Abelian Constructivist Lagrangian
†J. Chauca<EMAIL_ADDRESS>†R<EMAIL_ADDRESS>and †L.S.
Mendes<EMAIL_ADDRESS>
†Aprendanet, Petropolis, Brazil; Quarks, Petropolis, Brazil

###### Abstract

Constructivist lagrangian propiates a diverse approach to field theory.
Introduce the set action. Consider fields families under a same symmetry
group. The resulting fields set extends the standard atomist field theory to a
whole field theory.

An associative physics is proposed. The grouping physics. The relationship
between the part and the whole is considered. A third quantum type beyond
Planck granularity and quantum mechanics wave-particle is obtained. The
quantum inserted in the whole. Differentiated energy packets are formed. A
quantum system is constituted.

The abelian grouping physics is considered. The simplest whole unity. Set
action in terms of $U(1)$ symmetry. The correspondent constructivist
lagrangian is studied. A new type of individuation called whole quantum is
derived. An abelian quantum system is constituted. The usual atomist gauge
symmetry is preserved, but, constructivist properties are generated.

Quantum system with own norm is a necessary theoretical argument. More is
different, once time said P.W. Anderson. Physics is challenged to make the
passage from an isolated particle to a quantum system. A theory to describe
the physics of part in the whole. A challenge to be interpreted under gauge
symmetry. A symmetry of difference is proposed. Quantum diversities enlarging
the meanings of interaction, induction, connectivity.

A quantum system is introduced. Its elementarity is identified as a third
quantum type. It is called whole quantum or variety. This quantum of a many
particles system appears with a new physicality. The corresponding set action
derives antireductionist physical laws under gauge symmetry. Ruled by
associativity, set transformation, evolution. Associativitity providing
quantum under set, diversity, interdependence, nonlinearity, chance. Set
transformations performing a whole determinism with directive conducted by
gauge parameter and circumstances under lagrangian free coefficients.
Generating a set physics with growth, evolution, emergence, complexity. An
evolving quantum transforming their quantum numbers.

The abelian constructivist lagrangian is explored. A quantum system assembled
by fields families and gauge scalars is performed. It arises an environmental
physics. Gauge scalars form substrutures with realities and potentialities.
The volume of circumstances for each gauge scalar is calculated. Nonvirtual
relationships are derived. Physical entities as masses, charges, coupling
constants are expressed under constructivist properties. Functionalities will
lead them to a physical behaviour beyond four interactions.

##  1 Introduction

Quantum theory is a definitive step on the path of knowledgement about nature.
It fixes energy in small amounts. An history which may be divided in three
parts. The classical quantum theory started with Planck in 1900, electron,
photoelectic effect, Rutherford-Bohr atom, proton, and in 1923 concluded with
Compton words “… a radiation quantum carries with its momentum as well as
energy.” [1].

The second phase, initiated in 1924 with du Broglie wave-particle, 1926
through quantum mechanics, 1927 with Heisenberg uncertainity principle, Dirac
electromagnetic quantization, Jordan and Born relativistic wave equations [2].
They introduced quantum mechanics at old quantum theory. And in 1928, the
quanta theory was related with relativity through Dirac equation [3]. Since
then, physics has been developing an atomist quantum field theory
incorporating granularity, wave-particle and relativity. As consequence,
diverse theories have been developed as QED, QCD, electroweak,
grandunification, supersymmetry, supergravity, extra dimension, superstrings,
M-theory [4].

Nevertheless, a principle is missing in the quantum idea. The set notion. As
thermodynamics considers that universe events tend to entropy, the principle
of totality emphasizes the manifestation to association. Atoms,
molecules,…galaxies are grasping how the universe organizes different forms of
conglomerates. A perspective where the set would be the constitutive atom of
the universe.

At current physics prevaleces the reductionist cognition. Parts derived as
subunits summatory. However, the principles of associativity, confinement,
complexity lead physics to the set action. There is a quantum physics under
the whole unity to be understood. A performance beyond ultimate constituents.
Consider the set as the fundamental generator and the correspondent part
inserted in the whole. A quantum physics with a whole-part duality is
generated. As consequence, a new physical elementarity is introduced. Parts
become varieties of the whole environment. Variety defined as the whole
quantum inserted in a quantum system.

Differents origins coming from Kaluza-Klein, fibre bundle, supersymmetry, non-
linear $\sigma$-models, CP-models, degrees of freedom counting are showing the
presence of fields set $\\{A_{\mu I}\\}$ encoded in an abelian group [5].
These various demonstrations are supporting the totality principle in gauge
theory. Allowing, mathematically, the set notion and corresponding whole
quantum be stipulated through gauge symmetry.

A third quantum physics is expressed. There is a whole quantum field theory
beyond ultimate constituents to be constituted. An integral reality to be
taken. Quantum derived from the whole association. A grouping physics crossing
the Newton mechanics reductionism and quantum field theory atomism. A physics
of variety originated from the relationship between ultimate constituents and
set action.

Constructivist lagrangian turns out the formalism to prescribe the set action
and corresponding third quantum. Quantum in a whole enlarges systemically the
interaction, induction, connectivity. Antireductionist properties are
established. A physics under set, diversity, interdependence, non-linearity,
chance, directive and circumstance, growth, evolution, emergence, complexity.
Features producing functionalities beyond four interactions.

Thus, the research will be about the whole abelian quantum physics. Consider
an abelian lagrangian enlarged by the antireductionist gauge symmetry. A
symmetry identified first under a fields set $\\{A_{\mu I}\\}$ associated by a
common gauge parameter; rewritten, in terms of a constructor basis
$\\{D_{\mu},X_{\mu}^{i}\\}$; subsequently, do physics with a realistic quanta
set. Consider the constructivist lagrangian as exposed at Appendice A in terms
of a physical basis $\\{G_{\mu I}\\}$. On it, analyze the symmetry of
difference features.

##  2 Fields set $\\{G_{\mu I}\\}$

Considering the physical principle where nature acts through groupings, the
first purpose is to study an antireductionist gauge theory (AGT). Introduce an
abelian set of N-vector fields interconnected under a common gauge parameter.
The significance is that, one reinterprets a certain field $A_{\mu}$ within
the domain of other fields $A_{\mu I},A_{\mu}\to\\{A_{\mu I}\\},\text{where
}I\in 1...N$. This extension makes physics manifest the part in the whole.
Develop new physical properties.

###  2.1 Fields set basis

The possibility of expanding the gauge parameter for a systemic relationship,
results that, the corresponding physical properties can be seen under
different fields basis. Similarly, as physical laws are independent of
reference systems in space-time, AGT according to Salam-Kameguchi theorem can
be analyzed under different reference of fields such as $\\{A_{\mu
I}\\},\\{D_{\mu},X_{\mu}^{i}\\},\\{G_{\mu I}\\}$ [6].

Thus, the basis $\\{A_{\mu I}\\}$ associate a plurality of fields transforming
under the same parameter, $\\{A^{\prime}_{\mu I}\\}=\\{A_{\mu
I}\\}+k_{I}\partial_{\mu}\alpha$. The basis $\\{D_{\mu},X_{\mu}^{i}\\}$ is
useful for analizing explicitly the gauge invariance. Nevertheless, it is at
fields basis $\\{G_{\mu I}\\}$, where physics is done. The physical fields are
those that diagonalize the equations of motion. Defined as the poles at two
point Green’s function.

A rotation from the constructor basis has to be implemented to define the
physical fields $\\{G_{\mu I}\\}$. In terms of these fields that the
corresponding measurable entities of the model must be defined, as the
corresponding electric and magnetic fields. At Appendice A, one studies the
so-called constructor basis $\\{D_{\mu},X_{\mu}^{i}\\}$. It is the simplest
way to reveal on the physical abelian set gauge invariance.

###  2.2 Gauge transformation

The fields set approach is being studied. It provides quanta under the set
action. The physical basis $\\{G_{\mu I}\\}$ is defined in terms of the
constructor basis$\\{D_{\mu},X_{\mu i}\\}$ through the $\Omega$ matrix
rotation [7]. From Appendix A, one writes

$D_{\mu}=\Omega_{1I}G_{\mu}^{I}$ (2.1)

$X_{\mu i}=\Omega_{iI}G_{\mu}^{I},\text{ i= 2,...,N}$ (2.2)

where $\Omega$ is a transformation matrix defined in terms of lagrangian
coefficients. It does not depend on spacetime. It corresponds to the
transverse sector diagonalization. Index i means the flavour related to
$i=2,...,N$.

Thus, the physical basis is a $\Omega$ transformation from constructor basis.
The matrix $\Omega$ has the property of being invertible

${\Omega_{IK}\Omega_{KJ}^{-1}=\delta_{IJ}}$ (2.3)

It gives,

$\begin{pmatrix}G_{\mu 1}\\\ G_{\mu 2}\\\ \vdots\\\ G_{\mu
N}\end{pmatrix}=\begin{pmatrix}\Omega^{-1}_{11}&\Omega^{-1}_{12}&\dots&\Omega^{-1}_{1N}\\\
\Omega^{-1}_{21}&\Omega^{-1}_{22}&\dots&\Omega^{-1}_{2N}\\\
\vdots&\vdots&\ddots&\vdots\\\
\Omega^{-1}_{N1}&\Omega^{-1}_{N2}&\dots&\Omega^{-1}_{NN}\\\
\end{pmatrix}\begin{pmatrix}D_{\mu}\\\ X_{\mu 2}\\\ \vdots\\\ X_{\mu
N}\end{pmatrix}$ (2.4)

resulting

$\displaystyle{G_{\mu I}=\Omega^{-1}_{I1}D_{\mu}+\Omega^{-1}_{Ii}X_{\mu}^{i}}$
(2.5)

Thus the corresponding gauge transformation for every physical field envolved
in the abelian set is

$G_{\mu I}^{\prime}=G_{\mu I}+\Omega^{-1}_{I1}\partial_{\mu}\alpha$ (2.6)

where eq. (2.6) is expressing that each field inserted in the fields set
$\\{G_{\mu I}\\}$ is individualized by its own gauge transformation and with
their transversal poles diagonalized. It is identified as the symmetry of
difference.

##  3 Gauge invariance

###  3.1 Fields strengths

The interwined fields $\\{G_{\mu I}\\}$ determine various fields strengths
from eq. (2.6). They will have granular and collective nature, with
antisymmetric and symmetric properties. Considering first, the antisymmetry
sector, one rewrites Apendice A as below.

A. Antisymmetric sector

The corresponding granular fields strengths are given by

${G^{I}_{\mu\nu}}=\partial_{\mu}{G^{I}_{\nu}}-\partial_{\nu}{G^{I}_{\mu}}$

whose gauge invariance is supported by eq. (2.6)

$G^{I^{\prime}}_{\mu\nu}=G^{I}_{\mu\nu}$ (3.1)

Studying the collective field strength, one gets from Apendice A

$z_{[\mu\nu]}\equiv\gamma_{[IJ]}G_{\mu}^{I}G_{\nu}^{J},\textbf{
}\gamma_{[IJ]}\equiv\gamma_{[ij]}\Omega^{i}_{I}\Omega^{j}_{J}$ (3.2)

The collective antisymmetric field gauge invariance

$z_{[\mu\nu]}^{\prime}=\gamma_{[IJ]}{G^{I}_{\mu}}^{\prime}{G^{J}_{\nu}}^{\prime}=z_{[\mu\nu]}$

is derived on eq. (2.3) relationships

$\displaystyle\Omega^{i}_{I}\Omega^{j}_{J}\Omega^{-1J}_{1}=\Omega^{i}_{I}\delta_{j1}=0,\text{
where }j=2...N,$
$\displaystyle\Omega^{i}_{I}\Omega^{j}_{J}\Omega^{-1I}_{1}=\Omega^{i}_{I}\delta_{i1}=0,\text{
where }i=2...N,$
$\displaystyle\Omega^{i}_{I}\Omega^{j}_{J}\Omega^{-1I}_{1}\Omega^{-1J}_{1}=\delta_{i1}\delta_{j1}=0.$
(3.3)

The collective fields strength is generalized by introducing other potential
fields associations [8],

$\displaystyle s_{\mu\nu}=s_{IJ}G^{I}_{\mu}G^{J}_{\nu},\quad
s_{IJ}=\sum^{N}_{i,j=2}\Omega_{iI}\Omega_{jJ},$ (3.4) $\displaystyle
t_{\mu\nu}=t_{IJ}G^{I}_{\mu}G^{J}_{\nu},\quad
t_{IJ}=\sum^{N}_{K,L=1}\gamma_{KI}\gamma_{LJ},$ (3.5) $\displaystyle
u_{\mu\nu}=u_{IJ}G^{I}_{\mu}G^{J}_{\nu},\quad
u_{IJ}=\sum_{i,K}\Omega_{iI}\gamma_{KJ}$ (3.6)

which produces the generic gauge invariant collective field,

$\displaystyle e_{\mu\nu}=z_{\mu\nu}+t_{\mu\nu}+s_{\mu\nu}+u_{\mu\nu}$ (3.7)

where

$\displaystyle e_{\mu\nu}=e_{IJ}G^{I}_{\mu}G^{J}_{\nu}$ (3.8)

with

$\displaystyle e_{IJ}=\gamma_{IJ}+s_{IJ}+t_{IJ}+u_{IJ}$ (3.9)

where

$\displaystyle
e_{IJ}=\sum^{N}_{K,L=1}\sum^{N}_{\overset{i,j}{k,l}=2}\left(1+\gamma_{ij}+\gamma_{kj}\Omega_{kK}\right)\Omega_{iI}\Omega_{jJ}+$
$\displaystyle+\gamma_{ki}\gamma_{lj}\Omega_{kK}\Omega_{lL}\Omega_{iI}\Omega_{jJ}.$
(3.10)

Thus, the abelian fields set constructs the generic antisymmetric gauge
invariant tensor

$\displaystyle Z_{[\mu\nu]}=a_{I}G^{I}_{\mu\nu}+e_{[\mu\nu]}$ (3.11)

where

$\displaystyle a_{I}=d\Omega_{1I}+\alpha_{i}\Omega_{iI}$ (3.12)

and

$Z^{{}^{\prime}}_{[\mu\nu]}=Z_{[\mu\nu]}.$ (3.13)

B. Symmetric sector

The constructor basis produces the generic gauge invariant field strength

$Z_{(\mu\nu)}=\beta_{i}S^{i}_{\mu\nu}+\rho_{i}g_{\mu\nu}S^{\alpha
i}_{\alpha}+\gamma_{(ij)}X^{i}_{\mu}X^{j}_{\nu}+\tau_{(ij)}g_{\mu\nu}X^{i}_{\alpha}X^{\alpha
j}$ (3.14)

where

$\displaystyle
S^{i}_{\mu\nu}=\partial_{\mu}X^{i}_{\nu}+\partial_{\nu}X^{i}_{\mu}=\Omega^{i}_{I}S^{I}_{\mu\nu}$
(3.15)

and

$S^{\alpha i}_{\alpha}=\Omega^{i}_{I}S_{\alpha}^{\alpha I}$ (3.16)

The corresponding collective symmetric field strength is

$\displaystyle z_{(\mu\nu)}\equiv{\gamma_{(ij)}X_{\mu}^{i}X_{\nu}^{j}}$
$\displaystyle=$ $\displaystyle\gamma_{(IJ)}G_{\mu I}G_{\nu J}$ (3.17)

Similarly,

$\omega^{\alpha}_{\alpha}\equiv\tau_{(ij)}X^{i}_{\alpha}X^{\alpha
j}=\tau_{(IJ)}G_{\alpha}^{I}G^{\alpha J}$ (3.18)

Generalizing the symmetric collective field strenght,

$\displaystyle
e_{(\mu\nu)}=z_{(\mu\nu)}+s_{(\mu\nu)}+t_{(\mu\nu)}+u_{(\mu\nu)}$ (3.19)

where

$\displaystyle e_{(\mu\nu)}=e_{(IJ)}G_{\mu I}G_{\nu J}$ (3.20)

Substituting at eq. (3.14), in terms of physical fields, one gets the
following gauge invariant symmetric tensor

$Z_{(\mu\nu)}={\beta_{I}}S^{I}_{\mu\nu}+{\rho_{I}}g_{\mu\nu}S_{\alpha}^{\alpha
I}+\mathbf{e}_{(\mu\nu)}+g_{\mu\nu}\omega^{\alpha}_{\alpha}$ (3.21)

where

$\displaystyle S^{I}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}G^{I}_{\nu}+\partial_{\nu}G^{I}_{\mu}$ (3.22)
$\displaystyle\mathbf{e}_{(\mu\nu)}$ $\displaystyle=$
$\displaystyle\mathbf{e}_{(IJ)}G^{I}_{\mu}G^{J}_{\nu}$ (3.23)
$\displaystyle\omega_{(\mu\nu)}$ $\displaystyle=$
$\displaystyle\tau_{(IJ)}G^{I}_{\mu}G^{J}_{\nu}$ (3.24)

with the coefficients

$\displaystyle\beta_{I}=\beta_{i}\Omega^{i}_{I}\text{ ;
}\rho_{I}=\rho_{i}\Omega^{i}_{I}$ (3.26)
$\displaystyle\mathbf{e}_{(IJ)}=\gamma_{(IJ)}+s_{(IJ)}+t_{(IJ)}+u_{(IJ)}$
(3.27) $\displaystyle\tau_{(IJ)}=\tau_{(ij)}\Omega^{i}_{I}\Omega^{j}_{J}.$
(3.28)

###  3.2 Constructivist Abelian Lagrangian

A whole quanta process is expected from fields strengths through a
constructivist lagrangian. There is an abelian grouping fields to be explored.
A physics assembled by whole gauge symmetry. Introduce on fields
agglomerations and stipulate their corresponding quanta. Derive properties
inserted in a whole.

