###### Abstract

Field physics was founded by Faraday introducing magnetic fields (1831),
electric fields (1837) and light as an EM wave (1846), initiating the process
where nature is made by matter and fields. Consider that, ordinary space is
full of fields. The Faraday view is basis for modern quantum field theory.

The concept of fields set up a physicality in development. Physics would like
to know how far matter is created by fields. Generate matter from nonlinear
fields. Faraday lines of force relating physical entities as electric charge
and mass depending on fields.

Our purpose is on Faraday lines for nonlinear abelian electromagnetism.
Introduce the Four Bosons EM. The phenomenology of a generic charge
$\\{+,0,-\\}$ transmitted by four bosons
$\\{A_{\mu},U_{\mu},V_{\mu}^{\pm}\\}$. Nonlinear equations constituted. New
Faraday lines were introduced. The potentials fields of physics are developed.
Granular and collective fields strengths expressed. Four types of fields
charges are derived. They are electric charge, modulated, neutral, Bianchi.

This work introduces a systematic procedure of associative physics. Mass and
charge are generated due to the four fields interrelationships. Masses are
derived without spontaneous symmetry breaking. It is obtained naturally from
gauge symmetry, London, and mixing terms. Electric charge is written by fields
through the Noether theorem. EM interactions not necessarily coupled with
electric charge are proposed. An enlargement of EM energy is derived.

New Faraday lines through Four Bosons EM
R<EMAIL_ADDRESS>and L.S. Mendes†
<EMAIL_ADDRESS>

†Aprendanet, Petropolis, Brazil; Quarks, Petropolis, Brazil

Keywords: nonlinear electrodynamics; new Faraday lines; granular and
collective fields strengths; potential fields interacting with EM fields; self
interacting photons.

##  1 Introduction

Faraday pioneered the physics dependence on fields [1]. The outrageous
proposal of invisible lines discovered that physics is more than space, time,
and movement. It contains the fields of physicality. In 1911, Rutherford not
only discovered the atom as an empty space [2]. The greek atm would contain
nuclei, electrons, and field. In 1913, Bohr’s orbit theory postulated a
relationship between fields and matter in order to justify atom stability [3].
The assumption that there is physics more than matter became visible.

History shows that there is no complete nothingness. The bussole’s navigators
were guided by magnetic fields. Astrophysics discovered that the universe is
more than an empty space. It contains a large cosmic structure at $10^{17}km$
with weak magnetic fields [4]. A magnetar was detected with a strong magnetic
field at order $10^{19}T$ [5]. The QED Sauter-Schwinger effect says that
electron-positron pairs are spontaneously created in the presence of an
electric field [6].

Physics is more than matter. There is a field of physics to be exploited.
Since Faraday fields became the variables of physics equations. A quantum
field theory is ruling that particles are local vibrations of fields.
Providing a fields dynamics, with conservation laws and associated fields
charges. A realism to be taken.

Thus, a Faraday research is to understand how far the relationship between
matter and fields is still unknown in electromagnetic theory. Maxwell
equations are the origin of fields events [7]. It provides fields under Gauss
and Ampere dynamics, Faraday induction, and monopoles absence. However,
Maxwell is incomplete. It restricts fields to dependence on electric charge,
linearity, and subsidiary potential fields. On other hand, QED introduces the
particles creation and annihilation. Electron absorb or emit a photon;
electron and positron annihilate and create two photons. However, despite
$E=mc^{2}$, where matter is transformed into energy, but also, energy turn
into matter, it does not study charges transfer. The quantum mutation
phenomena responsible for the matter-energy transformation is not being
supported by a theory.

An electric charge transfer model beyond Maxwell and QED is expected. A
candidate is the so-called four bosons of electromagnetism [8]. Based on the
transfer of three electric charge flavors, it defines the generic electric
charge $\\{+,0,-\\}$. A charge transfer physics intermediate by four bosons
$\\{A_{\mu},U_{\mu},V_{\mu}^{\pm}\\}$ is proposed. It provides $A_{\mu}$ as
the usual photon, $U_{\mu}$ massive photon and $V_{\mu}^{\pm}$ charged
photons. These fields are interrelated by a non-univoque electromagnetic
symmetry [9].

A fundamental nonlinear four bosons electrodynamics is derived. New Faraday
lines are generated. Vector potentials become a real physical quantity that
conglomerates can be measured. Granular and collective fields strengths are
expressed. Potentials fields are working as their own sources and interacting
with granular and collective fields strengths. Based on non-linear abelian
fields, the quadruplet derives its own charges. They are electric charge,
modulated, neutral, and Bianchi. It yields a new relationship between mass,
charges, and fields to be studied.

##  2 Fields strengths

The quadruplet $\\{A_{\mu},U_{\mu},V_{\mu}^{\pm}\\}$ is associated by the
electromagnetic symmetry. Different possibilities to consider their
interrelations were studied [8-9]. Diverse EM physics conveyed by the
quadruplet may be derived. They share to differ common Lagrangian and Noether
theorem relationships, Lagrangian coefficients.

Our choice for the quadruplet interrelationship is the following gauge
transformation:

$\displaystyle A{{}^{\prime}}_{\mu}$ $\displaystyle=$ $\displaystyle
A_{\mu}+k_{1}\partial_{\mu}\alpha$ (2.1) $\displaystyle U{{}^{\prime}}_{\mu}$
$\displaystyle=$ $\displaystyle U_{\mu}+k_{2}\partial_{\mu}\alpha$ (2.2)
$\displaystyle V^{+^{\prime}}_{\mu}$ $\displaystyle=$ $\displaystyle
e^{iq\alpha(x)}V^{+}_{\mu}$ (2.3) $\displaystyle V^{-^{\prime}}_{\mu}$
$\displaystyle=$ $\displaystyle e^{-iq\alpha(x)}V^{-}_{\mu}$ (2.4)

At vectorial expression, one gets

It yields the following granular antisymmetric fields strengths

$\displaystyle F_{\mu\nu}$
$\displaystyle=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},$ $\displaystyle
F^{\prime}_{\mu\nu}=F_{\mu\nu}$ (2.5) $\displaystyle U_{\mu\nu}$
$\displaystyle=\partial_{\mu}U_{\nu}-\partial_{\nu}U_{\mu},$ $\displaystyle
U^{\prime}_{\mu\nu}=U_{\mu\nu}$ (2.6) $\displaystyle V_{\mu\nu}^{\pm}$
$\displaystyle=D_{\mu}V_{\nu}^{\pm}-D_{\nu}V_{\mu}^{\pm},$ $\displaystyle
V_{\mu\nu}^{\pm^{\prime}}=e^{\pm iq\alpha}V_{\mu\nu}^{\pm}$ (2.7)

where the covariant derivative is given by
$D_{\mu}=\partial_{\mu}+i(q_{1}A_{\mu}+q_{2}U_{\mu})$. The couplings $q_{1}$
and $q_{2}$ are modulated electric charge given by $q_{1}=a\cdot q;$
$q_{2}=b\cdot q$, where $ak_{1}+bk_{2}=-1$.

Collective fields strengths are expressed as

$\displaystyle\mathbf{e}^{[12]^{\prime}}_{[\mu\nu]}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathbf{e}_{[12]}\left(A_{\mu}U_{\mu}-U_{\mu}A_{\mu}\right)=\mathbf{e}^{[12]}_{[\mu\nu]}$
(2.8) $\displaystyle\mathbf{e}^{[+-]^{\prime}}_{[\mu\nu]}$ $\displaystyle=$
$\displaystyle
i\mathbf{e}_{[34]}\left(V^{+}_{\mu}V^{-}_{\nu}-V^{-}_{\mu}V^{+}_{\nu}\right)=\mathbf{e}^{[12]}_{[\mu\nu]}$
(2.9)

and

$\displaystyle\mathbf{e}^{[12+]}_{[\mu\nu]}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathbf{e}_{[12]}\left[\left(A_{\mu}-U_{\mu}\right)V^{+}_{\nu}+\left(A_{\nu}-U_{\nu}\right)V^{+}_{\mu}\right]$
(2.10) $\displaystyle\mathbf{e}^{[12-]}_{[\mu\nu]}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathbf{e}_{[12]}\left[\left(A_{\mu}-U_{\mu}\right)V^{-}_{\nu}+\left(A_{\nu}-U_{\nu}\right)V^{-}_{\mu}\right]$
(2.11)

transforming as

$\mathbf{e}^{[12+]^{\prime}}_{[\mu\nu]}=e^{iq\alpha(x)}\mathbf{e}^{[12+]}_{[\mu\nu]},$
(2.12)
$\mathbf{e}^{[12-]^{\prime}}=e^{-iq\alpha(x)}\mathbf{e}^{[12-]}_{[\mu\nu]}$
(2.13)

For the symmetric sector:

$S_{\mu\nu 1}=\partial_{\mu}A_{\nu}+\partial_{\nu}A_{\mu},S_{\mu\nu
2}=\partial_{\mu}U_{\nu}+\partial_{\nu}U_{\mu}.$ (2.14)

the invariance is under the condition

$S^{1^{\prime}}_{\mu\nu}+S^{2^{\prime}}_{\mu\nu}=S^{1}_{\mu\nu}+S^{2}_{\mu\nu}$
(2.15)

Charged fields are transforming as

$\displaystyle S_{\mu\nu}^{\pm}$
$\displaystyle=D_{\mu}V_{\nu}^{\pm}+D_{\nu}V_{\mu}^{\pm},$ $\displaystyle
S_{\mu\nu}^{\pm^{\prime}}=e^{\pm iq\alpha}S^{\pm}_{\mu\nu}$ (2.16)

Longitudinal terms are written as

$\displaystyle S_{\alpha 1}^{\alpha}$
$\displaystyle=2\partial_{\alpha}A^{\alpha},S_{\alpha 2}^{\alpha}$
$\displaystyle=2\partial_{\alpha}U^{\alpha}.$ (2.17)

with

$S^{\alpha 1^{\prime}}_{\alpha}+S_{\alpha}^{\alpha 2^{\prime}}=S^{\alpha
1}_{\alpha}+S^{\alpha 2}_{\alpha}$ (2.18)

and

$\displaystyle S_{\alpha}^{\alpha\pm}$
$\displaystyle=D_{\alpha}V^{\alpha\pm},$ $\displaystyle
S_{\alpha}^{\alpha\pm^{\prime}}=e^{\pm iq\alpha}S_{\alpha}^{\alpha\pm}$ (2.19)

