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\documentclass[12pt]{iopart}
\begin{document}

\title[Physical events]{Physical events and quantum field theory without Higgs}

\author{G. A. Quznetsov\dag  \footnote[3]{email: lak@cgu.chel.su, 
quznets@yahoo.com, gunn@mail.ru}
}

\address{\dag\ 454016, Chelyabinsk-16, yD.BET. $\Phi $ N 949892 , Russia}

\begin{abstract}
All physics events are expressed by particles which similar to well-known 
elementary particles - leptons, quarks and gauge bosons. Higgs is not 
necessary.

\end{abstract}

\pacs{03.70.+k, 11.10.-z, 12.10.Dm, 14.60.-z, 14.70.-e, 12.38}

\submitto{\JPA}

\maketitle

\section{Denotations:}

$\rm c=1$: the light velocity in vacuum;

$\alpha \cap \beta $ : conjunction of events $\alpha $ and $\beta $: ''$%
\alpha $ and $\beta $'';

$\alpha \cup \beta $ : disjunction of events $\alpha $ and $\beta $: ''$%
\alpha $ or/and $\beta $'';

$\overline{\alpha }$ : event, complementary to event $\alpha $: ''not $%
\alpha $'';

$\forall $ : generality quantifier: ''for every'';

$\exists $ : existenial quantifier: ''for some'';

$\mathbf{e}_1$, $\mathbf{e}_2$, $\mathbf{e}_3$ are the Cartesian basis
vectors;

$\mathbf{x}\stackrel{Def}{=}\left( x_1\mathbf{e}_1+x_2\mathbf{e}_2+x_3%
\mathbf{e}_3\right) $;

$x_0\stackrel{Def}{=}t$;

$\int d^3\mathbf{x}\stackrel{Def}{=}\int dx_1\int dx_2\int dx_3$;

$\partial _k\stackrel{Def}{=}\partial /\partial x_k$;

$\partial _t\stackrel{Def}{=}\partial _0\stackrel{Def}{=}\partial /\partial t
$;

$\partial _k^{\prime }\stackrel{Def}{=}\partial /\partial x_k^{\prime }$;


\[
\sum_{\mathbf{k}}\stackrel{Def}{=}\sum_{k_1=-\infty }^\infty 
\sum_{k_2=-\infty }^\infty \sum_{k_3=-\infty }^\infty \mbox{;} 
\]

\[
1_2\stackrel{Def}{=}\left[ 
\begin{array}{cc}
1 & 0 \\ 
0 & 1
\end{array}
\right] \mbox{, }0_2\stackrel{Def}{=}\left[ 
\begin{array}{cc}
0 & 0 \\ 
0 & 0
\end{array}
\right] \mbox{, }
\]

\[
\beta ^{\left[ 0\right] }\stackrel{Def}{=}-1_4\stackrel{Def}{=}-\left[ 
\begin{array}{cc}
1_2 & 0_2 \\ 
0_2 & 1_2
\end{array}
\right] \mbox{, }1_8\stackrel{Def}{=}\left[ 
\begin{array}{cc}
1_4 & 0_4 \\ 
0_4 & 1_4
\end{array}
\right] \mbox{,} 
\]


\[
\gamma ^{\left[ 5\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
1_2 & 0_2 \\ 
0_2 & -1_2
\end{array}
\right] \mbox{, }0_4\stackrel{Def}{=}\left[ 
\begin{array}{cc}
0_2 & 0_2 \\ 
0_2 & 0_2
\end{array}
\right] \mbox{,}
\]

the Pauli matrices:

\[
\sigma _1=\left( 
\begin{array}{cc}
0 & 1 \\ 
1 & 0
\end{array}
\right) \mbox{, }\sigma _2=\left( 
\begin{array}{cc}
0 & -\rmi \\ 
\rmi & 0
\end{array}
\right) \mbox{, }\sigma _3=\left( 
\begin{array}{cc}
1 & 0 \\ 
0 & -1
\end{array}
\right) \mbox{;} 
\]




\section{Introduction}
I denote {\it a point event} as an event which occures in single point 
space-time, and {\it a physics event} is an event which is formed by point
events.

Obviously, the double-slits experiment proves that an elementary particle
does not exist continuously, but it is a set of point events, bounded by 
probabilities \cite{Cp1}, \cite{Cp2},  \cite{W}, \cite{J}, \cite{Jd3}.

These probabilities are expressed by spinor functions and by {\it the 
creation} and {\it annihilation of probability operators} which similar
to the field operators of QFT. The motion equations in form of the Dirac 
equations with the additional fields are obtained for the spinor functions.
Some of these additional fields behave as mass members, and other - as gauge 
fields.
      
  The probability current 3+1 vectors and {\it the average} and {\it the local 
velocity of the probability propagation 3-vectors} are defined by the motion 
equations.

  A set $\widetilde{C}$ of complex $n\times n$ matrices is denoted as {\it 
Clifford's set} \cite{Md} if the following conditions are fulfilled:

if $\alpha _k\in \widetilde{C}$  $\alpha _r\in \widetilde{C}$ then
 $\alpha _k\alpha _r+\alpha _r\alpha _k=2\delta _{k,r}$;

if $\alpha _k\alpha _r+\alpha _r\alpha _k=2\delta _{k,r}$ 
for all elements of $\alpha _r$ ⢠ $\widetilde{C}$ then
$\alpha _k\in \widetilde{C}$.


If $n=4$ then Clifford's set either contains $3$ matrices (\textit{Clifford's 
triplet}) or contains $5$ matrices (\textit{Clifford's pentad}).

Only 6 Clifford's pentads exist \cite{Md}:

one \textit{light pentad} $\beta $:

\numparts
\begin{equation}
\beta ^{\left[ 1\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
\sigma _1 & 0_2 \\ 
0_2 & -\sigma _1
\end{array}
\right] \mbox{, }\beta ^{\left[ 2\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
\sigma _2 & 0_2 \\ 
0_2 & -\sigma _2
\end{array}
\right] \mbox{, }\beta ^{\left[ 3\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
\sigma _3 & 0_2 \\ 
0_2 & -\sigma _3
\end{array}
\right] \mbox{,} \label{lghr}
\end{equation}

\begin{equation}
\gamma ^{\left[ 0\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
0_2 & 1_2 \\ 
1_2 & 0_2
\end{array}
\right] \mbox{, } \label{lghr1}
\end{equation} 

\begin{equation}
\beta ^{\left[ 4\right] }\stackrel{Def}{=}\rmi\cdot
\left[ 
\begin{array}{cc}
0_2 & 1_2 \\ 
-1_2 & 0_2
\end{array}
\right] \mbox{;} \label{lghr2}
\end{equation} 
\endnumparts

three {\it chromatic} pentads (see Appendix): {\it red, blue} and {\it green};
and two {\it taste} pentad: {\it sweet} and {\it bitter} pentads.

At first the motion equations, held the light pentad elements, only, are 
considered. Such equations are denoted as {\it the leptonn motion equations}.

The Dirac equation contains four Clifford's pentads, only. Three of these 
elements \eref{lghr} accord to three space coordinates, and fourth element 
\eref{lghr1} either constitute the mass member or accords to the time 
coordinate. But Clifford's pentad holds five elements. Certainly fifth 
element \eref{lghr2} of pentad should be added to the motion equation. 
That is the Dirac equation mass part will hold two members. Moreover, if 
two additional quasi-space coordinates shall be put in accordance to these
two Clifford's mass pentads (\eref{lghr1} and \eref{lghr2}) then the 
homogeheous Dirac equation will be obtained. All five elements of Clifford's
pentads and all five space coordinates are held alike in this equation. 
All local velocities magnitudes equal to unit ($\rm c$) in such five-
dimensional space.

An equation of motion, overdefinited by the such ways, is invariant for 
a rotation in 2-space of fourth and fifth coordinates. This transformation 
defines a field, similar to the $B$-boson field.

Since values of probability are not defined absolutely exactly, the additional 
fields, forming mass members, are expressible by sufficiently fine stratums. 
And values of that two spatial coordinates, which are added for homogeneity of 
mass members, can be delimitted by a large number without the generality 
restriction. In this case a  masses spectrum is got discrete, and for every 
3+1 space-time points: either single mass is put in this point or this point 
is empty.

A mass is expressed by the 2-root from sum of two quadrates of integer numbers 
from two mass members of the motion equation:

\[
m_0=\sqrt{n_0^2+s_0^2} 
\]

But this motion equation is invariant for rotations in 2-space of fourth and 
fifth coordinates. That is a mass must be expressed by natural number, and 
numbers in the mass members must remain integer in these rotations.
 
Hence a mass and the mass members numbers, making this mass, must make up
Pythagorean triplet $\left\langle m_0;n_0,s_0\right\rangle $ \cite{Pf}.
Here $m_0$ is denoted as {\it a triplet father}. One triplet must be 
substituted by another one with the same father at rotations. That is 
{\it a family of Pythagorean triplets with the same father} for which one 
this family triplets is substituted by other triplet of this family must be 
exist for a setted calculation precision of the rotation angle. Fathers 
exist in the distant parts of the natural number array for a high precision. 
I believe, what these families of the Pythagorean triplets conform to the 
families of elementary particles.

The {\it particles creation and annihilation operators} are denoted as the
Fourier transformations of corresponding operators for probability, and 
antiparticles are denoted by the standard way.

Subsequently all unitary transformations on the two-masses functions, which 
retain the probability 4-vector, are considered. The transformations, adequate 
to electroweak gauge fields, exist among these unitary transformations. These
electroweak unitary transformations are expressed by rotations in 2-space of 
fourth and fifth coordinates, too. Particles, similar to neutrino ({\it 
neutrinno}), arise in these transformations. {\it Neutrinno} prove to be 
connected essentially with their {\it leptonns}.

The motions equations is invariant for these transformations, and fields, 
similar to the boson ones, arise in result. The massless field


\[
F_{\mu ,\nu }=\partial _\mu W_\nu -\partial _\nu W_\mu -i\frac{g_2}2\left(
W_\mu W_\nu -W_\nu W_\mu \right)
\]

denoted by usual way, but the Euler-Lagrange equations give result that 
though field $F_{\mu ,\nu }$ is massless, its generating fields  $W_\mu $ 
behave as fields with nonzero masses.

Massless field $A$ and the massive field $Z$ are denoted by standard way by 
fields $B$ and $W$.

The leptonn motion equation is invariant for rotations in 3-space of first 
usual spatial coordinates and and for Lorentz's rotations in 3+1 space-time.
The motion equations, maked by the chromatic pentads, are mixed between each 
other. That is particles, conformed to pentads of different colors, is 
unseparable in space-time (confinement). Two sorts of three colors of such 
particles are held in one family. Hence - six. I denote these particles as 
{\it quarrks}. 

\section{Events}

Let $\Omega _0$ be a finite set of $N`$ point-events $A_\iota \left(
t^{\left( \iota \right) },\mathbf{x}^{\left( \iota \right) }\right) $ ($%
\iota $ is a natural index number).

Let $\Omega $ be set of events for which:

1) $\Omega _0$ $\subseteq \Omega ;$

2) if $B\in \Omega $ and $C\in \Omega $ then $\left( B\cup C\right) \in
\Omega $ and $\left( B\cap C\right) \in \Omega $;

3) if $B\in \Omega $ then $\overline{B}\in \Omega $;

4) if $A_\iota \left( t^{\left( \iota \right) },\mathbf{x}^{\left( \iota
\right) }\right) \in \Omega $ then

$\left( \forall \mathbf{x}^{\left( \iota \right) }:A_\iota \left( t^{\left(
\iota \right) },\mathbf{x}^{\left( \iota \right) }\right) \right) \in \Omega 
$ and $\left( \exists \mathbf{x}^{\left( \iota \right) }:A_\iota \left(
t^{\left( \iota \right) },\mathbf{x}^{\left( \iota \right) }\right) \right)
\in \Omega $.

I denote elements of $\Omega $ as \textit{a physics events.}

Let:

\[
\alpha _0\left( t\right) \stackrel{Def}{=}\cap _{\iota =1}^{N`}\forall 
\mathbf{x}^{\left( \iota \right) }:\overline{A_\iota \left( t,\mathbf{x}%
^{\left( \iota \right) }\right) }\mbox{;} 
\]

\[
\begin{array}{c}
\alpha _1\left( t,\mathbf{x}^{\left( 1\right) }\right) \stackrel{Def}{=} \\ 
\stackrel{Def}{=}A_1\left( t,\mathbf{x}^{\left( 1\right) }\right) \cap
\left( \cap _{\iota \neq 1}^{N`}\forall \mathbf{x}^{\left( \iota \right) }:%
\overline{A_\iota \left( t,\mathbf{x}^{\left( \iota \right) }\right) }%
\right) \mbox{;}
\end{array}
\]

\[
\begin{array}{c}
\alpha _2\left( t,\mathbf{x}^{\left( 2\right) }\right) \stackrel{Def}{=} \\ 
\stackrel{Def}{=}A_2\left( t,\mathbf{x}^{\left( 2\right) }\right) \cap
\left( \cap _{\iota \neq 2}^{N`}\forall \mathbf{x}^{\left( \iota \right) }:%
\overline{A_\iota \left( t,\mathbf{x}^{\left( \iota \right) }\right) }%
\right) \mbox{;}
\end{array}
\]

$\ldots $

\[
\begin{array}{c}
\alpha _\iota \left( t,\mathbf{x}^{\left( \iota \right) }\right) \stackrel{%
Def}{=} \\ 
\stackrel{Def}{=}A_\iota \left( t,\mathbf{x}^{\left( \iota \right) }\right)
\cap \left( \cap _{\iota \neq \iota }^{N`}\forall \mathbf{x}^{\left( \iota
\right) }:\overline{A_\iota \left( t,\mathbf{x}^{\left( \iota \right)
}\right) }\right) \mbox{;}
\end{array}
\]

$\ldots $

\[
\alpha _{1,2}\left( t,\mathbf{x}^{\left( 1\right) },\mathbf{x}^{\left(
2\right) }\right) \stackrel{Def}{=} 
\]

\[
\stackrel{Def}{=}\left( A_1\left( t,\mathbf{x}^{\left( 1\right) }\right)
\cap A_2\left( t,\mathbf{x}^{\left( 2\right) }\right) \right) \cap \left(
\cap _{\iota \notin \left\{ 1,2\right\} }^{N`}\forall \mathbf{x}^{\left(
\iota \right) }:\overline{A_\iota \left( t,\mathbf{x}^{\left( \iota \right)
}\right) }\right) \mbox{;} 
\]

$\ldots $

\[
\alpha _{\iota _1,\iota _2,\ldots ,\iota _s}\left( t,\mathbf{x}^{\left(
\iota _1\right) },\mathbf{x}^{\left( \iota _2\right) },\ldots ,\mathbf{x}%
^{\left( \iota _s\right) }\right) \stackrel{Def}{=} 
\]

\[
\stackrel{Def}{=}\left( 
\begin{array}{c}
\left( A_{\iota _1}\left( t,\mathbf{x}^{\left( \iota _1\right) }\right) \cap
A_{\iota _2}\left( t,\mathbf{x}^{\left( \iota _2\right) }\right) \cap \cdots
\cap A_{\iota _s}\left( t,\mathbf{x}^{\left( \iota _s\right) }\right)
\right) \cap \\ 
\left( \cap _{\iota \notin \left\{ \iota _1,\iota _2,\ldots ,\iota
_s\right\} }^{N`}\left( \forall \mathbf{x}^{\left( \iota \right) }:\overline{%
A_\iota \left( t,\mathbf{x}^{\left( \iota \right) }\right) }\right) \right)
\end{array}
\right) \mbox{;} 
\]

Let $\mathbf{P}$ be a probability function, defined on $\Omega $.

Because $A_\iota \left( t^{\left( \iota \right) },\mathbf{x}^{\left( \iota
\right) }\right) $ is a point-event then for $m\neq n$:

\[
A_\iota \left( t_m^{\left( \iota \right) },\mathbf{x}_m^{\left( \iota
\right) }\right) \cap A_\iota \left( t_n^{\left( \iota \right) },\mathbf{x}%
_n^{\left( \iota \right) }\right) 
\]

is an impossible event. That is

\[
\mathbf{P}\left( A_\iota \left( t_m^{\left( \iota \right) },\mathbf{x}%
_m^{\left( \iota \right) }\right) \cap A_\iota \left( t_n^{\left( \iota
\right) },\mathbf{x}_n^{\left( \iota \right) }\right) \right) =0 
\]

for $m\neq n$. And by idempotence:

\[
\mathbf{P}\left( A_\iota \left( t^{\left( \iota \right) },\mathbf{x}^{\left(
\iota \right) }\right) \cap A_\iota \left( t^{\left( \iota \right) },\mathbf{%
x}^{\left( \iota \right) }\right) \right) =\mathbf{P}\left( A_\iota \left(
t^{\left( \iota \right) },\mathbf{x}^{\left( \iota \right) }\right) \right) 
\]

Let $\mathbf{H}$ be a Hilbert space.