The abelian constructive lagrangian will be studied in terms of fields
strenghts. It yields the following expression

$\mathscr{L}=Z_{[\mu\nu]}Z^{[\mu\nu]}+Z_{(\mu\nu)}Z^{(\mu\nu)}+\eta
Z_{[\mu\nu]}\tilde{Z}^{[\mu\nu]}+m^{2}_{II}G^{I}_{\mu}G^{\mu
I}+\xi_{IJ}(\partial_{\mu}G^{\mu I})(\partial_{\nu}G^{\nu J})$ (3.29)

including the dual tensor
$\tilde{Z}^{[\mu\nu]}=\epsilon_{\mu\nu\rho\sigma}Z^{[\rho\sigma]}$, with the
term $Z_{[\mu\nu]}\tilde{Z}^{[\mu\nu]}$, which is not exactly a total
derivative. It gives,

$\displaystyle
Z_{[\mu\nu]}\tilde{Z}^{[\mu\nu]}=\epsilon^{\mu\nu\rho\sigma}\left\\{2a_{I}\left[\partial_{\mu}(G^{I}_{\nu}Z_{[\rho\sigma]})-G_{\nu}^{I}\partial_{\mu}z_{[\rho\sigma]}\right]+z_{[\mu\nu]}Z_{[\rho\sigma]}\right\\}$
(3.30)

Eq. (3.30) is called as semitopological lagrangian [9].

Splitting the fields strengths, one gets the following non-linear abelian
Lagrangian, in terms of physical fields,

$\displaystyle\mathscr{L}(G)=a_{I}a_{J}G_{\mu\nu I}G^{\mu\nu J}+a_{I}G_{\mu\nu
I}\mathbf{e}^{[\mu\nu]}+\mathbf{e}_{[\mu\nu]}\mathbf{e}^{[\mu\nu]}$
$\displaystyle+\beta_{I}\beta_{J}S_{\mu\nu I}S^{\mu\nu
J}+2\beta_{I}\rho_{I}S_{\alpha I}^{\alpha}S^{\beta
I}_{\beta}+2\beta_{I}S_{\mu\nu}^{I}\mathbf{e}^{\mu\nu}$
$\displaystyle+2\beta_{I}S^{\alpha
I}_{\alpha}\omega_{\beta}^{\beta}+17\rho_{I}\rho_{J}S^{\alpha
I}_{\alpha}S^{\beta J}_{\beta}+2\rho_{I}S^{\alpha
I}_{\alpha}\mathbf{e}^{\alpha}_{\alpha}$ $\displaystyle 22\rho_{I}S^{\alpha
I}_{\alpha}\mathbf{\omega}^{\alpha}_{\alpha}+16\omega_{\alpha}^{\alpha}\omega_{\beta}^{\beta}+\partial_{\mu}\\{2\epsilon^{\mu\nu\rho\sigma}a_{I}G_{\nu}^{I}Z_{[\rho\sigma]}\\}+$
$\displaystyle
a_{I}\mathbf{e}_{[KL]}\epsilon^{\mu\nu\rho\sigma}G_{\mu}^{I}G_{\rho}^{K}\partial_{\nu}G_{\sigma}^{L}+2a_{K}\mathbf{e}_{[IJ]}\epsilon^{\mu\nu\rho\sigma}G_{\mu}^{I}G_{\nu}^{J}\partial_{\rho}G_{\sigma}^{K}$
$\displaystyle\mathbf{e}_{[IJ]}\mathbf{e}_{[KL]}\epsilon^{\mu\nu\rho\sigma}G_{\mu}^{I}G_{\nu}^{J}G_{\rho}^{K}G_{\sigma}^{L}$
(3.31)

Separating in pieces

$\displaystyle\mathscr{L}(G)=\mathscr{L}_{K}+\mathscr{L}_{I}+\mathscr{L}_{st}+\mathscr{L}_{M}+\mathscr{L}_{GF}$
(3.32)

The kinetic term is given by

$\mathscr{L}_{K}=a_{I}a_{J}G_{\mu\nu I}G^{\mu\nu
J}+\beta_{I}\beta_{J}S_{\mu\nu I}S^{\mu\nu J}+2\beta_{I}\rho_{I}S_{\alpha
I}^{\alpha}S^{\beta I}_{\beta}$ (3.33)

The interaction sector contains the abelian trilinear terms,

$\mathscr{L}^{3}_{I}=a_{IJK}\partial^{\mu}G^{\nu I}G^{J}_{\mu}G^{\nu
K}+4b_{IJK}\partial_{\mu}G^{I\mu}G^{J}_{\nu}G^{k\nu}$ (3.34)

or

$\mathscr{L}^{3}_{I}=(a_{IJK}+b_{JIK}+b_{KJI})\partial^{\mu}G^{\nu
I}G_{\mu}^{J}G_{\nu}^{K}$ (3.35)

and abelian four linear terms,

$\mathscr{L}^{4}_{I}=a_{IJKL}G_{\mu}^{I}G_{\nu}^{J}G^{\mu K}G^{\nu
L}+b_{IJKL}G_{\mu}^{I}G^{J\mu}G^{K}_{\nu}G^{L\nu}$ (3.36)

The semitopological term also receives tri and quadrilinear contribuitions,

$\displaystyle\mathscr{L}^{3}_{st}=a_{I}\mathbf{e}_{[KL]}\epsilon^{\mu\nu\rho\sigma}G_{\mu}^{I}G^{K}_{\rho}\partial_{\nu}G_{\sigma}^{L}+2a_{K}\mathbf{e}_{[IJ]}\epsilon^{\mu\nu\rho\sigma}G_{\mu}^{I}G^{J}_{\nu}\partial_{\rho}G_{\sigma}^{L}$
(3.37)

and

$\displaystyle\mathscr{L}^{4}_{st}=\mathbf{e}_{[IJ]}\mathbf{e}_{[KL]}\epsilon^{\mu\nu\rho\sigma}G_{\mu}^{I}G_{\nu}^{J}G_{\rho}^{K}G_{\sigma}^{L}$
(3.38)

In order to get a better underdstanding on the Lagrangian spectroscopy, one
should make a coefficients decomposition through Young Tableaux. It gives

$a_{IJ}=a_{[IJ]}+a_{(IJ)}$

where

$\displaystyle a_{(IJ)}=\tilde{a}_{IJ}+\frac{1}{N}\delta_{IJ}tr(a)\textbf{,
}tr(\tilde{a}_{IJ})=0,$ (3.39)

and

$\displaystyle
a_{IJK}=a_{[IJK]}+a_{[I(J)K]}+a_{[I(J)K]}+a_{[I(J)K]}+a_{(IJK)}$ (3.40)

where

$\displaystyle
a_{(IJK)}=\tilde{a}_{(IJK)}+\frac{1}{N+2}(\delta_{IJ}a_{K}+\delta_{JK}a_{I}+\delta_{IK}a_{J}),$
(3.41)

and

$\displaystyle a_{IJKL}$
$\displaystyle=a_{(IJKL)}+a_{(IJ[K)L]}+a_{(I[J)(K]L)}+a_{[IJ(K]L)}+a_{[IJKL]},$
(3.42)

as

$\displaystyle
a_{IJKL}=a_{[IJKL]}+a_{[I(J)KL]}+a_{[I(J)KL]}+a_{[I(J)KL]}+a_{[IJKL]}$ (3.43)

where

$\displaystyle
a_{(IJKL)}=\tilde{a}_{(IJKL)}+(\delta_{IJ}a_{KL}+\delta_{KL}a_{IJ}+\delta_{IK}a_{JL}+\delta_{JL}a_{IK}),$
(3.44)

where

$\displaystyle a_{(IJ[K)L]}=\tilde{a}_{(IJ[K)L]}+\delta_{IJ}a_{KL},$ (3.45)

$\displaystyle a_{(I[J)(K]L)}=\tilde{a}_{(I[J)(K]L)}+\delta_{IL}a_{JK},$
(3.46)

$\displaystyle a_{[IJ(K]L)}=\tilde{a}_{[IJ(K]L)}+\delta_{IL}a_{JK}$ (3.47)

Notice that eq.(3.33-3.36) shows a soft nonlinearity. Given than the
corresponding $\tilde{a}_{IJK},\tilde{b}_{IJK}...$ can be taken zero
separately without breaking gauge simmetry, it is possible, for instance, to
reduce the interaction term for $\tilde{a}_{IJK}$.

The fields set composes masses without requiring Higgs mechanism [10]. A mass
term more near to energy and Anderson prescription [11] than considering
degress of freedon transference as at symmetry breaking. It gives,

$\displaystyle\mathscr{L}_{m}$ $\displaystyle=$ $\displaystyle
m^{2}_{ij}X^{i}_{\mu}X^{\mu j}={m^{2}_{II}}G^{I}_{\mu}G^{I}_{\nu}$

or

$\mathscr{L}_{m}(G)=m^{2}_{II}G^{I}_{\mu}G^{\mu I}$ (3.48)

Given the existence of just one gauge parameter, there is just one gauge
fixing. It is written as a linear combination of the fields $G_{\mu I}$. From
Appendix A, it yields

$\mathscr{L}_{GF}=\xi_{IJ}\partial_{\alpha}G^{\alpha
I}\cdot\partial_{\beta}G^{\beta J}$ (3.49)

Thus, the abelian constructivist lagrangian is constituted. Eq. (3.32)
contains independent gauge invariants pieces. Their physical relevance are
terms providing own physics. Eqs. (3.33-3.38) develop isolated physical
conglomerates. Each of them works as a source to do nonvirtual physics.

##  4 Physical realities and potentialities

Reality in gauge theory is given by gauge scalars. Diverse from Yang-Mills
case [12], the abelian constructivist lagrangian is separated in different
sectors where each one is gauge invariant, as eq. (3.32) shows. It generates
diverse subunits of reality. They contain their own physics with
potentialities to be studied.

The constructivist abelian lagrangian is not monolithic. Usually gauge
theories develop lagrangians which are gauge invariant as a whole, i.e, their
separated pieces are virtual. As a new aspect, the fields set physics is
constituted by blocks where each of them englobes own opportunities. Each one
is individually able to do real physics. We list below these independent
structures. Gauge scalars subunits as source for interdependent
electromagnetic energy. They are

$\displaystyle G_{\mu\nu I}G^{\mu\nu I},$
$\displaystyle\epsilon_{\mu\nu\lambda\sigma}G^{\mu\nu I}G^{\lambda\rho I},$
$\displaystyle a_{IJ}S_{\mu\nu}^{I}S^{\mu\nu J},$ $\displaystyle
a_{IJ}\epsilon_{\mu\nu\lambda\rho}S^{\mu\nu I}S^{\lambda\rho J},$
$\displaystyle a_{J}G_{\mu\nu}^{I}S^{\mu\nu J}$ $\displaystyle
a_{J}\epsilon_{\mu\nu\lambda\rho}G^{\mu\nu I}S^{\lambda\rho J}$ $\displaystyle
G_{\mu\nu}^{I}\mathbf{e}^{[\mu\nu]}$
$\displaystyle\epsilon_{\mu\nu\lambda\rho}F^{\mu\nu
I}\mathbf{e}^{[\lambda\rho]},$
$\displaystyle\mathbf{e}_{[\mu\nu]}\mathbf{e}^{[\mu\nu]},$
$\displaystyle\epsilon_{\mu\nu\lambda\rho}\mathbf{e}^{[\mu\nu]}\mathbf{e}^{[\lambda\rho]},$
$\displaystyle a_{I}S_{\mu\nu}^{I}\mathbf{e}^{(\mu\nu)},$ $\displaystyle
a_{I}\epsilon_{\mu\nu\lambda\rho}S^{¨\mu\nu I}\mathbf{e}^{(\lambda\rho)},$
$\displaystyle\mathbf{e}_{(\mu\nu)}\mathbf{e}^{(\mu\nu)},$ $\displaystyle
a_{I}\epsilon_{\mu\nu\lambda\rho}\mathbf{e}^{(\mu\nu)}\mathbf{e}^{(\lambda\rho)},$
(4.1)

Nevertheless in terms of quanta and interactions it is more appropiate to
write eq. (3.29) in terms of eqs. (3.32). They will split on fields
associations potentialities. Take as example a three linear piece as
$a_{IJK}\partial^{\mu}G^{\nu I}G_{\mu}^{J}G^{K}_{\nu}$. It will produce its
own real physics. Their associations will contribute for a real trilinear
nonlinear term and for a nonvirtual vertice set of Feynman graphs.

The constructivist lagrangian is rewritten at table below. Diverse scalars
terms are catalogued. Each one contains parameters in front of it. They are
called as abelian symmetry free coefficients. Originated at Appendix A and
transcripted to eq. ( 3.29) they can take any value without violating the
gauge invariance. The number of free coefficients associated to each gauge
invariant term constitute the so-called volume of circumstances.