The collective symmetric fields strengths are

$\displaystyle\mathbf{e}_{(\mu\nu)}=\mathbf{e}_{(11)}A_{\mu}A^{\nu}+\mathbf{e}_{(12)}\left(A_{\mu}U_{\nu}+U_{\mu}A_{\nu}\right)+\mathbf{e}_{(22)}U_{\mu}U_{\nu}=\mathbf{e}^{\prime}_{(\mu\nu)}$
$\displaystyle\mathbf{e}^{(+-)}_{(\mu\nu)}=\mathbf{e}_{(34)}V^{+}_{\mu}V^{-}_{\nu}=\mathbf{e}^{(+-)^{\prime}}_{(\mu\nu)}$
(2.20)

and

$\displaystyle\mathbf{e}^{\alpha}_{\alpha}=\mathbf{e}_{(11)}A_{\alpha}A^{\alpha}+2\mathbf{e}_{(12)}A_{\alpha}U^{\alpha}+\mathbf{e}_{(22)}U_{\alpha}U^{\alpha}=\mathbf{e}^{\alpha^{\prime}}_{\alpha}$
(2.21)
$\displaystyle\mathbf{e}^{(+-)\alpha}_{\alpha}=\mathbf{e}_{(34)}V^{+}_{\alpha}V^{-\alpha}=\mathbf{e}^{(+-)\alpha^{\prime}}_{\alpha}$
(2.22)

Other collective fields strengths are included

$\displaystyle\mathbf{e}^{(12+)}_{(\mu\nu)}=\left[\mathbf{e}_{(11)}A_{\mu}+\mathbf{e}_{(12)}\left(A_{\mu}+U_{\mu}\right)+\mathbf{e}_{(22)}U_{\mu}\right]V^{+}_{\nu}$
(2.23)
$\displaystyle\mathbf{e}^{(12-)}_{(\mu\nu)}=\left[\mathbf{e}_{(11)}A_{\mu}+\mathbf{e}_{(12)}\left(A_{\mu}+U_{\mu}\right)+\mathbf{e}_{(22)}U_{\mu}\right]V^{-}_{\nu}$
(2.24)
$\displaystyle\mathbf{e}^{(++)}_{(\mu\nu)}=\frac{1}{2}\left(\mathbf{e}_{(33)}-\mathbf{e}_{(44)}\right)V_{\mu}^{+}V_{\nu}^{+}$
(2.25)
$\displaystyle\mathbf{e}^{(--)}_{(\mu\nu)}=\frac{1}{2}\left(\mathbf{e}_{(33)}-\mathbf{e}_{(44)}\right)V_{\mu}^{-}V_{\nu}^{-}$
(2.26)

transforming as

$\displaystyle\mathbf{e}^{(12+)^{\prime}}_{(\mu\nu)}$
$\displaystyle=e^{iq\alpha(x)}\mathbf{e}^{(12+)}_{(\mu\nu)},$
$\displaystyle\mathbf{e}^{(12-)^{\prime}}_{(\mu\nu)}=e^{-iq\alpha}\mathbf{e}^{(12-)}_{(\mu\nu)}$
(2.27) $\displaystyle\mathbf{e}^{(++)^{\prime}}_{(\mu\nu)}$
$\displaystyle=e^{2iq\alpha(x)}\mathbf{e}^{(++)^{\prime}}_{(\mu\nu)},$
$\displaystyle\mathbf{e}^{(--)^{\prime}}_{(\mu\nu)}=e^{-2iq\alpha(x)}\mathbf{e}^{(--)^{\prime}}_{(\mu\nu)}$
(2.28)

Similarly,

$\displaystyle\mathbf{e}^{(12+)\alpha}_{\alpha}=\left[\mathbf{e}_{(11)}A_{\alpha}+\mathbf{e}_{(12)}\left(A_{\alpha}+U_{\alpha}\right)+\mathbf{e}_{(22)}U_{\alpha}\right]V^{+\alpha},$
(2.29)
$\displaystyle\mathbf{e}^{(12-)\alpha}_{\alpha}=\left[\mathbf{e}_{(11)}A_{\alpha}+\mathbf{e}_{(12)}\left(A_{\alpha}+U_{\alpha}\right)+\mathbf{e}_{(22)}U_{\alpha}\right]V^{-\alpha}$
(2.30)
$\displaystyle\mathbf{e}^{(++)\alpha}_{\alpha}=\frac{1}{2}\mathbf{e}_{(34)}V_{\alpha}^{+}V^{\alpha+},\mathbf{e}^{(--)\alpha}_{\alpha}=\mathbf{e}_{(34)}V_{\alpha}^{-}V^{\alpha-}$
(2.31)

$\displaystyle\mathbf{e}^{(12+)\alpha}_{\alpha}=e^{iq_{2}\alpha(x)}\mathbf{e}^{(12+)\alpha}_{\alpha},\mathbf{e}^{(12-)\alpha}_{\alpha}=e^{-iq_{2}\alpha(x)}\mathbf{e}^{(12-)\alpha}_{\alpha}$
(2.32)
$\displaystyle\mathbf{e}^{(++)\alpha}_{\alpha}=e^{2iq_{2}\alpha(x)}\mathbf{e}^{(++)\alpha}_{\alpha},\mathbf{e}^{(--)\alpha}_{\alpha}=e^{-2iq_{2}\alpha}\mathbf{e}^{(--)\alpha}_{\alpha}$
(2.33)

The corresponding antisymmetric vectorial fields strengths are written as

$\displaystyle F^{I}_{\mu\nu}$
$\displaystyle=\partial_{\mu}A^{I}_{\nu}-\partial_{\nu}A^{I}_{\mu};$
$\displaystyle\vec{E}_{i}^{I}$ $\displaystyle=F^{I}_{0i};$
$\displaystyle\vec{B}_{i}^{I}$
$\displaystyle=\frac{1}{2}\epsilon_{ijk}F^{I}_{jk}$ (2.34)

where $A^{\mu}_{I}\equiv\left(\phi_{I},\vec{A}_{I}\right)$ and $I$ is a
flavour indice $I=1,...4$ corresponding to $A_{\mu},U_{\mu},V^{\pm}_{\mu}$

For the granular symmetric fields strengths, one gets

$\displaystyle S^{I}_{\mu\nu}$
$\displaystyle=\partial_{\mu}A^{I}_{\nu}+\partial_{\nu}A^{I}_{\mu};$
$\displaystyle S^{\alpha I}_{\alpha}$
$\displaystyle=2\partial_{\alpha}A^{\alpha I}$ (2.35) $\displaystyle
S^{I}_{0i}$ $\displaystyle=\partial_{0}A^{I}_{i}+\partial_{i}A^{I}_{0},$
$\displaystyle S^{I}_{ij}$
$\displaystyle=\partial_{i}A^{I}_{j}+\partial_{j}A^{I}_{i}$ (2.36)

The antisymmetric collective fields strengths are written as

$\displaystyle e_{[\mu\nu]}$
$\displaystyle=\mathbf{e}_{[IJ]}A^{I}_{\mu}A^{J}_{\nu},$
$\displaystyle\vec{\mathbf{e}}_{i}$ $\displaystyle=\mathbf{e}_{[0i]},$
$\displaystyle\vec{\mathbf{b}}_{i}$
$\displaystyle=\frac{1}{2}\epsilon_{ijk}\mathbf{e}_{[jk]}$ (2.37)

The simetric sector for collective

$\displaystyle s^{\alpha}_{\alpha IJ}$
$\displaystyle=\mathbf{e}_{(IJ)}A_{\alpha}^{I}A^{\alpha J}$ (2.38)
$\displaystyle
s_{ij}^{IJ}=\mathbf{e}_{(IJ)}\left(A_{i}^{I}A_{j}^{J}+U_{i}^{J}A_{j}^{J}\right)$
(2.39)

The following Lagrangian is composed by the above fields strengths

$L=L_{A}+L_{S}+L_{M}$ (2.40)

where the antisymmetric sector is

$L^{A}_{K}=a_{1}F_{\mu\nu}F^{\mu\nu}+U_{\mu\nu}U^{\mu\nu}+a_{3}V_{\mu\nu}V^{\mu\nu},$
(2.41) $\displaystyle
L^{A}_{3}=\left(a_{1}F_{\mu\nu}+a_{2}U_{\mu\nu}\right)\left(\mathbf{e}^{[12][\mu\nu]}+\mathbf{e}^{[+-][\mu\nu]}\right)+$
(2.42)
$\displaystyle+a_{3}V^{+}_{\mu\nu}\mathbf{e}^{[12-][\mu\nu]}+a_{3}V^{-}_{\mu\nu}\mathbf{e}^{[12+][\mu\nu]},$
$\displaystyle
L^{A}_{4}=\left(\mathbf{e}^{[12]}_{[\mu\nu]}+\mathbf{e}^{[+-]}_{[\mu\nu]}\right)^{2}+\mathbf{e}^{[12+][\mu\nu]}\mathbf{e}^{[12-][\mu\nu]}.$
(2.43)

The symmetric sector

$\displaystyle L^{S}_{K}=\left(S_{\mu\nu 1}+S_{\mu\nu 2}+g_{\mu\nu}S_{\alpha
1}^{\alpha}+g_{\mu\nu}S_{\alpha
2}^{\alpha}\right)^{2}+\beta_{3}S_{\mu\nu}S^{+}_{\mu\nu}S^{\mu\nu-}$ (2.44)
$\displaystyle
4\rho_{3}S^{\alpha+}_{\alpha}S^{\beta-}_{\beta}+\beta_{+}\rho_{-}S^{\alpha+}_{\alpha}S^{\beta-}_{\beta}+\beta_{-}\rho_{+}S^{\alpha-}_{\alpha}S^{\beta+}_{\beta},$
$\displaystyle L^{S}_{3}=\left(S_{\mu\nu 1}+S_{\mu\nu 2}+g_{\mu\nu}S_{\alpha
1}^{\alpha}+g_{\mu\nu}S_{\alpha
2}^{\alpha}\right)(\mathbf{e}^{(11)(\mu\nu)}+\mathbf{e}^{(12)}_{(\mu\nu)}+$
$\displaystyle\mathbf{e}^{(22)(\mu\nu)}+\mathbf{e}^{(+-)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}+g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}+g^{\mu\nu}\mathbf{e}^{(22)\alpha}_{\alpha}$
(2.45)
$\displaystyle+g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha})+(\beta_{+}S^{+}_{\mu\nu}+\rho_{+}g_{\mu\nu}S^{\alpha+}_{\alpha})(\mathbf{e}^{(12-)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(12-)\alpha}_{\alpha})$
$\displaystyle+(\beta_{-}S^{-}_{\mu\nu}+\rho_{-}g_{\mu\nu}S^{\alpha-}_{\alpha})(\mathbf{e}^{(12+)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(12+)\alpha}_{\alpha}),$

$\displaystyle
L^{S}_{4}=(\mathbf{e}^{(11)(\mu\nu)}+\mathbf{e}^{(12)}_{(\mu\nu)}+\mathbf{e}^{(22)(\mu\nu)}+\mathbf{e}^{(+-)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(11)\alpha}_{\alpha}$
$\displaystyle+g^{\mu\nu}\mathbf{e}^{(12)\alpha}_{\alpha}+g^{\mu\nu}\mathbf{e}^{(22)\alpha}_{\alpha}+g^{\mu\nu}\mathbf{e}^{(+-)\alpha}_{\alpha})^{2}+(\mathbf{e}^{(12-)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(12-)\alpha}_{\alpha})$
(2.46)
$\displaystyle(\mathbf{e}^{(12+)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(12+)\alpha}_{\alpha})+(\mathbf{e}^{(++)}_{(\mu\nu)}+g_{\mu\nu}\mathbf{e}^{(++)\alpha}_{\alpha})(\mathbf{e}^{(--)(\mu\nu)}+g^{\mu\nu}\mathbf{e}^{(--)\alpha}_{\alpha}).$