Let an linear operators $\psi _j^{\left( \iota \right) }\left( \mathbf{x}%
^{\left( \iota \right) }\right) $ ( $j\in \left\{ 1,2,3,4\right\} $) work on
elements of $\mathbf{H}$. And these operators have got the following
properties:

1. $\mathbf{H}$ contains the element $\Phi _0$ for which:

\[
\Phi _0^{\dagger }\Phi _0=1 
\]

and

\[
\psi _j^{\left( \iota \right) }\Phi _0=0\mbox{, }\Phi _0^{\dagger }\psi
_j^{\left( \iota \right) \dagger }=0\mbox{;} 
\]

2.

\[
\psi _j^{\left( \iota \right) }\left( \mathbf{x}^{\left( \iota \right)
}\right) \psi _j^{\left( \iota \right) }\left( \mathbf{x}^{\left( \iota
\right) }\right) =0 
\]

and

\[
\psi _j^{\left( \iota \right) \dagger }\left( \mathbf{x}^{\left( \iota
\right) }\right) \psi _j^{\left( \iota \right) \dagger }\left( \mathbf{x}%
^{\left( \iota \right) }\right) =0\mbox{;} 
\]

3.

\begin{equation}
\eqalign{
\left\{ \psi _{j^{\prime }}^{\left( \iota ^{\prime }\right) \dagger }\left( 
\mathbf{y}^{\left( \iota ^{\prime }\right) }\right) ,\psi _j^{\left( \iota
\right) }\left( \mathbf{x}^{\left( \iota \right) }\right) \right\} \stackrel{%
Def}{=}\\ 
\psi _{j^{\prime }}^{\left( \iota ^{\prime }\right) \dagger }\left( \mathbf{y%
}^{\left( \iota ^{\prime }\right) }\right) \psi _j^{\left( \iota \right)
}\left( \mathbf{x}^{\left( \iota \right) }\right) +\psi _j^{\left( \iota
\right) }\left( \mathbf{x}^{\left( \iota \right) }\right) \psi _{j^{\prime
}}^{\left( \iota ^{\prime }\right) \dagger }\left( \mathbf{y}^{\left( \iota
^{\prime }\right) }\right) = \\
=\delta _{\iota ^{\prime },\iota }\delta \left( \mathbf{y}^{\left( \iota
^{\prime }\right) }-\mathbf{x}^{\left( \iota \right) }\right) \delta
_{j^{\prime },j}\mbox{.}} \label{dddd} 
\end{equation}




I denote operator $\psi ^{\left( \iota \right) \dagger }\left( \mathbf{x}%
\right) $ as\textit{\ a creation operator} and $\psi ^{\left( \iota \right)
}\left( \mathbf{x}\right) $ -as \textit{an annihilation operator of the
event }$A_\iota $ \textit{probability in point} $\mathbf{x}$.

\section{An one event case}
\subsection{Hamiltonians}

Let $N`=1$.

Let real function $\rho \left( t,\mathbf{x}\right) $ be \textit{the
probability density of the event} $\alpha _1\left( t,\mathbf{x}\right) $.
That is for each domain $D$ ($D\subseteq R^3$):

\[
\int_Dd^3\mathbf{x}\cdot \rho \left( t,\mathbf{x}\right) \stackrel{Def}{=}%
\mathbf{P}\left( \exists \mathbf{x}\in D:\alpha _1\left( t,\mathbf{x}\right)
\right) \mbox{.} 
\]

Complex functions $\varphi _j\left( t,\mathbf{x}\right) $($j\in \left\{ 1,2,3,4\right\} $) exist for
which:

\begin{equation}
\rho \left( t,\mathbf{x}\right) =\sum_{j=1}^4\varphi _j^{*}\left( t,\mathbf{x%
}\right) \varphi _j\left( t,\mathbf{x}\right) \mbox{.}  \label{ro}
\end{equation}

Let us denote $\Psi \left( t,\mathbf{x}\right) $ as the following:

\begin{equation}
\Psi \left( t,\mathbf{x}\right) \stackrel{Def}{=}\sum_{j=1}^4\varphi
_j\left( t,\mathbf{x}\right) \psi _j^{\dagger }\left( \mathbf{x}\right) \Phi
_0  \label{Sat}
\end{equation}

From (\ref{dddd}):

\[
\Psi ^{\dagger }\left( t,\mathbf{x}^{\prime }\right) \Psi \left( t,\mathbf{x}%
\right) =\sum_{j=1}^4\varphi _j^{*}\left( t,\mathbf{x}^{\prime }\right)
\varphi _j\left( t,\mathbf{x}\right) \delta \left( \mathbf{x}^{\prime }-%
\mathbf{x}\right) \mbox{.} 
\]

I consider events, only, for which a tiny real positive number $h$,
fulfilled the following condition:

if $\left| x_r\right| \geq \frac \pi h$ ($r\in \left\{ 1,2,3\right\} $) then

\[
\varphi _j\left( t,\mathbf{x}\right) =0 
\]

exists, and $\left( V\right) $ be denoted as the following: $\mathbf{x}\in
\left( V\right) $ if and if, only, $\left| x_r\right| \leq \frac \pi h$ for $%
r\in \left\{ 1,2,3\right\} $. That is:

\[
\int_{\left( V\right) }d^3\mathbf{x}=\int_{-\frac \pi h}^{\frac \pi
h}dx_1\int_{-\frac \pi h}^{\frac \pi h}dx_2\int_{-\frac \pi h}^{\frac \pi
h}dx_3\mbox{.} 
\]

If

\[
\overline{\mathbf{x}}\left( t\right) =\int d^3\mathbf{x}\cdot \mathbf{x}\rho
\left( t,\mathbf{x}\right) 
\]

then $\overline{\mathbf{x}}\left( t\right) $ is \textit{the average
coordinate of the event} $\alpha _1$ and:

\[
\overline{\mathbf{x}}\left( t\right) =\int d^3\mathbf{x}\cdot \mathbf{x}%
\sum_{j=1}^4\varphi _j^{*}\left( t,\mathbf{x}\right) \varphi _j\left( t,%
\mathbf{x}\right) \mbox{,} 
\]

and \textit{the average velocity of the event} $\alpha _1$ {\it probability} is:

\begin{equation}
\eqalign{
\overline{\mathbf{v}}\left( t\right) =\partial _t\overline{\mathbf{x}}\left(
t\right) = \\ 
=\int d^3\mathbf{x}\cdot \mathbf{x}\sum_{j=1}^4\left( \left( \partial
_t\varphi _j\left( t,\mathbf{x}\right) \right) ^{*}\varphi _j\left( t,%
\mathbf{x}\right) +\varphi _j^{*}\left( t,\mathbf{x}\right) \left( \partial
_t\varphi _j\left( t,\mathbf{x}\right) \right) \right)}
\label{v}
\end{equation}

The following system of equations with 16 unknown complex functions $Q_{j,k}$
$\left( t,\mathbf{x}\right) $ ($Q_{j,k}\stackrel{Def}{=}-\varpi _{j,k}+%
\mathrm{i}\vartheta _{j,k}$ and $j\in \left\{ 1,2,3,4\right\} $, $k\in
\left\{ 1,2,3,4\right\} $)

\begin{equation}
\cases
{Q_{1,1}\varphi _1+Q_{1,2}\varphi _2+Q_{1,3}\varphi _3+Q_{1,4}\varphi _4= \\ 
=\partial _t\varphi _1-\sum_{k=1}^4\left( \beta _{1,k}^{\left[ 1\right]
}\partial _1+\beta _{1,k}^{\left[ 2\right] }\partial _2+\beta _{1,k}^{\left[
3\right] }\partial _3\right) \varphi _k; \\ 
Q_{2,1}\varphi _1+Q_{2,2}\varphi _2+Q_{2,3}\varphi _3+Q_{2,4}\varphi _4= \\ 
=\partial _t\varphi _2-\sum_{k=1}^4\left( \beta _{2,k}^{\left[ 1\right]
}\partial _1+\beta _{2,k}^{\left[ 2\right] }\partial _2+\beta _{2,k}^{\left[
3\right] }\partial _3\right) \varphi _k; \\
Q_{3,1}\varphi _1+Q_{3,2}\varphi _2+Q_{3,3}\varphi _3+Q_{3,4}\varphi _4= \\ 
=\partial _t\varphi _3-\sum_{k=1}^4\left( \beta _{3,k}^{\left[ 1\right]
}\partial _1+\beta _{3,k}^{\left[ 2\right] }\partial _2+\beta _{3,k}^{\left[
3\right] }\partial _3\right) \varphi _k; \\  
Q_{4,1}\varphi _1+Q_{4,2}\varphi _2+Q_{4,3}\varphi _3+Q_{4,4}\varphi _4= \\ 
=\partial _t\varphi _4-\sum_{k=1}^4\left( \beta _{4,k}^{\left[ 1\right]
}\partial _1+\beta _{4,k}^{\left[ 2\right] }\partial _2+\beta _{4,k}^{\left[
3\right] }\partial _3\right) \varphi _k; \\
Q_{1,1}^{*}=-Q_{1,1}; \\ 
Q_{2,2}^{*}=-Q_{2,2}; \\ 
Q_{3,3}^{*}=-Q_{3,3}; \\ 
Q_{4,4}^{*}=-Q_{4,4}; \\ 
Q_{1,2}^{*}=-Q_{2,1}; \\ 
Q_{1,3}^{*}=-Q_{3,1}; \\ 
Q_{1,4}^{*}=-Q_{4,4}; \\ 
Q_{2,3}^{*}=-Q_{3,2}; \\ 
Q_{2,4}^{*}=-Q_{4,2}; \\ 
Q_{3,4}^{*}=-Q_{4,3}} \label{sys}
\end{equation}   


is the linear real system of 24 equations with 32 real unknown functions $%
\varpi _{j,k}$ and $\vartheta _{j,k}$. This system matrices (the dilated
matrix and the basis matrix) ranks equal to 24. Hence this system has got
solutions for every $\varphi $ by the Kronecker-Capelli theorem.

Therefore:

\begin{equation}
\partial _t\varphi _j=\sum_{k=1}^4\left( \beta _{j,k}^{\left[ 1\right]
}\partial _1+\beta _{j,k}^{\left[ 2\right] }\partial _2+\beta _{j,k}^{\left[
3\right] }\partial _3+Q_{j,k}\right) \varphi _k\mbox{.}  \label{ham}
\end{equation}

In that case if

\[
\widehat{H}_{j,k}\stackrel{Def}{=}\mathrm{i}\left( \beta _{j,k}^{\left[
1\right] }\partial _1+\beta _{j,k}^{\left[ 2\right] }\partial _2+\beta
_{j,k}^{\left[ 3\right] }\partial _3+Q_{j,k}\right) 
\]

then $\widehat{H}$ is denoted as \textit{hamiltonian} of the moving with
equation (\ref{ham}).

If $\mathcal{H}\left( t,\mathbf{x}\right) $ is denoted as:

\begin{equation}
\mathcal{H}\left( t,\mathbf{x}\right) \stackrel{Def}{=}\sum_{j=1}^4\psi
_j^{\dagger }\left( \mathbf{x}\right) \sum_{k=1}^4\widehat{H}_{j,k}\left( t,%
\mathbf{x}\right) \psi _k\left( \mathbf{x}\right)  \label{hmm}
\end{equation}

then $\mathcal{H}\left( \tau ,\mathbf{x}\right) $ is called as \textit{the
hamiltonian density}. From (\ref{Sat}):

\[
-\mathrm{i}\int d^3\mathbf{x}\cdot \mathcal{H}\left( t,\mathbf{x}\right)
\Psi \left( t,\mathbf{x}_0\right) =\partial _t\Psi \left( t,\mathbf{x}%
_0\right) \mbox{.} 
\]

Because

\[
\beta _{j,k}^{\left[ s\right] *}=\beta _{k,j}^{\left[ s\right] } 
\]

and $\varphi _j\left( t,\pm \mathbf{\infty }\right) =0$ then from (\ref{v}):

\[
\overline{\mathbf{v}}\left( t\right) =-\int d^3\mathbf{x}\cdot \left( 
\begin{array}{c}
\mathbf{e}_1\sum_{k=1}^4\sum_{j=1}^4\varphi _j^{*}\beta _{j,k}^{\left[
1\right] }\varphi _k+ \\ 
+\mathbf{e}_2\sum_{k=1}^4\sum_{j=1}^4\varphi _j^{*}\beta _{j,k}^{\left[
2\right] }\varphi _k+ \\ 
+\mathbf{e}_3\sum_{k=1}^4\sum_{j=1}^4\varphi _j^{*}\beta _{j,k}^{\left[
3\right] }\varphi _k
\end{array}
\right) \mbox{.} 
\]

If denote:

\begin{equation}
\begin{array}{c}
j_1\stackrel{Def}{=}-\sum_{k=1}^4\sum_{j=1}^4\varphi _j^{*}\beta
_{j,k}^{\left[ 1\right] }\varphi _k\mbox{,} \\ 
j_2\stackrel{Def}{=}-\sum_{k=1}^4\sum_{j=1}^4\varphi _j^{*}\beta
_{j,k}^{\left[ 2\right] }\varphi _k\mbox{,} \\ 
j_3\stackrel{Def}{=}-\sum_{k=1}^4\sum_{j=1}^4\varphi _j^{*}\beta
_{j,k}^{\left[ 3\right] }\varphi _k
\end{array}
\label{j}
\end{equation}

then

\[
\mathbf{j}=j_1\mathbf{e}_1+j_2\mathbf{e}_2+j_3\mathbf{e}_3 
\]

is \textit{the probability current} for which:

\[
\overline{\mathbf{v}}\left( t\right) =\int d^3\mathbf{x}\cdot \mathbf{j} 
\]

If

\begin{equation}
\mathbf{j}\stackrel{Def}{=}\rho \mathbf{u}  \label{v2}
\end{equation}

then

\[
\overline{\mathbf{v}}\left( t\right) =\int d^3\mathbf{x}\cdot \rho \mathbf{u}%
\mbox{,} 
\]

that is $\mathbf{u}$ is \textit{the local velocity} of the $\alpha _1$
probability.

Formula (\ref{ham}) has got the following matrix form:

\begin{equation}
\partial _t\varphi =\left( \beta ^{\left[ 1\right] }\partial _1+\beta
^{\left[ 2\right] }\partial _2+\beta ^{\left[ 3\right] }\partial _3+\widehat{%
Q}\right) \varphi \mbox{,}  \label{ham1}
\end{equation}

here:

\[
\varphi =\left[ 
\begin{array}{c}
\varphi _1 \\ 
\varphi _2 \\ 
\varphi _3 \\ 
\varphi _4
\end{array}
\right] 
\]

and

\[
\widehat{Q}=\left[ 
\begin{array}{cccc}
\mathrm{i}\vartheta _{1,1} & \mathrm{i}\vartheta _{1,2}-\varpi _{1,2} & 
\mathrm{i}\vartheta _{1,3}-\varpi _{1,3} & \mathrm{i}\vartheta _{1,4}-\varpi
_{1,4} \\ 
\mathrm{i}\vartheta _{1,2}+\varpi _{1,2} & \mathrm{i}\vartheta _{2,2} & 
\mathrm{i}\vartheta _{2,3}-\varpi _{2,3} & \mathrm{i}\vartheta _{2,4}-\varpi
_{2,4} \\ 
\mathrm{i}\vartheta _{1,3}+\varpi _{1,3} & \mathrm{i}\vartheta _{2,3}+\varpi
_{2,3} & \mathrm{i}\vartheta _{3,3} & \mathrm{i}\vartheta _{3,4}-\varpi
_{3,4} \\ 
\mathrm{i}\vartheta _{1,4}+\varpi _{1,4} & \mathrm{i}\vartheta _{2,4}+\varpi
_{2,4} & \mathrm{i}\vartheta _{3,4}+\varpi _{3,4} & \mathrm{i}\vartheta
_{4,4}
\end{array}
\right] 
\]

Let $\Theta _0$, $\Theta _2$, $\Upsilon _0$ and $\Upsilon _3$ are the
solution of the following system of equations:

\[
\cases{ 
-\Theta _0+\Theta _3-\Upsilon _0+\Upsilon _3=\vartheta _{1,1}\mbox{,} \\ 
-\Theta _0-\Theta _3-\Upsilon _0-\Upsilon _3=\vartheta _{2,2}\mbox{,} \\ 
-\Theta _0-\Theta _3+\Upsilon _0+\Upsilon _3=\vartheta _{3,3}\mbox{,} \\ 
-\Theta _0+\Theta _3+\Upsilon _0-\Upsilon _3=\vartheta _{4,4};
}
\]

$\Theta _1$ and $\Upsilon _1$ are the solution of the following system of
equations:

\[
\cases{ 
\Theta _1+\Upsilon _1=\vartheta _{1,2}\mbox{,} \\ 
-\Theta _1+\Upsilon _1=\vartheta _{3,4}\mbox{;}
}
\]

$\Theta _2$ and $\Upsilon _2$ are the solution of the following system of
equations:

\[
\cases{ 
-\Theta _2-\Upsilon _2=\varpi _{1,2}\mbox{,} \\ 
\Theta _2-\Upsilon _2=\varpi _{3,4}\mbox{.}
}
\]

and further:

\[
\cases{ 
M_0+M_{3,0}=\vartheta _{1,3}\mbox{,} \\ 
M_0-M_{3,0}=\vartheta _{2,4}\mbox{,}
}
\]

\[
\cases{ 
M_4+M_{3,4}=\varpi _{1,3}\mbox{,} \\ 
M_4-M_{3,4}=\varpi _{2,4}\mbox{,}
}
\]

\[
\cases{ 
M_{1,0}-M_{2,4}=\vartheta _{1,4}, \\ 
M_{1,0}+M_{2,4}=\vartheta _{2,3},
}
\]

\[
\cases{ 
M_{1,4}-M_{2,0}=\varpi _{1,4}\mbox{,} \\ 
M_{1,4}+M_{2,0}=\varpi _{2,3}
}
\]

then from (\ref{ham1}):


\begin{equation}
\eqalign{
\left(\partial _t+\mathrm{i}\Theta _0+\mathrm{i}\Upsilon _0\gamma ^{\left[
5\right] }\right) \varphi = \\
= 
\left(
\begin{array}{c}
\sum_{k=1}^3\beta ^{\left[ 2\right] }\left( \partial _k+\mathrm{i}\Theta _k+%
\mathrm{i}\Upsilon _k\gamma ^{\left[ 5\right] }\right) +\mathrm{i}M_0\gamma
^{\left[ 0\right] }+\mathrm{i}M_4\beta ^{\left[ 4\right] } \\ 
-\mathrm{i}M_{1,0}\gamma _\zeta ^{\left[ 0\right] }-\mathrm{i}M_{1,4}\zeta
^{\left[ 4\right] }- \\ 
-\mathrm{i}M_{2,0}\gamma _\eta ^{\left[ 0\right] }-\mathrm{i}M_{2,4}\eta
^{\left[ 4\right] }- \\ 
-\mathrm{i}M_{3,0}\gamma _\theta ^{\left[ 0\right] }-\mathrm{i}M_{3,4}\theta
^{\left[ 4\right] }
\end{array}
\right) \varphi} 
\label{ham0}
\end{equation}


Here summands

\[
\begin{array}{c}
-\mathrm{i}M_{1,0}\gamma _\zeta ^{\left[ 0\right] }-\mathrm{i}M_{1,4}\zeta
^{\left[ 4\right] }- \\ 
-\mathrm{i}M_{2,0}\gamma _\eta ^{\left[ 0\right] }-\mathrm{i}M_{2,4}\eta
^{\left[ 4\right] }- \\ 
-\mathrm{i}M_{3,0}\gamma _\theta ^{\left[ 0\right] }-\mathrm{i}M_{3,4}\theta
^{\left[ 4\right] }\mbox{.}
\end{array}
\]

contain the chromatic pentads elements and

\[
\sum_{k=1}^3\beta ^{\left[ 2\right] }\left( \partial _k+\mathrm{i}\Theta _k+%
\mathrm{i}\Upsilon _k\gamma ^{\left[ 5\right] }\right) +\mathrm{i}M_0\gamma
^{\left[ 0\right] }+\mathrm{i}M_4\beta ^{\left[ 4\right] } 
\]

contain the light pentad elements, only. I denote the following sum

\[
\widehat{H}_l\stackrel{Def}{=}\sum_{k=1}^3\beta ^{\left[ 2\right] }\left( 
\mathrm{i}\partial _k-\Theta _k-\Upsilon _k\gamma ^{\left[ 5\right] }\right)
-M_0\gamma ^{\left[ 0\right] }-M_4\beta ^{\left[ 4\right] } 
\]

as \textit{the leptonn hamiltonian}.