Table 4.1: Relationship between scalar gauge and Nº of free coefficients Scalar | Free Coefficients | Nº of Free Coefficients
---|---|---
$G_{\mu}^{I}\Box\theta^{\mu\nu}G_{\nu}^{J}$ | $d^{2}\Omega_{1I}\Omega_{1J}+2d\alpha_{i}\Omega_{1I}\Omega^{i}_{J}+\Omega_{J}^{j}$ | $\frac{4N(N+1)+N^{2}(N-1)}{8}\cdot$
| $+2(\alpha_{i}\alpha_{j}\beta_{i}\beta_{j})\Omega_{I}^{i}$ | $\cdot[N^{4}+5N^{2}+2(N^{2}-(N-1))]$
$G_{\mu}^{I}\Box\omega_{\mu\nu}G^{L\nu}$ | $(\beta_{i}\beta_{j}+2\rho_{i}\beta_{j}+\rho_{i}\rho_{j})\Omega_{I}^{i}\Omega_{J}^{j}$ | $\frac{N^{3}(N^{2}-1)}{16}[N^{2}(N+1)+2]$
$G_{\mu}^{I}G^{I\mu}$ | $m^{2}_{ij}\Omega_{I}^{i}\Omega_{I}^{j}$ | $\frac{N^{3}(N-1)^{3}}{2}$
$\partial^{\mu}G^{\nu I}G^{J}_{\mu}G^{\nu K}$ | $d\mathbf{e}_{[ij]}\Omega_{1i}\Omega^{i}_{J}\Omega^{j}_{K}+$ | $N^{3}(N-1)^{3}\cdot$
| $+(\alpha_{i}\mathbf{e}_{[kj]}+\beta_{i}\mathbf{e}_{[kj]})\Omega_{I}^{i}\Omega_{J}^{j}\Omega_{K}^{k}$ | $\cdot[(N-2)+N(N-1)]$
$\partial_{\mu}G^{\mu I}G^{J}_{\nu}G^{K\nu}$ | $(\beta_{i}\tau_{(kj)}+\rho_{i}\gamma_{(kj)}+\rho_{i}\tau_{(kj)})\Omega_{I}^{i}\Omega_{J}^{j}\Omega_{K}^{k}$ | $\frac{3N^{9}(N-1)^{5}}{2}$
$G_{\mu}^{I}G^{J}_{\nu}G^{\mu K}G^{\nu L}$ | $(\mathbf{e}_{(ij)}\mathbf{e}_{(kl)}+\mathbf{e}_{(ij)}\mathbf{e}_{(kl)})\Omega_{I}^{i}\Omega_{J}^{j}\Omega_{K}^{k}\Omega_{L}^{l}$ | $\frac{N^{4}(N-1)^{4}}{2}[N(N-1)+3]$
$G_{\mu}^{I}G^{J\mu}G^{K}_{\nu}G^{L\nu}$ | $(\mathbf{e}_{(ij)}\mathbf{e}_{(kl)}+\tau_{(ij)}\tau_{(kl)})\Omega_{I}^{i}\Omega_{J}^{j}\Omega_{K}^{k}\Omega_{L}^{l}$ | $\frac{N^{4}(N-1)^{4}}{8}[3N(N-1)+2]$

Table (4.1) expresses the reality of the abelian constructivist lagrangian
subunits. It catalogues each physical term potentiality. Eq. (3.29) is not a
rigid physics. Their fields associations constitute subunits of reality,
independent terms due to their gauge invariances and interdependent by sharing
common fields. Their potentiality means the volume of circumstances
propitiated by gauge symmetry to each gauge scalar term. So, given by the
number of free coefficients associated to every scalar term, a physics with
free will is introduced. A new way to express energy.

Summarizing, the constructivist lagrangian contain terms with own realities
and potentialities. And so, the phenomena may be analyzed by different
lagrangian pieces where each one contains its own reality and potentiality.
Physical entities as energy, mass, coupling constants, conservation laws,
Feynman graphs will be depending on each of these separated lagrangian blocks.

##  5 Associative physics

Constructivist lagrangian establishes physics under fields set. The grouping
physics. A significance under associativity. The quantum is no longer an
enterprise from itself, but of a set. A perspective where the corresponding
properties will be ruled under environment.

Physics laws derived from fields environment is a study to be considered. The
fields set $\\{G_{\mu I}\\}$ does not consider ultimate constituents as
definitive. They produce an environmental physics. Consequently, the search
for elementarity is not only through reductionist subunits, but also, by
inserting parts in the whole. Instead of studying an atomist gauge theory like
the Yang-Mills [13], whose objective is to reveal the fundamental
interactions, it turns out to consider an integral physics A constructive
lagrangian where particles are described by a third quantum depending on
fields set.

Nature is more than playing with ultimate constituents. It also processes the
associativity through conglomerate’s determination. The whole-part duality. A
complementarity relating the quantum and to the whole. Nature also works as a
collective manifestation. A quantum system physics to be studied beyond four
interactions.

###  5.1 Ultimate constituents and two quantum formations

Subunit and environment are two nature constructions. The atomist and grouping
behaviours. Physics followed the first, while biology found the second. The
physics postulate has been that subunit precedes wholeness. It concerns on how
phenomena are atomistically constituted. Consider on Big Bang and the
formation of particles. Let aside the existence of conglomerates.

Nevertheless nature provides groupings. A set physics supported by general
principles as associativity, confinement, complexity. Introducing a
relationship between the part and the whole. For instance, particles getting
together at eightfoldway and atoms at periodic table. They are relating how
systems are constituted from constituents. Quarks and protons creating
groupings. Showing that, subunits are not only working for composing
particles, as gathering them into conglomerates.

Thus, there are two ways for expressing the nature of things. Possibilites on
quantum formation. Reductionistic and antireductionistic. Procedures from
ultimate constituents. On one hand subunit, another set. The first one, based
on building blocks (as quarks and leptons); other, on associations
(conglomerates). Two performances for expressing the quantum.

The 1940s and 50s decades led physics to witness the proliferation of
elementary particles. At that time, it was considered as a crisis in physics
such diversity of particles. However, the discovery of the omega minus
particle in 1964, definitively, came to show that these particles were not
isolated, but associated. There was an eightfoldway grouping physics promoted
by ultimate constituents called quarks.

Thus, at 1960s physics encountered a relationship between the greek atom and
the notion of set. After Planck and quantum mechanics, came out the existence
of a new type of quantum. The eightfoldway showed that, proton, neutron and so
forth are not isolated particles. They are formed by quarks, and also,
associated by quarks. The Quark Model provides particles and patterns of
particles. Showing that, nature contains ultimate subunits as quarks, and,
develops conglomerates unities as eightfoldway representations. Consequently,
instead of isolated particles, nature produces particles in associations.

A third quantum physics emerges. Particles physics is not a zoo of dissociated
particles, but, under groupings associations. Eightfoldway shows that the
ultimate constituents produce particles and environment of particles.
Particles and systems are created. Therefore, beyond the greek atm there
another perspective to be token. Ultimate constituents are formatting the
environmental physics.

There is an unexplored physics. The relationship between ultimate constituent
and conglomerate. A new elementarity has to be proposed. The quantum related
to the system which is involved. Composed by ultimate constituents and
respective grouping physics this quantum is more than a property of itself.
The part in the whole turns up as the third quantum configuration. In
reference to biology, this whole quantum will be denominated as variety. The
elementarity constitued by the fields set environment.

Nevertheless, particle physics has always been impregnated by atomism. The
Quark Model subproducts, protons and neutrons, appeared constituted by quarks
but as dissociated particles. Physics just study their strong interactions.
Something is lost. Incorporate their associative physics. Consider the quantum
dependence on ultimate constituents and whole associations.

The Quark Model atomist philosophy did not registered the conglomerated
physics. The search for the greek atom over the last 2500 years has diverted
physics from set notion. As result, current physics interprets nature under
interactions and not associations. Subtracting the meaning that, the universe
is an association derived from ultimate constituents, and, as a natural order,
providing the development of patterns.

Physics has to include the existence of variety. A new elementarity under the
set action. The quantum associated to a fields environment. A system with an
own physics to be expressed generating distinct energy packets. The proton
identified as an eightfoldway variety. There are new physical properties to be
investigated under the conglomerate view.

Physisc has to express the variety. Based on associativity, eq. (3.32)
definies a new energy packet. An antireductionist quantum is constituted. A
definition where physics contains another perspective for describing the
phenomena. The relationship between ultimate constituents and conglomerates
cause another type of procedure. Near to biology, it will identified as
constructivist physics.

###  5.2 Constructivist physics

A constructivist physics is identified. A physics beyond atomism. Since 1900
with Planck granularity and 1927 with quantum field theory, the quantum world
became the lecture of greek atm. However, the atomism is not the last quantum
performance. The principles of associativity, confinement and complexity are
requiring the existence of a grouping physics. Physics under the set action.
Consequently, the part in the whole generates environmental properties to be
explored.

A third quantum should exist. A comprehension about the quantum behaviour
within of the whole proceed. Consider set physics constituted by Lorentz group
and gauge symmetry. While the usual reductionist methodology defines particles
interacting with each other mediated by gauge bosons, the antireductionist way
will relate them through a systemic behaviour. Consider each Lorentz group
representation associated with a fields set. For example, in the vector case,
the $A_{\mu}$ field representation $(\frac{1}{2},\frac{1}{2})$ is extended to
the fields set $\\{A_{\mu I}\\}$ [14].

Gauge symmetry is also reinterpreted. Instead of establishing just fundamental
interactions, its new physical proposal is to assemble fields. Establish
nature in associations through antireductionist gauge symmetry. An
environmental physics emerges. Quanta insert in the whole is what the
antireductionist gauge symmetry provides. Given eq. (3.32), the fields set
$\\{G_{\mu I}\\}$ provides $(2N-1)$ interconnected quanta to be compiled [15].

Thus, the constructivist physics interprets nature as a whole atom. Establish
a systemic physics by groupings the Lorentz transformations $A_{\mu
I}=\Lambda_{\mu}^{\nu}A_{\mu I}$ and the gauge symmetry by eq. (2.6). A whole
behaviour to be analyzed. Instead of looking for isolate subunits address the
principle of wholeness. Elementary particles and respective quantum numbers
constituted from the whole which are envolved. Whole-part duality and quanta
network constitute a new physics.

The content of grouping physics provokes a new understanding. A performance
where the quantum is not expressed isolatedely. A new construction from
ultimate constituents is identified. Constructivist lagrangian provides a
systemic quantum packet. The whole quantum associativity. Ultimate
constituents forming the whole-part duality are causing an environmental
physics.

###  5.3 Quantum system: environment, variety and functionality

A perspective not well exploited by quantum field theory is that physics
includes the grouping manifestation. An integral activity moved not only by
fundamental interactions, but firstly, under associativity. Dirac equation
relating electrons and positrons and eightfoldway are exemplifying on this
associative behaviour. They make physics join to biology on nature
constructivism. Specie in biology and whole quantum in physics share the
meaning of environment. Based on that, both should be subjects of nature
associative behaviour.

Biology managed to privilege environment, variety and functionality over
ultimate constituents and interactions. Biology takes variety from
environment. Darwin, through the differentiation of species, found the
evolutionary timeline. Introduced the notion of adaptive variety. However,
given that biology and physics are natural sciences, this line of thought
should be extended into physics. As biology, physics working out as a subject
proposing environment under fields set. Consider that physical bodies, as
particles, contain a behaviour major than just interactions.

The interconnection between physics and biology should be analysed. Darwin in
1859 understood the relationship between species (varieties) and environment
[16]. Similarly, Gell-Mann and Ne’eman in 1961 joined particles in
eightfoldway groupings [17]. Later in 1937, Dobzhansky introduced the
relationship between genes, environment and varieties [18]. Similarly, in 1964
Gell-Mann and Zweig introduced quarks at group representations 3, 3* and
developing conglomerates through representations 8,10,15,… [19]. Demonstrating
that, physics also contains a relationship between ultimate constituents and
grouping. Formatting particles as varieties. Therefore, another consequence
from quarks is working as subunits constituting varieties.

Thus, a third phase on quantum story is the variety physics. After granular
and wave-particle behaviours the performance is on the quantum association.
The Quark Model generating interlinked composite particles through
conglomerates brings a new perfomance. Ultimate constituents forming a set
action and introducing as whole quantum. It shows unlike restricting physics
to isolated reductionist processes, one gets that, environment, variety and
functionalities are establishing a new possibility to do physics. A third type
of energy packet processed with antireductionist properties.

A new situation is stated. The association physics enlarges phenomena beyond
the interactive atomism. The relationship between ultimate constituent and
environmental varieties generates a next perspective. Eq. (2.6) symmetry
introduces a fields set and eq. (3.23) a set action. Interaction is enlarged
to interrelationship. Grouping arrangements producing antireductionist
properties beyond isolated particles. The atomist four interactions are
prolonged to antireductionist quantum functionalities.

###  5.4 Grouping properties

A new physics appears. The atomism is extended to constructivism through gauge
symmetry. A third quantum is established under a whole field theory. Inserted
in the whole adds new physicalities. The fields set produces relationships
beyond the atomist perspective. Maintaining the original reductionism
symmetry, and so, preserving the four fundamental forces a quantum system is
constituted. New physical relationships are performed. Antireductionist
properties depending on conglomerate are produced.

The physics of a single particle is extended for a system. A field $G_{\mu I}$
is incorporated at $\\{G_{\mu I}\\}$. A quantum system is established. An
extension where the single particle physics is preserved. The covariant
derivative is preserved, and so, the original atomist interaction is not
modified. But, new relationships based on interlinked fields are developed.
The aggregation enlarges physical laws on interaction, induction and
connection. That is the theme to be explored.

Three types of physics emerge from the set action. Associativity, set
determinism, complexity. Through associativity appears set, diversity,
interdependence, nonlinearity, chance. The resulting conglomerate unity
introduces a set transformation physics. A set determinism under directive and
circumstance, growth, evolution, emergence. Features which will constitute a
multitude acting together. Promoting complexity.

Associativity introduces the grouping behaviour. The antireductionist gauge
symmetry provides the whole quantum system under difference and regularity.
Whole-quantum duality and quanta network are constituted. Varieties under
diversity. Kinematically, an interconnectivity identified by quantum numbers.
The respective quanta are not independent of each other, but interrelated by
common quantum numbers. Dynamically, potential fields interdependences are
expressed through equations of motion, Faraday laws, conservation laws,
Noether laws, constructive Lorentz force.

Thus, before considering the interaction between particles, we must observe
their interrelationships. For example, when assuming the photon and $Z_{0}$
inserted in the same set, it turns out that, before supposing them as
independent carriers of the electromagnetic and weak interactions, we must
start from the fact that they are associated particles. Making
electromagnetism and weak interaction interrelated.

Nonlinearity and chance are also properties beyond the quantum atomism. The
first one, establishes a self organizing system. A dynamics without depending
on external sources. The concept of chance is derived from the potentialities
that each gauge scalar term produce. It yields the physics of opportunities.
The quantum is no more a fixed structure, but an entity which properties are
under chance. Decisions depending on each lagrangian gauge invariant term.
Even at abelian case, tri-and-quadrilinear chances are expressed at
eqs.(3.34-3.38). Nonvirtual opportunities quantified by the volume of
circumstances.

Thus, a quantum system is formed. These five environmental properties takes
physics to a new behaviour. The whole unity expressed by the symmetry of
difference leads physics to evolve in time through set transformations.
Introducing growth, evolution, emergence, complexity. Growth from potentiality
and nonlinearity. Evolution means that nature allows on quantum individuation
through growth with chance. Emergence is evolution under circumstance.
Complexity is everything entrelaced.

##  6 Variety physics

Historically, physics has being an atomist subject looking for the greek atm.
Nevertheless conglomerates lead physics to the whole meaning. Nature develops
systems and patterns resulting another type of individuation. The so-called
third quantum. The quantum inserted in the whole.

Physics is not only the search for smaller and smaller things. Something
together happens. Quantum systems states physics. Organizations follow under
the antireductionist principle. Physical laws are more than ultimate
constituents interactions. There is the diverse and how nature produces
itself. Phase transitions show on laws depending on collective groupings.

The constructivism introduces a set action physics to be investigated. The
variety physics. Conglomerates form units generating own physics. New laws are
constituted. A protagonism beyond reductionist subunits. Diversely from
atomist physics based on interaction matter contains different levels of
association to be considered. A systemic behaviour under interrelationship.

Nature performs varieties. A physics will environmental behaviour. Entities
acting in grouping provide a second type of individuation. For instance,
hydrogen and oxygen gases establishing water are showing that, isolated
structures are diverse from grouping behaviour. Water is a variety derived
from basic constituents with another properties.

Physics is more than greek atms. Varieties introduces a way for understanding
nature oppositedely than to break into little pieces. Differently from usual
reductionist subunits, objects together are more. The novelty is that ultimate
constituents does not work just as matryoshkas. Varieties in environment are
constituing another type of elementarity. Nature also contains a determination
derived from the fields set elementarities to be studied.

Varieties happens in all nature places. The elementarity related to a
conglomerate. Proposing ultimate constituents not as lego but as parts in
environment. Nature works out with a whole elementarity derived from
associations. Physics and biology show conglomerates by composing matter and
species. Generating differentiated structures with own character. Biology with
genes individualizing species. The physics counter part should be preons and
strings individualizing objects as varieties at eightfoldway, molecules and so
forth.