A gauge invariant mass sector is introduced without requiring Higgs mechanism
or dynamical symmetry breaking. It gives,

$L_{M}=\mathbf{m}^{2}_{U}U_{\mu}U^{\mu}+\mathbf{m}^{2}_{V}V_{\mu}^{+}V^{\mu-}$
(2.47)

Thus, the quadruplet interrelationships are expressing a four bosons
lagrangian. A structure to be investigated

##  3 Constitutive equations

The quadruplet $\\{A_{\mu},U_{\mu},V_{\mu}^{\pm}\\}$ is constituted by fields
interrelations. It yields a physics given by constitutive fields equations.
They join the corresponding Euler-Lagrange equation, algebraic relationships
as $\partial_{\nu}S^{\nu\mu I}=\partial_{\nu}F^{\nu\mu
I}+g^{\nu\mu}\partial_{\mu}S^{\alpha}_{\alpha I}$,
$\partial_{\nu}\mathbf{e}^{(\nu\mu)}=\frac{1}{2}\mathbf{e}_{IJ}S^{\alpha
I}_{\alpha}+\mathbf{e}_{(IJ)}S^{\nu\mu
I}A_{\mu}^{J}-\frac{1}{2}\partial^{\mu}\mathbf{e}^{\alpha}_{\alpha}$ and
Noether theorem. A whole fields dynamics is derived. For the spin-1 sector, it
gives the following non-linear abelian equations.

For $A^{\mu}_{T}$:

The Gauss constitutive law is

$\vec{\nabla}\cdot\\{4a_{1}\vec{E}_{A}+a_{2}(\vec{\mathbf{e}}_{AU}+\vec{\mathbf{e}}_{+-})\\}=2a_{3}\mathbf{m}^{2}_{U}\phi_{U}+\rho_{AT}$
(3.1)

with the fields charge density expressed as

$\rho_{AT}\equiv\rho_{AT}+a_{3}\rho_{UT}+\rho_{NT}$ (3.2)

where

$\displaystyle\rho_{AT}\equiv\mathbf{e}_{[12]}[(a_{1}\vec{E}_{A}+E_{U})\cdot\vec{U}+2(\vec{E}^{+}\cdot\vec{V}^{-}+\vec{E}^{+}\cdot\vec{E}^{-})+4(\vec{\mathbf{e}}_{AU}+\vec{\mathbf{e}}_{+-})$
$\displaystyle+(\vec{\mathbf{e}}_{AU-}\cdot\vec{V}^{+}+\vec{\mathbf{e}}_{AU+}\cdot\vec{V}^{-})]+2iq_{1}(\frac{\partial\phi^{+}}{\partial
t}\phi^{-}+\frac{\partial\vec{V}^{-}}{\partial
t}\vec{V}^{+}-\frac{\partial\phi^{+}}{\partial t}\phi^{-}$ (3.3)
$\displaystyle-\vec{\nabla}\phi^{+}\cdot\vec{V}^{-}+\frac{\partial\vec{V}^{-}}{\partial
t}\cdot\vec{V}^{+}-\vec{\nabla}\cdot\phi^{-}\cdot\vec{V}^{+})-q_{1}^{2}(2\phi^{+}\phi^{-}\phi_{A}+2\vec{V}^{+}\cdot\vec{V}^{-}\phi_{A}$
$\displaystyle-\vec{V}^{+}\cdot\vec{A}\phi^{-}-\phi^{-}\phi_{A}\phi^{+}-\vec{V}^{-}\cdot\vec{A}\phi^{+})$

and

$\displaystyle\rho_{UT}\equiv\mathbf{e}_{[12]}[(a_{1}\vec{E}_{A}+E_{U})\cdot\vec{A}+2(\vec{E}^{+}\cdot\vec{V}^{-}+\vec{E}^{+}\cdot\vec{V}^{-})+4(\vec{\mathbf{e}}_{AU}+\vec{\mathbf{e}}_{+-})$
$\displaystyle+(\vec{\mathbf{e}}_{AU-}\cdot\vec{V}^{+}+\vec{\mathbf{e}}_{AU+}\cdot\vec{V}^{-})]+2iq_{2}(\frac{\partial\phi^{+}}{\partial
t}\phi^{-}+\frac{\partial\vec{V}^{-}}{\partial
t}\vec{V}^{+}-\frac{\partial\phi^{+}}{\partial t}\phi^{-}$ (3.4)
$\displaystyle-\vec{\nabla}\phi^{+}\cdot\vec{V}^{-}+\frac{\partial\vec{V}^{-}}{\partial
t}\cdot\vec{V}^{+}-\vec{\nabla}\cdot\phi^{-}\cdot\vec{V}^{+})-q_{2}^{2}(2\phi^{+}\phi^{-}\phi_{U}+2\vec{V}^{+}\cdot\vec{V}^{-}\phi_{U}$
$\displaystyle-\vec{V}^{+}\cdot\vec{U}\phi^{-}-\phi^{-}\phi_{U}\phi^{+}-\vec{V}^{-}\cdot\vec{U}\phi^{+})$
(3.5)

are derived from Euler-Lagrange equations. Noether theorem provides the
conserved electric charge $\rho_{NT}$ expressed as

$\displaystyle\rho_{NT}=iq\\{[v_{1}\vec{E}^{-}+v_{2}\vec{e}_{AU-}]\cdot\vec{V}^{+}-[v_{1}\vec{E}^{+}+v_{2}\vec{e}_{AU+}]\cdot\vec{V}^{-}\\}$
(3.6)

The Ampère constitutive law is

$\displaystyle\vec{\nabla}\times\\{4a_{1}\vec{B}_{A}+2a_{2}(\vec{b}_{AU}+\vec{b}_{+-})\\}-\frac{\partial}{\partial
t}\\{4a_{1}\vec{E}_{A}+2a_{2}(\vec{\mathbf{e}}_{AU}+\vec{\mathbf{e}}_{+-})\\}=2\mathbf{m}^{2}_{U}\vec{U}+\vec{J}_{AT}$
(3.7)

with

$\vec{J}_{AT}=\vec{j}_{AT}+a_{3}\vec{j}_{UT}+\vec{j}_{NT}$ (3.8)

where

$\displaystyle\vec{j}_{AT}\equiv\mathbf{e}_{[12]}[(a_{1}\vec{E}_{A}+u_{1}\vec{E}_{u}+2\vec{\mathbf{e}}_{AU}+2\vec{\mathbf{e}}_{+-})\phi_{U}+(a_{1}\vec{B}_{A}+u_{1}\vec{B}_{u}$
$\displaystyle+\vec{b}_{AU}+\vec{b}_{+-})\times\vec{U}-(\vec{E}_{+}+\vec{\mathbf{e}}_{AU+})\phi^{-}-(\vec{B}_{+}+\vec{b}_{AU+})\times\vec{V}^{-}$
$\displaystyle-(\vec{E}_{-}+\vec{\mathbf{e}}_{AU-})\phi^{+}-(\vec{B}_{-}+\vec{b}_{AU-})\times\vec{V}^{+}]+2iq_{1}(\nabla\phi^{+}\phi^{-}+$
$\displaystyle\vec{\nabla}\cdot\vec{V}^{+}\cdot\vec{V}^{-}-\frac{\partial\vec{V}^{+}}{\partial
t}\phi^{-}+\nabla\phi^{-}\phi^{+}+\vec{\nabla}\cdot\vec{V}^{-}\vec{V}^{+}-\frac{\partial\vec{V}^{-}}{\partial
t}\phi^{+}$
$\displaystyle-\vec{\nabla}\cdot\vec{V}^{+})-q_{1}^{2}(2\phi^{+}\phi^{-}\vec{A}+2\vec{V}^{+}\cdot\vec{V}^{-}\vec{A}-\phi^{+}\phi_{A}\vec{V}^{-}-\vec{V}^{+}\cdot\vec{A}\vec{V}^{-}$
$\displaystyle-\phi^{-}\phi_{A}\vec{V}^{-}-(\vec{V}^{-}\cdot\vec{A})\vec{V}^{+})$
(3.9)

and

$\displaystyle\vec{j}_{UT}\equiv\mathbf{e}_{[12]}[(a_{1}\vec{E}_{A}+u_{1}\vec{E}_{u}+2\vec{\mathbf{e}}_{AU}+2\vec{\mathbf{e}}_{+-})\phi_{A}+(a_{1}\vec{B}_{A}+u_{1}\vec{B}_{u}$
$\displaystyle+\vec{b}_{AU}+\vec{b}_{+-})\times\vec{A}-(\vec{E}_{+}+\vec{\mathbf{e}}_{AU+})\phi^{-}-(\vec{B}_{+}+\vec{b}_{AU+})\times\vec{V}^{-}$
$\displaystyle-(\vec{E}_{-}+\vec{\mathbf{e}}_{AU-})\phi^{+}-(\vec{B}_{-}+\vec{b}_{AU-})\times\vec{V}^{+}]+2iq_{2}(\nabla\phi^{+}\phi^{-}+$
$\displaystyle\vec{\nabla}\cdot\vec{V}^{+}\cdot\vec{V}^{-}-\frac{\partial\vec{V}^{+}}{\partial
t}\phi^{-}+\nabla\phi^{-}\phi^{+}+\vec{\nabla}\cdot\vec{V}^{-}\vec{V}^{+}-\frac{\partial\vec{V}^{-}}{\partial
t}\phi^{+}$
$\displaystyle-\vec{\nabla}\cdot\vec{V}^{+})-q_{2}^{2}(2\phi^{+}\phi^{-}\vec{U}+2\vec{V}^{+}\cdot\vec{V}^{-}\vec{U}-\phi^{+}\phi_{U}\vec{V}^{-}-\vec{V}^{+}\cdot\vec{U}\vec{V}^{-}$
$\displaystyle-\phi^{-}\phi_{U}\vec{V}^{-}-\vec{V}^{-}\cdot\vec{U}\vec{V}^{+})$
(3.10)

The Noether current is

$\displaystyle\vec{j}_{NT}=iq\\{[v_{1}\vec{B}^{-}+v_{2}\vec{b}_{AU-}]\times\vec{V}^{+}+[[v_{1}\vec{E}^{-}+v_{2}\vec{e}_{AU-}]\phi^{+}]$
$\displaystyle-[v_{1}\vec{B}^{+}+v_{2}\vec{b}_{AU+}]\times\vec{V}^{-}+[[v_{1}\vec{E}^{+}+v_{2}\vec{e}_{AU+}]\phi^{-}]\\}$
(3.11)

The Faraday law for $A^{\mu}_{T}$ is

$\displaystyle\vec{\nabla}\times\vec{E}_{A}=-\frac{\partial\vec{B}_{A}}{\partial
t}$ (3.12)

with the Gauss law

$\displaystyle\vec{\nabla}\cdot\vec{B}_{A}=0$ (3.13)

The above expressions are expressing a fields EM behavior. In 1820, Oersted
discovered a link between electricity and magnetism by observing the current
in a wire would move a magnetic needle. A result later explored by Ampère [10]
and concluded by Maxwell. However, when one moves from linear to nonlinear EM,
new properties emerge. Fields are generating fields and matter. Potential
fields physicalities appear. Interactions beyond electric charge are inserted
as eqs. (3.1) and (3.7) are showing.