\subsection{Rotation of $x_5Ox_4$ and $B$-bosonn}

If denote (\ref{j}):

\[
j_4\stackrel{Def}{=}-\varphi ^{*}\beta ^{\left[ 4\right] }\varphi 
\mbox{ and
}j_5\stackrel{Def}{=}-\varphi ^{*}\gamma ^{\left[ 0\right] }\varphi 
\]

and (\ref{v2}):

\[
\rho u_4\stackrel{Def}{=}j_4\mbox{ and }\rho u_5\stackrel{Def}{=}j_5\mbox{,} 
\]

then

\[
u_1^2+u_2^2+u_3^2+u_4^2+u_5^2=1\mbox{.} 
\]

Hence of only all five elements of the Clifford pentad lends the entire kit
of the velocity components and, for the completeness, yet two ''space''
coordinates $x_5$ and $x_4$ should be added to our three $x_1,x_2,x_3$.

Let us denote: 


\begin{eqnarray}
\widetilde{\varphi }\left( t,x_1,x_2,x_3,x_5,x_4\right) 
 \stackrel{Def}{=}\varphi \left( t,x_1,x_2,x_3\right) \cdot \nonumber \\ 
\cdot \left( \exp \left( -\mathrm{i}\left( x_5M_0\left( t,x_1,x_2,x_3\right)
+x_4M_4\left( t,x_1,x_2,x_3\right) \right) \right) \right) \mbox{.} \nonumber
\end{eqnarray}

In this case from (\ref{ham0}) for

\[
\begin{array}{c}
M_{1,0}=0\mbox{, }M_{1,4}=0\mbox{,} \\ 
M_{2,0}=0\mbox{, }M_{2,4}=0\mbox{,} \\ 
M_{3,0}=0\mbox{\textrm{, }}M_{3,4}=0
\end{array}
\]

the motion equation for the leptonn hamiltonian is the following:

\begin{equation}
\left( \sum_{\mu =0}^3\beta ^{\left[ \mu \right] }\left( \mathrm{i}\partial
_\mu -\Theta _\mu -\Upsilon _\mu \gamma ^{\left[ 5\right] }\right) +\gamma
^{\left[ 0\right] }\mathrm{i}\partial _5+\beta ^{\left[ 4\right] }\mathrm{i}%
\partial _4\right) \widetilde{\varphi }=0.  \label{gkk}
\end{equation}

Let $g_1$ be a positive real number and for $\mu \in \left\{ 0,1,2,3\right\} 
$: $F_\mu $ and $B_\mu $ be the solutions of the following systems of the
equations:

\[
\cases{ 
-0.5g_1B_\mu +F_\mu =-\Theta _\mu -\Upsilon _\mu, \\ 
-g_1B_\mu +F_\mu =-\Theta _\mu +\Upsilon _\mu.
}  
\]

Let \textit{the charge matrix} be denoted as the following:

\[
Y\stackrel{Def}{=}-\left[ 
\begin{array}{cc}
1_2 & 0_2 \\ 
0_2 & 2\cdot 1_2
\end{array}
\right] \mbox{.} 
\]

Hence from (\ref{gkk}):

\begin{equation}
\left( \sum_{\mu =0}^3\beta ^{\left[ \mu \right] }\left( \mathrm{i}\partial
_\mu +F_\mu +0.5g_1YB_\mu \right) +\gamma ^{\left[ 0\right] }\mathrm{i}%
\partial _5+\beta ^{\left[ 4\right] }\mathrm{i}\partial _4\right) \widetilde{%
\varphi }=0\mbox{.}  \label{gkB}
\end{equation}

Let $\chi \left( t,x_1,x_2,x_3\right) $ be a real function and:

\[
\widetilde{U}\left( \chi \right) \stackrel{Def}{=}\left[ 
\begin{array}{cc}
\exp \left( \mathrm{i}\frac \chi 2\right) 1_2 & 0_2 \\ 
0_2 & \exp \left( \mathrm{i}\chi \right) 1_2
\end{array}
\right] \mbox{.} 
\]

Because

\[
\partial _\mu \widetilde{U}=-\mathrm{i}\frac{\partial _\mu \chi }2Y%
\widetilde{U} 
\]

and

%\begin{equation}
\[
\eqalign{
\widetilde{U}^{\dagger }\gamma ^{\left[ 0\right] }\widetilde{U}=\gamma
^{\left[ 0\right] }\cos \frac \chi 2+\beta ^{\left[ 4\right] }\sin \frac
\chi 2\mbox{,} \\ 
\widetilde{U}^{\dagger }\beta ^{\left[ 4\right] }\widetilde{U}=\beta
^{\left[ 4\right] }\cos \frac \chi 2-\gamma ^{\left[ 0\right] }\sin \frac
\chi 2\mbox{,} \\ 
\widetilde{U}^{\dagger }\widetilde{U}=1_4\mbox{,} \\ 
\widetilde{U}^{\dagger }Y\widetilde{U}=Y\mbox{,} \\ 
\beta ^{\left[ k\right] }\widetilde{U}=\widetilde{U}\beta ^{\left[ k\right]}}
\]
%\end{equation}

for $k\in \left\{ 1,2,3\right\} $

then the motion equation (\ref{gkB}) is invariant for the following
transformation (rotation of $x_4Ox_5$):


\begin{equation}
\eqalign{
x_4\rightarrow x_4^{\prime }=x_4\cos \frac \chi 2-x_5\sin \frac \chi 2%
\mbox{;}\\ 
x_5\rightarrow x_5^{\prime }=x_5\cos \frac \chi 2+x_4\sin \frac \chi 2%
\mbox{;}\\ 
x_\mu \rightarrow x_\mu ^{\prime }=x_\mu \mbox{ for }\mu \in \left\{
0,1,2,3\right\} \mbox{;}\\ 
Y\rightarrow Y^{\prime }=\widetilde{U}^{\dagger }Y\widetilde{U}=Y\mbox{;}\\
\widetilde{\varphi }\rightarrow \widetilde{\varphi }^{\prime }=\widetilde{U}%
\widetilde{\varphi }\mbox{,}\\ 
B_\mu \rightarrow B_\mu ^{\prime }=B_\mu -\frac 1{g_1}\partial _\mu \chi %
\mbox{,}\\ 
F_\mu \rightarrow F_\mu ^{\prime }=F_\mu \mbox{.}}
\label{T}
\end{equation}

\subsection{Masses}

Let a hamiltonian be denoted as \textit{a Planck hamiltonian } if a tiny
positive real number $h$ and functions $N_\vartheta \left(
t,x_1,x_2,x_3\right) $ and $N_\varpi \left( t,x_1,x_2,x_3\right) $, which
have got a range of values in the set of the integer numbers, exist for
which:

\[
M_0=N_\vartheta h\mbox{ and }M_4=N_\varpi h\mbox{.} 
\]

Let

\[
-\frac \pi h\leq x_5\leq \frac \pi h,-\frac \pi h\leq x_4\leq \frac \pi h%
\mbox{,} 
\]

\[
\widetilde{\varphi }\left( t,x_1,x_2,x_3,\pm \frac \pi h,x_4\right) =0%
\mbox{
and }\widetilde{\varphi }\left( t,x_1,x_2,x_3,x_5,\pm \frac \pi h\right) =0%
\mbox{.} 
\]

In that case the Fourier series for $\widetilde{\varphi }$ is of the
following form:


\begin{eqnarray}
\widetilde{\varphi }\left( t,x_1,x_2,x_3,x_5,x_4\right)  
=\varphi \left( t,x_1,x_2,x_3\right) \cdot \nonumber\\ 
\cdot \sum_{n,s}\delta _{-n,N_\vartheta \left( t,\mathbf{x}\right) }\delta
_{-s,N_\varpi \left( t,\mathbf{x}\right) }\exp \left( -\mathrm{i}h\left(
nx_5+sx_4\right) \right) \nonumber
\end{eqnarray}

with

\begin{eqnarray*}
\delta _{-n,N_\vartheta } &=&\frac h{2\pi }\int_{-\frac \pi h}^{\frac \pi
h}\exp \left( \mathrm{i}h\left( nx_5\right) \right) \exp \left( \mathrm{i}%
N_\vartheta hx_5\right) dx_5=\frac{\sin \left( \pi \left( n+N_\vartheta
\right) \right) }{\pi \left( n+N_\vartheta \right) }\mbox{,} \\
\delta _{-s,N_\varpi } &=&\frac h{2\pi }\int_{-\frac \pi h}^{\frac \pi
h}\exp \left( \mathrm{i}h\left( sx_4\right) \right) \exp \left( \mathrm{i}%
N_\varpi hx_4\right) dx_4=\frac{\sin \left( \pi \left( s+N_\varpi \right)
\right) }{\pi \left( s+N_\varpi \right) }\mbox{.}
\end{eqnarray*}

If denote:

\[
\phi \left( t,\mathbf{x},-n,-s\right) \stackrel{Def}{=}\varphi \left( t,%
\mathbf{x}\right) \delta _{n,N_\vartheta \left( t,\mathbf{x}\right) }\delta
_{s,N_\varpi \left( t,\mathbf{x}\right) } 
\]

then 

\begin{equation}
\eqalign{
\widetilde{\varphi }\left( t,\mathbf{x},x_5,x_4\right) = \\ 
=\sum_{n,s}\phi \left( t,\mathbf{x},n,s\right) \exp \left( -\mathrm{i}%
h\left( nx_5+sx_4\right) \right) \mbox{.}} \label{lt}
\end{equation}

The integer numbers $n$ and $s$ be denoted as \textit{the mass numbers}.

From the properties of $\delta $: in every point $\left\langle t,\mathbf{x}%
\right\rangle $: either

\[
\widetilde{\varphi }\left( t,\mathbf{x},x_5,x_4\right) =0 
\]

or an integer numbers $n_0$ and $s_0$ exist for which:

\begin{equation}
\eqalign{
\widetilde{\varphi }\left( t,\mathbf{x},x_5,x_4\right) = \\ 
=\phi \left( t,\mathbf{x},n_0,s_0\right) \exp \left( -\mathrm{i}h\left(
n_0x_5+s_0x_4\right) \right) \mbox{.}} \label{dlt}
\end{equation}

Here if

\[
m_0\stackrel{Def}{=}\sqrt{n_0^2+s_0^2} 
\]

then

\[
m\stackrel{Def}{=}hm_0 
\]

is denoted as \textit{a mass} of $\widetilde{\varphi }$.

That is for the every space-time point: either this point is empty or single
mass is placed in this point.

The motion equation (\ref{gkB}) under the transformation (\ref{T}) has got
the following form:

\begin{eqnarray}
\sum_{n^{\prime },s^{\prime }}\left( \sum_{\mu =0}^3\beta ^{\left[ \mu
\right] }\left( \mathrm{i}\partial _\mu +F_\mu +0.5g_1YB_\mu \gamma ^{\left[
5\right] }\right) +\gamma ^{\left[ 0\right] }\mathrm{i}\partial _5^{\prime
}+\beta ^{\left[ 4\right] }\mathrm{i}\partial _4^{\prime }\right) \cdot \nonumber\\ 
\cdot \exp \left( -\mathrm{i}h\left( n^{\prime }x_5+s^{\prime }x_4\right)
\right) \widetilde{U}\phi =0 \nonumber
\end{eqnarray}

with:

\[
\begin{array}{c}
n^{\prime }=n\cos \frac \chi 2-s\sin \frac \chi 2\mbox{,} \\ 
s^{\prime }=n\sin \frac \chi 2+s\cos \frac \chi 2\mbox{.}
\end{array}
\]

But $s$ and $n$ are an integer numbers and $s^{\prime }$ and $n^{\prime }$
must be an integer numbers, too.

A triplet $\left\langle \lambda ,n,s\right\rangle $ of integer numbers is 
\textit{a Pythagorean triple} \cite{Pf}, if

\[
\lambda ^2=n^2+s^2\mbox{.} 
\]

Let $\varepsilon $ be any tiny positive real number. An integer number $%
\lambda $ is \textit{a father number with a precise }$\varepsilon $ if for
each real number $\chi $ and for every Pythagorean triple $\left\langle
\lambda ,n,s\right\rangle $ a Pythagorean triple $\left\langle \lambda
,n^{\prime },s^{\prime }\right\rangle $ exists for which:

\begin{eqnarray}
\left| -s\sin \frac \chi 2+n\cos \frac \chi 2-n^{\prime }\right|
<\varepsilon \mbox{,} \nonumber \\ 
\left| s\cos \frac \chi 2+n\sin \frac \chi 2-s^{\prime }\right| <\varepsilon %
\mbox{.} \nonumber
\end{eqnarray}

\textit{For every }$\varepsilon $\textit{: denumerable many of father
numbers with a precise }$\varepsilon $\textit{\ exist.}

Let on the space of spinors $\widetilde{\varphi }$ the scalar product $%
\widetilde{\varphi }*\widetilde{\chi }$ be denoted as the following:

\[
\widetilde{\varphi }*\widetilde{\chi }\stackrel{Def}{=}\left( \frac h{2\pi
}\right) ^2\int_{-\frac \pi h}^{\frac \pi h}dx_5\int_{-\frac \pi h}^{\frac
\pi h}dx_4\cdot \left( \widetilde{\varphi }^{\dagger }\cdot \widetilde{\chi }%
\right) \mbox{.} 
\]

In this case:

\[
\widetilde{\varphi }*\beta ^{\left[ k\right] }\widetilde{\varphi }=\varphi
^{\dagger }\beta ^{\left[ k\right] }\varphi \mbox{.} 
\]

for $k\in \left\{ 1,2,3\right\} $.