A qualitative change happens. The existence of antireductionist laws due to
varieties. A collective physics where such elementarities and not ultimate
constituents are responsible for another nature process. A systemic behaviour
on set, diversity, interdependence, nonlinear, chance is expressed. A set
dynamics is accomplished. The new physicality is that the whole unity is that
evolves in time. Grouping physics under set transformations generates the
variety dynamics. Determining on nature growth, evolution, emergence,
complexity.

Evolution appears as a branch of the physics of varieties. Nature evolution is
processed not through ultimate constituents but under the environment
elementarities. The prescription of evolution was given by darwin. A
determination under environment diversity, growth, chance, adaptaion. However,
current physics is a research restricted to determine bodies under space-time
dynamics. It does not face on mutation. Particles are not considered as corps
in transformations. The quantum physics at $20^{th}$ century was limited to
search for ultimate particles and their interactions. The relationship
expressed between ultimate constituent, environment, variety and evolution
still has to be constituted.

Complexity is another set consequence derived from the physics of varieties. A
process where properties are depending on context. It is not described by
fundamental subunits, but through cohesive sets. The reductionist approach
does not work anymore. A multitude of facts integrated in a set action has to
be considered. Quantum with complex synapses are expected.

Under this context, physics has to write down the mathematics of variety. New
systemic properties about the whole quantum have to be expressed analytically.
Gauge symmetry reinterpreted under antireducionist content propitiates a
formalism. By studying the abelian constructivist lagrangian, one gets the
existence of whole equations. A number of equations which have to be
interpreted in a certain way. The grouping physics and corresponding set
action express mathematically antireductionist properties as diversity,
nonlinearity, interdependence, chance. They will be leading to set
transformations, as evolution.

The variety mathematical expression is a study to follow. Eq. (3.32) contains
a formulation to be expanded. Its proposal is to look analytically for the set
action. Determine the relationship between the part and the whole. Describe a
quantum system through constructivist equations. Express mathematically the
presence of a third quantum.

##  7 Set physics

A set action is proposed by inserting a given field $G_{\mu I}$ in the fields
set $\\{G_{\mu I}\\}$. Eqs. (2.6) and (3.29) are composing a constructivist
physics based on an abelian antireductionist gauge symmetry. A third type of
quantum is derived. An energy packet with antireductionist properties is
composed.

A third quantum is being studied. It makes the atomist physics move for
grouping physics. Five grouping properties will promote a whole quantum in
network. They are set, diversity, interconnectivity, nonlinearity, chance.
Introducing whole-parts relationships expecting to be expressed mathematized
through the antireductionist gauge symmetry. A physics to be done.

The model contains the internal consistencies of locality, positive
Hamiltonian, renormalizability and definite unitarity [20]. Given that, one
can analyse the correspondent physical manifestations of such fields set
action. The presence of an association with health quantum numbers.

##  8 Diversity physics

The second grouping physics property is diversity. Analyse how far eq. (2.6)
contains a set action under diversity. A symmetry establishing a pluriformity
of diversities. Consider that, from a single gauge parameter, it is possible
to gather particles with different quantum numbers.

The thesis is that the set action provides a symmetry of diference. Consider
that the quanta inserted in the abelian fields set are processing differences.
The respective quantum numbers are multiple differentiated. They constitute a
quantum system where spin, mass, charges, discrete symmetries are under a
diversity. A quanta variety to be demonstrated at this section.

###  8.1 Spin diversity

####  8.1.1 Sectors of spin-1 e spin-O

Given the field $A_{\mu I}$ belong to the representation of the Lorentz group
$(j,k)=\left(\frac{1}{2},\frac{1}{2}\right)$, of the expression
$s=[j-k]...(j+k)$ we obtain the information that this representation contains
two spin-1 and spin-0. Therefore, it is deduced that the set$\\{A_{\mu I}\\}$
contains N-vector particles and $(N-1)$-scalars (one of which must be
eliminated by gauge fixing) [21].

For understanding the variety of spins predicted from Lorentz group, one
should analyze the Lagrangian. Rewriting L(G) in terms of its antisymmetric
and symmetric sectors, we have:

$\mathscr{L}(G)=\mathscr{L}_{A}+\mathscr{L}_{S}+\mathscr{L}_{st}$ (8.1)

where

$\mathscr{L}_{A}=b_{I}b_{J}G_{\mu\nu}^{I}G^{\mu\nu
J}+2b_{I}e_{[JK]}G_{\mu\nu}^{I}G^{\mu J}G^{\nu
K}+e_{[IJ]}e_{[KL]}G_{\mu}^{I}G_{\nu}^{J}G_{\mu K}G_{\nu L},$ (8.2)

$\displaystyle\mathscr{L}_{S}$ $\displaystyle=$
$\displaystyle(\beta_{I}\beta_{J})S_{\mu\nu}^{I}S^{\mu\nu
J}+(2\beta_{I}\rho_{J}+4\rho_{I}\rho_{J})S^{\mu I}_{\mu}S^{\nu
J}_{\nu}+(2\beta_{I}e_{(JK)})S_{\mu\nu}^{I}G^{\mu J}G^{\nu K}+$
$\displaystyle+$ $\displaystyle(2\beta_{I}e_{(JK)})S_{\mu\nu}^{I}G^{\mu
J}G^{\nu
K}+(2\beta_{I}\tau_{(JK)}+2\rho_{I}e_{(JK)}+8\rho_{I}\tau_{(JK)})S_{\mu}^{\mu
I}S_{\nu}^{J}S^{\nu K}+$ $\displaystyle+$
$\displaystyle(e_{(IK)}e_{(JL)}+2e_{(IJ)}\tau_{(KL)}+4\tau_{(IJ)}\tau_{(KL)})G_{\mu}^{I}G^{\mu
J}G_{\nu}^{K}G^{\nu L}$

and

$\mathscr{L}_{st}=\eta\epsilon_{\mu\nu\rho\sigma}(2b_{I}\mathbf{e}_{[JK]}G^{\mu\nu
I}G^{\rho J}G^{\sigma k}+\mathbf{e}_{[IJ]}\mathbf{e}_{[KL]}G^{\mu I}G^{\nu
J}G^{\rho K}G^{\sigma L})$ (8.3)

Nevertheless eq (8.1) show a model mixing spin-1 and spin-0. While $L_{A}$
carries spin-l, the symmetric and semi-topological sectors are mixing vectors
and scalars. Therefore, in order to obtain a spectroscopy chemically pure, we
must separate the kinetic term through transverse and longitudinal parts. The
spins sectors involved in $\mathscr{L}(G)$ as

$\mathscr{L}(G)=\mathscr{L}_{T}+\mathscr{L}_{L}+\mathscr{L}_{I}$ (8.4)

The spin-1 sector given by

$\mathscr{L}_{T}=G_{\mu\nu I}G^{\mu\nu I}$ (8.5)

The spin-0 sector by

$L_{L}=\beta_{I}\beta_{J}S_{\mu}^{\mu I}S_{\nu}^{\nu
J}+\beta_{I}S_{\mu\nu}^{I}(\rho_{J}g^{\mu\nu}S_{\rho}^{\rho
J})+\rho_{I}g_{\mu\nu}S_{\rho}^{\rho I}(\beta_{J}S^{\mu\nu
J}+\rho_{J}g^{\mu\nu}S_{\sigma}^{\sigma J})$ (8.6)

and the interaction term as

$\mathscr{L}_{I}=\mathscr{L}_{I}^{A}+\mathscr{L}_{I}^{S}+\mathscr{L}_{I}^{ST}$
(8.7)

where

$\displaystyle\mathscr{L}_{I}^{A}=(b_{I}G_{\mu\nu}^{I}+Z_{[\mu\nu]})\mathbf{e}_{[\mu\nu]}$
(8.8)
$\displaystyle\mathscr{L}_{I}^{S}=(\beta_{I}S_{\mu\nu}^{I}+\rho_{I}g_{\mu\nu}S_{\rho}^{\rho
I}+Z_{(\mu\nu)})(\mathbf{e}^{(\mu\nu)}+g^{\mu\nu}\omega_{\rho}^{\rho})$ (8.9)
$\displaystyle\mathscr{L}_{I}^{ST}=\eta\epsilon_{\mu\nu\rho\sigma}(2b_{I}\mathbf{e}_{[JK]}G^{\mu\nu
I}G^{\rho J}G^{\sigma k}+\mathbf{e}_{[IJ]}\mathbf{e}_{[KL]}G^{\mu I}G^{\nu
J}G^{\rho K}G^{\sigma L})$

Thus, eq. (8.4) decants the first quantum number associated at the set
$\\{G_{\mu I}\\}$. Since spin is the most direct physical entity provided by
field theory, eqs.(8.5 e 8.6) respectively separate spin-1 and spin-0 sectors.
Consequently, with defined spins, we will be able to follow about other
quantum numbers being proposed by this systemic model [22].

A further discussion is how far one can separate the spin sectors without
violating localizability. Decomposing $A_{\mu I}\equiv A_{\mu I}^{T}+A_{\mu
I}^{L}$ where $A_{\mu}^{T}=\theta_{\mu\nu}A^{\nu}$,
$A_{\mu}^{L}=\omega_{\mu\nu}A^{\nu}$ and considering the Lorentz
transformation $x^{\prime}_{\mu}=\Lambda_{\mu}^{\nu}x_{\nu}$, one gets the
relationships

$\displaystyle(A^{\prime}_{T})_{\mu}=(\Lambda^{-1})^{\rho}_{\mu}(A_{T})_{\rho}\text{;
}(A^{\prime}_{L})_{\mu}=(\Lambda^{-1})^{\rho}_{\mu}(A_{L})_{\rho}$ (8.11)

Eq.(8.11) is showing that the fields transverse and longitudinal
decompositions are consistent with Lorentz covariance. This guarantee these
two sectors under different Klein Gordon equations without violating Poincarè.
They will carry different dynamics preserving the original symmetries.

###  8.2 Mass diversity

Mass is the next quantum number to be understood. Considering the spin
sectors, and given the possibility two separate them consistently, next will
be to the understand their masses constitutions. Split in transverse and
longitudinal masses.

Physical masses are obtained by decomposing the kinetic part in terms of
transverse and longitudinal projectors. According to Appendix A, it yields

$L_{K}=L_{K}^{T}+L_{K}^{L}=\frac{1}{2}G^{t}_{\mu}(\Box+m^{2}_{T})\theta^{\mu\nu}G_{\nu}+\frac{1}{2}G^{t}_{\mu}(\tilde{B}\Box+m^{2}_{T})\omega^{\mu\nu}G_{\nu}$
(8.12)

where $G^{t}_{\mu}=(G_{\mu 1},\dots,G_{\mu n})$,
$\theta_{\mu\nu}-\frac{\partial_{\mu}\partial_{\nu}}{\Box}$ e
$\omega_{\mu\nu}=\eta_{\mu\nu}\frac{\partial_{\mu}\partial_{\nu}}{\Box}$.

The abelian symmetry enlargement is introduced. While the standard atomist
gauge theory states that the number of gauge bosons is determined by the
number of group theory generators, a variety of quanta is expressed at eq.
(8.12). Analizying the correspondent spin-1 and spin-0 sectors, one gets
masses $m^{2}_{T}$ and $m^{2}_{L}$. Apendix A is showing transverse
$m^{2}_{T}$ and longitudinal masses $m^{2}_{L}$ as eigenvalues of matrizes
$K^{-1}_{T}M^{2}$ and $(K_{L}+G_{F})^{-1}M^{2}$ respectively.

####  8.2.1 Transverse masses

As stipulated by the $\Omega$ matriz transverse diagonalization, it fixes the
physical fields $\\{G_{\mu I}\\}$ associated with spin-1 and corresponding
physical mass, $m^{2}_{T}$. The transverse propagates are expressed as
$<G_{\mu I}G_{\nu J}>=\frac{i\delta_{IJ}}{\kappa^{2}-m^{2}_{T,IJ}}$. It yields
that $\mathscr{L}_{T}(G)$ contains a diagonalized mass matrix $m^{2}_{T}$
given by

$m^{2}_{T}=\begin{pmatrix}0&0&...&0\\\ 0&m^{2}_{T,22}&...&0\\\
\vdots&\vdots&&\vdots\\\ 0&0&\dots&m^{2}_{T,NN}\end{pmatrix}$ (8.13)

Eq. (8.13) shows the mass spectroscopy corresponding to spin-1 sector. Notice
that the system necessarily contains a mandatory zero mass and others whose
value depends on free coefficients analyzed at section 4.

The physical choice was to make the transverse sector completely diagonalized.
The relationships $M^{2}=(\Omega^{2})^{-1}m^{2}\Omega^{-1}$ and
$\Omega^{t}K_{T}\ =\mathbb{I}$ yields that $\Omega\Omega^{t}=K^{-1}_{T}$,
which gives $K_{T}^{-1}m^{2}=\Omega m\Omega^{-1}$. This expression says that
the matrices that contain the poles in both parametrization systems,
$\\{D_{\mu},X_{\mu}^{i}\\}$ and $\\{G_{\mu\pm}\\}$ are related by a similarity
transformation. A result showing the poles invariance under fields
transformation. Another possibility would be diagonalize the longitudinal
sector. Apendice – works on that.

###  8.3 Longitudinal masses

We must now discuss the question of the poles in the L-sector of the field
propagators $G_{\mu I}$. These are given at expression
$(\tilde{B}\Box+m^{2}_{T})^{-1}\omega_{\mu\nu}$ where
$\tilde{B}=\Omega^{t}(K_{L}+G_{F})\Omega$. Therefore, we must ensure that the
matrix $(\tilde{B}\Box+m^{2}_{T})$ is invertible. However, $m^{2}_{T}$ can
have one or more zero eigenvalues, so it is not, in general, invertible. So,
knowing that ’the sum of two matrices is invertible, if and only if one of the
matrices is invertible,’ we have that, the invertibility of
$(\tilde{B}\Box+m^{2}_{T})$ is due, in general, to the invertibility of
$\tilde{B}$. Thus, returning to the propagators at L-sector, we obtain

$<G_{\mu I}G_{\nu
J}>_{L}=\left[\frac{i}{\Box+\tilde{B}^{-1}m^{2}_{T}}\tilde{B}^{-1}\right]_{IJ}\omega_{\mu\nu}$
(8.14)

resulting that the poles of the longitudinal sector are obtained as the
eigenvalues of $\tilde{B}^{-1}m^{2}_{T}$. It should be noted that the
$\tilde{B}$ matrix also contains the gauge-fixing parameters, so in principle
its poles should be bringing a dependency on the gauge parameter. It yields
(N-1) quanta where one is suppressed by the gauge fixing term.

The following step to analyze is that the T-sectors and L-sectors are not
completely independent. There is a relationship between the masses of both
sectors given by the expression $det(\tilde{B}^{-1}m^{2}_{T})=\frac{det\text{
}m^{2}_{T}}{det\text{ }\tilde{B}}$, which results,

$(m^{2}_{1}\times m^{2}_{2}\times\dots m^{2}_{N})=det\tilde{B}\cdot
det(\tilde{B}^{-1}m^{2}_{T})$ (8.15)

Therefore, given $det\tilde{B}\neq 0$ due to $\tilde{B}$ being invertible, eq.
(7.15) is that given that necessarily $m^{2}_{T,11}=0$, we have

$det(\tilde{B}^{-1}m^{2}_{T})=0=det(m^{2}_{L})$ (8.16)

Thus, a question is the degenerescence degree correspondence between a null
$m_{T}^{2}$ and $m_{L}^{2}$ eigenvalues. For understanding, consider at sector
T, that between N-fields $G_{\mu I}$ with M independent fields $(M\leq N)$
containing zero mass. Considering that vectors $G_{\mu 1},...,G_{\mu N}$ are
also independent eigenvectors of $(\tilde{B}^{-1})^{t}m^{2}$, one gets

$\displaystyle m_{L}^{2}G_{\mu 1}=0,...,m_{L}^{2}G_{\mu M}=0$ (8.17)

where eq. (8.17) shows that the degenerescence degree for null $m_{T}^{2}$
eigenvalues.

Thus eq (7.15-7.17) are expressions to characterize mass spectroscopy. Saying
that, if there is a zero mass in the T-sector, it will correspond to another
zero in the L-sector. Physically, a zero-mass spin-1 photon will be associated
with another zero-mass scalar photon.