For $U^{\mu}_{T}$, the constitutive Gauss equation is

$\displaystyle\vec{\nabla}\cdot\\{4u_{1}\vec{E}_{U}+2u_{2}\left(\vec{\mathbf{e}}_{AU}+\vec{\mathbf{e}}_{+-}\right)\\}-2\mathbf{m}^{2}_{U}\phi_{U}=\rho_{UT}$
(3.14)

where

$\rho_{UT}\equiv u_{3}\rho_{AT}+\rho_{UT}+\rho_{NT}$ (3.15)

The corresponding Ampère law is

$\displaystyle\vec{\nabla}\times\\{4u_{1}\vec{B}_{U}+u_{2}(\vec{b}_{AU}+\vec{b}_{+-})\\}-\frac{\partial}{\partial
t}\\{4u_{2}\vec{E}_{U}+u_{2}(\vec{\mathbf{e}}_{AU}+\vec{\mathbf{e}}_{+-})\\}=$
$\displaystyle=-2\mathbf{m}^{2}_{U}\vec{U}+\vec{J}_{UT}$ (3.16)

where

$\displaystyle\vec{J}_{UT}\equiv u_{3}\vec{j}_{AT}+\vec{j}_{UT}+\vec{j}_{NT}$
(3.17)

The Faraday law for $U^{\mu}_{T}$ is given for

$\displaystyle\vec{\nabla}\times\vec{E}_{U}=-\frac{\partial\vec{B}_{U}}{\partial
t}$ (3.18)

with the Gauss law for

$\displaystyle\vec{\nabla}\cdot\vec{B}_{U}=0$ (3.19)

For $V^{\pm}_{\mu}$, the constitutive Gauss law is

$\displaystyle\vec{\nabla}\cdot\\{2(v_{1}+\beta_{3})\vec{E}^{\pm}+2\vec{\mathbf{e}}_{AU\pm}\\}-\mathbf{m}^{2}_{V}\phi^{\pm}=\rho^{\pm}_{VT}$
(3.20)

where

$\displaystyle\rho_{VT}^{\pm}\equiv
2v_{1}\vec{E}^{\pm}\cdot(\vec{A}+\vec{U})-q_{1}^{2}(\phi^{\pm}\vec{A}\cdot\vec{A}-\vec{V}^{\pm}\cdot\vec{A}\phi_{A})$
$\displaystyle-q^{2}_{2}(\phi^{\pm}\vec{U}\cdot\vec{U}-\vec{V}^{\pm}\cdot\vec{U}\phi_{U}-4i\mathbf{e}_{[34]}(\vec{\mathbf{e}}_{AU}+\vec{\mathbf{e}}_{+-})\cdot\vec{V}^{\pm}$
(3.21)
$\displaystyle-2iq_{1}(\vec{\nabla}\phi^{\pm}\cdot\vec{A}+\frac{\partial}{\partial
t}\vec{V}^{\pm}\cdot\vec{A})-2iq_{2}(\vec{\nabla}\phi^{\pm}\cdot\vec{U}+\frac{\partial}{\partial
t}\vec{V}^{\pm}\cdot\vec{U})$

The corresponding Ampère law is

$\displaystyle\vec{\nabla}\times\\{2(v_{1}+\beta_{3})\vec{B}^{\pm}+2v_{2}\vec{b}_{AU\pm}\\}-\frac{\partial}{\partial
t}\\{2(v_{1}+\beta_{3})\vec{E}^{\pm}$
$\displaystyle+2v_{2}\vec{\mathbf{e}}_{AU\pm}\\}-\mathbf{m}^{2}_{V}\vec{V}^{\pm}=\vec{J}^{\pm}_{VT}$
(3.22)

where

$\displaystyle\vec{J}^{\pm}_{VT}\equiv
2v_{1}\vec{B}^{\pm}\times(\vec{A}+\vec{U})+2v_{1}\vec{1}\vec{E}^{\pm}\cdot(\phi_{A}+\phi_{U})-$
$\displaystyle-
q_{1}^{2}(\vec{V}^{\pm}\phi_{A}\phi_{A}+\vec{V}^{\pm}\cdot\vec{A}\vec{A}-\phi^{\pm}\phi_{A}\vec{A}-\vec{V}^{\pm}\vec{A}\cdot\vec{A})-$
$\displaystyle-
q_{2}^{2}(\vec{V}^{\pm}\phi_{U}\phi_{U}+\vec{V}^{\pm}\cdot\vec{U}\vec{U}-\phi^{\pm}\phi_{U}\vec{U}-\vec{V}^{pm}\vec{U}\cdot\vec{U})-$
$\displaystyle-4i\mathbf{e}_{[34]}[(\vec{b}_{AU}+\vec{b}_{+-})\times\vec{V}^{\pm}+(\vec{\mathbf{e}}_{AU}+\vec{\mathbf{e}}_{+-})\phi^{\pm}]-$
$\displaystyle-2iq_{1}(\frac{\partial}{\partial
t}\vec{V}^{\pm}\phi_{A}-\vec{\nabla}\phi^{\pm}\phi_{A})-2iq_{2}(\frac{\partial}{\partial
t}\vec{V}^{\pm}\phi_{U}-\vec{\nabla}\phi^{\pm}\phi_{U})$ (3.23)

Faraday law for $V^{\pm}_{\mu}$

$\displaystyle\vec{\nabla}\times\vec{E}^{\pm}+\frac{\partial\vec{B}^{\pm}}{\partial
t}=iq_{1}\\{\phi_{A}\vec{B}^{\pm}+2\vec{A}\cdot\vec{E}^{\pm}\\}+iq_{2}\\{\phi_{U}\vec{B}^{\pm}+2\vec{U}\cdot\vec{E}^{\pm}\\}$
(3.24)

with the Gauss law

$\displaystyle\vec{\nabla}\cdot\vec{B}^{\pm}=iq_{1}\vec{A}\cdot\vec{B}^{\pm}+iq_{2}\vec{U}\cdot\vec{B}^{\pm}$
(3.25)

where eqs. (3.24-3.25) are introducing sources.

Thus, the spin-1 sector, introduces nonlinear Faraday lines of force. The
couplings between potential fields and fields strengths are the origins for
the electromagnetic flux. A nonlinear fields dynamics with conservation laws
and fields charges are introduced. Potential fields are explicated at
equations of motion and two types of conserved charges beyond electric charge
are obtained.

For the spin-0 longitudinal sector:

For $A^{\mu}_{L}$:

$\displaystyle\partial^{0}\\{s_{1}S^{\alpha
1}_{\alpha}+c_{1}\mathbf{e}^{\alpha}_{\alpha}\\}-2t_{1}\mathbf{m}_{U}U^{0}=J^{0}_{AL}$
(3.26)

with

$\displaystyle J^{0}_{AL}=j^{0}_{AL}+t_{1}j^{0}_{UL}-(t_{1}+1)J^{0}_{L}$
(3.27)

and

For $U^{\mu}_{L}$:

$\partial^{0}\\{s_{2}S^{\alpha}_{\alpha
2}+c_{2}s^{\alpha}_{\alpha}\\}-2m_{U}U^{\mu}_{L}=J^{0}_{UL}$ (3.28)

For $V_{L}^{\pm}$:

$\displaystyle\partial^{0}\\{s_{\pm}S^{\alpha\pm}+c_{\pm}s^{(12\pm)\alpha}_{\alpha}\\}-\mathbf{m}^{2}_{V}V^{0}=J^{0\pm}_{VL}$
(3.29)

and

$\displaystyle\partial^{i}\\{s_{\pm}S^{\alpha\pm}+c_{\pm}s^{(12\pm)\alpha}_{\alpha}\\}-\mathbf{m}^{2}_{V}V^{i}=J^{i\pm}_{VL}$
(3.30)

##  4 Collective Bianchi identities

The quadruplet replies with the usual Bianchi identities. Four granular
identities as
$\partial_{\mu}F^{I}_{\nu\rho}+\partial_{\rho}F^{I}_{\mu\nu}+\partial_{\nu}F^{I}_{\rho\mu}$
are obtained for
$F^{\pm}_{\mu\nu}\equiv\\{F_{\mu\nu},U_{\mu\nu},V_{\mu\nu}^{\pm}\\}$.
Nevertheless, due to the quadruplet interrelationships, collective identities
are also developed. They are for the pair $\\{\vec{e}_{AU},\vec{b}_{AU}\\}:$

$\vec{\nabla}\times\vec{\mathbf{e}}_{AU}+\frac{\partial}{\partial
t}\vec{b}_{AU}=\mathbf{e}_{[12]}\left(\vec{A}\times\vec{E}_{U}-\vec{U}\times\vec{E}_{A}+\phi_{A}\vec{B}_{U}-\phi_{U}\vec{B}_{A}\right)$
(4.1)

and

$\vec{\nabla}\cdot\vec{b}_{AU}=\mathbf{e}_{[12]}\left(-\vec{A}\cdot\vec{B}_{U}-\vec{U}\cdot\vec{B}_{A}\right),$
(4.2)

For $\\{\vec{\mathbf{e}_{+-}},\vec{b}_{+-}\\}$:

$\vec{\nabla}\times\vec{\mathbf{e}_{+-}}+\frac{\partial}{\partial
t}\vec{b}_{+-}=-i\mathbf{e}_{[34]}\\{\vec{V}^{+}\times\vec{E}_{-}-\vec{V}^{-}\times\vec{E}_{+}-\phi^{+}\vec{B}_{-}+\phi^{-}\vec{B}_{+}\\}$
(4.3)
$\vec{\nabla}\cdot\vec{b}_{+-}=i\mathbf{e}_{[+-]}\\{\vec{V}^{+}\cdot\vec{B}_{-}-\vec{V}^{-}\cdot\vec{B}_{+}\\},$
(4.4)

For $\\{\vec{e}_{AU+}+\vec{e}_{AU-},\vec{b}_{AU+}+\vec{b}_{AU-}\\}$:

$\displaystyle\vec{\nabla}\times(\vec{b}_{AU+}+\vec{b}_{AU-})+\frac{\partial}{\partial
t}(\vec{\mathbf{e}}_{AU+}+\vec{\mathbf{e}}_{AU-})=Re\\{\mathbf{e}_{[12]}(\vec{A}\times\vec{E}_{+}$
$\displaystyle-\vec{V}^{+}\times\vec{E}_{A}+\phi_{A}\vec{B}_{+}-\phi_{+}\vec{B}_{A})\\}+Re\\{\mathbf{e}_{[12]}\big{(}\vec{U}\
\times\vec{E}_{+}+$
$\displaystyle-\vec{V}^{+}\times\vec{E}_{U}+\phi_{U}\vec{B}_{+}-\phi_{+}\vec{B}_{U})\\}+\mathbf{e}_{[12]}\\{Req_{1}[\phi^{+}\vec{A}\times\vec{A}$
$\displaystyle+2\phi_{A}\vec{A}\times\vec{V}^{+}+2\phi_{A}\vec{A}\times\vec{V}^{+}+\phi_{A}\phi_{A}\vec{V}^{+}+2\phi_{A}\phi^{+}\vec{A}$
$\displaystyle-\phi_{U}\phi_{A}\phi^{+}\vec{U}-\phi_{U}\phi^{+}\vec{V}^{+}]+Req_{2}[\phi^{+}\vec{U}\times\vec{U}$
$\displaystyle+2\phi_{U}\vec{U}\times\vec{V}^{+}+2\phi_{U}\vec{U}\times\vec{V}^{+}+\phi_{U}\phi_{U}\vec{V}^{+}+2\phi_{U}\phi^{+}\vec{A}$
$\displaystyle-\phi_{U}\phi_{A}\phi^{+}\vec{A}-\phi_{A}\phi^{+}\vec{U}^{+}]\\}$
(4.5)

$\displaystyle\vec{\nabla}\cdot\left(\vec{b}_{+AU}+\vec{b}_{-AU}\right)=Re\mathbf{e}_{[12]}\\{(-\vec{A}\cdot\vec{B}_{+}+\vec{V}^{+}\cdot\vec{B}_{A})$
$\displaystyle+(-\vec{U}\cdot\vec{B}_{+}+\vec{V}^{+}\cdot\vec{B}_{U})\\}+Re\mathbf{e}_{[12]}\\{q_{1}[2\phi_{A}\vec{A}\cdot\vec{V}^{+}+\phi^{+}\vec{A}\cdot\vec{A}$
$\displaystyle-\phi_{+}\vec{A}\cdot\vec{U}-\phi_{A}\vec{U}\cdot\vec{V}^{+}]+q_{2}[2\phi_{U}\vec{U}\cdot\vec{V}^{+}+\phi^{+}\vec{U}\cdot\vec{U}-\phi^{+}\vec{U}\cdot\vec{A}-$
$\displaystyle\phi_{U}\vec{A}\cdot\vec{V}^{+}-\phi_{A}\vec{U}\cdot\vec{V}^{+}]\\},$
(4.6)

For symmetric collective fields strengths, with
$\\{s^{\mu\nu}_{AA}+s^{\mu\nu}_{UU}+s^{\mu\nu}_{AU},\vec{s}_{AA}+\vec{s}_{UU}+\vec{s}_{AU}\\}$:

$\displaystyle\partial^{i}(s^{0j}_{AA}+s^{0j}_{UU}+s^{0j}_{AU})+\partial^{j}(s^{i0}_{AA}+s^{i0}_{UU}+s^{i0}_{AU})+\partial^{0}(s^{ij}_{AA}+\partial^{0}s^{ij}_{UU}+s^{ij}_{AU})=$
$\displaystyle+\mathbf{e}_{(11)}\left\\{A^{i}S^{0j}_{A}+A^{j}S^{0i}_{A}+A^{0}S^{ij}\right\\}+\mathbf{e}_{(22)}\left\\{U^{i}S^{0j}_{U}+U^{j}S^{0i}_{U}+U^{0}S^{ij}\right\\}+$
$\displaystyle+\mathbf{e}_{(12)}\left\\{A^{i}S^{0j}_{U}+A^{j}S^{0i}_{U}+A^{0}S^{ij}_{U}+U^{i}S^{0j}_{A}+U^{j}S^{0i}_{A}+U^{0}S^{ij}_{A}\right\\}$
(4.7)

$\displaystyle\frac{\partial}{\partial
t}(\vec{s}_{AA}+\vec{s}_{UU}+\vec{s}_{AU})=\mathbf{e}_{(11)}\left\\{\phi_{A}\cdot\vec{S}_{A}+\vec{A}S_{A}\right\\}+\mathbf{e}_{(22)}\left\\{\phi_{U}\cdot\vec{S}_{U}+\vec{U}S_{U}\right\\}$
$\displaystyle+\mathbf{e}_{(12)}\left\\{\phi_{A}\vec{S}_{U}+\vec{A}S_{U}+\phi_{U}\vec{S}_{A}+\vec{U}S_{A}\right\\}$
(4.8)

For

$\displaystyle\frac{\partial}{\partial
t}\vec{s}_{+-}=\mathbf{e}_{(34)}Re\left\\{\phi_{+}\vec{S}_{+}+\vec{V}^{+}S_{-}\right\\}+iq_{1}\\{A_{0}V_{i}^{+}V_{j}^{-}+$
$\displaystyle
V_{0}^{+}A_{i}V_{j}^{-}+V_{0}^{-}A_{i}V_{j}^{+}\\}+iq_{2}\\{U_{0}V_{i}^{+}V_{j}^{-}+V_{0}^{+}U_{i}V_{j}^{-}+V_{0}^{-}U_{i}V_{j}^{+}\\},$
(4.9)

$\displaystyle\partial^{i}s^{0j}_{+-}+\partial^{j}s^{i0}_{+-}+\partial^{0}s^{ij}_{+-}=\mathbf{e}_{(34)}Re\left\\{V^{i+}S^{0j}_{-}+V^{j+}S^{0i}_{-}+A^{0}S^{ij}_{-}\right\\}+$
$\displaystyle+iq_{1}\\{A_{i}V_{j}^{+}V_{k}^{-}+V_{i}^{+}A_{j}V_{k}^{-}+V_{i}^{-}A_{j}V_{k}^{+}\\}+iq_{2}\\{U_{i}V_{j}^{+}V_{k}^{-}+$
$\displaystyle V_{i}^{+}U_{j}V_{k}^{-}+V_{i}^{-}U_{j}V_{k}^{+}\\}$ (4.10)

For $\\{s^{\mu\nu}_{AU+},s^{\mu\nu}_{AU-}\\}$:

$\displaystyle\partial^{i}(s^{0j}_{AU+}+s^{0j}_{AU-})+\partial^{j}(s^{i0}_{A+}+s^{i0}_{A-})\partial^{0}(s^{ij}_{A+}+s^{ij}_{A-}+)=\mathbf{e}_{(12)}Re\\{V^{i+}S^{0j}_{A}+$
$\displaystyle
V^{j+}S^{0i}_{A}+V^{0+}S^{ij}_{A}+A^{i}S^{0j}_{+}+A^{j}S^{0i}_{+}+A^{0}S^{ij}_{+}+V^{i+}S^{0j}_{U}+V^{j+}S^{0i}_{U}+V^{0+}S^{ij}_{U}+$
$\displaystyle
U^{i}S^{0j}_{+}+U^{j}S^{0i}_{+}+U^{0}S^{ij}_{+}\\}+q_{1}\\{2(A_{i}A_{i}V_{k}^{+}+V_{k}^{+}A_{i}A_{j})+A_{i}U_{j}V^{+}_{k}$
$\displaystyle+U_{i}A_{j}V_{k}^{+}+V_{i}^{+}U_{j}A_{k}\\}+q_{2}\\{2(U_{i}U_{j}V_{k}^{+}+V^{+}_{i}U_{j}A_{k})$
$\displaystyle+A_{i}U_{j}V^{+}_{k}+U_{i}A_{j}V_{k}^{+}+V^{+}_{i}U_{j}A_{k}\\}$
(4.11)

and

$\displaystyle\frac{\partial}{\partial
t}\left(\vec{s}_{AU+}+\vec{s}_{AU-}\right)=\mathbf{e}_{(34)}[Re\\{\phi_{+}\vec{S}_{A}+\vec{V}^{+}S_{A}+\phi_{A}\vec{S}_{+}+\vec{A}S_{+},$
$\displaystyle\phi_{+}\vec{S}_{U}+\vec{V}^{+}S_{U}+\phi_{U}\vec{S}_{+}+\vec{U}S_{+}\\}+Re\\{V^{i+}S^{0j}_{U}+V^{j+}S^{0i}_{U}+$
$\displaystyle
V^{0+}S^{ij}_{U}+U^{i}S^{0j}_{+}+U^{j}S^{0i}_{+}+U^{0}S^{ij}_{+}\\}]+q_{1}\\{2(\phi_{A}A_{i}V_{j}^{+}$
$\displaystyle+\phi^{+}A_{i}A_{j})+\phi_{A}U_{i}V^{+}_{j}+\phi_{U}A_{i}V_{j}^{+}+\phi^{+}U_{i}A_{j}\\}+q_{2}\\{2(\phi_{U}U_{i}V_{j}^{+}$
$\displaystyle+\phi^{+}U_{i}A_{j})+\phi_{A}U_{i}V^{+}_{j}+\phi_{U}A_{i}V_{j}^{+}+\phi^{+}U_{i}A_{j}\\}$
(4.12)

##  5 Matter from fields associations

A new Faraday lines physics arises from nonlinearity. Nonlinear fields work as
sources for producing matter. Mass and electric charge are derived from fields
associations. This leads us to rewrite the previous constitutive equations of
motion through London, conglomerate, and current terms.

An origin for mass is obtained from fields conglomerates. Fields condensations
as $A_{\mu}A^{\mu}$, $U_{\mu}U^{\mu}$, $A_{\mu}U^{\mu}$ and so on are
constituting masses terms. They redefine the $A_{\mu}^{T}$ Gauss law as

$\displaystyle\vec{\nabla}\cdot\\{4a_{1}\vec{E}_{A}+a_{2}(\vec{\mathbf{e}}_{AU}+\vec{\mathbf{e}}_{+-})\\}+l^{0}_{AT}+c^{0}_{AT}=2a_{3}\mathbf{m}^{2}_{U}\phi_{U}+\rho_{AT}+\rho_{NT}$
(5.1)

where the London term is

$\displaystyle
l_{AT}^{0}=\\{4\mathbf{e}_{[12]}(\phi_{U}\phi_{U}+\vec{U}\cdot\vec{U})-2a_{3}(q_{2}^{2}+q_{1}q_{2})(\phi^{+}\phi^{-}+\vec{V}^{+}\vec{V}^{-})\phi_{A}\\}$
$\displaystyle\\{4a_{3}\mathbf{e}_{[12]}(\phi_{A}\phi_{A}+\vec{A}\cdot\vec{A})-2a_{3}(q_{1}^{2}+q_{1}q_{2})(\phi^{+}\phi^{-}+\vec{V}^{+}\vec{V}^{-})\phi_{U}\\}$
(5.2)

and the conglomerate association

$\displaystyle
c_{AT}^{0}=4\mathbf{e}_{[12]}[(\phi_{A}\phi_{U}+\vec{A}\cdot\vec{U})(\phi_{U}+a_{3}\phi_{A})+1/2(1+a_{3})(\vec{\mathbf{e}}_{AU-}\cdot\vec{V}^{+}$
$\displaystyle+\vec{\mathbf{e}}_{AU+}\cdot\vec{V}^{-})]-2q_{1}q_{2}[(\phi^{+}\phi_{U}+\vec{V}^{+}\cdot\vec{U})\phi^{-}+a_{3}(\phi^{+}\phi_{A}+\vec{V}^{+}\cdot\vec{A})\phi^{-}$
$\displaystyle+(\phi^{-}\phi_{U}+\vec{V}^{-}\cdot\vec{U})\phi^{+}+a_{3}(\phi^{-}\phi_{A}+\vec{V}^{-}\cdot\vec{A})\phi^{+}]-q_{1}^{2}[(\phi^{+}\phi_{A}+\vec{V}^{+}\cdot\vec{A})\phi^{-}$
$\displaystyle+(\phi^{-}\phi_{A}+\vec{V}^{-}\cdot\vec{A})\phi^{+}]-2a_{3}q_{2}^{2}[(\phi^{+}\phi_{U}+\vec{V}^{+}\cdot\vec{U})\phi^{-}$
$\displaystyle+(\phi^{-}\phi_{U}+\vec{V}^{-}\cdot\vec{U})\phi^{+}],$ (5.3)