Hence from (\ref{j}):

\[
\cases{ 
\widetilde{\varphi }*\widetilde{\varphi }=\rho \mbox{,} \\ 
-\widetilde{\varphi }*\beta ^{\left[ 1\right] }\widetilde{\varphi }=j_1%
\mbox{,} \\ 
-\widetilde{\varphi }*\beta ^{\left[ 2\right] }\widetilde{\varphi }=j_2%
\mbox{,} \\ 
-\widetilde{\varphi }^{\dagger }*\beta ^{\left[ 3\right] }\widetilde{\varphi 
}=j_3.
}
\]

\subsubsection{The one-mass state}

Let $\epsilon _\mu $ ($\mu \in \left\{ 1,2,3,4\right\} $) be a basis in
which the light pentad has got a form (\ref{lghr}) and let (\ref{lt}):

\[
\widetilde{\varphi }\left( t,\mathbf{x},x_5,x_4\right) =\exp \left( -\mathrm{%
i}hnx_5\right) \sum_{j=1}^4\phi _j\left( t,\mathbf{x},n,0\right) \epsilon _j%
\mbox{.} 
\]

In that case the hamiltonian has got the following form (from \ref{gkB}):

\[
\widehat{H}=\sum_{k=1}^3\beta ^{\left[ k\right] }\mathrm{i}\partial
_k+hn\gamma ^{\left[ 0\right] }+\widehat{G} 
\]

with

\[
\widehat{G}\stackrel{Def}{=}\sum_{\mu =0}^3\beta ^{\left[ \mu \right]
}\left( F_\mu +0.5g_1YB_\mu \right) \mbox{.} 
\]

If

\begin{equation}
\widehat{H}_0\stackrel{Def}{=}\sum_{k=1}^3\beta ^{\left[ k\right] }\mathrm{i}%
\partial _k+hn\gamma ^{\left[ 0\right] }  \label{hmm0}
\end{equation}

then the functions

\[
u_1\left( \mathbf{k}\right) \exp \left( -\mathrm{i}h\mathbf{kx}\right) 
\mbox{
and }u_2\left( \mathbf{k}\right) \exp \left( -\mathrm{i}h\mathbf{kx}\right) 
\]

with

\[
u_1\left( \mathbf{k}\right) \stackrel{Def}{=}\frac 1{2\sqrt{\omega \left( 
\mathbf{k}\right) \left( \omega \left( \mathbf{k}\right) +n\right) }}\left[ 
\begin{array}{c}
\omega \left( \mathbf{k}\right) +n+k_3 \\ 
k_1+\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n-k_3 \\ 
-k_1-\mathrm{i}k_2
\end{array}
\right] 
\]

and

\[
u_2\left( \mathbf{k}\right) \stackrel{Def}{=}\frac 1{2\sqrt{\omega \left( 
\mathbf{k}\right) \left( \omega \left( \mathbf{k}\right) +n\right) }}\left[ 
\begin{array}{c}
k_1-\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n-k_3 \\ 
-k_1+\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n+k_3
\end{array}
\right] 
\]

are the eigenvectors of $\widehat{H}_0$ with the eigenvalue $\omega \left( 
\mathbf{k}\right) \stackrel{Def}{=}\sqrt{\mathbf{k}^2+n^2}$, and the
functions

\[
u_3\left( \mathbf{k}\right) \exp \left( -\mathrm{i}h\mathbf{kx}\right) 
\mbox{
and }u_4\left( \mathbf{k}\right) \exp \left( -\mathrm{i}h\mathbf{kx}\right) 
\]

with

\[
u_3\left( \mathbf{k}\right) \stackrel{Def}{=}\frac 1{2\sqrt{\omega \left( 
\mathbf{k}\right) \left( \omega \left( \mathbf{k}\right) +n\right) }}\left[ 
\begin{array}{c}
-\omega \left( \mathbf{k}\right) -n+k_3 \\ 
k_1+\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n+k_3 \\ 
k_1+\mathrm{i}k_2
\end{array}
\right] 
\]

and

\[
u_4\left( \mathbf{k}\right) \stackrel{Def}{=}\frac 1{2\sqrt{\omega \left( 
\mathbf{k}\right) \left( \omega \left( \mathbf{k}\right) +n\right) }}\left[ 
\begin{array}{c}
k_1-\mathrm{i}k_2 \\ 
-\omega \left( \mathbf{k}\right) -n-k_3 \\ 
k_1-\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n-k_3
\end{array}
\right] 
\]

are the eigenvectors of $\widehat{H}_0$ with the eigenvalue $-\omega \left( 
\mathbf{k}\right) $.

Here $u_\mu \left( \mathbf{k}\right) $ form an orthonormal basis in the
space, spanned on vectors $\epsilon _\mu $.

Let :

\[
b_{r,\mathbf{k}}\stackrel{Def}{=}\left( \frac h{2\pi }\right)
^3\sum_{j^{\prime }=1}^4\int_{\left( V\right) }d^3\mathbf{x}^{\prime }\cdot
e^{\mathrm{i}h\mathbf{kx}^{\prime }}u_{r,j^{\prime }}^{*}\left( \mathbf{k}%
\right) \psi _{j^{\prime }}\left( \mathbf{x}^{\prime }\right) 
\]

In that case because

\[
\sum_{r=1}^4u_{r,j}^{*}\left( \mathbf{k}\right) u_{r,j^{\prime }}\left( 
\mathbf{k}\right) =\delta _{j,j^{\prime }} 
\]

then

\begin{equation}
\psi _j\left( \mathbf{x}\right) =\sum_{\mathbf{k}}e^{-\mathrm{i}h\mathbf{kx}%
}\sum_{r=1}^4b_{r,\mathbf{k}}u_{r,j}\left( \mathbf{k}\right)  \label{bb}
\end{equation}

and

\begin{equation}
\eqalign{
\left\{ b_{s,\mathbf{k}^{\prime }}^{\dagger },b_{r,\mathbf{k}}\right\}
=\left( \frac h{2\pi }\right) ^3\delta _{s,r}\delta _{\mathbf{k},\mathbf{k}%
^{\prime }}\mbox{,} \\ 
\left\{ b_{s,\mathbf{k}^{\prime }}^{\dagger },b_{r,\mathbf{k}}^{\dagger
}\right\} =0=\left\{ b_{s,\mathbf{k}^{\prime }},b_{r,\mathbf{k}}\right\} %
\mbox{,} \\ 
b_{r,\mathbf{k}}\Phi _0=0\mbox{.}}
\label{bubu}
\end{equation}


The hamiltonian density (\ref{hmm}) for $\widehat{H}_0$ is the following:

\[
\mathcal{H}_0\left( \mathbf{x}\right) =\sum_{j=1}^4\psi _j^{\dagger }\left( 
\mathbf{x}\right) \sum_{k=1}^4\widehat{H}_{0,j,k}\psi _k\left( \mathbf{x}%
\right) \mbox{.} 
\]

Hence from (\ref{bb}):

\[
\int_{\left( V\right) }d^3\mathbf{x}\cdot \mathcal{H}_0\left( \mathbf{x}%
\right) =\left( \frac{2\pi }h\right) ^3\sum_{\mathbf{k}}h\omega \left( 
\mathbf{k}\right) \cdot \left( \sum_{r=1}^2b_{r,\mathbf{k}}^{\dagger }b_{r,%
\mathbf{k}}-\sum_{r=3}^4b_{r,\mathbf{k}}^{\dagger }b_{r,\mathbf{k}}\right) 
\]

Let a Fourier transformation for $\varphi $ be the following:

\[
\varphi _j\left( t,\mathbf{x}\right) =\sum_{\mathbf{p}}\sum_{r=1}^4c_r\left(
t,\mathbf{p}\right) u_{r,j}\left( \mathbf{p}\right) e^{-\mathrm{i}h\mathbf{px%
}} 
\]

with

\[
c_r\left( t,\mathbf{p}\right) \stackrel{Def}{=}\left( \frac h{2\pi }\right)
^3\sum_{j^{\prime }=1}^4\int_{\left( V\right) }d^3\mathbf{x}^{\prime }\cdot
u_{r,j^{\prime }}^{*}\left( \mathbf{p}\right) \varphi _{j^{\prime }}\left( t,%
\mathbf{x}^{\prime }\right) e^{\mathrm{i}h\mathbf{px}^{\prime }} 
\]

I denote a function $\varphi _j\left( t,\mathbf{x}\right) $ as \textit{%
ordinary function} if a real positive number $K$ exists for which:

if $\left| p_1\right| >K$ or/and $\left| p_2\right| >K$ or/and $\left|
p_3\right| >K$ then $c_r\left( t,\mathbf{p}\right) =0$.

In that case I denote:

\[
\sum_{\mathbf{p\in \Xi }}\stackrel{Def}{=}\sum_{p_1=-L}^L\sum_{p_2=-L}^L%
\sum_{p_3=-L}^L 
\]

If $\varphi _j\left( t,\mathbf{x}\right) $ is an ordinary function then:

\[
\varphi _j\left( t,\mathbf{x}\right) =\sum_{\mathbf{p\in \Xi }%
}\sum_{r=1}^4c_r\left( t,\mathbf{p}\right) u_{r,j}\left( \mathbf{p}\right)
e^{-\mathrm{i}h\mathbf{px}}\mbox{.} 
\]

Hence from (\ref{Sat}):

\[
\Psi \left( t,\mathbf{x}\right) =\sum_{\mathbf{p}}\sum_{r=1}^4\sum_{\mathbf{k%
}}\sum_{r^{\prime }=1}^4c_r\left( t,\mathbf{p}\right) e^{\mathrm{i}h\left( 
\mathbf{k}-\mathbf{p}\right) \mathbf{x}}\sum_{j=1}^4u_{r^{\prime
},j}^{*}\left( \mathbf{k}\right) u_{r,j}\left( \mathbf{p}\right)
b_{r^{\prime },\mathbf{k}}^{\dagger }\Phi _0 
\]

and

\[
\int_{\left( V\right) }d^3\mathbf{x}\cdot \Psi \left( t,\mathbf{x}\right)
=\left( \frac{2\pi }h\right) ^3\sum_{\mathbf{p}}\sum_{r=1}^4c_r\left( t,%
\mathbf{p}\right) b_{r,\mathbf{p}}^{\dagger }\Phi _0 
\]

If denote:

\[
\widetilde{\Psi }\left( t,\mathbf{p}\right) \stackrel{Def}{=}\left( \frac{%
2\pi }h\right) ^3\sum_{r=1}^4c_r\left( t,\mathbf{p}\right) b_{r,\mathbf{p}%
}^{\dagger }\Phi _0 
\]

then

\[
\int_{\left( V\right) }d^3\mathbf{x}\cdot \Psi \left( t,\mathbf{x}\right)
=\sum_{\mathbf{p}}\widetilde{\Psi }\left( t,\mathbf{p}\right) 
\]

and

\[
H_0\widetilde{\Psi }\left( t,\mathbf{p}\right) =\left( \frac{2\pi }h\right)
^3\sum_{\mathbf{k}}h\omega \left( \mathbf{k}\right) \cdot \left(
\sum_{r=1}^2c_r\left( t,\mathbf{k}\right) b_{r,\mathbf{k}}^{\dagger }\Phi
_0-\sum_{r=3}^4c_r\left( t,\mathbf{k}\right) b_{r,\mathbf{k}}^{\dagger }\Phi
_0\right) 
\]

On set of ordinary functions $H_0$ equivalent to operator:

\[
\stackrel{\Xi }{H}_0\stackrel{Def}{=}\left( \frac{2\pi }h\right) ^3\sum_{%
\mathbf{k\in \Xi }}h\omega \left( \mathbf{k}\right) \cdot \left(
\sum_{r=1}^2b_{r,\mathbf{k}}^{\dagger }b_{r,\mathbf{k}}-\sum_{r=3}^4b_{r,%
\mathbf{k}}^{\dagger }b_{r,\mathbf{k}}\right) \mbox{.} 
\]

Because (from (\ref{bubu}))

\[
b_{r,\mathbf{k}}^{\dagger }b_{r,\mathbf{k}}=\left( \frac h{2\pi }\right)
^3-b_{r,\mathbf{k}}b_{r,\mathbf{k}}^{\dagger } 
\]

then

\begin{equation}
\stackrel{\Xi }{H}_0=\left( \frac{2\pi }h\right) ^3\sum_{\mathbf{k\in \Xi }%
}h\omega \left( \mathbf{k}\right) \left( \sum_{r=1}^2b_{r,\mathbf{k}%
}^{\dagger }b_{r,\mathbf{k}}+\sum_{r=3}^4b_{r,\mathbf{k}}b_{r,\mathbf{k}%
}^{\dagger }\right) -h\sum_{\mathbf{k\in \Xi }}\omega \left( \mathbf{k}%
\right) \mbox{.}  \label{uW}
\end{equation}

Let:

\begin{equation}
\eqalign{
v_{\left( 1\right) }\left( \mathbf{k}\right) \stackrel{Def}{=}\gamma
^{\left[ 0\right] }u_3\left( \mathbf{k}\right) \mbox{,} \\ 
v_{\left( 2\right) }\left( \mathbf{k}\right) \stackrel{Def}{=}\gamma
^{\left[ 0\right] }u_4\left( \mathbf{k}\right) \mbox{,} \\ 
u_{\left( 1\right) }\left( \mathbf{k}\right) \stackrel{Def}{=}u_1\left( 
\mathbf{k}\right) \mbox{,} \\ 
u_{\left( 2\right) }\left( \mathbf{k}\right) \stackrel{Def}{=}u_2\left( 
\mathbf{k}\right) \mbox{.}}
\label{xa}
\end{equation}

and let:

\[
\begin{array}{c}
d_1\left( \mathbf{k}\right) \stackrel{Def}{=}-b_3^{\dagger }\left( -\mathbf{k%
}\right) \mbox{,} \\ 
d_2\left( \mathbf{k}\right) \stackrel{Def}{=}-b_4^{\dagger }\left( -\mathbf{k%
}\right) \mbox{.}
\end{array}
\]

In that case:

\begin{center}
\[
\psi _j\left( \mathbf{x}\right) =\sum_{\mathbf{k}}\sum_{\alpha =1}^2\left(
e^{-\mathrm{i}h\mathbf{kx}}b_{\alpha ,\mathbf{k}}u_{\left( \alpha \right)
,j}\left( \mathbf{k}\right) +e^{\mathrm{i}h\mathbf{kx}}d_{\alpha ,\mathbf{k}%
}^{\dagger }v_{\left( \alpha \right) ,j}\left( \mathbf{k}\right) \right) 
\]
\end{center}

and from (\ref{uW}) the Wick-ordering hamiltonian has got the following form:

\[
:\stackrel{\Xi }{H}_0:=\left( \frac{2\pi }h\right) ^3h\sum_{\mathbf{k\in \Xi 
}}\omega \left( \mathbf{k}\right) \sum_{\alpha =1}^2\left( b_{\alpha ,%
\mathbf{k}}^{\dagger }b_{\alpha ,\mathbf{k}}+d_{\alpha ,\mathbf{k}}^{\dagger
}d_{\alpha ,\mathbf{k}}\right) \mbox{.} 
\]

Here $b_{\alpha ,\mathbf{k}}^{\dagger }$ are \textit{creation operators},
and $b_{\alpha ,\mathbf{k}}$ are \textit{annihilation operators} of $n$-%
\textit{leptonn} with \textit{a momentum} $\mathbf{k}$ and \textit{a spin
index} $\alpha $; $d_{\alpha ,\mathbf{k}}^{\dagger }$ are \textit{creation
operators}, and $d_{\alpha ,\mathbf{k}}$ are \textit{annihilation operators}
of \textit{anti}-$n$-\textit{leptonn} with \textit{a momentum} $\mathbf{k}$
and \textit{a spin index} $\alpha $.

Functions:

\[
u_{\left( 1\right) }\left( \mathbf{k}\right) \exp \left( -\mathrm{i}h\mathbf{%
kx}\right) \mbox{
and }u_{\left( 2\right) }\left( \mathbf{k}\right) \exp \left( -\mathrm{i}h%
\mathbf{kx}\right) 
\]

are \textit{the basic} $n$-\textit{leptonn functions} with momentum $\mathbf{%
k}$, and

\[
v_{\left( 1\right) }\left( \mathbf{k}\right) \exp \left( \mathrm{i}h\mathbf{%
kx}\right) \mbox{
and }v_{\left( 2\right) }\left( \mathbf{k}\right) \exp \left( \mathrm{i}h%
\mathbf{kx}\right) 
\]

are \textit{the anti-}$n$\textit{-leptonn basic functions} with momentum $%
\mathbf{k}$.

\subsubsection{The bi-mass state}

Let

\begin{equation}
\eqalign{
\widetilde{\varphi }\left( t,\mathbf{x},x_5,x_4\right) = \\ 
=\exp \left( -\mathrm{i}hsx_4\right) \sum_{r=1}^4\phi _r\left( t,\mathbf{x}%
,s,0\right) \epsilon _r \\ 
+\exp \left( -\mathrm{i}hnx_5\right) \sum_{j=1}^4\phi _j\left( t,\mathbf{x}%
,n,0\right) \epsilon _j\} \mbox{.}} \label{bir}
\end{equation} 

Hence in the basis

\[
\begin{array}{c}
\langle \exp \left( -\mathrm{i}hsx_4\right) \epsilon _1,\exp \left( -\mathrm{%
i}hsx_4\right) \epsilon _2,\exp \left( -\mathrm{i}hsx_4\right) \epsilon
_3,\exp \left( -\mathrm{i}hsx_4\right) \epsilon _4, \\ 
\exp \left( -\mathrm{i}hnx_5\right) \epsilon _1,\exp \left( -\mathrm{i}%
hnx_5\right) \epsilon _2,\exp \left( -\mathrm{i}hnx_5\right) \epsilon
_3,\exp \left( -\mathrm{i}hnx_5\right) \epsilon _4\rangle
\end{array}
\]

a 8-components bi-spinor:

\begin{equation}
\xi =\left[ 
\begin{array}{c}
\phi _1\left( 0,s\right) \\ 
\phi _2\left( 0,s\right) \\ 
\phi _3\left( 0,s\right) \\ 
\phi _4\left( 0,s\right) \\ 
\phi _1\left( n,0\right) \\ 
\phi _2\left( n,0\right) \\ 
\phi _3\left( n,0\right) \\ 
\phi _4\left( n,0\right)
\end{array}
\right]  \label{br}
\end{equation}

corresponds to $\widetilde{\varphi }$.

From (\ref{dlt}): in every point $\left\langle t,\mathbf{x}\right\rangle $:

\begin{equation}
\mbox{either }\xi =\left[ 
\begin{array}{c}
\phi _1\left( 0,s\right) \\ 
\phi _2\left( 0,s\right) \\ 
\phi _3\left( 0,s\right) \\ 
\phi _4\left( 0,s\right) \\ 
0 \\ 
0 \\ 
0 \\ 
0
\end{array}
\right] \mbox{ or }\xi =\left[ 
\begin{array}{c}
0 \\ 
0 \\ 
0 \\ 
0 \\ 
\phi _1\left( n,0\right) \\ 
\phi _2\left( n,0\right) \\ 
\phi _3\left( n,0\right) \\ 
\phi _4\left( n,0\right)
\end{array}
\right]  \label{dl2}
\end{equation}

of $\delta $ characteristics.