The next step to ensure the physicality of the model is that the physical
masses $m^{2}_{T}$ and $m^{2}_{L}$ of the transversal and longitudinal sectors
do not contain tachyons. Control the relations $m^{2}_{T}.\geq 0,m^{2}_{L}\geq
0$ through the free coefficients of the theory. These relationships will be
controlled by the free coefficients of theory.

###  8.4 Charges diversity.

The set action produces three types of charges. First, corresponding to the
2N-equations of motion derived from eqs. (8.5-8.6). Second, the Noether charge
associated to gauge symmetry. Third, that one related to the symmetric Bianchi
identity given by eq. (10.3).

A physics beyond electric charge is developed. It shows, that from a common
abelian symmetry it is possible to generate $(2N+2)$ conserved charges. They
are differentiated but with interdependent nature given by the fields set
$\\{G_{\mu I}\\}$. An elecromagnetism with conservation laws beyond electric
charge is derived to be analysed.

Thus, the N-vector field system $\\{G_{\mu I}\\}$ constitutes $(2N+2)$
conserved charges. The basic charge term is potential fields interacting with
fields strengths. They are observed at equations of motion, Noether, Bianchi
identities. Their expressions make diverse compositions of the $G_{\mu I}$
fields and fields strengths with granular, collective antisymmetric, symmetric
nature. It yields abelian fields selfinteractions working as sources for
fields charges.

###  8.5 Discrete symmetries

The next study on quantum diversity through the set notion is to consider the
discrete symmetries [23].

####  8.5.1 P, C, T and PCT symmetries

Given the fields set $\\{G_{\mu I}\\}$, one has to understand on the
possibilities of allocating a diversity of discrete symmetries. Starting from
$L(G)$ seen by eq. (3.7),

$\displaystyle L=$ $\displaystyle\frac{1}{2}$ $\displaystyle
G_{\mu}^{t}(\Box+m^{2}_{T})\theta^{\mu\nu}G_{\nu}+\frac{1}{2}G_{\mu}^{t}(\tilde{B}\Box+m^{2}_{T})\omega^{\mu\nu}G_{\nu}$
(8.18) $\displaystyle+$
$\displaystyle\partial_{\mu}G_{\nu}^{t}\tilde{\lambda}G^{\mu}G^{\nu}+G^{t}_{\mu}G^{t}_{\nu}\tilde{\Lambda}G^{\mu}G^{\nu}$

where $G_{\mu}^{t}=(G_{\mu 1},\dots,G_{\mu N})$ and
$m^{2}_{T}=diag(m_{11}^{2}=0,\dots,m_{NN}^{2})$.

The purpose is to observe the discreted symmetries P, C, T for diverse whole
quanta [23], without breaking the PCT structure characterized by the theorem
due to Luders, Pauli and Schwinger [24].

Starting from parity [25],

$G_{\mu I}(x)\stackrel{{\scriptstyle P}}{{\to}}UG_{\mu
I}(x)U^{-1}=G^{\prime}_{\mu I}(x^{\prime})=\eta_{P}G_{I}^{\mu}(x)$ (8.19)

where $t^{\prime}=t,\vec{x^{\prime}}=-\vec{x}$ end
$(\eta_{P})_{IJ}=\eta_{P}\delta_{IJ}$ is the intrinsic parity. For,
$\eta_{P}=1$, implies vector, $\eta_{P}=-1$, pseudo-vector. Replacing (7.19)
in eq. (7.18), we obtain

$\displaystyle
L^{{}^{\prime}}=\frac{1}{2}G_{\mu}^{t}(\Box+m^{2}_{T})\theta^{\mu\nu}G_{\nu}+\frac{1}{2}G_{\mu}^{t}(\eta_{P}\tilde{B}\eta_{P}+m^{2}_{T})\omega^{\mu\nu}G_{\nu}$
$\displaystyle\partial_{\mu}G_{\nu}^{t}\eta_{P}\tilde{\lambda}\eta_{P}\eta_{P}G_{\mu}G_{\nu}+G_{\mu
t}G_{\nu t}\eta_{P}\eta_{P}\tilde{\lambda}\eta_{P}\eta_{P}G_{\mu}G_{\nu}$
(8.20)

which yields the following conditions for P-invariance:

$\displaystyle\eta_{P}\tilde{B}\eta_{P}=\tilde{B}$ or
$\displaystyle\tilde{B}_{IJ}=\eta_{IM}\tilde{B}_{MN}\eta_{NJ}$
$\displaystyle\eta_{P}\tilde{\lambda}\eta_{P}\eta_{P}=\tilde{\lambda}$ or
$\displaystyle\tilde{\lambda}_{IJK}=\eta_{IM}\tilde{\lambda}_{MNP}\eta_{MJ}\eta_{PQ}$
$\displaystyle\eta_{P}\eta_{P}\tilde{\Lambda}\eta_{P}\eta_{P}=\tilde{\Lambda}$
or
$\displaystyle\tilde{\lambda}_{IJKL}=\eta_{IM}\eta_{JN}\tilde{\Lambda}_{MNPQ}\eta_{PQ}\eta_{LQ}$
(8.21)

Equity (8.21) shows that parity is naturally conserved when all $G_{\mu
I}^{\prime}$ are vectors $(\eta_{IM}=-\delta_{IM})$. However, for pseudo-
vectors $(\eta_{IM}=\delta_{IM})$, the trilinear term is not satisfied. For
mixed case, one should study the restrictions of eq. (7.21). For example, for
the case of three fields with two vectors
$(A^{\prime}_{\mu}(x^{\prime})=A_{\mu}(x))$ and a pseudo-vector
$(A^{\prime}_{\mu}(x^{\prime})=-A_{\mu}(x))$, $\eta$ $\eta_{p}=diag(1,1,-1)$.
For three linear term one must explicitly derive the relationships between the
coefficients of the theory. For four linear terms both cases are directive
preserved.

The second discrete symmetry is charge conjugation [26]. It reverses the signs
of all non-zero generalized charges of a given particle, transforming it into
the corresponding antiparticle. Considering complex fields $G_{\mu I}$,

$G_{\mu I}(x)\stackrel{{\scriptstyle C}}{{\to}}U_{C}G_{\mu
I}(x)U_{C}^{-1}=G^{\prime}_{\mu I}(x^{\prime})=\eta_{C}G_{\mu I}^{*}(x)$
(8.22)

The transverse kinetic term is automatically invariant, $\eta_{C}^{2}=1$ and
$\eta_{c}m^{2}_{T}\eta_{c}=m^{2}_{T}$. But, for other terms in $L$ one must
establish the corresponding conditions. Substituting (8.22) into (8.13), we
get

$\displaystyle L\stackrel{{\scriptstyle C}}{{\to}}L^{\prime}=L$ (8.23)

for

$\displaystyle\tilde{B}=\eta_{C}\tilde{B}\eta_{C},$
$\displaystyle\tilde{\lambda}=\eta_{C}\tilde{\lambda}\eta_{C}\eta_{C}$
$\displaystyle\tilde{\Lambda}=\eta_{C}\eta_{C}\tilde{\Lambda}\eta_{C}\eta_{C}$
(8.24)

Equity (7.25) shows that a C-invariance is obtained when all $G_{\mu
I}^{\prime}s$ have $\eta_{C}=1$. In the case when the $G_{\mu I}^{\prime}s$
have $\eta_{C}=-1$ (photonic case), we have $\lambda=0$, which means that the
Lagrangian does not consider trilinear interactions for the same field, as
prescribed by Furry’s theorem [27].

In this context, it is observed that L and separately the fields can break the
parity and charge conjugation symmetries. However, it is also possible that
the combined operation PC is invariant.

The third case is the anti-unit operation of time inversion [28],

$G_{\mu I}(x)\stackrel{{\scriptstyle T}}{{\to}}U_{T}G_{\mu
I}(x)U_{T}^{-1}=G^{\prime}_{\mu I}(x^{\prime})=\eta_{T}G_{\mu I}(x)$ (8.25)

where
$x^{\prime\mu}=x^{\mu},\partial^{\prime}_{\mu}=\partial_{\mu},\Box^{\prime}=-\Box,\theta^{\prime}_{\mu\nu}=\theta_{\mu\nu},\omega^{\prime}_{\mu\nu}=\omega_{\mu\nu}$

Considering that $m^{2}_{T},\tilde{B},\tilde{\lambda}$ and $\tilde{\Lambda}$
are real matrices ($\Omega$ is real), the transverse term is naturally
invariant while the longitudinal and interaction sectors depend on the
relations

$\displaystyle\tilde{B}=\eta_{T}\tilde{B}\eta_{T},$
$\displaystyle\tilde{\lambda}=-\eta_{T}\tilde{\lambda}\eta_{T}\eta_{T}$
$\displaystyle\tilde{\Lambda}=\eta_{T}\eta_{T}\tilde{\Lambda}\eta_{T}\eta_{T}$
(8.26)

where eq. (7.25) means that trilinear terms are prohibited for all $G_{\mu
I}^{\prime}s$ with $\eta_{T}=1$. In the case where all of them have
$\eta_{T}=-1$, the T-invariance is automatic.

The operation PCT should be performed for proofing the model consistency.
Based on the so-called PCT theorem, due to Lürders, Pauli and Schwinger,which
states that the product of the transformations P,C,T considered in any order,
is always a symmetry of a quantum theory, when the Lagrangian that defines it
is: local, hermitian, invariant under proper Lorentz transformation and if the
fields are quantized in accord with the usual spin-statistics correction. Now,
one has to verify how this PCT operation acts for a whole system.

The PCT symmetry is given by

$G_{\mu I}(x)\stackrel{{\scriptstyle PCT}}{{\to}}U_{T}U_{C}U_{P}G_{\mu
I}(x)U_{T}^{-1}U_{C}^{-1}U_{P}^{-1}=G^{\prime}_{\mu
I}(x^{\prime})=\eta_{PCT}G^{*}_{\mu I}(x)$ (8.27)

where $\eta_{PCT}=\eta_{P}\eta_{C}\eta_{T}$ with
$x^{\prime\mu}=x^{\mu},\partial^{\prime}_{\mu}=\partial_{\mu}$

$\displaystyle L\stackrel{{\scriptstyle PCT}}{{\to}}L^{\prime}$
$\displaystyle=\frac{1}{2}G_{\mu}^{t}\eta_{PCT}(\Box+m_{T}^{2})\theta^{\mu\nu}\eta_{PCT}G_{\nu}+$
(8.28)
$\displaystyle+\frac{1}{2}G_{\mu}^{t}\eta_{PCT}(\tilde{B}\Box+m_{T}^{2})\omega^{\mu\nu}\eta_{PCT}G_{\nu}+$
$\displaystyle+\partial_{\mu}G_{\nu}^{t}\eta_{PCT}\tilde{\xi}\eta_{PCT}G^{\mu}\eta_{PCT}G^{\nu}+$
$\displaystyle+G_{\mu}^{t}\eta_{PCT}G_{\nu}^{t}\eta_{PCT}\tilde{\Lambda}\eta_{PCT}G^{\mu}\eta_{PCT}G^{\nu}$

Thus, the transverse term is automatically invariant
($\eta_{PCT}^{2}=1,\eta_{PCT}m^{2}_{T}\eta_{PCT}=m^{2}_{T}$) Meanwhile, the
other terms require the following conditions:

$\displaystyle\eta_{PCT}\tilde{B}\eta_{PCT}=\tilde{B}$
$\displaystyle\eta_{PCT}\tilde{\lambda}\eta_{PCT}\eta_{PCT}=-\tilde{\lambda}$
$\displaystyle\eta_{PCT}\eta_{PCT}\tilde{\Lambda}\eta_{PCT}\eta_{PCT}=\tilde{\Lambda}$
(8.29)

Consequently the transverse term is invariant, but the longitudinal depends on
eqs. (8.29). However, as we know, PCT theorem requires a closure not related
to theory free coefficients. Then, in order to express it in a more generic
way, not depending on these coefficients, we are going to take
$V_{\mu}$-referential system. From Eq. (10) and $G_{\mu}=\Omega^{-1}V_{\mu}$,

$\displaystyle V_{\mu}(x)\stackrel{{\scriptstyle
PCT}}{{\to}}U_{PCT}V_{\mu}(x)U_{PCT}^{-1}=V^{\prime}_{\mu}=\mathscr{N}V_{\mu}(x),$
(8.30)

where $\mathscr{N}\equiv\Omega\eta_{PCT}\Omega^{-1}$. Substituing Eq. (12) in
Eq. (2), one derives the following relationships:
$\mathscr{N}^{t}K\mathscr{N}=K$ and $\mathscr{N}^{t}M^{2}\mathscr{N}=m^{2}$
showing that transverse part is PCT invariant. Similarly the longitudinal part
is invariant through $\mathscr{N}^{t}B\mathscr{N}=B$, trilinear term for
$\mathscr{N}^{t}\lambda\mathscr{N}\mathscr{N}=-\lambda$ and quadrilinear for
$\mathscr{N}^{t}\mathscr{N}^{t}\Lambda\mathscr{N}\mathscr{N}=-\Lambda$. These
relationships show that PCT teorem is explicity verified reference $V_{\mu}$.
Consequently, transporting the PCT-invariance at reference $V_{\mu}$ at
reference $G_{\mu I}$, it provides PCT-invariance not depending on any
freecoefficient.

Thus, PCT theorem is verified. As expected it depends only on Lagrangian
nature not on its expression. In summary, $\mathscr{L}\stackrel{{\scriptstyle
PCT}}{{\to}}\mathscr{L}^{\prime}=\mathscr{L}$, and a consistency for this
whole abelian model is obtained.

Considering that$L(G)$ is local and hermitian the CPT invariance should
obtained as a whole [24]. Nevertheless, the appearance of these invariance
conditions is yet to be understood. It was expected that given the invariance
conditions under Poincaré, locality and hermitity for an L means that it
should automatically respect CPT.

The above equations are showing that the discrete symmetries of the transverse
sector are regulated by the coefficients $\eta_{p},\eta_{c},\eta_{T}$ while
those of the longitudinal sector are regulated by the coefficients of the
matrix $\tilde{B}$. Allowing a given $G_{\mu I}$ field to accommodate
different discrete symmetries for its vector and scalar quanta. In this
scenario, the $\\{G_{\mu I}\\}$ set contains groups of spin-1 and spin-0
quanta featuring a variety of discrete symmetries.

###  8.6 Symmetry of difference

At previous sections, a constructivist physics was proposed. We are assuming
the nature set action. The relationship between the part and the whole
introduced a third quantum. It includes another a physics perspective. Quanta
diversity and antireductionist properties are new aspects to include at
quantum physics. Two features requiring to be covered by a symmetry. It will
be called as the symmetry of difference.

Eq. (2.6) supports a new symmetry. The atomism is replaced by constructivism.
A physics between set and diversity is constituted. The antireductionist gauge
symmetry proposes differences acting together. A new quantum creation is
introduced. Gauge symmetry with the sense of unity at difference. A unity
producing a conglomerate physics with own properties.

The symmetry of difference is introduced by constructivist Lagrangian. By
analyzing the quanta spectroscopy at set $\\{G_{\mu I}\\}$, a diversity is
observed. As studied, the quantum numbers corresponding to each quanta
embedded in the fields set $\\{G_{\mu I}\\}$ are differentiated. The
correspondent spin, mass, charges; P, C, T are under a diversity without
violating the gauge symmetry and PCT theorem. Consequently, the third quantum
existence is not only based on the part inserted in the whole, but under
quantum diversity.

A quantum system is established under the unity of difference. The set action
$\\{G_{\mu I}\\}$ contains multiple quanta. The quanta pluriformity. A whole
with differentiated parts. A new elementarity is proposed. Quantum systems
with varieties. Differentiated quanta and whole physical properties are
established by the antireductionist gauge symmetry. The meaning of variety is
established. There is a new way to do physics. Instead of wave function
consider variety. A construction where the fields set action differentiates
parts under gauge symmetry. A quantum system is constituted by interconnected
fields.