The above equations are constituting mass condensates terms. Expliciting
bilinear scalar terms. The corresponding fields charge density is

$\displaystyle\rho_{AT}=\mathbf{e}_{[12]}[(a_{1}\vec{E}_{A}+u_{1}\vec{E}_{U})\cdot(\vec{U}+a_{3}\vec{A})+1/2(1+a_{3})(\vec{E}^{+}\cdot\vec{V}^{-}+\vec{E}^{-}\cdot\vec{V}^{+})]$
$\displaystyle-(2iq_{1}+a_{3}q_{2})(\vec{E}^{+}\cdot\vec{V}^{-}+\vec{E}^{-}\cdot\vec{V}^{+}).$
(5.4)

The Amperè law for $A_{\mu}^{T}$ becomes

$\displaystyle\vec{\nabla}\times[4a_{1}\vec{B}_{A}+a_{2}(\vec{b}_{AU}+\vec{b}_{+-})]-\frac{\partial}{\partial
t}[4a_{1}\vec{E}_{A}+2a_{2}(\vec{\mathbf{e}}_{AU}+\vec{\mathbf{e}}_{+-})]+\vec{l}_{AT}$
$\displaystyle+\vec{c}_{AT}=2a_{3}\mathbf{m}_{U}\vec{U}+\vec{J}_{AT}+\vec{J}_{NT}$
(5.5)

where

$\displaystyle\vec{l}_{AT}=\\{4\mathbf{e}_{[12]}(\phi_{U}\phi_{U}+\vec{U}\cdot\vec{U})-2a_{3}(q_{2}^{2}+q_{1}q_{2})(\phi^{+}\phi^{-}+\vec{V}^{+}\vec{V}^{-})\\}\vec{A}$
$\displaystyle\\{4a_{3}\mathbf{e}_{[12]}(\phi_{A}\phi_{A}+\vec{A}\cdot\vec{A})-2a_{3}(q_{1}^{2}+q_{1}q_{2})(\phi^{+}\phi^{-}+\vec{V}^{+}\vec{V}^{-})\\}\vec{U},$
(5.6)

and

$\displaystyle\vec{c}_{AT}=4\mathbf{e}_{[12]})[(\phi_{A\phi_{U}}+\vec{A}\cdot\vec{U})(\vec{U}+a_{3}\vec{A})+1/2(1+a_{3})(\vec{b}_{AU-}\times\vec{V}^{+}+\phi^{+}\vec{\mathbf{e}}_{AU-}$
$\displaystyle+\vec{b}_{AU+}\times\vec{V}^{-}+\phi^{-}\vec{\mathbf{e}}_{AU+})]-2q_{1}q_{2}[(\phi^{+}\phi_{U}+\vec{V}^{+}\cdot\vec{U})\vec{V}^{-}+(\phi^{-}\phi_{U}+\vec{V}^{-}\cdot\vec{U})\vec{V}^{+}$
$\displaystyle+(\phi^{+}\phi_{A}+\vec{V}^{-}\cdot\vec{A})\vec{V}^{+}]-q_{1}^{2}[(\phi^{+}\phi_{A}+\vec{V}^{+}\cdot\vec{A})\vec{V}^{-}+(\phi^{-}\phi_{A}+\vec{V}^{-}\cdot\vec{A})\vec{V}^{+}]+$
$\displaystyle-2a_{3}q_{2}^{2}[(\phi^{+}\phi_{U}+\vec{V}^{+}\cdot\vec{U})\vec{V}^{-}+(\phi^{-}\phi_{A}+\vec{V}^{-}\cdot\vec{A})\vec{V}^{+}],$
(5.7)

and

$\displaystyle\vec{J}_{AT}=\mathbf{e}_{[12]}[(a_{a}\vec{B}_{A}+u_{1}\vec{B}_{U})\times(a_{3}\vec{A}+\vec{U})+(a_{1}\vec{E}_{A}+u_{2}\vec{E}_{U})(a_{3}\phi_{A}+\phi_{U})]+$
$\displaystyle+1/2(1+a_{3}-4iq_{1}+4ia_{3}q_{2})(\vec{B}^{+}\times\vec{V}^{-}+\vec{B}^{-}\times\vec{V}^{+}+\vec{E}^{+}\phi^{-}+\vec{E}^{-}\phi^{+}),$

Eq. (5.1) is showing the neutral field physics. Differently from usual photons
that just transport EM fields the above equations are showing properties of
photon-generating EM fields, mass, and neutral couplings.

The condensated Gauss law for $U^{T}_{\mu}$:

$\displaystyle\vec{\nabla}\cdot[4u_{1}\vec{E}_{U}+2u_{2}(\vec{\mathbf{e}}_{AU}+\vec{\mathbf{e}}_{+-})]-2\mathbf{m}^{2}_{U}+l^{0}_{UT}+c^{0}_{UT}=\rho_{UT}+\rho_{NT}$
(5.9)

where

$\displaystyle
l_{UT}^{0}=\\{4\mathbf{e}_{[12]}(\phi_{A}\phi_{A}+\vec{A}\cdot\vec{A})-2u_{3}(q_{2}^{2}+q_{1}q_{2})(\phi^{+}\phi^{-}+\vec{V}^{+}\vec{V}^{-})\phi_{U}\\}$
$\displaystyle\\{4a_{3}\mathbf{e}_{[12]}(\phi_{U}\phi_{U}+\vec{U}\cdot\vec{U})-2u_{3}(q_{1}^{2}+q_{1}q_{2})(\phi^{+}\phi^{-}+\vec{V}^{+}\vec{V}^{-})\phi_{A}\\},$
(5.10)

and

$\displaystyle
c_{UT}^{0}=4\mathbf{e}_{[12]}[(\phi_{U}\phi_{A}+\vec{U}\cdot\vec{A})(\phi_{A}+u_{3}\phi_{U})+1/2(1+u_{3})(\vec{\mathbf{e}}_{AU-}\cdot\vec{V}^{+}$
$\displaystyle+\vec{\mathbf{e}}_{AU-}\cdot\vec{V}^{+})]-2q_{1}q_{2}[(\phi^{+}\phi_{A}+\vec{U}\cdot\vec{A})\phi^{-}+u_{3}(\phi^{+}\phi_{U}+\vec{U}\cdot\vec{U})\phi^{-}$
$\displaystyle+(\phi^{-}\phi_{A}+\vec{V}^{-}\cdot\vec{A})\phi^{+}+(\phi^{-}\phi_{U}+\vec{V}^{-}\cdot\vec{U})\phi^{+}]-q_{2}^{2}[(\phi^{+}\phi_{U}+$
$\displaystyle+$
$\displaystyle\vec{V}^{+}\cdot\vec{U})\phi^{-}+(\phi^{-}\phi_{U}+\vec{V}^{-}\cdot\vec{U})\phi^{+}]-2u_{3}q_{1}^{2}[(\phi_{+}\phi_{A}+\vec{V}^{+}\vec{A})\phi^{-}$
(5.11) $\displaystyle+(\phi_{-}\phi_{A}+\vec{V}^{-}\vec{A})\phi^{+}],$

and

$\displaystyle\rho_{UT}=\mathbf{e}_{[12]}[(a_{1}\vec{E}_{A}+u_{2}\vec{E}_{U})\cdot(\vec{A}+u_{3}\vec{U})+1/2(1+a_{3})(\vec{E}^{+}\cdot\vec{V}^{-}+\vec{E}^{-}\cdot\vec{V}^{+})]+$
$\displaystyle-2i(u_{3}q_{1}+q_{2})(\vec{E}^{+}\cdot\vec{V}^{-}+\vec{E}^{-}\cdot\vec{V}^{+}),$
(5.12)

Ampère law for $U_{\mu}^{T}$:

$\displaystyle\vec{\nabla}\times[4u_{1}\vec{B}_{U}+u_{2}(\vec{b}_{AU}+\vec{b}_{+-})]-\frac{\partial}{\partial
t}[u_{1}\vec{E}_{A}+2u_{2}(\vec{\mathbf{e}}_{AU}+\vec{\mathbf{e}}_{+-})]-2\mathbf{m}^{2}_{U}\vec{U}+\vec{l}_{UT}+$
$\displaystyle\vec{c}_{UT}=\vec{J}_{UT}+\vec{J}_{NT}$ (5.13)

where

$\displaystyle\vec{l}_{UT}=\\{4\mathbf{e}_{[12]}(\phi_{A}\phi_{A}+\vec{A}\cdot\vec{A})-2u_{3}(q_{2}^{2}+q_{1}q_{2})(\phi^{+}\phi^{-}+\vec{V}^{+}\vec{V}^{-})\\}\vec{U}$
$\displaystyle\\{4a_{3}\mathbf{e}_{[12]}(\phi_{U}\phi_{U}+\vec{U}\cdot\vec{U})-2u_{3}(q_{1}^{2}+q_{1}q_{2})(\phi^{+}\phi^{-}+\vec{V}^{+}\vec{V}^{-})\\}\vec{A},$
(5.14)

and

$\displaystyle\vec{c}_{UT}=4\mathbf{e}_{[12]}[(\phi_{A}\phi_{U}+\vec{A}\cdot\vec{U})(u_{3}\vec{U}+\vec{A})+1/2(1+u_{3})(\vec{b}_{AU-}\times\vec{V}^{+}$
$\displaystyle+\vec{b}_{AU+}\times\vec{V}^{-}+\phi^{+}\vec{\mathbf{e}}_{AU-}+\phi^{-}\mathbf{e}_{AU+})]-2q_{1}q_{2}[(\phi^{+}\phi_{U}+\vec{V}^{+}\cdot\vec{U})\vec{V}^{-}$
$\displaystyle+(\phi^{-}\phi_{U}+\vec{V}^{-}\cdot\vec{U})\vec{V}^{+}]-u_{3}q_{1}^{2}[(\phi^{+}\phi_{A}+\vec{V}^{+}\cdot\vec{A})\vec{V}^{-}+(\phi^{-}\phi_{A}+\vec{V}^{-}\cdot\vec{A})\vec{V}^{-}]$
$\displaystyle
2q_{2}^{2}[(\phi^{+}\phi_{U}+\vec{V}^{+}\cdot\vec{U})\vec{V}^{-}+(\phi^{-}\phi_{U}+\vec{V}^{-}\cdot\vec{U})\vec{V}^{-}],$
(5.15)

and

$\displaystyle\vec{J}_{UT}=\mathbf{e}_{[12]}[(a_{1}\vec{B}_{A}+u_{1}\vec{B}_{U})\times(\vec{A}+u_{3}\vec{U})+(a_{1}\vec{E}_{A}+u_{2}\vec{B}_{U})\cdot(u_{3}\phi_{U}+\phi_{A})$
$\displaystyle+1/2(1+u_{3})(\vec{B}^{+}\times\vec{V}^{-}+\vec{B}^{-}\times\vec{V}^{+}+\vec{E}^{+}\phi^{-}+\vec{E}^{-}\phi^{+})]$
$\displaystyle-2i(q_{1}+u_{3}q_{2})(\vec{B}^{+}\times\vec{V}^{-}+\vec{B}^{-}\times\vec{V}^{+}+\vec{E}^{+}\phi^{-}+\vec{E}^{-}\phi^{+}).$
(5.16)