Let us denote:

\[
\phi _L\stackrel{Def}{=}\phi _1\epsilon _1+\phi _2\epsilon _2\mbox {
and }\phi _R\stackrel{Def}{=}\phi _3\epsilon _3+\phi _4\epsilon _4\mbox {.} 
\]

Hence from (\ref{bir}):

\begin{equation}
\eqalign{
\widetilde{\varphi }\left( x_5,x_4\right) =\exp \left( -\mathrm{i}%
hsx_4\right) \left( \phi _L\left( 0,s\right) +\phi _R\left( 0,s\right)
\right) + \\ 
+\exp \left( -\mathrm{i}hnx_5\right) \left( \phi _L\left( n,0\right) +\phi
_R\left( n,0\right) \right)} \label{bir1}
\end{equation} 

and $\xi $ can be denoted as the following:

\begin{equation}
\xi =\left( 
\begin{array}{c}
\phi _L\left( 0,s\right) \\ 
\phi _R\left( 0,s\right) \\ 
\phi _L\left( n,0\right) \\ 
\phi _R\left( n,0\right)
\end{array}
\right) \mbox{.}  \label{fif}
\end{equation}

If use denotation:

\[
\underline{\varsigma }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
\varsigma & 0_4 \\ 
0_4 & \varsigma
\end{array}
\right] \mbox {,} 
\]

then from (\ref{bir1}) :

\[
\cases{ 
\xi ^{\dagger }\xi =\rho \mbox{,} \\ 
-\xi ^{\dagger }\underline{\beta ^{\left[ 1\right] }}\xi =j_1\mbox{,} \\ 
-\xi ^{\dagger }\underline{\beta ^{\left[ 2\right] }}\xi =j_2\mbox{,} \\ 
-\xi ^{\dagger }\underline{\beta ^{\left[ 3\right] }}\xi =j_3\mbox{.} 
} 
\]

If $U$ is an 8$\times $8 complex matrix, $\xi ^{\prime }\stackrel{Def}{=}%
U\xi $ and

\begin{equation}
\cases{ 
\xi ^{\prime \dagger }\xi ^{\prime }=\rho \mbox{,} \\ 
-\xi ^{\prime \dagger }\underline{\beta ^{\left[ 1\right] }}\xi ^{\prime
}=j_1\mbox{,} \\ 
-\xi ^{\prime \dagger }\underline{\beta ^{\left[ 2\right] }}\xi ^{\prime
}=j_2\mbox{,} \\ 
-\xi ^{\prime \dagger }\underline{\beta ^{\left[ 3\right] }}\xi ^{\prime
}=j_3}  \label{uni}
\end{equation}

then for $k\in \left\{ 1,2,3\right\} $:

\[
U^{\dagger }\underline{\beta ^{\left[ k\right] }}U=\underline{\beta ^{\left[
k\right] }} 
\]

and

\[
U^{\dagger }U=1_8\mbox{.} 
\]

In that case real functions $\chi \left( t,\mathbf{x}\right) $, $\alpha
\left( t,\mathbf{x}\right) $, $a\left( t,\mathbf{x}\right) $, $b\left( t,%
\mathbf{x}\right) $, $c\left( t,\mathbf{x}\right) $, $q\left( t,\mathbf{x}%
\right) $, $u\left( t,\mathbf{x}\right) $, $v\left( t,\mathbf{x}\right) $, $%
k\left( t,\mathbf{x}\right) $, $s\left( t,\mathbf{x}\right) $ exist for
which:

\[
U\stackrel{Def}{=}\underline{\widetilde{U}}U^{\left( \alpha \right)
}U^{\left( -\right) }U^{\left( +\right) }\mbox{,} 
\]

with

\[
U^{\left( \alpha \right) }\stackrel{Def}{=}\exp \left( \mathrm{i}\alpha
\right) 1_8\mbox{,} 
\]

\[
U^{\left( -\right) }\stackrel{Def}{=}\left[ 
\begin{array}{cccc}
\left( a+\mathrm{i}b\right) 1_2 & 0_2 & \left( c+\mathrm{i}q\right) 1_2 & 0_2
\\ 
0_2 & 1_2 & 0_2 & 0_2 \\ 
\left( -c+\mathrm{i}q\right) 1_2 & 0_2 & \left( a-\mathrm{i}b\right) 1_2 & 
0_2 \\ 
0_2 & 0_2 & 0_2 & 1_2
\end{array}
\right] \mbox{,} 
\]

with

\[
a^2+b^2+c^2+q^2=1 
\]

and

\begin{equation}
U^{\left( +\right) }\stackrel{Def}{=}\left[ 
\begin{array}{cccc}
1_2 & 0_2 & 0_2 & 0_2 \\ 
0_2 & \left( u+\mathrm{i}v\right) 1_2 & 0_2 & \left( k+\mathrm{i}s\right) 1_2
\\ 
0_2 & 0_2 & 1_2 & 0_2 \\ 
0_2 & \left( -k+\mathrm{i}s\right) 1_2 & 0_2 & \left( u-\mathrm{i}v\right)
1_2
\end{array}
\right] \mbox{,}  \label{upls}
\end{equation}

with

\[
u^2+v^2+k^2+s^2=1\mbox{.} 
\]

\subsubsection{The global $U^{\left( -\right) }$ transformation}

The $U^{\left( -\right) }$ eigenvalues are:

\[
w_{\circ }=a+\mathrm{i}\sqrt{\left( 1-a^2\right) }\mbox{ and }w_{*}=a-%
\mathrm{i}\sqrt{\left( 1-a^2\right) }\mbox{.} 
\]

Let us denote:

\[
\ell _{\circ }\stackrel{Def}{=}\frac 1{2\sqrt{\left( 1-a^2\right) }}\left[ 
\begin{array}{cc}
\left( b+\sqrt{\left( 1-a^2\right) }\right) 1_4 & \left( q-\mathrm{i}%
c\right) 1_4 \\ 
\left( q+\mathrm{i}c\right) 1_4 & \left( \sqrt{\left( 1-a^2\right) }%
-b\right) 1_4
\end{array}
\right] \mbox{,} 
\]

\[
\ell _{*}\stackrel{Def}{=}\frac 1{2\sqrt{\left( 1-a^2\right) }}\left[ 
\begin{array}{cc}
\left( \sqrt{\left( 1-a^2\right) }-b\right) 1_4 & \left( -q+\mathrm{i}%
c\right) 1_4 \\ 
\left( -q-\mathrm{i}c\right) 1_4 & \left( b+\sqrt{\left( 1-a^2\right) }%
\right) 1_4
\end{array}
\right] \mbox{.} 
\]

These operators are fulfilled to the following conditions:

\begin{eqnarray}
\ell _{\circ }\ell _{\circ }=\ell _{\circ }\mbox{, }\ell _{*}\ell _{*}=\ell
_{*}\mbox{;} \nonumber\\ 
\ell _{\circ }\ell _{*}=0=\ell _{*}\ell _{\circ }\mbox{,} \nonumber\\ 
\left( \ell _{\circ }-\ell _{*}\right) \left( \ell _{\circ }-\ell
_{*}\right) =1_8\mbox{,} \nonumber\\ 
\ell _{\circ }+\ell _{*}=1_8\mbox{,} \nonumber
\end{eqnarray}

\begin{eqnarray}
\ell _{\circ }\underline{\gamma ^{\left[ 0\right] }}=\underline{\gamma
^{\left[ 0\right] }}\ell _{\circ }\mbox{, }\ell _{*}\underline{\gamma
^{\left[ 0\right] }}=\underline{\gamma ^{\left[ 0\right] }}\ell _{*}\mbox{,}
\nonumber\\ 
\ell _{\circ }\underline{\beta ^{\left[ 4\right] }}=\underline{\beta
^{\left[ 4\right] }}\ell _{\circ }\mbox{, }\ell _{*}\underline{\beta
^{\left[ 4\right] }}=\underline{\beta ^{\left[ 4\right] }}\ell _{*} \nonumber
\end{eqnarray}

and


\begin{equation}
\eqalign{
U^{\left( -\right) \dagger }\underline{\gamma ^{\left[ 0\right] }}U^{\left(
-\right) }=a\underline{\gamma ^{\left[ 0\right] }}-\left( \ell _{\circ
}-\ell _{*}\right) \sqrt{1-a^2}\underline{\beta ^{\left[ 4\right] }}\mbox{,}
\\ 
U^{\left( -\right) \dagger }\underline{\beta ^{\left[ 4\right] }}U^{\left(
-\right) }=a\underline{\beta ^{\left[ 4\right] }}+\left( \ell _{\circ }-\ell
_{*}\right) \sqrt{1-a^2}\underline{\gamma ^{\left[ 0\right] }}\mbox{.}} \label{gaa}
\end{equation}

From (\ref{gkB}) the leptonn motion equation:

\begin{equation}
\left( \sum_{\mu =0}^3\beta ^{\left[ \mu \right] }\left( \mathrm{i}\partial
_\mu +F_\mu +0.5g_1YB_\mu \right) +\left( \gamma ^{\left[ 0\right] }\mathrm{i%
}\partial _5+\beta ^{\left[ 4\right] }\mathrm{i}\partial _4\right) \right) 
\widetilde{\varphi }=0\mbox{.}  \label{mme}
\end{equation}

If

\[
\partial _kU^{\left( -\right) \dagger }=U^{\left( -\right) \dagger }\partial
_k 
\]

for $k\in \left\{ 0,1,2,3,4,5\right\} $ then

\[
\left( 
\begin{array}{c}
U^{\left( -\right) \dagger }\mathrm{i}\sum_{\mu =0}^3\underline{\beta
^{\left[ \mu \right] }}\left( \mathrm{i}\partial _\mu +F_\mu +0.5g_1%
\underline{Y}B_\mu \right) \\ 
+\underline{\gamma ^{\left[ 0\right] }}U^{\left( -\right) \dagger }\mathrm{i}%
\partial _5+\underline{\beta ^{\left[ 4\right] }}U^{\left( -\right) \dagger }%
\mathrm{i}\partial _4
\end{array}
\right) U^{\left( -\right) }\widetilde{\varphi }=0\mbox{.} 
\]

Hence from (\ref{gaa}):

\[
U^{\left( -\right) \dagger }\left( 
\begin{array}{c}
\sum_{\mu =0}^3\underline{\beta ^{\left[ \mu \right] }}\left( \mathrm{i}%
\partial _\mu +F_\mu +0.5g_1\underline{Y}B_\mu \right) \\ 
+\underline{\gamma ^{\left[ 0\right] }}\mathrm{i}\left( a\partial _5-\left(
\ell _{\circ }-\ell _{*}\right) \sqrt{1-a^2}\partial _4\right) \\ 
+\underline{\beta ^{\left[ 4\right] }}\mathrm{i}\left( \sqrt{1-a^2}\left(
\ell _{\circ }-\ell _{*}\right) \partial _5+a\partial _4\right)
\end{array}
\right) U^{\left( -\right) }\widetilde{\varphi }=0\mbox{.} 
\]

Thus, if to denote:

\begin{eqnarray}
x_4^{\prime }=\left( \ell _{\circ }+\ell _{*}\right) ax_4+\left( \ell
_{\circ }-\ell _{*}\right) \sqrt{1-a^2}x_5 \nonumber\\ 
x_5^{\prime }=\left( \ell _{\circ }+\ell _{*}\right) ax_5-\left( \ell
_{\circ }-\ell _{*}\right) \sqrt{1-a^2}x_4 \nonumber
\end{eqnarray}

then

\begin{equation}
\left( \sum_{\mu =0}^3\underline{\beta ^{\left[ \mu \right] }}\left( \mathrm{%
i}\partial _\mu +F_\mu +0.5g_1\underline{Y}B_\mu \right) +\left( \underline{%
\gamma ^{\left[ 0\right] }}\mathrm{i}\partial _5^{\prime }+\underline{\beta
^{\left[ 4\right] }}\mathrm{i}\partial _4^{\prime }\right) \right) 
\widetilde{\varphi }^{\prime }=0\mbox{.}  \label{me8}
\end{equation}

with

\[
\widetilde{\varphi }^{\prime }=U^{\left( -\right) }\widetilde{\varphi }%
\mbox{.} 
\]

That is the leptonn hamiltonian is invariant for the following global
transformation:


\begin{equation}
\eqalign{
\widetilde{\varphi }\rightarrow \widetilde{\varphi }^{\prime }=U^{\left(
-\right) }\widetilde{\varphi }\mbox{,} \nonumber\\ 
x_4\rightarrow x_4^{\prime }=\left( \ell _{\circ }+\ell _{*}\right)
ax_4+\left( \ell _{\circ }-\ell _{*}\right) \sqrt{1-a^2}x_5\mbox{,} \nonumber\\ 
x_5\rightarrow x_5^{\prime }=\left( \ell _{\circ }+\ell _{*}\right)
ax_5-\left( \ell _{\circ }-\ell _{*}\right) \sqrt{1-a^2}x_4\mbox{,} \nonumber\\ 
x_\mu \rightarrow x_\mu ^{\prime }=x_\mu \mbox{.}} \label{glb}
\end{equation}

\subsubsection{Neutrinno}

From (\ref{bir1}) for $s=n$:

\begin{eqnarray}
\widetilde{\varphi }=\exp \left( -\mathrm{i}hnx_4\right) \left( \phi
_L\left( 0,n\right) +\phi _R\left( 0,n\right) \right) + \nonumber\\ 
+\exp \left( -\mathrm{i}hnx_5\right) \left( \phi _L\left( n,0\right) +\phi
_R\left( n,0\right) \right) \mbox{.} \nonumber
\end{eqnarray}

If from (\ref{dl2}): $\phi _L\left( 0,n\right) =0$ and $\phi _R\left(
0,n\right) =0$ then from (\ref{fif}):

\[
\xi =\left( 
\begin{array}{c}
0 \\ 
0 \\ 
\phi _L\left( n,0\right) \\ 
\phi _R\left( n,0\right)
\end{array}
\right) \mbox{.} 
\]

Let

\[
\underline{\widehat{H}}_{0,4}\stackrel{Def.}{=}\sum_{k=1}^3\underline{\beta
^{\left[ k\right] }}\mathrm{i}\partial _k+hn\left( \underline{\gamma
^{\left[ 0\right] }}+\underline{\beta ^{\left[ 4\right] }}\right) \mbox{.} 
\]

The 8-vectors

\[
\underline{u}_1\left( \mathbf{k}\right) \stackrel{Def}{=}\frac 1{2\sqrt{%
\omega \left( \mathbf{k}\right) \left( \omega \left( \mathbf{k}\right)
+n\right) }}\left[ 
\begin{array}{c}
0 \\ 
0 \\ 
0 \\ 
0 \\ 
\omega \left( \mathbf{k}\right) +n+k_3 \\ 
k_1+\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n-k_3 \\ 
-k_1-\mathrm{i}k_2
\end{array}
\right] 
\]

and

\[
\underline{u}_2\left( \mathbf{k}\right) \stackrel{Def}{=}\frac 1{2\sqrt{%
\omega \left( \mathbf{k}\right) \left( \omega \left( \mathbf{k}\right)
+n\right) }}\left[ 
\begin{array}{c}
0 \\ 
0 \\ 
0 \\ 
0 \\ 
k_1-\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n-k_3 \\ 
-k_1+\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n+k_3
\end{array}
\right] 
\]

correspond to the eigenvectors of $\underline{\widehat{H}}_{0,4}$ with
eigenvalue $\omega \left( \mathbf{k}\right) =\sqrt{\mathbf{k}^2+n^2}$, and
the 8-vectors

\[
\underline{u}_3\left( \mathbf{k}\right) \stackrel{Def}{=}\frac 1{2\sqrt{%
\omega \left( \mathbf{k}\right) \left( \omega \left( \mathbf{k}\right)
+n\right) }}\left[ 
\begin{array}{c}
0 \\ 
0 \\ 
0 \\ 
0 \\ 
-\omega \left( \mathbf{k}\right) -n+k_3 \\ 
k_1+\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n+k_3 \\ 
k_1+\mathrm{i}k_2
\end{array}
\right] 
\]

and

\[
\underline{u}_4\left( \mathbf{k}\right) \stackrel{Def}{=}\frac 1{2\sqrt{%
\omega \left( \mathbf{k}\right) \left( \omega \left( \mathbf{k}\right)
+n\right) }}\left[ 
\begin{array}{c}
0 \\ 
0 \\ 
0 \\ 
0 \\ 
k_1-\mathrm{i}k_2 \\ 
-\omega \left( \mathbf{k}\right) -n-k_3 \\ 
k_1-\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n-k_3
\end{array}
\right] 
\]

correspond to the eigenvectors of $\underline{\widehat{H}}_{0,4}$ with
eigenvalue $-\omega \left( \mathbf{k}\right) $.