Three achievements are obtained. First, the symmetry of difference replace the
spontaneous symmetry breaking and grand unification. Second, it yields a third
quantum physics. The fields set action interrelates equations of motion,
Bianchi identities, conservation laws, Lorentz forces. Third, quantum systems
are redefined by substituting quantum mechanics through gauge symmetry.

The set action turns out a new physical performance to be studied. The
symmetry of difference constitutes an alternative to grand unification and
spontaneous symmetry breaking. It minimizes the physical conditions to carry
physical entities. It does not follow the group theory rule to gauge theory
established by the Göttingen group since 1920s.

The symmetry of difference breaks the symmetry relationship established by
group theory. Two statements were constituted. On the degeneracy of physical
states and the number of intermediate bosons. The quantum mechanics
development in the 1920s created a standard for the use of symmetry in
physics. To assert whether G is the symmetry group of a quantum system, the
states of this system are grouped into representations of G called
degeneracies. For example, according to the SO(4) group, the hydrogen atom
(quantum system) has $2N^{2}$ degenerate states where N is the energy level
[29].

In particle physics, instead of degenerate states, the presence of particles
is introduced. And so, from the point of view of relativity, $E=mc^{2}$,
degeneracy is characterized by the existence of particles with the same mass.
This proposal was considered in 1932 by Heisenberg when he included the proton
and the neutron (mass difference $0.1\%$) in a some multiplet and established
the isospin symmetry. This idea was followed by Yang-Mills in 1954 where the
number of associated gauge bosons must be equal to the number of group
generators [12-13]. For instance, SU(5) requires 24 intermediated particles
[30]. It also simplifies on mass introduction. A second aspect of the symmetry
of difference is that it may be manifested without implemeting an ad hoc a
scalar potential. Higgs fields are not necessary [30].

####  8.6.1 Example: symmetry of difference for a quantum system with five
bosons

Quantum system featuring a quantum diversity must be exemplified. Consider
five $G_{\mu I}$ fields constituting the set $\\{G_{\mu 1},G_{\mu 2},G_{\mu
3},G_{\mu 4},G_{\mu 5}\\}$. To examine the inclusion of quantum numbers
differentiations with respect to different masses, coupling constants and
invariances C, P, T, and CPT, let us consider the following possibilities:

  1. (i)

$J^{PC}=1^{--}$, like the photon, $J/\psi$, $\rho(770)$. Identify such
particles as $G_{\mu 1},G_{\mu 2},G_{\mu 3}$.

  2. (ii)

$J^{PC}=1^{+-}$, such as $b_{1}(1235)$. associate particle as $G_{\mu 4}$.

  3. (iii)

$J^{PC}=1^{++}$, like the $X_{1}(3510)$. Identify the particle as $G_{\mu 5}$.

Thus, $L(G)$ at eq. (8.2) accomodates the spin-1 and spin-0 sector with
different masses and coupling constants. Then, we are going study on discrete
symmetry. In addition, to the usual discrete individual symmetries of each
$G_{\mu I}$ field, there are discrete systemic symmetries for grouping
$\\{G_{\mu I}\\}$. The perspective what counts is the symmetry of $L(G)$.
Physically, discrete symmetry can act differently for each $G_{\mu I}$ field
interaction. Implying that, there is the set invariance and a diversity of
individual discrete symmetries.

Consequently, the set $\\{G_{\mu},G_{\mu 2},G_{\mu 3},G_{\mu 4},G_{\mu 5}\\}$
formed by the composition of these particles will preserve CPT, but promoting
interactions which preserve or not the individual discrete symmetries.
Discrete symmeties may considered individualy or as a whole $\\{G_{\mu},G_{\mu
2},G_{\mu 3},G_{\mu 4},G_{\mu 5}\\}$. For example, as a whole it is invariant
under parity, although individually several of its fields are not.

##  9 Nonlinear physics

A whole quantum physics is being developed. The usual relativistic fields
equations are developed in a fields set minimal action, $\delta S=0$. It gives
nonlinear a dynamics constituted by coupled equations of motion [31].

$\displaystyle\partial_{\mu}\\{(G^{\mu\nu}_{I}+a_{I}\mathbf{e}^{[\mu\nu]})+\eta\epsilon^{\mu\nu\rho\sigma}(G_{\rho\sigma
I}+a_{I}\mathbf{e}_{[\rho\sigma]})+$
$\displaystyle+\beta_{I}(\beta_{J}S^{\mu\nu J}+\rho_{J}g^{\mu\nu}S^{\alpha
J}_{\alpha}+\mathbf{e}^{(\mu\nu)}+g^{\mu\nu}\mathbf{\omega}^{\alpha}_{\alpha})\\}+$
$\displaystyle+\partial^{\nu}\\{(\beta_{I}\beta_{J}+\beta_{I}\rho_{J}+\beta_{J}\rho_{J}+\frac{1}{4}\xi_{IJ})S^{\alpha
J}_{\alpha}+\rho_{I}(\mathbf{e}^{\alpha}_{\alpha}+4\omega^{\alpha}_{\alpha})\\}$
$\displaystyle+\frac{1}{2}\mathbf{m}_{II}^{2}G^{\nu I}=J^{\nu}_{I}(G)$ (9.1)

where

$\displaystyle J^{\nu}_{I}(G)=J^{A\nu}_{I}(G)+J^{S\nu}_{I}(G)+J^{st\nu}(G)$

are expressing a nonlinearity

$\displaystyle J^{A\nu}_{I}(G)=\mathbf{e}_{[IJ]}G^{J}_{\mu}Z^{[\mu\nu]},$
$\displaystyle
J^{S\nu}_{I}(G)=\mathbf{e}_{(IJ)}G_{\mu}^{J}Z^{(\mu\nu)}+\tau_{(IJ)}G^{\nu
J}Z_{\alpha}^{\alpha}$ $\displaystyle
J^{st\nu}_{I}(G)=\eta\mathbf{e}_{[IJ]}\epsilon^{\mu\nu\rho\sigma}G_{\mu}^{J}Z_{[\rho\sigma]}+\eta
b_{I}\epsilon^{\mu\nu\rho\sigma}\partial_{\nu}\mathbf{e}_{[\rho\sigma]}$ (9.2)

Eq. (9.1) can be reduced by using the kinetic identity
$\partial_{\mu}S^{\mu\nu}_{I}=\partial_{\mu}G^{\mu\nu}_{I}+\partial^{\nu}S^{\mu}_{\mu
I}$. It will suppress the fields strenghts $S^{\mu\nu}_{I}$ and $e^{\mu\nu}$
from the dynamics. The spin-1 and spin-0 sectors are separated.

Eq. (9.2) are sources carrying the fields set $\\{G_{\mu I}\\}$. A selfprocess
where fields strengths $Z_{\mu\nu}$ work as own sources. Through its own
dynamics. $Z_{\mu\nu}$ is moving at space-time and growing. Developing a
quantum self creation own internal sources.

##  10 Systemic physics

Considering eq. (2.6) where fields set $\\{G_{\mu I}\\}$ is sharing a common
gauge symmetry, an interrelatioship is expected. A system working under a
fields set unity. Fields are interwined by collective fields strengths and by
equations of motion, Faraday laws, conservation laws, constructive Lorentz
forces. It yields a systemic physics producing antireductionist interaction,
induction, connection [32].

Separating eq. (9.1) in transverse and longitudinal sectors, it gives

For spin 1:

$\displaystyle\partial_{\mu}\\{G^{\mu\nu}_{I}+a_{I}\mathbf{e}^{[\mu\nu]}\\}+\frac{1}{2}\mathbf{m}^{2}_{II}G^{\nu}_{I}=J^{\nu}_{I}(G)$
(10.1)

For spin-0:

$\displaystyle\partial^{\nu}\\{(\beta_{I}\beta_{J}+\beta_{I}\rho_{J}+\rho_{I}\beta_{J}+\rho_{I}\rho_{J}+\xi_{IJ})S^{\alpha
J}_{\alpha}+(-\frac{1}{2}\beta_{I}+\rho_{I})\mathbf{e}^{\alpha}_{\alpha}+$
$\displaystyle+(\beta_{I}+4\rho_{I})\omega^{\alpha}_{\alpha}\\}+\frac{1}{2}\mathbf{m}^{2}_{II}G^{\nu}_{I}=J^{\nu}(G)_{I}-\frac{1}{2}\mathbf{e}_{IJ}S^{\alpha}_{\alpha
J}G^{\mu}_{I}$ $\displaystyle-\mathbf{e}_{(IJ)}S^{\nu\mu}_{J}G_{\nu I}$ (10.2)

The above differential equations system explicit the $\\{G_{\mu I}\\}$
interrelationships.

The Bianchi identities improve the above systemic interrelationships. They
provide the inductions below

$\displaystyle\partial_{\mu}G^{I}_{\nu\rho}+\partial_{\nu}G^{I}_{\rho\mu}+\partial_{\rho}G_{\mu\nu}^{I}=0$
$\displaystyle\partial_{\mu}\mathbf{e}_{[\nu\rho]}+\partial_{\nu}\mathbf{e}_{[\rho\mu]}+\partial_{\rho}\mathbf{e}_{[\mu\nu]}=\mathbf{e}_{[IJ]}G_{\nu}^{I}G^{J}_{\mu\rho}+\mathbf{e}_{[IJ]}G_{\mu}^{I}G^{J}_{\rho\nu}+\mathbf{e}_{[IJ]}G_{\rho}^{I}G^{J}_{\mu\nu}$
$\displaystyle\partial_{\mu}\mathbf{e}_{(\nu\rho)}+\partial_{\nu}\mathbf{e}_{(\rho\mu)}+\partial_{\rho}\mathbf{e}_{(\mu\nu)}=\mathbf{e}_{(IJ)}G_{\nu}^{I}S^{J}_{\mu\rho}+\mathbf{e}_{(IJ)}G_{\mu}^{I}S^{J}_{\rho\nu}+\mathbf{e}_{(IJ)}G_{\rho}^{I}S^{J}_{\mu\nu}$
$\displaystyle\partial_{\mu}\omega^{\nu}_{\nu}+2\partial_{\nu}\omega_{\nu}^{\nu}=\tau_{(IJ)}G^{I}_{\mu}S^{\nu
J}_{\nu}+2\tau_{(IJ)}G^{I}_{\nu}S^{\mu J}_{\mu}$ (10.3)

Eqs. (10.3) retreat the granular and collective, antisymmetric and collective
symmetric Bianchi identities. It yields the following conservation law
$\partial_{\mu}J^{\mu}_{B}=0$ where
$J^{\mu}_{B}=\varepsilon^{\mu\nu\rho\sigma}e_{[IJ]}G^{I}_{\nu}G^{J}_{\rho\sigma}$.

The three Noether identities are

$\displaystyle\partial_{\mu}J^{\mu}_{N}=0$
$\displaystyle\Omega_{I1}\partial_{\nu}\frac{\partial
L}{\partial(\partial_{\nu}G_{\mu I})}+J^{\mu}_{N}=0$
$\displaystyle\Omega_{I1}\frac{\partial L}{\partial(\partial_{\mu}G_{\nu
I})}\partial_{\mu}\partial_{\nu}\alpha=0$ (10.4)

Considering the energy-momentum tensor, one gets another relationships through
the conservation law [33].

$\displaystyle\frac{\partial U}{\partial t}+\vec{\nabla}\cdot\vec{S}=0,$
(10.5) $\displaystyle\frac{\partial S_{i}}{\partial t}-\partial_{j}T_{ij}=0$
(10.6)

The Lorentz constructivist force between the fields inserted at $\\{G_{\mu
I}\\}$ is extended for the following grouping interrelationships [22]

$\displaystyle f^{\nu}=f^{\nu}_{L}+f^{\nu}_{M}+f^{\nu}_{E}$ (10.7)

with

$\displaystyle f^{\nu}_{L}=4G^{\nu\mu I}j_{\mu I},\text{
}f^{\nu}_{M}=-2m_{I}^{2}G^{\nu}_{I}(\partial^{\mu}G_{\mu}^{I}),\text{
}f^{\nu}_{E}=4G^{\nu I}(\partial^{\mu}J_{\mu I}(G))$ (10.8)

where $j_{\mu I}$ is an external source and $J_{\mu I}(G)$ defined at eqs
(9.2). At vectorial shape,

$\displaystyle\vec{f}_{L}=4\rho_{I}\vec{E}^{I}+4\vec{j}_{I}\times\vec{B}^{I},$
(10.9)
$\displaystyle\vec{f}_{M}=-2m^{2}_{I}\vec{G}_{I}(\frac{\partial}{\partial
t}\phi^{I}+\vec{\nabla}\cdot\vec{G}^{I}),$ (10.10)
$\displaystyle\vec{f}_{E}=4\vec{G}_{I}(\frac{\partial}{\partial
t}\rho^{I}(G)+\vec{\nabla}\cdot\vec{J}^{I}(G))$ (10.11)

The above equations are showing how fields inserted in a fields set $\\{G_{\mu
I}\\}$ are producing constructivist fields forces. Lorentz force is reproduced
at eq. (10.9) and extended for a fields set. Eqs. (10.9-10.11) are introducing
a force dependence on masses and fields. It yields a perspective different
from usual atomist gauge theory which is just providing their interactions
with external sources.

Summarizing, a quantum system is formulated mathematically. The above
relationships between the whole and the part (whole quantum or variety)
originated from eq. (2.6) produce a classical structure with $3N+11$
equations. These equations are showing a set physics. A systemic interdepent
determinism where the individual, collective and whole are related together.

##  11 Chance physics

A grouping physics is being derived. A new whole quantum significance
proposed. Instead of ultimate constituents its definition comes out from the
set. The quanta expressed from the fields set $\\{G_{\mu I}\\}$ introduce new
perspectives. Next consider on the quantum chance.

Let us start with the spin-1 sector. The matrix $K_{T}$ expressed at (C.8) can
be separated into four NxN matrices. It gives,

$K_{T}=A+B+C+D$ (11.1)

$A=\begin{bmatrix}a&0&\dots&0\\\ \vdots&&\ddots&\vdots\\\ 0&0&\dots&0\\\
\end{bmatrix}_{NxN}B=\begin{bmatrix}0&\alpha_{1}&\dots&d\alpha_{N}\\\
d\alpha_{2}&0&\dots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
d\alpha_{N}&0&\dots&0\\\ \end{bmatrix}_{NxN}$ (11.2)

$C=\begin{bmatrix}0&\dots&\dots&\dots&0\\\
\vdots&\alpha^{2}_{2}&\alpha_{2}\alpha_{3}&\dots&\alpha_{2}\alpha_{N}\\\
\vdots&\vdots&&\ddots&\vdots\\\
0&\alpha_{N}\alpha_{3}&\dots&\dots&\alpha^{2}_{N}\end{bmatrix}_{NxN}D=\begin{bmatrix}0&\dots&\dots&\dots&0\\\
\vdots&\beta^{2}_{2}&\beta_{2}\beta_{3}&\dots&\beta_{2}\beta_{N}\\\
\vdots&\vdots&&\ddots&\vdots\\\
0&\beta_{N}\beta_{3}&\dots&\dots&\beta^{2}_{N}\end{bmatrix}_{NxN}$ (11.3)

which yields the following volume of circumstance for $K_{T}$:

$K_{T}\to 1+(N-1)(N+1)\text{ free coefficients}$ (11.4)

Introducing the mass term

$M_{T}\equiv\begin{bmatrix}0&\dots&\dots&\dots&0\\\
\vdots&m_{22}^{2}&m_{23}^{2}&\dots&m_{2N}^{2}\\\
\vdots&\vdots&&\ddots&\vdots\\\
0&m_{N2}^{2}&\dots&\dots&m_{NN}^{2}\end{bmatrix}_{NxN}$ (11.5)

one gets the following volume of opportunites for quanta masses at spin-1
sector

$K_{T}+M^{2}_{T}\to 1+(N-1)(\frac{3}{2}N+1)\text{free coefficients}$ (11.6)

Considering the spin-0 sector, the $K_{L}$ matrix expressed at (C.9) is
rewritten as

$K_{L}=E+F+G$ (11.7)

$E=\begin{bmatrix}0&\dots&\dots&\dots&0\\\
\vdots&8\beta^{2}_{2}&8\beta_{2}\beta_{3}&\dots&\beta_{2}\beta_{N}\\\
\vdots&\vdots&&\ddots&\vdots\\\
0&8\beta_{N}\beta_{2}&\dots&\dots&\beta^{2}_{N}\end{bmatrix}_{NxN}F=\begin{bmatrix}0&\dots&\dots&\dots&0\\\
\vdots&\beta_{2}\rho_{2}&\beta_{2}\rho_{3}&\dots&\beta_{2}\rho_{N}\\\
\vdots&\vdots&&\ddots&\vdots\\\
0&\beta_{N}\rho_{2}&\dots&\dots&\beta_{N}\rho_{N}\end{bmatrix}_{NxN}$ (11.8)

and

$G=\begin{bmatrix}0&\dots&\dots&\dots&0\\\
\vdots&8\rho^{2}_{2}&8\rho_{2}\rho_{3}&\dots&\rho_{2}\rho_{N}\\\
\vdots&\vdots&&\ddots&\vdots\\\
0&8\rho_{N}\rho_{2}&\dots&\dots&\rho^{2}_{N}\end{bmatrix}_{NxN}$ (11.9)

which gives,

$\displaystyle K_{L}\to(N-1)(2N-1)\text{ free coefficients}$ (11.10)

Adding the longitudinal mass matrix, one gets

$K_{L}+M_{L}^{2}\to(N-1)(\frac{5}{2}-1)\text{ free coefficients}$ (11.11)

Expressions (4.7) and (4.12) are showing the chances which the constructivist
abelian model derived from eq. (2.6) provides for accommodating the quanta
diversities.

