skobov and Minguzzi [93]. Evidence for light by light scattering
in heay-ion collisions is being explored at ATLAS [94]. The confirmation
happened in 2017, when LHC measured the photon-photon interaction [95].
However, through one loop correction. Something is missing. Primordial photon
should be manifested at fundamental physics. Scatterings
$\gamma\gamma\to\gamma\gamma$ and $\gamma\gamma\to f\bar{f}$ at tree level.

Astrophysics [96], photonics process in intense fields [97], nonlinear optics
[98] and lasers [99] are studying on light non-linearity. Nonlinear photons
may also be derived from intense EM fields as in pulsars ($10^{8}T$),
magnetars ($10^{11}T$). Photonic processes under intense EM fields are making
physics return to later Delbruck scattering 85 years later [100]. Different
propositions are being studied as the vacuum birefringence [101]. Lasers
experiment in Padova [102], the Photon Collider in Germany (Vaccum Hohlraum),
Star (BNL), Shangai Super-Laser (SULF) [103] and other lasers are being
developed around the world. In 2026 the Xcels in Russia expect to look for the
primordial photon.

A new EM formulation with the electron being substituted by the photon is
expected. Photonics be constituted. Eq. (68) considers nonlinear photon field
carrying its own charge, producing proper electric and magnetic fields and
self-interactions without the electric charge presence. A fundamental NLED
candidate to provide theoretical support to the deflection of light by light
[104], quantum computer [105], two and three photons bound states, light
molecules [106], liquid photons [107].

The last EM regime is electroweak. Interpret weak interaction as an EM sector.
Photon, $Z_{0},W^{\pm}$ be included together without asking for a Standard
Model. Consider just charge transference physics and identify the
corresponding four bosons as $\gamma,Z^{0},W^{\pm}$. Recover the Abelian
Standard Model version $U(1)\times SU(2)\stackrel{{\scriptstyle
SSB}}{{\rightarrow}}U_{em}$ through $U(1)\times SO(2)_{global}$. However,
instead of finalizing with $U_{em}(1)$ symmetry, as SM does, the four bosons
EM stars with abelian symmetry since the beginning. Take as guidance principle
a modified electric charge conservation. And so, as an alternative to the
Standard Model, the four bosons electromagnetism introduces a kinetic
longitudinal sector with scalar photons [108], mass without Higgs [52], photon
including small mass [109], interaction $\gamma-Z_{0}$, anomalous triple gauge
bosons coupling [110], spin (g-2) terms $F_{\mu\nu}Z^{\mu}Z^{\nu}$,
$Z_{\mu\nu}A^{\mu}A^{\nu}$ and self-interacting photon at tree level. New
scales of intensities and range are derived by intermediate fields through
coupled nonlinear equations, mass terms, and coupling constants beyond
electric charge [111].

The third EM meaning from charge transfer is on electric charge physics.
Electromagnetism is being considered as anything that acts on an electric
charge. However, it should primordially be a consequence of electric charge
symmetry. The Symmetry $U(1)\times SO(2)_{global}$ moves the Maxwell charge
physics from interactions and conservation law to embracing a new performance.
It shows that electric charge physics is more primitive than being the source
for electric-magnetic fields or acting as a coupling constant. Electric charge
originally function is associative. Its symmetry physics is to integrate
fields. As a consequence, the EM energy is derived from electric charge
symmetry, without necessarily depending on its presence.

This discrepancy between electric charge and EM energy is a fact already
constituted in Maxwell. There, the energy flux at the electric circuit does
not follow in wires. Yet, Faraday law does not depend on electric charge.
Therefore, charge transfer extends but does not modify Maxwell’s structure. An
EM energy flux based on fields is processed. A field’s physicality preceding
matter is derived. Each potential field, $A_{\mu I}$, provides its own
continuity flowing not directly related to electric charge. It yields physical
laws based on electric charge as a quantum number but beyond electric charge
as coupling constant.

A perspective for EM energy based on a general electric charge physics is
envisaged. Electric charge symmetry is the prior definition for what EM is.
The charge transfer symmetry rules new sectors and states an EM energy without
the electric charge presence. Maxwell is rewritten as part of enlarged
electromagnetism. A physics to be explored just based on fields. Faraday lines
of force antecede the presence of physical constants as permissively and
permeability. There is a third electromagnetic historical period to be
conquested.

## Acknowledgment

Luís Santiago Mendes would like to express his gratitute to FAPERJ (Rio de
Janeiro state foundation) for supporting with an innovation scholarship.

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