Let

\begin{eqnarray}
\underline{\widehat{H}}_{0,4}^{\prime }\stackrel{Def}{=}U^{\left( -\right) }%
\underline{\widehat{H}}_{0,4}U^{\left( -\right) \dagger }\mbox{,} \nonumber\\ 
\underline{u}_\mu ^{\prime }\left( \mathbf{k}\right) \stackrel{Def}{=}%
U^{\left( -\right) }\underline{u}_\mu \left( \mathbf{k}\right) \mbox{.} \nonumber
\end{eqnarray}

That is

\[
\underline{u}_1^{\prime }\left( \mathbf{k}\right) =\frac 1{2\sqrt{\omega
\left( \mathbf{k}\right) \left( \omega \left( \mathbf{k}\right) +n\right) }%
}\left[ 
\begin{array}{c}
\left( c+\mathrm{i}q\right) \left( \omega \left( \mathbf{k}\right)
+n+k_3\right) \\ 
\left( c+\mathrm{i}q\right) \left( k_1+\mathrm{i}k_2\right) \\ 
0 \\ 
0 \\ 
\left( a-\mathrm{i}b\right) \left( \omega \left( \mathbf{k}\right)
+n+k_3\right) \\ 
\left( a-\mathrm{i}b\right) \left( k_1+\mathrm{i}k_2\right) \\ 
\omega \left( \mathbf{k}\right) +n-k_3 \\ 
-k_1-\mathrm{i}k_2
\end{array}
\right] \mbox{,} 
\]

\[
\underline{u}_2^{\prime }\left( \mathbf{k}\right) =\frac 1{2\sqrt{\omega
\left( \mathbf{k}\right) \left( \omega \left( \mathbf{k}\right) +n\right) }%
}\left[ 
\begin{array}{c}
\left( c+\mathrm{i}q\right) \left( k_1-\mathrm{i}k_2\right) \\ 
\left( c+\mathrm{i}q\right) \left( \omega \left( \mathbf{k}\right)
+n-k_3\right) \\ 
0 \\ 
0 \\ 
\left( a-\mathrm{i}b\right) \left( k_1-\mathrm{i}k_2\right) \\ 
\left( a-\mathrm{i}b\right) \left( \omega \left( \mathbf{k}\right)
+n-k_3\right) \\ 
-k_1+\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n+k_3
\end{array}
\right] \mbox{,} 
\]

\[
\underline{u}_3^{\prime }\left( \mathbf{k}\right) =\frac 1{2\sqrt{\omega
\left( \mathbf{k}\right) \left( \omega \left( \mathbf{k}\right) +n\right) }%
}\left[ 
\begin{array}{c}
-\left( c+\mathrm{i}q\right) \left( \omega \left( \mathbf{k}\right)
+n-k_3\right) \\ 
\left( c+\mathrm{i}q\right) \left( k_1+\mathrm{i}k_2\right) \\ 
0 \\ 
0 \\ 
-\left( a-\mathrm{i}b\right) \left( \omega \left( \mathbf{k}\right)
+n-k_3\right) \\ 
\left( a-\mathrm{i}b\right) \left( k_1+\mathrm{i}k_2\right) \\ 
\omega \left( \mathbf{k}\right) +n+k_3 \\ 
k_1+\mathrm{i}k_2
\end{array}
\right] \mbox{,} 
\]

\[
\underline{u}_4^{\prime }\left( \mathbf{k}\right) =\frac 1{2\sqrt{\omega
\left( \mathbf{k}\right) \left( \omega \left( \mathbf{k}\right) +n\right) }%
}\left[ 
\begin{array}{c}
\left( c+\mathrm{i}q\right) \left( k_1-\mathrm{i}k_2\right) \\ 
-\left( c+\mathrm{i}q\right) \left( \omega \left( \mathbf{k}\right)
+n+k_3\right) \\ 
0 \\ 
0 \\ 
\left( a-\mathrm{i}b\right) \left( k_1-\mathrm{i}k_2\right) \\ 
-\left( a-\mathrm{i}b\right) \left( \omega \left( \mathbf{k}\right)
+n+k_3\right) \\ 
k_1-\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n-k_3
\end{array}
\right] \mbox{.} 
\]

Here $\underline{u}_1^{\prime }\left( \mathbf{k}\right) $ and $\underline{u}%
_2^{\prime }\left( \mathbf{k}\right) $ correspond to the eigenvectors of $%
\underline{\widehat{H}}_{0,4}^{\prime }$ with eigenvalue $\omega \left( 
\mathbf{k}\right) =\sqrt{\mathbf{k}^2+n^2}$, and $\underline{u}_3^{\prime
}\left( \mathbf{k}\right) $ and $\underline{u}_4^{\prime }\left( \mathbf{k}%
\right) $ correspond to the eigenvectors of $\underline{\widehat{H}}_{0,4}$
with eigenvalue $-\omega \left( \mathbf{k}\right) $.

Let as in (\ref{xa}):

\begin{eqnarray}
\underline{v}_{\left( 1\right) }\left( \mathbf{k}\right) \stackrel{Def}{=}%
\underline{\gamma ^{\left[ 0\right] }}\underline{u}_3^{\prime }\left( 
\mathbf{k}\right) \mbox{,} \nonumber\\ 
\underline{v}_{\left( 2\right) }\left( \mathbf{k}\right) \stackrel{Def}{=}%
\underline{\gamma ^{\left[ 0\right] }}\underline{u}_4^{\prime }\left( 
\mathbf{k}\right) \mbox{,} \nonumber\\ 
\underline{u}_{\left( 1\right) }\left( \mathbf{k}\right) \stackrel{Def}{=}%
\underline{u}_1^{\prime }\left( \mathbf{k}\right) \mbox{,} \nonumber\\ 
\underline{u}_{\left( 2\right) }\left( \mathbf{k}\right) \stackrel{Def}{=}%
\underline{u}_2^{\prime }\left( \mathbf{k}\right) \mbox{.} \nonumber
\end{eqnarray}

Hence

\[
\underline{v}_{\left( 1\right) }\left( \mathbf{k}\right) =\frac 1{2\sqrt{%
\omega \left( \mathbf{k}\right) \left( \omega \left( \mathbf{k}\right)
+n\right) }}\left[ 
\begin{array}{c}
0 \\ 
0 \\ 
-\left( c+\mathrm{i}q\right) \left( \omega \left( \mathbf{k}\right)
+n-k_3\right) \\ 
\left( c+\mathrm{i}q\right) \left( k_1+\mathrm{i}k_2\right) \\ 
\omega \left( \mathbf{k}\right) +n+k_3 \\ 
k_1+\mathrm{i}k_2 \\ 
-\left( a-\mathrm{i}b\right) \left( \omega \left( \mathbf{k}\right)
+n-k_3\right) \\ 
\left( a-\mathrm{i}b\right) \left( k_1+\mathrm{i}k_2\right)
\end{array}
\right] 
\]

and

\[
\underline{v}_{\left( 2\right) }\left( \mathbf{k}\right) =\frac 1{2\sqrt{%
\omega \left( \mathbf{k}\right) \left( \omega \left( \mathbf{k}\right)
+n\right) }}\left[ 
\begin{array}{c}
0 \\ 
0 \\ 
\left( c+\mathrm{i}q\right) \left( k_1-\mathrm{i}k_2\right) \\ 
-\left( c+\mathrm{i}q\right) \left( \omega \left( \mathbf{k}\right)
+n+k_3\right) \\ 
k_1-\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n-k_3 \\ 
\left( a-\mathrm{i}b\right) \left( k_1-\mathrm{i}k_2\right) \\ 
-\left( a-\mathrm{i}b\right) \left( \omega \left( \mathbf{k}\right)
+n+k_3\right)
\end{array}
\right] \mbox{.} 
\]

$\underline{u}_{\left( \alpha \right) }^{\prime }\left( \mathbf{k}\right) $
are denoted as \textit{bi-}$n$-\textit{leptonn} and $\underline{v}_{\left(
\alpha \right) }\left( \mathbf{k}\right) $ - as \textit{bi-anti-}$n$\textit{%
-leptonn} basic vectors with momentum $\mathbf{k}$ and spin index $\alpha $.

Hence bi-anti-$n$-leptonn basic vectors are the result of the acting of $%
U^{\left( +\right) }$ (\ref{upls}).

The vectors

\begin{eqnarray*}
l_{n,\left( 1\right) }\left( \mathbf{k}\right) &=&\left[ 
\begin{array}{c}
\left( a-\mathrm{i}b\right) \left( \omega \left( \mathbf{k}\right)
+n+k_3\right) \\ 
\left( a-\mathrm{i}b\right) \left( k_1+\mathrm{i}k_2\right) \\ 
\omega \left( \mathbf{k}\right) +n-k_3 \\ 
-k_1-\mathrm{i}k_2
\end{array}
\right] \mbox{ and } \\
l_{n,\left( 2\right) }\left( \mathbf{k}\right) &=&\left[ 
\begin{array}{c}
\left( a-\mathrm{i}b\right) \left( k_1-\mathrm{i}k_2\right) \\ 
\left( a-\mathrm{i}b\right) \left( \omega \left( \mathbf{k}\right)
+n-k_3\right) \\ 
-k_1+\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n+k_3
\end{array}
\right]
\end{eqnarray*}

are denoted as \textit{\ leptonn components} of bi-$n$-leptonn basic
vectors, and vectors

\[
\nu _{n,\left( 1\right) }\left( \mathbf{k}\right) =\left[ 
\begin{array}{c}
\omega \left( \mathbf{k}\right) +n+k_3 \\ 
k_1+\mathrm{i}k_2 \\ 
0 \\ 
0
\end{array}
\right] \mbox{ and }\nu _{n,\left( 2\right) }\left( \mathbf{k}\right)
=\left[ 
\begin{array}{c}
k_1-\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n-k_3 \\ 
0 \\ 
0
\end{array}
\right] 
\]

are denoted as \textit{\ neutrinno components} of bi-$n$-leptonn basic
vectors.

The vectors

\begin{eqnarray*}
\overline{l}_{n,\left( 1\right) }\left( \mathbf{k}\right) &=&\left[ 
\begin{array}{c}
\omega \left( \mathbf{k}\right) +n+k_3 \\ 
k_1+\mathrm{i}k_2 \\ 
-\left( a-\mathrm{i}b\right) \left( \omega \left( \mathbf{k}\right)
+n-k_3\right) \\ 
\left( a-\mathrm{i}b\right) \left( k_1+\mathrm{i}k_2\right)
\end{array}
\right] \mbox{ and } \\
\overline{l}_{n,\left( 2\right) }\left( \mathbf{k}\right) &=&\left[ 
\begin{array}{c}
k_1-\mathrm{i}k_2 \\ 
\omega \left( \mathbf{k}\right) +n-k_3 \\ 
\left( a-\mathrm{i}b\right) \left( k_1-\mathrm{i}k_2\right) \\ 
-\left( a-\mathrm{i}b\right) \left( \omega \left( \mathbf{k}\right)
+n+k_3\right)
\end{array}
\right]
\end{eqnarray*}

are denoted as \textit{\ leptonn components} of anti-bi-$n$-leptonn basic
vectors, and vectors

\[
\overline{\nu }_{n,\left( 1\right) }\left( \mathbf{k}\right) =\left[ 
\begin{array}{c}
0 \\ 
0 \\ 
-\left( \omega \left( \mathbf{k}\right) +n-k_3\right) \\ 
k_1+\mathrm{i}k_2
\end{array}
\right]
\]

and 

\[
\overline{\nu }_{n,\left( 2\right) }\left( \mathbf{k}%
\right) =\left[ 
\begin{array}{c}
0 \\ 
0 \\ 
k_1-\mathrm{i}k_2 \\ 
-\left( \omega \left( \mathbf{k}\right) +n+k_3\right)
\end{array}
\right] 
\]

are denoted as \textit{\ neutrinno components} of anti-bi-$n$-leptonn basic
vectors.

\subsubsection{Local $U^{\left( -\right) }$, $W$, $Z$ and $A$-bozons}

From (\ref{mme}):

\[
\left( \sum_{\mu =0}^3\underline{\beta ^{\left[ \mu \right] }}\left( \mathrm{%
i}\partial _\mu +F_\mu +0.5g_1\underline{Y}B_\mu \right) +\underline{\gamma
^{\left[ 0\right] }}\mathrm{i}\partial _5+\underline{\beta ^{\left[ 4\right]
}}\mathrm{i}\partial _4\right) \widetilde{\varphi }=0\mbox{.} 
\]

Let:

\begin{equation}
\ \ K\stackrel{Def}{=}\sum_{\mu =0}^3\underline{\beta ^{\left[ \mu \right] }}%
\left( F_\mu +0.5g_1\underline{Y}B_\mu \right) \mbox{.}  \label{kdf}
\end{equation}

In that case the motion equation has got the following form:

\begin{equation}
\left( K+\sum_{\mu =0}^3\underline{\beta ^{\left[ \mu \right] }}\mathrm{i}%
\partial _\mu +\underline{\gamma ^{\left[ 0\right] }}\mathrm{i}\partial _5+%
\underline{\beta ^{\left[ 4\right] }}\mathrm{i}\partial _4\right) \widetilde{%
\varphi }=0\mbox{.}  \label{me81}
\end{equation}

Hence for the following transformations:

\begin{equation}
\eqalign{
\widetilde{\varphi }\rightarrow \widetilde{\varphi }^{\prime }\stackrel{Def}{%
=}U^{\left( -\right) }\widetilde{\varphi }\mbox{,} \\ 
x_4\rightarrow x_4^{\prime }\stackrel{Def}{=}\left( \ell _{\circ }+\ell
_{*}\right) ax_4+\left( \ell _{\circ }-\ell _{*}\right) \sqrt{1-a^2}x_5%
\mbox{,} \\ 
x_5\rightarrow x_5^{\prime }\stackrel{Def}{=}\left( \ell _{\circ }+\ell
_{*}\right) ax_5-\left( \ell _{\circ }-\ell _{*}\right) \sqrt{1-a^2}x_4%
\mbox{,} \\ 
x_\mu \rightarrow x_\mu ^{\prime }\stackrel{Def}{=}x_\mu \mbox{, for }\mu
\in \left\{ 0,1,2,3\right\} \mbox{,} \\ 
K\rightarrow K^{\prime }
}
\label{gll}
\end{equation}

with

\[
\begin{array}{c}
\partial _4U^{\left( -\right) }=U^{\left( -\right) }\partial _4\mbox{ and }%
\partial _5U^{\left( -\right) }=U^{\left( -\right) }\partial _5
\end{array}
\]

this equation has got the following form:

\begin{equation}
\left( 
\begin{array}{c}
U^{\left( -\right) \dagger }K^{\prime }U^{\left( -\right) }+ \\ 
+\sum_{\mu =0}^3\underline{\beta ^{\left[ \mu \right] }}\mathrm{i}\left(
\partial _\mu +U^{\left( -\right) \dagger }\left( \partial _\mu U^{\left(
-\right) }\right) \right) +\underline{\gamma ^{\left[ 0\right] }}\mathrm{i}%
\partial _5+\underline{\beta ^{\left[ 4\right] }}\mathrm{i}\partial _4
\end{array}
\right) \widetilde{\varphi }=0\mbox{.}  \label{me82}
\end{equation}

Therefore if

\begin{equation}
K^{\prime }=K-\mathrm{i}\sum_{\mu =0}^3\underline{\beta ^{\left[ \mu \right]
}}\left( \partial _\mu U^{\left( -\right) }\right) U^{\left( -\right)
\dagger }  \label{ksht}
\end{equation}

then the equation (\ref{me81}) is invariant for the local transformation (%
\ref{gll}).

Let $g_2$ be a positive real number.

If design ($a,b,c,q$ form $U^{\left( -\right) }$):

\[
\begin{array}{c}
W_{0,\mu }\stackrel{Def}{=}-2\frac 1{g_2q}\left( 
\begin{array}{c}
q\left( \partial _\mu a\right) b-q\left( \partial _\mu b\right) a+\left(
\partial _\mu c\right) q^2+ \\ 
+a\left( \partial _\mu a\right) c+b\left( \partial _\mu b\right) c+c^2\left(
\partial _\mu c\right)
\end{array}
\right) \\ 
W_{1,\mu }\stackrel{Def}{=}-2\frac 1{g_2q}\left( 
\begin{array}{c}
\left( \partial _\mu a\right) a^2-bq\left( \partial _\mu c\right) +a\left(
\partial _\mu b\right) b+ \\ 
+a\left( \partial _\mu c\right) c+q^2\left( \partial _\mu a\right) +c\left(
\partial _\mu b\right) q
\end{array}
\right) \\ 
W_{2,\mu }\stackrel{Def}{=}-2\frac 1{g_2q}\left( 
\begin{array}{c}
q\left( \partial _\mu a\right) c-a\left( \partial _\mu a\right) b-b^2\left(
\partial _\mu b\right) - \\ 
-c\left( \partial _\mu c\right) b-\left( \partial _\mu b\right) q^2-\left(
\partial _\mu c\right) qa
\end{array}
\right)
\end{array}
\]

and

\[
W_\mu \stackrel{Def}{=}\left[ 
\begin{array}{cccc}
W_{0,\mu }1_2 & 0_2 & \left( W_{1,\mu }-\mathrm{i}W_{2,\mu }\right) 1_2 & 0_2
\\ 
0_2 & 0_2 & 0_2 & 0_2 \\ 
\left( W_{1,\mu }+\mathrm{i}W_{2,\mu }\right) 1_2 & 0_2 & -W_{0,\mu }1_2 & 
0_2 \\ 
0_2 & 0_2 & 0_2 & 0_2
\end{array}
\right] 
\]

then

\begin{equation}
-\mathrm{i}\left( \partial _\mu U^{\left( -\right) }\right) U^{\left(
-\right) \dagger }=\frac 12g_2W_\mu \mbox{,}  \label{w}
\end{equation}

and from (\ref{w}), (\ref{kdf}), (\ref{ksht}), (\ref{me81}):

\begin{equation}
\left( 
\begin{array}{c}
\sum_{\mu =0}^3\underline{\beta ^{\left[ \mu \right] }}\mathrm{i}\left(
\partial _\mu -\mathrm{i}0.5g_1B_\mu \underline{Y}-\mathrm{i}\frac
12g_2W_\mu -\mathrm{i}F_\mu \right)  \\ 
+\underline{\gamma ^{\left[ 0\right] }}\mathrm{i}\partial _5^{\prime }+%
\underline{\beta ^{\left[ 4\right] }}\mathrm{i}\partial _4^{\prime }
\end{array}
\right) \widetilde{\varphi }^{\prime }=0\mbox{.}  \label{hW}
\end{equation}



Let

\[
U^{\prime }\stackrel{Def}{=}\left[ 
\begin{array}{cccc}
\left( a^{\prime }+\mathrm{i}b^{\prime }\right) 1_2 & 0_2 & \left( c^{\prime
}+\mathrm{i}q^{\prime }\right) 1_2 & 0_2 \\ 
0_2 & 1_2 & 0_2 & 0_2 \\ 
\left( -c^{\prime }+\mathrm{i}q^{\prime }\right) 1_2 & 0_2 & \left(
a^{\prime }-\mathrm{i}b^{\prime }\right) 1_2 & 0_2 \\ 
0_2 & 0_2 & 0_2 & 1_2
\end{array}
\right] \mbox{.} 
\]