Chance is a property under directive and circunstance. The directive is given
by the minimum action principle and gauge symmetry determinations as Noether
theorem, tensor energy-momentum invariance, gauge fixing, Ward identity. The
circumstance by the free coefficients attached to the Lagrangian gauge
scalars. As biology provides accidents and quantum mechanics probability, a
free will is expressed by the constructivist volume of circumstances.

Consider the mass case. Given that the physical mass is determined
perturbativily as the pole of a two point Green’s function, the corresponding
fields masses quanta are expressed through directive and circumstance. Eq.
(4.1) contains as directive the presence of a massless quantum and as
circumstance the existence of massive quanta depending on the theory free
parameters. It is mandatory a massless gauge boson, $\mathbf{m}_{11}=0$, due
to the directive gauge symmetry. Circumstantial masses are determined
depending on the theory free coefficients paraments as $m_{ij}^{2}$, $a_{I}$,
$\mathbf{e}_{IJ}$ and so on. Physical masses with chance of taking different
values without violating gauge invariance.

##  12 Evolution physics

Nature is made by transformations. However, more than an atomism devoted
ultimate constituents and respective interactions, its objective is to
individualize objects. Physics has to understand this determination. How to
give character to constituents. A requirement asking for what is the dynamics
able to do that.

A physics is being constructed in terms of set action. An associative physics
where the variable is the variety. A perspective where physics converges to
biology. While biology produces varieties (species) through genes and
environment, physics generates (particles, etc) from last constituents and
fields set. Both converging to the individuation proposed under environment.

Physics is not only expressed through isolated dynamics but also under a set
determinism. The fields set approach is expliciting an environmental physics.
Showing that $\\{G_{\mu I}\\}$ interrelationships develops granular and
collective fields strengths under an antireductionist dynamics. An
associativity physics prospected by systemic equations.

Quantum evolution is introduced by quantum systems evolving through set
determinism. Quantum constituted through fields set environment. And so, the
whole quantum associated to a given field $G_{\mu I}$ inserted in the set
$\\{G_{\mu I}\\}$ is modifying according to the constructivist Lagrangian L(G)
and correspondent classical equations studied at section 10.

A quantum system is generated under symmetry of difference. An environmental
bioogical physics is proposed. A whole quantum is individualized over
constructivist lagrangian. A set determinism is established introducing
varieties with diversity, growth, and chance. The meanings of evolution is
introduced. A new picture to be understood from set action.

Eq. (2.6) provides physics with environment. Quantum systems with
individualized quanta are formed. Differentiated energy packets are
constituted. Acording to biology the corresponding diversity, growth and
chance propitiate evolution. The physics of evolution is produced by varieties
in quantum systems. Emergence follows as evolution under circumstance.
Complexity as the final product of the evolution manifestation.

###  12.1 Quantic evolution

Particle physics has been collecting elementary particles. However, nature is
more than particles and interactions. There is a physics beyond the excelse
standard model particles protagonism. A grouping physics which associativities
produce realistic varieties. Elementarities responsible for matter behaviour.
Performing evolution.

At usual physics, a field is a function of space-time with an equation of
motion which describes how this function varies. A particle is defined as its
smallest vibration (quantum). So relativistic fields are the fundamentals
elements of reality. However, at present scope they are just retreating on the
particles dynamics and not their transformation.

Quantic evolution should be identified as particles transformations. They are
verified at many particles reactions. Physics provides three perspectives for
particle evolution qualified through quantum numbers modifications. First,
physical entities changing under the renormalization group equation [34].
Second, fields modifications from constitutive fields equations coming from
the fields set $\\{G_{\mu I}\\}$ interdependence [35]. Third, from a
constructivist lagrangian, eq. (3.25) working as a quantum tree.

Thus, there is still an evolutive timeline to be understood. Understand field
theory more than a fixed quantum moving in space-time. Introduce a dynamics
mutating the quantum numbers values. A first perspective is with the
renormalization group equation. To define a theory we most specificy the
energy scale, say $\mu$, under the condition that physics can not depended on
$\mu$. It yields the following equation.

$\displaystyle\frac{d}{d\mu}P(g,\mu)=(\mu\frac{\partial}{\partial\mu}+\beta(g)\frac{\partial}{\partial\beta})P(g,\mu)$
(12.1)

where $P(g,\mu)$ is a physical funcation and $\beta$ the beta function given
by $\beta(g)=\mu\frac{\partial}{\partial\mu}g$.

It develops parameters flowing with energy scale $m(\mu)$ and e$(\mu)$.
Showing how the fine structure constant $\alpha=\frac{1}{137}$ suffer
modifications.

By second, physical changes are also configurated through the constructivist
equations of motion. At eq (9.1) a given field $G_{\mu I}$ is transforming
under the fields set interdependence. Fields interdependence modifies the
original reductionist field equation. For instance, consider the Maxwell
photon.

Thus, there is the set action principle of evolution to be studied. It
incorporates two features beyond the renormalization group. The fields set
$\\{G_{\mu I}\\}$ modified constitutive equations and the quantum numbers
modifications stipulated by L(G) according to previous section. A third
evolutive possibility is to consider the constructivist Lagrangian $L(G)$. The
evolutive feature is a an inner dynamics in L(G) transforming their quantum
numbers. It develops quantum numbes under environment, diversity, growth,
chance. The conditions to provide individuations according to biology.

A constitutive photon equation was obtained [35]. A new photon equation is
obtained rewriting eq. (10.1) in terms of masses and currents. It yields the
following photon field. It modified equation for an electromagnetic quantum
system involving the electric charge intermediated by four intermediated
bosons.

$\displaystyle\partial_{\nu}\left\\{4a_{1}F^{\nu\mu}+a_{2}\left(\mathbf{e}^{[12][\nu\mu]}+\mathbf{e}^{[+-][\nu\mu]}\right)\right\\}+l^{\mu}_{AT}+c^{\mu}_{AT}=$
$\displaystyle=2a_{3}\mathbf{m}^{2}_{U}U^{\mu}+j^{\mu}_{AT}+j^{\mu}_{UT}+J^{\mu}_{NT}$
(12.2)

The above equation does not consider on isolated photon but inserted in a
whole. It retreats the photon in an electromagnetic set. But, preserving the
original QED symmetry. The coupling constant $\alpha$ between the electron and
photon is preserved.

A mass type term appears. It is through the London term, $l^{\mu}_{AT}$. It
constituted by fields association expressed as

$\displaystyle
l^{\mu}_{AT}=\\{4\mathbf{e}_{[12]}U_{\nu}U^{\nu}-2a_{3}(q_{2}^{2}+q_{1}q_{2})V_{\nu}^{+}V^{\nu-}\\}A^{\mu}+$
$\displaystyle\\{4a_{3}\mathbf{e}_{[12]}A_{\nu}A^{\nu}-2(q_{1}q_{2}+q^{2}_{1})V_{\nu}^{+}V^{\nu-}\\}U^{\mu}$
(12.3)

with $A_{\nu I}A^{\nu I}=\phi_{I}$, introducing means a scalar field with
$mass^{2}$ dimension.

Term $c^{\mu}_{AT}$ means the conglomerate $mass$. It involves the
conglomerates generated by fields

$\displaystyle
c^{\mu}_{AT}=4\mathbf{e}_{[12]}[A_{\nu}U^{\nu}U^{\mu}+a_{3}A_{\nu}U^{\nu}A^{\mu}+1/2(1+a_{3})(\mathbf{e}^{[12-][\mu\nu]}V_{\nu}^{+}$
$\displaystyle+\mathbf{e}^{[12+][\mu\nu]}V_{\nu}^{-})]-2q_{1}q_{2}(+V_{\nu}^{+}U^{\nu}V^{\mu-}+a_{3}V_{\nu}^{+}A^{\nu}V^{\mu-}+V_{\nu}^{+}U^{\nu}V^{\mu-}$
$\displaystyle+a_{3}V_{\nu}^{+}A^{\nu}V^{\mu-})+2\mathbf{e}_{[12]}(1+a_{3})(\mathbf{e}^{[12-][\mu\nu]}V_{\nu}^{+}+\mathbf{e}^{[12+][\mu\nu]}V_{\nu}^{-})$
$\displaystyle-
q_{1}^{2}V_{\nu}^{+}A^{\nu}V^{\mu-}-q_{1}^{2}V^{-}_{\nu}A^{\nu}V^{\mu+}$
$\displaystyle-2a_{3}q_{2}^{2}(V_{\nu}^{+}U^{\nu}V^{\mu-}+V^{-}_{\nu}U^{\nu}V^{\mu+})$
(12.4)

with $A_{\nu I}A^{\nu}_{J}=c_{IJ}$, introducing another kind of scalar field
with $mass^{2}$ dimension. Notice that, eqs (12.2 - 12.3) are like the Higgs.
They are introducing a mass perception through scalar fields.

The influence of these effective masses, $l_{\mu}$ and $c_{\mu}$, on the
photon should be studied by the dispersion relation corresponding to eq.
(12.1). Although the classically the photon has mass zero and quantically
protected by the Ward identify, there is for such abelian fields set a
contribution to be understood through the correspondent dispersion relation
[36].

The current term is expressed through the interaction between fields strengths
and potential fields. It gives,

$\displaystyle
j^{\mu}_{AT}=\mathbf{e}_{[12]}[(a_{1}F^{\mu\nu}+u_{2}U^{\mu\nu})(U_{\nu}+a_{3}A_{\nu})+1/2\mathbf{e}_{[12]}(1+a_{3})(V^{\mu\nu+}V_{\nu}^{-}+V^{\mu\nu-}V_{\nu}^{+})]$
$\displaystyle-(2iq_{1}+a_{3}q_{2})(\partial^{\mu}V_{\nu}^{+}V^{\nu-}-\partial_{\nu}V_{\mu+}V^{\nu-}+\partial^{\mu}V_{\nu}^{-}V^{\nu+}-\partial_{\nu}V^{\mu-}V^{\nu+})$
(12.5)

and

$\displaystyle
j^{\mu}_{UT}=\mathbf{e}_{[12]}[(a_{1}F^{\mu\nu}+u_{2}U^{\mu\nu})(A_{\nu}+u_{3}U_{\nu})+1/2\mathbf{e}_{[12]}(1+u_{3})(V^{\mu\nu+}V_{\nu}^{-}+V^{\mu\nu-}V_{\nu}^{+})]$
$\displaystyle-(2iq_{1}+u_{3}q_{2})(\partial^{\mu}V_{\nu}^{+}V^{\nu-}-\partial_{\nu}V_{\mu+}V^{\nu-}+\partial^{\mu}V_{\nu}^{-}V^{\nu+}-\partial_{\nu}V^{\mu-}V^{\nu+}).$
(12.6)

Eq. (12.4-12.5) are showing an electromagnetism with potential fields
physicalities. A nonlinearity with potential fields interacting with granular
and collective fields. A physics expected by Faraday, Bohm-Aharanov and
Aharanov-Casher effects [37].

Noether current is related as

$\displaystyle J^{\mu}_{NT}\equiv
iq\\{V_{\nu}^{+}[v_{1}V^{\mu\nu-}+v_{2}\mathbf{e}^{[12-][\mu\nu]}]-V_{\nu}^{-}[v_{1}V^{\mu\nu+}+v_{2}\mathbf{e}^{[12+][\mu\nu]}]\\}$
(12.7)

where $q$ means the electric charge. Notice also the presence of uncharged
fields coupled to electric charge.

Thus, eq. (12.1) is expressing the photon constructivist transformation. A
photonics evolutive abelian fields dynamics. A whole quantum photon evolving
at space-time. Although a massless photon is preserved by gauge invariance,
constructivist effects are added on its behaviour. New terms on mass and
charge are introduced through classical interdependent equations. The
electron-photon interaction is preserved, but, the photon behaviour is added
by introducing an induced mass through London and conglomerate terms and with
interactions beyond electric charge and potential fields. There is a mass term
eq (12.2) to be verified by the respective photon dispersion relation. Perhap
a darwinian type quantum evolution

A quantum with potential to become something else is proposed. A mutating
physics proposed under set action. While at reductionist methodology a thing
is what it is, and the corresponding equations of motion just act on their
space-time dynamics, the constructivist lagrangian introduces a new
performance. The whole quantum is expressed in terms of diversity,
nonlinearity, growth interdependence, chance, directive and circumstances. A
similarity to Darwin evolution.

Thus, a given field $G_{\mu I}$ dynamics is not to be interpreted just as a
wave equation. It contains a quantum numbers environment which is changing
under a set action space-time dynamics. From biology, evolution happens
through cambial exchange between generations. Here, the part-whole duality
generates a conglomerates lineage. Instead of generations, it yields
successive sets through. The biology conditions to propiate evolution -
diversity, growth, chance - appears in physics. Modifying the particle
structures by as biology do with species. And so, physical entities as masses,
charges, coupling contant will be changing according to the fields environment
context.

Darwin shows that evolution arises from environment. It yields ingredients as
variety, diversity, interdependence, self-interaction, growth chance,
adaptation. The elements of the darwinian type of evolution that make up
natural selection. A comparision to be investigated. Extra polet whether the
darwinian natural solecation is the nature standard way to move from single to
complex. Mutatis mtuandis, the challenge for physics should be to find quantum
evolution within darwinian standards.

Considering that biology and physics are natural sciences, there must be a
correlation between the formation and determination of species and particles.
The difference is that in his two books Darwin did not use mathematics to show
the evolution of species [16]. Meanwhile, physics has to structure these
concepts analytically. Contemplate evolution not by figures, but
mathematically.

In the symmetry of difference physics finds its mathematical prescription for
describing evolution. Based on the antireductionist gauge theory, it
elaborates the physicality of the elements that lead to Darwin’s natural
selection. It develops an environment through the set of fields $\\{G_{\mu
I}\\}$, differentiated quanta produced by L(G), evolutionary determinism
expressed by respective differential equations.

In this way, the symmetry of difference is a candidate to develop the notion
of evolution in physics. Sections 8-11 mathematize evolutionary precepts. A
systemic dynamics in space-time is provoking the sense of transformation of
the field itself. This would be the perspective of a constructivist
lagrangian. Constituting changes in its quantum numbers. Promoting a new
order, where, instead of the space-time dynamics of particles, introduce the
physics of particles evolution.