The condensate Gauss law for $V_{\mu}^{\pm}$ is

$\displaystyle\vec{\nabla}\cdot\\{2(v_{1}+\beta_{3})\vec{E}^{+}+2v_{2}\vec{e}_{AU\pm}\\}-\mathbf{m}^{2}_{V}\phi^{\pm}+l_{VT}^{0\pm}+c_{VT}^{0\pm}=\rho^{VT}$
(5.17)

where

$\displaystyle
l_{VT}^{0\pm}=-4i\mathbf{e}_{[34]}\\{(\phi^{\pm}\phi^{\pm}+\vec{V}^{\pm}\cdot\vec{V}^{\pm})\phi^{\mp}+\vec{e}_{AU}\cdot\vec{V}^{\pm}\\}$
$\displaystyle-
q_{1}^{2}\\{\phi_{A}\phi_{A}+\vec{A}\cdot\vec{A}\\}\phi^{\pm}-q_{2}^{2}\\{\phi_{U}\phi_{U}+\vec{U}\cdot\vec{U}\\}\phi^{\pm},$
(5.18)

$\displaystyle
c^{0\pm}_{VT}=-4i\mathbf{e}_{[34]}\\{(\phi^{\pm}\phi^{\mp}+\vec{V}^{\pm}\cdot\vec{V}^{\mp})\phi^{\pm}+\mathbf{e}_{AU}\cdot\vec{V}^{\pm}\\}$
$\displaystyle-2q_{1}q_{2}\\{(\phi_{A}\phi_{U}+\vec{A}\cdot\vec{U})\phi^{\pm}-(\phi^{\pm}\phi_{U}+\vec{V}^{\pm}\cdot\vec{U})\phi_{A}\\}+$
$\displaystyle
q_{1}^{2}\\{\phi^{\pm}\phi_{A}+\vec{V}^{\pm}\cdot\vec{A}\\}\phi_{A}+q_{2}^{2}\\{\phi^{\pm}\phi_{U}+\vec{V}^{\pm}\cdot\vec{U}\\}\phi_{U},$
(5.19)

$\displaystyle\rho^{\pm}_{VT}=2v_{1}\vec{E}^{\pm}\cdot(\vec{A}+\vec{U})-2iq_{1}\vec{E}^{\pm}\cdot\vec{A}-2iq_{2}\vec{E}^{\pm}\cdot\vec{U}$
(5.20)

The associated $V_{\mu}^{\pm}$ Ampère law is

$\displaystyle\vec{\nabla}\times\\{2(v_{1}+\beta_{3})\vec{B}^{\pm}+2\vec{b}_{AU\pm}\\}-\frac{\partial}{\partial
t}\\{2(v_{1}+\beta_{3})\vec{E}^{\pm}+2\vec{e}_{AU\pm}\\}-\mathbf{m}^{2}_{V}\vec{V}^{\pm}$
$\displaystyle+\vec{l}_{VT}^{\pm}+\vec{c}^{\pm}_{VT}=\vec{J}^{\pm}_{VT}$
(5.21)

where

$\displaystyle\vec{l}_{VT}^{\pm}=-4i\mathbf{e}_{[34]}\\{(\phi^{\pm}\phi^{\pm}+\vec{V}^{\pm}\cdot\vec{V}^{\pm})\vec{V}^{\mp}+\vec{e}_{AU}\phi^{\pm}+\vec{b}_{AU}\times\vec{V}\\}$
$\displaystyle-
q_{1}^{2}\\{\phi_{A}\phi_{A}+\vec{A}\cdot\vec{A}\\}\vec{V}^{\pm}-q_{2}^{2}\\{\phi_{U}\phi_{U}+\vec{U}\cdot\vec{U}\\}\vec{V}^{\pm},$
(5.22)

and

$\displaystyle\vec{c}^{\pm}_{VT}=-4i\mathbf{e}_{[34]}\\{(\phi^{\pm}\phi^{\mp}+\vec{V}^{\pm}\cdot\vec{V}^{\mp})\vec{V}^{\pm}+\mathbf{e}_{AU}\phi^{\pm}+\vec{b}_{AU}\times\vec{V}^{\pm}\\}$
$\displaystyle-2q_{1}q_{2}\\{(\phi_{A}\phi_{U}+\vec{A}\cdot\vec{U})\vec{V}^{\pm}-(\phi^{\pm}\phi_{U}+\vec{V}^{\pm}\cdot\vec{U})\vec{A}\\}+$
$\displaystyle
q_{1}^{2}\\{\phi^{\pm}\phi_{A}+\vec{V}^{\pm}\cdot\vec{A}\\}\vec{A}+q_{2}^{2}\\{\phi^{\pm}\phi_{U}+\vec{V}^{\pm}\cdot\vec{U}\\}\vec{U},$
(5.23)

and

$\displaystyle\vec{J}^{\pm}_{VT}=2v_{1}[\vec{B}^{\pm}\times(\vec{A}+\vec{U})+\vec{E}^{\pm}(\phi_{A}+\phi_{U})]-2iq_{1}(\vec{B}^{\pm}\times\vec{A}+\vec{E}^{\pm}\phi_{a})$
$\displaystyle-2iq_{2}(\vec{B}^{\pm}\times\vec{U}+\vec{E}^{\pm}\phi_{A})$
(5.24)

For longitudinal sector For $A^{\mu}_{L}$

$\displaystyle\partial^{0}\\{s_{1}S^{\alpha
1}_{\alpha}+c_{1}\mathbf{e}^{\alpha}_{\alpha}\\}+l_{AL}^{0}+c_{AL}^{0}=2t_{1}\mathbf{m}^{2}_{U}U^{0}+J^{0}_{AL}+(t_{1}+1)J^{0}_{NL}$
(5.25)

$\displaystyle
l_{ATL}^{0}=4\mathbf{e}_{(11)}\mathbf{e}_{(11)}[\mathbf{e}^{(11)(00)}A_{0}+\mathbf{e}^{(11)(0i)}A_{i}+4(\mathbf{e}^{(11)\alpha}_{\alpha}+\mathbf{e}^{(12)\alpha}_{\alpha})A^{0}]$
$\displaystyle+2\mathbf{e}_{(12)}(2\mathbf{e}^{(12)(00)}+\mathbf{e}^{(22)(00)})U_{0}+(2\mathbf{e}^{(12)(0i)}+\mathbf{e}^{(22)(0i)})U_{i}+$
$\displaystyle+4(\mathbf{e}^{(12)(00)}A_{0}+\mathbf{e}^{(12)(0i)}A_{i})+\mathbf{e}^{(11)\alpha}_{\alpha}A^{0}+\mathbf{e}^{(22)\alpha}_{\alpha}U^{0})+$
$\displaystyle+4t_{1}[(\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)\alpha}_{\alpha})U^{0}+\mathbf{e}^{(22)(00)}U_{0}+\mathbf{e}^{(22)(0i)}U_{i}]+$
$\displaystyle+2t_{1}\mathbf{e}_{(12)}[2\mathbf{e}^{(11)(00)}A_{0}+\mathbf{e}^{(11)(0i)}A_{i}+\mathbf{e}^{(12)(00)}U_{0}+\mathbf{e}^{(12)(0i)}U_{i}+$
$\displaystyle+(\mathbf{e}^{(22)\alpha}_{\alpha}+\mathbf{e}^{(12)\alpha}_{\alpha})U^{0}]-q_{1}^{2}(\phi^{+}\phi^{-}+\vec{V}^{+}\cdot\vec{V}^{-})A^{0}$
$\displaystyle+2q_{2}^{2}t_{2}(\phi^{+}\phi^{-}+\vec{V}^{+}\cdot\vec{V}^{-})U^{0}-2q_{1}q_{2}(\phi^{+}\phi^{-}+\vec{V}^{+}\cdot\vec{V}^{-})(A^{0}+U^{0})$

and

$\displaystyle
c^{0}_{AL}=2\mathbf{e}_{(12)}[\mathbf{e}^{(\alpha)}_{\alpha}(U^{0}+A^{0})+(\mathbf{e}^{(11)(00)}+\mathbf{e}^{(+-)(00)})U_{0}+$
$\displaystyle+(\mathbf{e}^{(11)(0i)}+\mathbf{e}^{(+-)(0i)})U_{i}+\mathbf{e}^{(+-)\alpha}_{\alpha}U^{0}]+\mathbf{e}_{(11)}(\mathbf{e}^{(12)(00)}$
$\displaystyle+\mathbf{e}^{(+-)(00)})A_{0}+(\mathbf{e}^{(22)(0i)}+\mathbf{e}^{(+-)(0i)})A_{i}+8\mathbf{e}^{(+-)\alpha}_{\alpha}A^{0}]$
$\displaystyle+2\mathbf{e}_{(34)}(\mathbf{e}^{(12+)(00)}V_{0}^{-}+\mathbf{e}^{(12-)(00)}V_{0}^{+}+\mathbf{e}^{(12+)(0i)}V_{i}^{-}+\mathbf{e}^{(12-)(0i)}V_{i}^{+}$
$\displaystyle+\mathbf{e}^{(12+)\alpha}_{\alpha}V^{0-}+\mathbf{e}^{(12-)\alpha}_{\alpha}V^{0+})+4t_{1}\mathbf{e}_{(22)}[(\mathbf{e}^{(11)(00)}+\mathbf{e}^{(+-)(00)})U_{0}$
$\displaystyle+(\mathbf{e}^{(11)(0i)}+\mathbf{e}^{(+-)(0i)})A_{i}+\mathbf{e}^{(+-)\alpha}_{\alpha}U^{i}]+2t_{1}\mathbf{e}_{12}[\mathbf{e}^{(12)\alpha}_{\alpha}A^{0}+\mathbf{e}^{(12)\alpha}_{\alpha}U^{0}]$
$\displaystyle+2t_{1}\mathbf{e}_{(34)}(\mathbf{e}^{(12+)(00)}V_{0}^{-}+\mathbf{e}^{(12-)(00)}V_{0}^{+}+\mathbf{e}^{(12+)(0i)}V_{i}^{-}+\mathbf{e}^{(12-)(0i)}V_{i}^{+}$
$\displaystyle+\mathbf{e}^{(12+)\alpha}_{\alpha}V^{0-}+\mathbf{e}^{(12-)\alpha}_{\alpha}V^{0+})+2t_{1}q_{2}^{2}[(V_{0}^{+}A^{0}+V_{i}^{+}A^{i})V^{0-}+$
$\displaystyle+(V_{0}^{-}A^{0}+V_{i}^{-}A^{i})V^{0+}]-2q_{1}q_{2}[t_{1}(V_{0}^{+}A^{0}+V_{i}^{+}A^{i})V^{0-})V^{0-}$
$\displaystyle+(V_{0}^{-}A^{0}+V_{i}^{-}A^{i})V^{0+}+(V_{0}^{+}U^{0}+V_{i}^{+}U^{i})V^{0-}+(V_{0}^{-}U^{0}+V_{i}^{-}U^{i})V^{0+}]$