In this case if

\[
U^{\prime \prime }\stackrel{Def}{=}U^{\prime }U^{\left( -\right) } 
\]

then $U^{\prime \prime }$ has got the similar form:

\[
U^{\prime \prime }=\stackrel{Def}{=}\left[ 
\begin{array}{cccc}
\left( a^{\prime \prime }+\mathrm{i}b^{\prime \prime }\right) 1_2 & 0_2 & 
\left( c^{\prime \prime }+\mathrm{i}q^{\prime \prime }\right) 1_2 & 0_2 \\ 
0_2 & 1_2 & 0_2 & 0_2 \\ 
\left( -c^{\prime \prime }+\mathrm{i}q^{\prime \prime }\right) 1_2 & 0_2 & 
\left( a^{\prime \prime }-\mathrm{i}b^{\prime \prime }\right) 1_2 & 0_2 \\ 
0_2 & 0_2 & 0_2 & 1_2
\end{array}
\right] \mbox{.} 
\]

If

\[
\ell _{\circ }^{\prime \prime }\stackrel{Def}{=}\frac 1{2\sqrt{\left(
1-a^{\prime \prime 2}\right) }}\left[ 
\begin{array}{cc}
\left( b^{\prime \prime }+\sqrt{\left( 1-a^{\prime \prime 2}\right) }\right)
1_4 & \left( q^{\prime \prime }-\mathrm{i}c^{\prime \prime }\right) 1_4 \\ 
\left( q^{\prime \prime }+\mathrm{i}c^{\prime \prime }\right) 1_4 & \left( 
\sqrt{\left( 1-a^{\prime \prime 2}\right) }-b^{\prime \prime }\right) 1_4
\end{array}
\right] \mbox{,} 
\]

\[
\ell _{*}^{\prime \prime }\stackrel{Def}{=}\frac 1{2\sqrt{\left( 1-a^{\prime
\prime 2}\right) }}\left[ 
\begin{array}{cc}
\left( \sqrt{\left( 1-a^{\prime \prime 2}\right) }-b^{\prime \prime }\right)
1_4 & \left( -q^{\prime \prime }+\mathrm{i}c^{\prime \prime }\right) 1_4 \\ 
\left( -q^{\prime \prime }-\mathrm{i}c^{\prime \prime }\right) 1_4 & \left(
b^{\prime \prime }+\sqrt{\left( 1-a^{\prime \prime 2}\right) }\right) 1_4
\end{array}
\right] \mbox{;} 
\]

\begin{equation}
\eqalign{
\widetilde{\varphi }\rightarrow \widetilde{\varphi }^{\prime \prime }%
\stackrel{Def}{=}\stackrel{,}{U}\widetilde{\varphi }\mbox{,} \\ 
x_4\rightarrow x_4^{\prime \prime }\stackrel{Def}{=}\left( \ell _{\circ
}^{\prime \prime }+\ell _{*}^{\prime \prime }\right) a^{\prime \prime
}x_4+\left( \ell _{\circ }^{\prime \prime }-\ell _{*}^{\prime \prime
}\right) \sqrt{1-a^{\prime \prime 2}}x_5\mbox{,} \\ 
x_5\rightarrow x_5^{\prime \prime }\stackrel{Def}{=}\left( \ell _{\circ
}^{\prime \prime }+\ell _{*}^{\prime \prime }\right) a^{\prime \prime
}x_5-\left( \ell _{\circ }^{\prime \prime }-\ell _{*}^{\prime \prime
}\right) \sqrt{1-a^{\prime \prime 2}}x_4\mbox{,} \\ 
x_\mu \rightarrow x_\mu ^{\prime \prime }\stackrel{Def}{=}x_\mu \mbox{, for }%
\mu \in \left\{ 0,1,2,3\right\} \mbox{,} \\ 
K\rightarrow K^{\prime \prime }\stackrel{Def}{=}\sum_{\mu =0}^3\underline{%
\beta ^{\left[ \mu \right] }}\left( F_\mu +0.5g_1\underline{Y}B_\mu +\frac
12g_2W_\mu ^{\prime \prime }\right)
}
\label{tt2}
\end{equation}

then from (\ref{w}):

\[
W_\mu ^{\prime \prime }=-\frac{2i}{g_2}\left( \partial _\mu \left( U^{\prime
}U^{\left( -\right) }\right) \right) \left( U^{\prime }U^{\left( -\right)
}\right) ^{\dagger }\mbox{.} 
\]

Hence:

\[
W_\mu ^{\prime \prime }=-\frac{2i}{g_2}\left( \partial _\mu U^{\prime
}\right) U^{\prime \dagger }-\frac{2i}{g_2}\stackrel{,}{U}\left( \partial
_\mu U^{\left( -\right) }\right) U^{\left( -\right) \dagger }U^{\prime
\dagger }\mbox{,} 
\]

i.e.:

\[
W_\mu ^{\prime \prime }=U^{\prime }W_\mu U^{\prime \dagger }-\frac{2i}{g_2}%
\left( \partial _\mu U^{\prime }\right) U^{\prime \dagger }\mbox{.} 
\]

If

\[
F_{\mu ,\nu }=\left( \partial _\mu W_\nu -\partial _\nu W_\mu -i\frac{g_2}%
2\left( W_\mu W_\nu -W_\nu W_\mu \right) \right) 
\]

then

\[
F_{\mu ,\nu }^{\prime \prime }=\partial _\mu W_\nu ^{\prime \prime
}-\partial _\nu W_\mu ^{\prime \prime }-i\frac{g_2}2\left( W_\mu ^{\prime
\prime }W_\nu ^{\prime \prime }-W_\nu ^{\prime \prime }W_\mu ^{\prime \prime
}\right) =U^{\prime }F_{\mu ,\nu }^{\prime }U^{\prime \dagger }\mbox{.} 
\]

Therefore $F_{\mu ,\nu }$ is invariant for the transformation (\ref{tt2}).

The Lagrangian for $F_{\mu ,\nu }$:

\[
\mathcal{L}_F=\left( -\frac 14\sum_{\mu ,\nu }F^{\mu ,\nu }F_{\mu ,\nu
}\right) \mbox{.} 
\]

Hence the Euler-Lagrange equations for $W_\mu $ are the following:

\[
\sum_\nu \partial ^\nu \left( \partial _\mu W_\nu -\partial _\nu W_\mu -i%
\frac{g_2}2\left[ W_\mu ,W_\nu \right] \right) =0\mbox{.} 
\]

For the components:

\begin{eqnarray}
\sum_\nu \partial ^\nu \partial _\nu W_{0,\mu }=g_2\sum_\nu \partial ^\nu
\left( W_{1,\mu }W_{2,\nu }-W_{2,\mu }W_{1,\nu }\right) +\partial _\mu
\sum_\nu \partial ^\nu W_{0,\nu }\mbox{,} \nonumber\\ 
\sum_\nu \partial ^\nu \partial _\nu W_{1,\mu }=g_2\sum_\nu \partial ^\nu
\left( W_{0,\nu }W_{2,\mu }-W_{0,\mu }W_{2,\nu }\right) +\partial _\mu
\sum_\nu \partial ^\nu W_{1,\nu }\mbox{,} \nonumber\\ 
\sum_\nu \partial ^\nu \partial _\nu W_{2,\mu }=g_2\sum_\nu \partial ^\nu
\left( W_{0,\mu }W_{1,\nu }-W_{0,\nu }W_{1,\mu }\right) +\partial _\mu
\sum_\nu \partial ^\nu W_{2,\nu }\mbox{.} \nonumber
\end{eqnarray}

Let:

\begin{eqnarray}
\alpha _{0,\mu ,\nu }\stackrel{Def}{=}\partial _\nu W_{0,\mu }-g_2\left(
W_{1,\mu }W_{2,\nu }-W_{2,\mu }W_{1,\nu }\right) \mbox{,} \nonumber\\ 
\alpha _{1,\mu ,\nu }\stackrel{Def}{=}\partial _\nu W_{1,\mu }-g_2\left(
W_{0,\nu }W_{2,\mu }-W_{0,\mu }W_{2,\nu }\right) \mbox{,} \nonumber\\ 
\alpha _{2,\mu ,\nu }\stackrel{Def}{=}\partial _\nu W_{2,\mu }-g_2\left(
W_{0,\mu }W_{1,\nu }-W_{0,\nu }W_{1,\mu }\right) \mbox{.} \nonumber
\end{eqnarray}

if $\sum_\nu \partial _\nu W_\nu =0$ and

\[
\alpha _{\mu ,\nu }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
\alpha _{0,\mu ,\nu } & \alpha _{1,\mu ,\nu }-i\alpha _{2,\mu ,\nu } \\ 
\alpha _{1,\mu ,\nu }+i\alpha _{2,\mu ,\nu } & -\alpha _{0,\mu ,\nu }
\end{array}
\right] 
\]

then:

\begin{eqnarray}
\sum_\nu \partial _\nu \partial _\nu W_\mu =-g_2^2W_\mu \sum_\nu W_\nu ^2+
\nonumber\\ 
+\frac{g_2^2}2\sum_\nu \left( W_\nu W_\mu +W_\mu W_\nu \right) W_\nu -i\frac{%
g_2^2}2\sum_\nu \left( \alpha _{\mu ,\nu }W_\nu -W_\nu \alpha _{\mu ,\nu
}\right) \mbox{.} \nonumber
\end{eqnarray}

It is the motion equation for the field $W_\mu $ which has got a less than
unit $\mathrm{c}=1$ velocity. That is this field does not behave as a
massless field.

Hence although $F_{\mu ,\nu }$ is a massless field but its components $W_\mu 
$ do not behave like a massless fields.

If

\[
\sum_\nu \left( W_\nu \frac{\partial W_\nu }{\partial W_\mu }+\frac{\partial
W_\nu }{\partial W_\mu }W_\nu \right) =0 
\]

then a real $\upsilon $ exists for which

\[
\upsilon =\left( 2\sum_\nu W_\nu ^2\right) ^{\frac 12} 
\]

and

\[
\partial _{W_\mu }\upsilon =0\mbox{.} 
\]

Hence the Lagrangian of $W_\mu $ is:

\begin{eqnarray}
\widehat{\mathcal{L}}=\sum_\nu \left( \partial _\nu W_\mu \right) \left(
\partial _\nu W_\mu \right) -g_2^2\frac{\upsilon ^2}2W_\mu ^2+ \nonumber\\ 
+\frac{g_2^2}4\sum_\nu \left( W_\nu W_\mu +W_\mu W_\nu \right) ^2- \nonumber\\ 
-i\frac{g_2^2}2\left( \left( \sum_\nu \left[ \alpha _{\mu ,\nu },W_\nu
\right] \right) W_\mu +W_\mu \left( \sum_\nu \left[ \alpha _{\mu ,\nu
},W_\nu \right] \right) \right) \mbox{.} \nonumber
\end{eqnarray}

It is a lagrangian of a field with mass

\[
M=g_2\frac \upsilon {\sqrt{2}} 
\]

and $M>0$.

Let $A_\mu $ and $Z_\mu $ are a fields for which \cite{Kn}:

\[
Z_\mu \stackrel{Def}{=}\frac 1{\sqrt{g_1^2+g_2^2}}\left( g_2W_{0,\mu
}-g_1B_\mu \right) \mbox{, }A_\mu \stackrel{Def}{=}\frac 1{\sqrt{g_1^2+g_2^2}%
}\left( g_2B_\mu +g_1W_{0,\mu }\right) 
\]

and

\[
\sum_\nu \partial ^\nu \partial _\nu A_\mu =0\mbox{.} 
\]

In that case the moving equation for $Z_\mu $:

\begin{eqnarray}
\sum_\nu \partial ^\nu \partial _\nu Z_\mu =-\frac 12\frac{\upsilon ^2}%
2\left( g_1^2+g_2^2\right) Z_\mu + \nonumber\\ 
+\frac 12\sqrt{g_1^2+g_2^2}\left( \frac 1{g_2}\Lambda -\frac 1{g_1}\left(
\sum_\nu \partial ^\nu \partial _\nu B_\mu +g_1^2\frac{\upsilon ^2}2B_\mu
\right) \right) \mbox{.} \nonumber
\end{eqnarray}

That is $Z_\mu $ has got the mass:

\[
M_Z=\frac \upsilon 2\sqrt{g_1^2+g_2^2}\mbox{.} 
\]

\subsection{Rotations of the Cartesian frame and quarrks}

Let:

\begin{equation}
\eqalign{
\mathbf{e}_1^{\prime }=\cos \left( \alpha \right) \mathbf{e}_1-\sin \left(
\alpha \right) \mathbf{e}_2\mbox{;} \\ 
\mathbf{e}_2^{\prime }=\sin \left( \alpha \right) \mathbf{e}_1+\cos \left(
\alpha \right) \mathbf{e}_2\mbox{,}
}
\label{rot}
\end{equation}

that is if

\begin{eqnarray*}
\mathbf{x} &=&x_1\mathbf{e}_1+x_2\mathbf{e}_2+x_3\mathbf{e}_3 \\
&&\mbox{and} \\
\mathbf{x} &=&x_1^{\prime }\mathbf{e}_1^{\prime }+x_2^{\prime }\mathbf{e}%
_2^{\prime }+x_3^{\prime }\mathbf{e}_3^{\prime }
\end{eqnarray*}

then

\begin{eqnarray*}
x_1^{\prime }=x_1\cos \left( \alpha \right) -x_2\sin \left( \alpha \right) %
\mbox{;} \\ 
x_2^{\prime }=x_1\sin \left( \alpha \right) +x_2\cos \left( \alpha \right) %
\mbox{;} \\ 
x_3^{\prime }=x_3\mbox{;}
\end{eqnarray*}

and for any function $\varphi $:

\begin{equation}
\eqalign{
\partial _1^{\prime }\varphi =\left( \partial _1\varphi \cdot \cos \alpha
-\partial _2\varphi \cdot \sin \alpha \right) \mbox{;} \\ 
\partial _2^{\prime }\varphi =\left( \partial _2\varphi \cdot \cos \alpha
+\partial _1\varphi \cdot \sin \alpha \right) \mbox{;} \\ 
\partial _3^{\prime }\varphi =\partial _3\varphi \mbox{.}
}
\label{d}
\end{equation}

From (\ref{j}):

\begin{eqnarray*}
j_1^{\prime }=-\varphi ^{\dagger }\left( \beta ^{\left[ 1\right] }\cos
\left( \alpha \right) -\beta ^{\left[ 2\right] }\sin \left( \alpha \right)
\right) \varphi \mbox{;} \\ 
j_2^{\prime }=-\varphi ^{\dagger }\left( \beta ^{\left[ 1\right] }\sin
\left( \alpha \right) +\beta ^{\left[ 2\right] }\cos \left( \alpha \right)
\right) \varphi \mbox{;} \\ 
j_3^{\prime }=-\varphi ^{\dagger }\beta ^{\left[ 3\right] }\varphi \mbox{.}
\end{eqnarray*}

Hence if for $\varphi ^{\prime }$:


\begin{eqnarray*}
j_1^{\prime }=-\varphi ^{\prime \dagger }\beta ^{\left[ 1\right] }\varphi
^{\prime }\mbox{;} \\ 
j_2^{\prime }=-\varphi ^{\prime \dagger }\beta ^{\left[ 2\right] }\varphi
^{\prime }\mbox{;} \\ 
j_3^{\prime }=-\varphi ^{\prime \dagger }\beta ^{\left[ 3\right] }\varphi
^{\prime }
\end{eqnarray*}

and

\[
\varphi ^{\prime }=U_{1,2}\left( \alpha \right) \varphi 
\]

then

\begin{equation}
\eqalign{
U_{1,2}^{\dagger }\left( \alpha \right) \beta ^{\left[ 1\right]
}U_{1,2}\left( \alpha \right) =\beta ^{\left[ 1\right] }\cos \alpha -\beta
^{\left[ 2\right] }\sin \alpha \mbox{;} \\ 
U_{1,2}^{\dagger }\left( \alpha \right) \beta ^{\left[ 2\right]
}U_{1,2}\left( \alpha \right) =\beta ^{\left[ 2\right] }\cos \alpha +\beta
^{\left[ 1\right] }\sin \alpha \mbox{;} \\ 
U_{1,2}^{\dagger }\left( \alpha \right) \beta ^{\left[ 3\right]
}U_{1,2}\left( \alpha \right) =\beta ^{\left[ 3\right] }\mbox{;}
}
\label{con1}
\end{equation}

from (\ref{ro}): because

\[
\rho =\varphi ^{\dagger }\varphi =\varphi ^{\prime \dagger }\varphi ^{\prime
} 
\]

then

\begin{equation}
U_{1,2}^{\dagger }\left( \alpha \right) U_{1,2}\left( \alpha \right) =1_4%
\mbox{.}  \label{con2}
\end{equation}

If

\[
U_{1,2}\left( \alpha \right) =\cos \frac \alpha 2\cdot 1_4-\sin \frac \alpha
2\cdot \beta ^{\left[ 1\right] }\beta ^{\left[ 2\right] } 
\]

then $U_{1,2}\left( \alpha \right) $ fulfils to all these conditions ((\ref
{con1}), (\ref{con2})). Moreover:

\begin{equation}
\eqalign{
U_{1,2}^{\dagger }\left( \alpha \right) \beta ^{\left[ 4\right]
}U_{1,2}\left( \alpha \right) =\beta ^{\left[ 4\right] }\mbox{;} \\ 
U_{1,2}^{\dagger }\left( \alpha \right) \gamma ^{\left[ 0\right]
}U_{1,2}\left( \alpha \right) =\gamma ^{\left[ 0\right] }\mbox{;}
}
\label{con3}
\end{equation}

and

\[
U_{1,2}^{\dagger }\left( \alpha \right) \gamma ^{\left[ 5\right]
}U_{1,2}\left( \alpha \right) =\gamma ^{\left[ 5\right] }\mbox{.} 
\]

Let $\widehat{H}_l^{\prime }$ be the result of substitution $\beta ^{\left[
k\right] }$ by $\beta ^{\left[ k\right] \prime }=U_{1,2}^{\dagger }\left(
\alpha \right) \beta ^{\left[ k\right] }U_{1,2}\left( \alpha \right) $ and $%
\partial _k$ by $\partial _k^{\prime }=\frac \partial {\partial x_k^{\prime
}}$ in $\widehat{H}_l$.