The symmetry of difference not only brings togetherness and diversity. It
provides an arrow of time with the sense of evolution. Introducing another
feasible modification of quantum physics. The correspondent fields equations
express joint action, interdependence between fields, growth through non-
linearity, chance under free coefficients. Adaptation is driven by boundary
conditions and energy minimization.

An evolutive quantum system is performed. The fields set $\\{G_{\mu I}\\}$
generates a whole quantum evolving their quantum numbers through a space-time
dynamics. Producing a physics with particles variations and corresponding
palaeontology. Since 1930s physics knows that particles are not only
interacting and moving at space-time. While in the past physics moved from
mechanics to field theory, now we have to introduce an associative field
theory mutate under a quantum tree.

###  12.2 Quantum palaeontology

A particle contains an history to be told. It is not a fixed object moving
through space-time. Particles have an origin and a process. Since Big Bang
have been forming diverse structures. And so, the particles quantum numbers
transformations is a physics to be interrogated.

Nature is the science of changing. The second Newton’s law narrates the space-
time dynamics of a body. However, there is another aspect of modification
which is the inner transformation. When this body is replaced by a nature
internal transformation. A quantum mutation to be understood by field theory.
Physics has to investigate how a certain quantum transforms its structure.

The LHC is projected to discover particles. Its essence is the particles
production. However, evolutive laws has to be considered. Look at LHC
experiments not only as the search for new particles, but also, regarding how
particles are developed along time. Search for the tracks left by their
mutations. Particle physics contains an evolutive perspective to be studied.
The quantum is not an entity repeating the same interactions. It contains an
inner dynamics. Understand how the formative quantum numbers are modified. A
change to be registired by a quantum numbers paleontology.

A register for quantic evolution is expected. Identify the evolution tracks
left by eq. (3.29). Antireductionist properties were qualified at previous
sections. A study was expressed about the whole quantum transformations. The
difference between a field treated by an isolated differential equation and
inserted in an interdependent fields system. Undersand the evolutive
environment expressed by a coupled differential equations system.

A quantum system is being proposed through constructivist lagrangian. The
symmetry of difference creates an environment with quanta diversity working as
a set. The associated fields develops a self-organizing system. An inner whole
space-time dynamics is constituted. Particles mutations are derived.
Properties as diversity, interdependence, nonlinearity, chance, directive and
circumstance are promoting internal transformations. A new physics is
provoked. It is to register on the particle structure modifications.

We should explore on quantum paleontology. Consider the conglomerate physics.
Following the dynamics expressed from a constructivist abelian lagrangian, it
provides set transformations features. Fields strengths at eq. (9.1) are
acting as sources for modifying their own values. A selfgrowth set is produced
under potentiality and nonlinearity expliciting a development to be analysed
by inner quantum numbers modifications.

Physics has to identify on elementary particles palaeontology. Quantum numbers
are transformed by a set action and lineage. An evolutive quantum system is
formed instead of an amorphous concept of the electron, an appendage of
elementarity, physics makes the electron change from ultimate constituent to
variety. Associating the single electron with quantum systems as leptonic
families, Cooper pairs and so forth. A new electronic expression in nature
depending on set action.

Nature evolution can be detected microscopically. There is a quantum physics
to be registered by palaeontology. The whole quantum variety dynamics under
potentialities and realities produces quantum transformations. A quantum
historical process is defined. Eq. (26) involves set properties composing the
quantum evolution. The correspondent quanta diversiy introduces an inner
space-time transformations. Quantum numbers changing are leaving
palaeontological tracks to be investigated.

###  12.3 Complexity

Experimental physics detected the universe between $10^{-18}m$ (quarks) until
$10^{26}m$ (distant galaxies). A long road containning a process between
fundamental physics and complexity. Since an atom or a cell the universe
produce a linage of conglomerates. A process achieving solar systems and
species. A spectrum challenging physics to understand how the universe moves
from simplicity to complexity.

Physics and biology are natural sciences. And so, whole quantum should share
the meaning of variety in biology. A clue for, while the theory of evolution
in biology is based on environment and varieties, the physical world is in
terms of fields set and whole quanta. As a cell, the whole quantum contains a
whole unity. Parts in the whole are working with diversity and
interdependence. Making the atomist quantum move from force and interaction to
quantum systems and interrelationships. Producing multiplication under
nonlinearity and opportunities. And, introducing complexity as a next theme
from conglomerate physics.

The complexity scenario emerges. Physics has to explain how nature evolves
from a single particle to complex systems. There is a complex get together
developed by constructivist lagrangian. Quanta under antireductionist
coalization. Parts in the whole are candidates to express physics under
complexity. Expressing functionalities under antireductionist laws.

Thus, complexity leads physics to new properties than fundamental physics.
Features which will allow assemble diverse quanta in a physical structure
beyond fundamental law. Further consequence from evolution, quantum complexity
context is the next topic to be developed by quantum field theory. Introduce a
quantum under the synapsis presence.

##  13 Quantum system from gauge symmetry

An associative quantum field theory is proposed. It leads physics beyond the
usual atomist quantum field theory initiated in 1927 by Dirac [38]. Introduces
through gauge symmetry the passage from a single quantum to a quantum system.
Discover a continuity between reductionism and antireductionism.

A third type of quantum is expressed. It is called whole quantum. An
excitation derived from a field set. An energy signad made from a quantum
system. While fundamental physics is based on ultimate constituents a quantum
system develops varieties as elementaries. As biology, from proton onwards,
physics would be moved by varieties inserted in conglomerates.

single quantumquantum systemgaugesymmetry Figure 13.1: Constructivist
Lagrangian makes the passage from a single quantum to a quantum system.

The P.W. Anderson lemma ’more is different’ is still to be understood [39].
Quantum systems are well identified structures in nature by constructivist
lagrangian. Enlarging field theory from Dirac to Anderson. Consider a quantum
of associated quantum system with whole properties.

Diverse realistic situations provide quantum systems. For electron, one gets
the following cases: superconductivity with Cooper pairs [40], electronic
collective excitations separating the electron charge and spin and introducing
the so-called holon and spinon [41], plasmons in plasma physics [42], polarons
[43], polariton [44], excitons [45]. For spin, waves as phonons [40]. For
photon, quantum system in crystals [47]. For electromagnetism, four bosons
with an electromagnetism based on four photons associations (massless,
massive, charged) [33]. For EM phenomena, Hall effect [48].

Thus, physics should study quantum system through gauge symmetry and the
challenge is how to move from a single to a system of particles. The
discussion is not on their existence but about the approach. Current physics
states that a particle in a quantum system instead of classical properties
exibits a behaviour ruled by quantum mechanics laws. Described by wave
functions under the prescription of possibility measurement.

The fields set action introduces a new interpretation for quantum system.
Field theory appeared as an extension to quantum mechanics but was not able to
describe quantum mechanics but was not able describe quantum systems. The
associated quantum field theory appears to cover this gap. The pluriformity of
quantum diversities analysed at this work support a quantum system formulation
without requiring quantum mechanics. Diversely from wave function its
description is in terms of variables coming from mechanics and waves. Based on
symmetry of difference a quantum system is proposed where conglomerates are
described by measurable variables.

This is the physics promoted by eq. (2.6). The correspondent antireductionist
gauge symmetry relates the single and the set as required. A way where the
particle is inserted in the whole, preserving its atomist symmetry, and with
new collective properties. Also, maintaining the Planck quantum variables. The
a quantum system as a single quantum with antireductionist physicalities.

Four features make such passage from an isolated particle to a system
particles. First, the enlargement from atomism to constructivism preserving
the original gauge symmetry. Second, a quantum diversity without increasing
the group theory rank, or, requiring spontaneous symmetry breaking. Third, a
collective physical description in terms of the physical variables that
constitute the du Broglie quantum. Finally, a whole quantum with
constructivist properties unrealized by quantum mechanics.

Thus, an evolutive quantum system is proposed. As limit, their features
preserve the atomist quantum symmetry, and, includes antieductionist
properties. Proposing a condensed matter physics without requering quantum
mechanics. Considering the whole as fundamental unity, it introduces a set
physics under diversity, interdependence nonlinearity, chance. A quantum
system with space-time transformation involving growth, evolution, emergence,
complexity.

##  14 Conclusion

At last 2500 years physics has been searching for the greek atom. Understand
nature under ultimate constituents. Over this perspective, quantum physics
became the study of the energy packet. Qualifying the minimum energy structure
under Planck constant. However, nature performed by quantum mechanics is
unsatisfactory. It does not reach its full maturity. It appears as framework
for subatomic particles, but contains various not understandable fuzzy
aspects. The quantum world still requires a further explanation.

The quantum realm remains a debate. Nature features as associativity,
confinement and complexity lead physics to a set order. Gauge symmetry
assembling fields $\\{G_{\mu I}\\}$ is a new procedure. Constructivist
Lagrangian as the framework for set action. The quantum numbers that
constitute a quantum no more specified by reductionist properties. Determined
in terms of the involved whole.

Thus, the quantum system may be interpreted beyond quantum mechanics. A view
pointing out to a third moment on quantum story. As relativity extends
classical mechanics, the current reductionism and quantum mechanics should be
transposed to constructivism. The last constituents not only form particles,
but make quantum systems as nuclei, atoms, molecules. A perspective for a
third quantum be introduced by an associative physics enlarging interactions
to interrelationships.

Three types of quantum are classified. Granular, quantic and whole. The first
two have historic signature. However, nature from its capacity to generate
systems and patterns, expresses a further energy packet. The whole quantum as
a next possibility. Distinct arguments may select its existence. Three thesis
we select to support a third quantum physics. Aristotle, quantum mechanics,
variety. They produce arguments from where the quantum should not be restrict
to Planck and wave-particle.

Aristotle divided three modes of being. Really being, not-being, potentially-
being. Given such generic statement, let us consider its application on
quantum physics. Understand on the individualization possibilities. In terms
of quantum, the first one was detected experimentally and expressed
theoretically by the atomist quantum field theory. Quarks under confinement
introduce the not-being, a quantum detected indirectly. The potentially-being
is left for constructivism.

On that, there are two methodologies for understanding the quantum nature.
Atomist and constructivist. Atomism prevaleces at current physics. Ultimate
constituents, as quarks and leptons, working like matryoshkas make physics as
the summatory of parts. Atomism describes the first two Aristotles
assumptions. On the other hand, parts associated in the whole introduce
another type of energy packing. Therefore, for answering the third Aristotle
philosophical inquiring, one deduces that, there is a quantum constructivism
to be investigated.

More is different, and so, the constructivist physics should be taken.
Consider on the quantum associativity. An extension where the $A_{\mu I}$
field will work inserted at fields set $\\{A_{\mu I}\\}$. A passage from
reductionism to antireductionism preserving the original symmetry. An
enlargement with interactions preserving the corresponding covariant
derivative, but, establishing new antireductionists properties for the field
$A_{\mu I}$. For instance, at QED the electron-photon interaction is preserved
by $(\partial_{\mu}+ieA_{\mu})\psi$, but the photon properties differentiated
from usual electromagnetism as section 5 studied.

A second indication on a third quantum physics would be due to quantum
mechanics uncompleteness. A precious debate happened in 1935 with the
Einstein, Podolski, Rosen paradox. The paper title was ’Can Quantum-Mechanical
Description of Physical Reality Be Considered Complete?’ [49]. The answer was
no. In back, Bohr said yes [50]. A discussion followed, Schrödinger introduced
the cat [51]. Later on, other contribuitions came from Bohm [52], Bell [53]
and others.

Physics is waiting for an undestanding of the laws of nature beyond wave
function. Quantum mechanics effects as probability [54], tunneling [55],
entanglement [56], coherence [57] are still expecting a better comprehension.
Instead of describing the quantum phenomenon through the mathematical entity
psi, which does not have a direct physics, consider explanations through
measurable variables. The quantum description under a new perspective.
Particles quantum numbers as spin, mass, charges, discrete symmetries
expressed over acessible variables than quantum mechanics operators
methodology.

Quantum systems should be performed under the symmetry of difference acessible
physical entities. The quantum particles behaviour not necessarily described
by Schrödinger equation. Replace the abstract wave psi in phase space by the
whole quantum. The physical quantum picture of reality expressed by dynamical
readable variables. Something is expected. Quantum effects written in terms of
du Broglie entities as energy-momentum, wavelength. For instance,
entanglement, where two particles can be linked together even separated at
large distance, treated with a dispersion relation depending on palpaple
entities [36].

The third support for a next quantum physics is the variety physics. A whole
physics introducing a second type of individuation. The whole elementarity
called as varieties. The associative principle makes the ultimate constituents
express a constructivist behaviour. Moving physics from excelse ultimate
constituents to ordinary varieties description. Introducing varieties with
antireductionist properties. A qualitative change happens with respect to
classical physical and quantum mechanics. A variety physics with new quantum
features.

A quantum system is defined through varieties. The crucial difference for the
atomist quantum physics is a set action producing. Five aspects not expressed
by the atomist quantum are constructing a quantum system through gauge
symmetry. It includes five features. They are set action, antireductionist
properties, nonvirtuality, functionality, evolution.

Thus, a new definition on quantum system is given by constructivist physics.
Firstly, the set action introduces the meaning of associativity. The ultimate
constituents not only form particles as quantum systems based on varieties.
Secondly, such quanta webfication introduces antireductionist laws.
Relationships as interaction, induction, connection are enlarging reality
through the set action. Deriving a whole quantum under a set determinism.
Providing set, diversity, interdependence, nonlinearity, chance. Propitiating
growth, evolution, emergence, complexity.

A third feature is nonvirtuality. Physics is being based on virtual particles.
There is a not very clean mixing effects between virtual and real in quantum
field theory. Although the lagrangian is real, the related vaccuum and the
corresponding Feynman graphs are performed under virtualities. Examples are
photon-photon scattering through QED and Higgs decay through Standard Model.

Thus, a constructivist lagrangian is augmenting reality. It provides a more
accurated realism than atomist lagrangians. A pertubative quantum reality is
propiated through gauge invariant terms. Realistic Feynman graphs are
performed. Working on it, with pieces of reality which are nonvirtual gauge
invariant terms, it develops a real pertubative theory. Events described by
Feynman graphs based at reality.

Fourth, we would observe that a quantum embedded in a whole provides quantum
functionalities. Physical entities as spin, mass, charges and so forth, are
not in service of four interactions. They are working as funcionalities.
Depending on how the quantum is associated in the whole. A physics where
quantum functionalities replaces four interactions to a constructive scope.

Finally, the constructive physics provides nature with its most relevant
behaviour, which is evolution. Evolution is not an usual concept at current
physics. Equations are just describing entities moving at space-time. They did
not consider on how particles are modified internally. A difference appears
through grouping physics. The fields set action develops the quantum mutation.
It provides a set dynamics transforming their elementarities (varieties)
internal struture. Evolving the correspondent quantum numbers. A physics where
ultimate constituents are preserved and varieties modified.

The constructivist physics provides the ingredients for quantum evolution.
They are environment (set action), set arrow (directive), interdependence
(interwined potential fields), growth (potentiality, nonlinearity), chance
(free will, volume of circumstances), adaptation (minimal energy, boundary
condition). The antireducticionist gauge symmetry determines an environmental
physics providing the quantum internal change.

A quantum evolution happens. Eq. (2.6) introduces the meaning of evolution. A
physics further than usual determinism. It flavors the quantum take decisions
on its formation inside of a fields set environment. Fields equations are more
than moving at space-time they carry quantum numbers. Evolution objective is
the matter transformation. Give choice for objects existence. Manifest
individuation. A physics establishing quantum mutation and correspondent
quantum numbers paleontogy. It provides an individuation dynamics where each
corresponding quantum evolves through an individuation.

Concluding, a third quantum physics is expressed by an associative quantum
field theory. It introduces a physics beyond the usual atomist quantum field
theory. Its objective is to replace quantum mechanics. A perpective is
expected for covering quantum mechanics properties without the psi wave
function. Introduce quantum evolution through gauge symmetry. Instead of wave
function it introduces the antireductionist symmetry determines. It the
symmetry of difference. The whole-part complementarity becomes a quantum
physics to be studied. The whole quantum is qualified with antireductionist
properties without requering quantum mechanics.

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