and

$\displaystyle
J^{0}_{AL}=2(\beta_{1}S^{001}+\beta_{2}S^{002})[\mathbf{e}_{(11)}(1+t_{1})A_{0}+\mathbf{e}_{(11)}U_{0}+t_{1}\mathbf{e}_{(22)}U_{0}]$
$\displaystyle+2(\beta_{1}S^{0i1}+\beta_{2}S^{0i2})[\mathbf{e}_{(11)}(1+t_{1})A_{i}+\mathbf{e}_{(12)}U_{i}+t_{1}\mathbf{e}_{(12)}U_{i}]+$
$\displaystyle 2[(\beta_{1}+5\rho_{1})S^{\alpha
1}_{\alpha}+(\beta_{2}+5\rho_{2})S^{\alpha
2}_{\alpha}][\mathbf{e}_{(11)}(1+t_{1})A^{0}+\mathbf{e}_{12}(1+t_{1})U^{0}+$
$\displaystyle+\mathbf{e}_{(22}U^{0}]+\mathbf{e}_{(34)}(1+t_{1})[\beta_{1}(S^{00+}V_{0}^{-}+S^{0i+}V_{i}^{+})+\beta_{-}(S^{00-}V_{0}^{+}+$
$\displaystyle+S^{0i-}V_{i}^{+})+\rho_{+}S^{\alpha+}_{\alpha}V^{0-}+S^{\alpha-}_{\alpha}V^{0+}]$
(5.28)

and

$\displaystyle\partial^{i}\\{s_{1}S^{\alpha
1}_{\alpha}+c_{1}\mathbf{e}^{\alpha}_{\alpha}\\}+l_{AL}^{i}+c_{AL}^{i}=2t_{1}\mathbf{m}^{2}_{U}U^{i}+J^{i}_{AL}+(t_{1}+1)J^{i}_{NL}$
(5.29)

For $U^{\mu}_{L}$

$\displaystyle\partial^{0}\\{s_{2}S^{\alpha
2}_{\alpha}+c_{2}\mathbf{e}^{\alpha}_{\alpha}\\}+l_{UL}^{0}+c_{UL}^{0}-2\mathbf{m}^{2}_{U}U^{0}=J^{0}_{UL}+(t_{2}+1)J^{0}_{NL}$
(5.30)

And

$\displaystyle\partial^{i}\\{s_{2}S^{\alpha
2}_{\alpha}+c_{2}\mathbf{e}^{\alpha}_{\alpha}\\}-2\mathbf{m}^{2}_{U}U^{i}+l_{AL}^{i}+c_{AL}^{i}=J^{i}_{AL}+(t_{1}+1)J^{i}_{NL}$
(5.31)

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

$\displaystyle\partial^{0}\\{s_{\pm}S^{\alpha\pm}_{\alpha}+c_{1}\mathbf{e}^{(12\pm)\alpha}_{\alpha}\\}+l_{VL}^{0\pm}+c^{0}_{VL}-\mathbf{m}^{2}_{V}V^{0\pm}=J^{0\pm}_{VL}$
(5.32)
$\displaystyle\partial^{i}\\{s_{\pm}S^{\alpha\pm}_{\alpha}+c_{1}\mathbf{e}^{(12\pm)\alpha}_{\alpha}\\}+l_{VL}^{i\pm}+c^{i}_{VL}-\mathbf{m}^{2}_{V}V^{i\pm}=J^{i\pm}_{VL}$
(5.33)

The above expressions are showing dynamically on EM fields associations. Next
step will be to understand show these fields products are transformed into a
condensation mechanism.

##  6 Conclusion

Faraday anteceded Einstein on Newton extension. Introduced a physics depending
on fields that Newton epoch did not recognize. EM would be guided by variables
diverse from mechanics. Nature provides not only matter as fields. Maxwell
equations projected fields as variables and matter as sources.

Faraday used field lines to signal that there was something. His law of
induction was definitive. The lines of force became tangible effects. EM is
redefined as a flow of fields. Maxwell states it in terms of divergent and
rotational operators. The question would be whether these fields formed a
substance of their own. The limitation in Faraday is the EM fields depending
on electrical charge. Therefore, they would be consequential.

History grows by, and, in current days, physics is based on fields. The fields
play the role of the variables of quantum field theory. Faraday fields led to
a viewpoint where quantum fields precede particles. The next stage is to
explore nonlinear fields dynamics. Different subjects such as plasma,
astrophysics, and condensated matter are requiring nonlinear extensions. EM
literature provides 15 nonlinear models [11]. Most of them are effective
theories, as Born-Infeld and Euler-Heisenberg. In this work, a nonlinear
abelian gauge theory is considered. The simplest nonlinear fields dynamics
without requiring non-linear Yang-Mills is achieved by introducing different
potential fields rotating under the same group [12].

A four bosons EM is proposed. A fundamental nonlinear electromagnetism based
on the generic electric charge $\\{+,0,-\\}$ transmitted by a quadruplet
$\\{A_{\mu},U_{\mu},V_{\mu}^{\pm}\\}$ connected under an abelian gauge
symmetry, eqs. (2.1-2.4). This association of new EM provides physicalities.
Fields equations are providing potential fields with dynamics, conserved
charges and couplings. A fields charge physics is derived. Non linear fields
are emitting Faraday lines of force. A new texture appears through the
proposal of abelian non-linear electrodynamics. Its novelty compared to
Faraday lies in the existence of potential fields acting as their own sources
Equations of motion show fields charges as a function of fields. Gauge
invariant, potential fields conglomerates emerge. The photonic field observed
in interference and diffraction, introduced by hand in quantum mechanics and
detected in the Ahranov-Bohm and Ahranov-Casher experiments, gains presence by
fundamental principles.

An EM model with nonlinear abelian potential fields is proposed. Potential
fields are no more subsidiary as in Maxwell. They work as their own sources
and interact with granular and collective EM fields strengths. Their explicit
participation in fields equations is consistent with the Faraday experiment in
1841 when light polarization was shifted by a magnetic field [13], and
according to Aharonov-Bohm and Aharonov-Casher experiments [14]. Originating
mass through London and conglomerated terms. Electric charge is no more the
only conserved charge and coupling constant. Neutral couplings with granular
and collective fields strengths are also ruling the model.

Four bosons EM leads to a physics based on the potential fields physicality.
Instead of each point in space is associated with E-B, that is, configured in
6 numbers, a new determination is observed. The EM origin is given by the
cluster ${A_{\mu},U_{\mu},V_{\mu}^{\pm}}$where each point P is described by
4x4 numbers. Therefore, these 16 numbers define the EM field substance.

Thus, preserving the two Maxwell postulates, which are light invariance and
electric charge conservation, the four bosons electromagnetism redefines what
electromagnetic energy is. Potential fields conglomerates, granular and
collective fields strengths are extending the EM observable beyond the usual
pair $\vec{E}$-$\vec{B}$. New sectors are ascribed beyond Maxwell. Analyzing
the corresponding energy expression in [15], one notices other terms than the
usual Maxwell $\vec{E}^{2}+\vec{B}^{2}$. Nonlinearity, neutral, weak
interaction, spintronics, and photonics regimes are introduced [16].

There is an extended EM energy to be understood. Missing energy is always a
present theme in physics. In 1927, Ellis and Wooster by studying the average
distribution of $\beta$-radium understood that something was missing [17].
Later Pauli identified as neutrinos. Actually, physics discusses dark energy
and matter. Something again is absent. Various hypotheses have been
considered. Perhaps one should include an EM energy extension.

Concluding, there is a deeper relationship between matter and fields to be
investigated by a nonlinear Faraday lines of force. A physics that antecedes
matter with dynamics, conservation and charge expressed by fields. Consider
particles production by electric, magnetic, and gravitational fields [18]. A
subject that has been studied since 1931 by Sauter, Breit-Wheeler in (1934),
Schwinger (1951), Sexl (1969). There is a new Faraday field physics to be
included in electromagnetic theory.

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The second Chern class can be rewritten as:

$\displaystyle c_{2}=\dfrac{1}{2}\sum_{i\neq
j}x_{i}x_{j}=\dfrac{1}{2}\sum_{i,j}x_{i}x_{j}-\dfrac{1}{2}\sum_{i}x_{i}^{2}=\dfrac{1}{2}\,c_{1}^{2}-\dfrac{1}{2}\sum_{i}x_{i}^{2}~{}.$
(A.4)

In a similar spirit the third Chern class can also be rewritten. When
inserting

$\displaystyle\sum_{i\neq
j}x_{i}x_{j}^{2}=\sum_{i,j}x_{i}x_{j}^{2}-\sum_{i}x_{i}^{3}$ (A.5)

in

$\displaystyle\Big{(}\sum_{i}x_{i}^{3}\Big{)}^{3}=\sum_{i}x_{i}^{3}+3\sum_{i\neq
j}x_{i}x_{j}^{2}+\sum_{i\neq j\neq k}x_{i}x_{j}x_{k}~{},$ (A.6)

one finds the identity

$\displaystyle
c_{1}^{3}=-2\sum_{i}x_{i}^{3}+c_{1}\sum_{j}x_{j}^{2}+3!\,c_{3}~{}.$ (A.7)

If the first Chern class vanishes, i.e. $c_{1}=0$, the relations (A.4) and
(A.7) imply that

$\displaystyle c_{2}=-\dfrac{1}{2}\sum_{i}x_{i}^{2}~{},\qquad
c_{3}=\dfrac{1}{3}\sum_{i}x_{i}^{3}~{}.$ (A.8)

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