From (\ref{d}), (\ref{con1}), (\ref{con2}) and (\ref{con3}):

\[
\widehat{H}_l^{\prime }=\mathrm{i}\left( 
\begin{array}{c}
\beta ^{\left[ 1\right] }\left( 
\begin{array}{c}
\partial _1+\mathrm{i}\left( \Theta _1^{\prime }\cos \left( \alpha \right)
+\Theta _2^{\prime }\sin \left( \alpha \right) \right) + \\ 
+\mathrm{i}\left( \Upsilon _1^{\prime }\cos \left( \alpha \right) +\Upsilon
_2^{\prime }\sin \left( \alpha \right) \right) \gamma ^{\left[ 5\right] }
\end{array}
\right) + \\ 
+\beta ^{\left[ 2\right] }\left( 
\begin{array}{c}
\partial _2+\mathrm{i}\left( -\Theta _1^{\prime }\sin \left( \alpha \right)
+\Theta _2^{\prime }\cos \left( \alpha \right) \right) + \\ 
+\mathrm{i}\left( -\Upsilon _1^{\prime }\sin \left( \alpha \right) +\mathrm{i%
}_2^{\prime }\cos \left( \alpha \right) \right) \gamma ^{\left[ 5\right] }
\end{array}
\right) + \\ 
+\beta ^{\left[ 3\right] }\left( \partial _3+\mathrm{i}\Theta _3^{\prime }+%
\mathrm{i}\Upsilon _3^{\prime }\gamma ^{\left[ 5\right] }\right) + \\ 
+\mathrm{i}M_0^{\prime }\gamma ^{\left[ 0\right] }+\mathrm{i}M_4\beta
^{\left[ 4\right] }\mbox{.}
\end{array}
\right) 
\]

Therefore if

\begin{eqnarray*}
\Theta _0^{\prime }=\Theta _0\mbox{,} \\ 
\Theta _1^{\prime }=\Theta _1\cos \left( \alpha \right) -\Theta _2\sin
\left( \alpha \right) \mbox{,} \\ 
\Theta _2^{\prime }=\Theta _1\sin \left( \alpha \right) +\Theta _2\cos
\left( \alpha \right) \mbox{,} \\ 
\Theta _3^{\prime }=\Theta _3
\end{eqnarray*}

and the same formulas hold for $\left\langle \Upsilon _0,\Upsilon
_1,\Upsilon _2,\Upsilon _3\right\rangle $ then $\widehat{H}_l^{\prime }=%
\widehat{H}_l$ for the Cartesian frame reference rotation (\ref{rot}).

But:

\begin{equation}
\eqalign{
U_{1,2}^{\dagger }\left( \alpha \right) \zeta ^{\left[ 1\right]
}U_{1,2}\left( \alpha \right) =\zeta ^{\left[ 1\right] }\cos \alpha -\eta
^{\left[ 2\right] }\sin \alpha \mbox{;} \\ 
U_{1,2}^{\dagger }\left( \alpha \right) \eta ^{\left[ 2\right]
}U_{1,2}\left( \alpha \right) =\eta ^{\left[ 2\right] }\cos \alpha +\zeta
^{\left[ 1\right] }\sin \alpha \mbox{;}
}
\label{conk}
\end{equation}

\begin{equation}
\eqalign{
U_{1,2}^{\dagger }\left( \alpha \right) \zeta ^{\left[ 2\right]
}U_{1,2}\left( \alpha \right) =\zeta ^{\left[ 2\right] }\cos \alpha -\eta
^{\left[ 1\right] }\sin \alpha \mbox{;} \\ 
U_{1,2}^{\dagger }\left( \alpha \right) \eta ^{\left[ 1\right]
}U_{1,2}\left( \alpha \right) =\eta ^{\left[ 1\right] }\cos \alpha +\zeta
^{\left[ 2\right] }\sin \alpha \mbox{;}
}
\label{conk1}
\end{equation}

\begin{eqnarray*}
U_{1,2}^{\dagger }\left( \alpha \right) \zeta ^{\left[ 3\right]
}U_{1,2}\left( \alpha \right) =\zeta ^{\left[ 3\right] }\mbox{;} \\ 
U_{1,2}^{\dagger }\left( \alpha \right) \eta ^{\left[ 3\right]
}U_{1,2}\left( \alpha \right) =\eta ^{\left[ 3\right] }\mbox{;}
\end{eqnarray*}

\begin{equation}
\eqalign{
U_{1,2}^{\dagger }\left( \alpha \right) \gamma _\zeta ^{\left[ 0\right]
}U_{1,2}\left( \alpha \right) =\gamma _\zeta ^{\left[ 0\right] }\cos \alpha
-\gamma _\eta ^{\left[ 0\right] }\sin \alpha \mbox{;} \\ 
U_{1,2}^{\dagger }\left( \alpha \right) \gamma _\eta ^{\left[ 0\right]
}U_{1,2}\left( \alpha \right) =\gamma _\eta ^{\left[ 0\right] }\cos \alpha
+\gamma _\zeta ^{\left[ 0\right] }\sin \alpha \mbox{;} \\ 
U_{1,2}^{\dagger }\left( \alpha \right) \gamma _\theta ^{\left[ 0\right]
}U_{1,2}\left( \alpha \right) =\gamma _\theta ^{\left[ 0\right] }\mbox{;}
}
\label{conk2}
\end{equation}

\begin{equation}
\eqalign{
U_{1,2}^{\dagger }\left( \alpha \right) \zeta ^{\left[ 4\right]
}U_{1,2}\left( \alpha \right) =\zeta ^{\left[ 4\right] }\cos \alpha +\eta
^{\left[ 4\right] }\sin \alpha \mbox{;} \\ 
U_{1,2}^{\dagger }\left( \alpha \right) \eta ^{\left[ 4\right]
}U_{1,2}\left( \alpha \right) =\eta ^{\left[ 4\right] }\cos \alpha -\zeta
^{\left[ 4\right] }\sin \alpha \mbox{;} \\ 
U_{1,2}^{\dagger }\left( \alpha \right) \theta ^{\left[ 4\right]
}U_{1,2}\left( \alpha \right) =\theta ^{\left[ 4\right] }.
}
\label{conk3}
\end{equation}


Terefore from (\ref{conk}), (\ref{conk1}), (\ref{conk2}), (\ref{conk3}):

$\widehat{H}\left( \zeta \right) $ is mixed with $\widehat{H}\left( \eta
\right) $ for this rotation. For other rotations of the frame reference: the
leptonn hamiltonians do persist, but the chromatic hamiltonians are mixed
with other chromatic hamiltonians$.$

Therefore the chromatic triplet elements can not be separated in the space.
These elements must be localized in the identical place.

Each chromatic pentad contains two mass elements. Hence, every family
contains two sorts of the chromatic particles of tree colors. I denote these
particles as \textit{quarrks}.

\section{The two-event case}

Let $N`=2$ and

\[
\int_{D_1}d^3\mathbf{x}^{\left( 1\right) }\int_{D_2}d^3\mathbf{x}^{\left(
2\right) }\cdot \rho \left( t,\mathbf{x},\mathbf{\ y}\right) \stackrel{Def}{=%
}\mathbf{P}\left( \exists \mathbf{x}\in D_1:\exists \mathbf{y}\in D_2:\alpha
_{1,2}\left( t,\mathbf{x},\mathbf{y}\right) \right) 
\]

Complex functions $\varphi _{j_1,j_2}\left( t,\mathbf{x},\mathbf{y}\right) $
($j_k\in \left\{ 1,2,3,4\right\} $) exist for which:

\[
\rho \left( t,\mathbf{x},\mathbf{\ y}\right)
=4\sum_{j_1=1}^4\sum_{j_2=1}^4\varphi _{j_1,j_2}^{*}\left( t,\mathbf{x},%
\mathbf{y}\right) \varphi _{j_1,j_2}\left( t,\mathbf{x},\mathbf{y}\right) %
\mbox{.} 
\]

If


\begin{eqnarray*}
\Psi \left( t,\mathbf{x},\mathbf{y}\right) \stackrel{Def}{=} \\ 
=\sum_{j_1=1}^4\sum_{j_2=1}^4\varphi _{j_1,j_2}\left( t,\mathbf{x},\mathbf{y}%
\right) \left( \psi _{j_1}^{\left( 1\right) \dagger }\left( \mathbf{x}%
\right) \psi _{j_2}^{\left( 2\right) \dagger }\left( \mathbf{y}\right) -\psi
_{j_2}^{\left( 2\right) \dagger }\left( \mathbf{y}\right) \psi
_{j_1}^{\left( 1\right) \dagger }\left( \mathbf{x}\right) \right) \Phi _0
\end{eqnarray*}
then

\begin{eqnarray*}
\Psi ^{\dagger }\left( t,\mathbf{x},\mathbf{y}\right) \Psi \left( t,\mathbf{x%
},\mathbf{y}\right) = \\ 
=4\sum_{j_1=1}^4\sum_{j_2=1}^4\varphi _{j_1,j_2}^{\left( 1,2\right) *}\left(
t,\mathbf{x}^{\prime },\mathbf{y}^{\prime }\right) \varphi
_{j_1,j_2}^{\left( 1,2\right) }\left( t,\mathbf{x},\mathbf{y}\right) \cdot
\delta \left( \mathbf{x}-\mathbf{x}^{\prime }\right) \delta \left( \mathbf{y}%
-\mathbf{y}^{\prime }\right) \mbox{.}
\end{eqnarray*}

Like to (\ref{sys}): the system with unknown complex functions $%
Q_{j_1,k_1}^{\left( 1\right) }$, $Q_{j_2,k_2}^{\left( 2\right) }$, $%
Q_{j_1,k_1;j_2,k_2}^{\left( 1,2\right) }$:

\[
\cases{ 
\sum_{k_1=1}^4Q_{j_1,k_1}^{\left( 1\right) }\varphi
_{k_1,j_2}+\sum_{k_2=1}^4Q_{j_2,k_2}^{\left( 2\right) }\varphi _{j_1,k_2}+
%\\ 
\sum_{k_1=1}^4\sum_{k_2=1}^4Q_{j_1,k_1;j_2,k_2}^{\left( 1,2\right) }\varphi
_{k_1,k_2}= \\ 
=\partial _t\varphi _{j_1,j_2}-\sum_{r=1}^3\left( \sum_{k_1=1}^4\beta
_{j_1,k_1}^{\left[ r\right] }\frac \partial {\partial x_r}\varphi
_{k_1,j_2}+\sum_{k_2=1}^4\beta _{j_2,k_2}^{\left[ r\right] }\frac \partial
{\partial y_r}\varphi _{j_1,k_2}\right) ; \\ 
Q_{k_1,j_1}^{\left( 1\right) *}=-Q_{j_1,k_1}^{\left( 1\right) }; \\ 
Q_{k_2,j_2}^{\left( 2\right) *}=-Q_{j_2,k_2}^{\left( 2\right) }; \\ 
Q_{k_1,j_1;j_2,k_2}^{\left( 1,2\right) *}=-Q_{j_1,k_1;j_2,k_2}^{\left(
1,2\right) }; \\ 
Q_{j_1,k_1;k_2,j_2}^{\left( 1,2\right) *}=-Q_{j_1,k_1;j_2,k_2}^{\left(
1,2\right) }
}
\]

has got a solution.

et cetera...

\section{Conclusion}

Therefore all physical events are interpreted by well-known elementary
particles - leptons, quarks and gauge bosons. And if anybody will claim 
that he has found Higgs then not believe - it is not Higgs.


\ack

I'm grateful to all discussions on this paper participants.

\clearpage


\section*{Appendix}
\label{ap}
\appendix



Three {\it chromatic} pentads:
\textit{red pentad} $\zeta $:

\[
\zeta ^{\left[ 1\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
-\sigma _1 & 0_2 \\ 
0_2 & \sigma _1
\end{array}
\right] \mbox{, }\zeta ^{\left[ 2\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
-\sigma _2 & 0_2 \\ 
0_2 & -\sigma _2
\end{array}
\right] \mbox{, }\zeta ^{\left[ 3\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
\sigma _3 & 0_2 \\ 
0_2 & \sigma _3
\end{array}
\right] \mbox{,} 
\]

\[
\gamma _\zeta ^{\left[ 0\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
0_2 & -\sigma _1 \\ 
-\sigma _1 & 0_2
\end{array}
\right] \mbox{, }\zeta ^{\left[ 4\right] }\stackrel{Def}{=}-\mathrm{i}\cdot
\left[ 
\begin{array}{cc}
0_2 & \sigma _1 \\ 
-\sigma _1 & 0_2
\end{array}
\right] 
\]

\textit{green pentad} $\eta $:

\[
\eta ^{\left[ 1\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
\sigma _1 & 0_2 \\ 
0_2 & \sigma _1
\end{array}
\right] \mbox{, }\eta ^{\left[ 2\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
-\sigma _2 & 0_2 \\ 
0_2 & \sigma _2
\end{array}
\right] \mbox{, }\eta ^{\left[ 3\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
-\sigma _3 & 0_2 \\ 
0_2 & -\sigma _3
\end{array}
\right] \mbox{,} 
\]

\[
\gamma _\eta ^{\left[ 0\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
0_2 & -\sigma _2 \\ 
-\sigma _2 & 0_2
\end{array}
\right] \mbox{, }\eta ^{\left[ 4\right] }\stackrel{Def}{=}\mathrm{i}\cdot
\left[ 
\begin{array}{cc}
0_2 & \sigma _2 \\ 
-\sigma _2 & 0_2
\end{array}
\right] \mbox{,} 
\]

\textit{blue pentad} $\theta $:

\[
\theta ^{\left[ 1\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
-\sigma _1 & 0_2 \\ 
0_2 & \sigma -_1
\end{array}
\right] \mbox{, }\theta ^{\left[ 2\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
\sigma _2 & 0_2 \\ 
0_2 & \sigma _2
\end{array}
\right] \mbox{, }\theta ^{\left[ 3\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
-\sigma _3 & 0_2 \\ 
0_2 & \sigma _3
\end{array}
\right] \mbox{, } 
\]

\[
\gamma _\theta ^{\left[ 0\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
0_2 & -\sigma _3 \\ 
-\sigma _3 & 0_2
\end{array}
\right] \mbox{, }\theta ^{\left[ 4\right] }\stackrel{Def}{=}-\mathrm{i}\cdot
\left[ 
\begin{array}{cc}
0_2 & \sigma _3 \\ 
-\sigma _3 & 0_2
\end{array}
\right] \mbox{;} 
\]

one \textit{light pentad} $\beta $:

%\begin{equation}
\[
\beta ^{\left[ 1\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
\sigma _1 & 0_2 \\ 
0_2 & -\sigma _1
\end{array}
\right] \mbox{, }\beta ^{\left[ 2\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
\sigma _2 & 0_2 \\ 
0_2 & -\sigma _2
\end{array}
\right] \mbox{, }\beta ^{\left[ 3\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
\sigma _3 & 0_2 \\ 
0_2 & -\sigma _3
\end{array}
\right] \mbox{,} 
%\end{equation}
\]

\[
\gamma ^{\left[ 0\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
0_2 & 1_2 \\ 
1_2 & 0_2
\end{array}
\right] \mbox{, }\beta ^{\left[ 4\right] }\stackrel{Def}{=}\mathrm{i}\cdot
\left[ 
\begin{array}{cc}
0_2 & 1_2 \\ 
-1_2 & 0_2
\end{array}
\right] 
\]

and two \textit{taste pentads}:

\textit{sweet pentad} \underline{$\Delta $}:

\[
\underline{\Delta }^{\left[ 1\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
0_2 & -\sigma _1 \\ 
-\sigma _1 & 0_2
\end{array}
\right] \mbox{, }\underline{\Delta }^{\left[ 2\right] }\stackrel{Def}{=}%
\left[ 
\begin{array}{cc}
0_2 & -\sigma _2 \\ 
-\sigma _2 & 0_2
\end{array}
\right] \mbox{, }\underline{\Delta }^{\left[ 3\right] }\stackrel{Def}{=}%
\left[ 
\begin{array}{cc}
0_2 & -\sigma _3 \\ 
-\sigma _3 & 0_2
\end{array}
\right] \mbox{, } 
\]

\[
\underline{\Delta }^{\left[ 0\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
-1_2 & 0_2 \\ 
0_2 & 1_2
\end{array}
\right] \mbox{, }\underline{\Delta }^{\left[ 4\right] }\stackrel{Def}{=}%
\mathrm{i}\cdot \left[ 
\begin{array}{cc}
0_2 & 1_2 \\ 
-1_2 & 0_2
\end{array}
\right] \mbox{, } 
\]

\textit{bitter pentad} \underline{$\Gamma $}:

\[
\underline{\Gamma }^{\left[ 1\right] }\stackrel{Def}{=}\mathrm{i}\cdot
\left[ 
\begin{array}{cc}
0_2 & -\sigma _1 \\ 
\sigma _1 & 0_2
\end{array}
\right] \mbox{, }\underline{\Gamma }^{\left[ 2\right] }\stackrel{Def}{=}%
\mathrm{i}\cdot \left[ 
\begin{array}{cc}
0_2 & -\sigma _2 \\ 
\sigma _2 & 0_2
\end{array}
\right] \mbox{, }\underline{\Gamma }^{\left[ 3\right] }\stackrel{Def}{=}%
\mathrm{i}\cdot \left[ 
\begin{array}{cc}
0_2 & -\sigma _3 \\ 
\sigma _3 & 0_2
\end{array}
\right] \mbox{, } 
\]

\[
\underline{\Gamma }^{\left[ 0\right] }\stackrel{Def}{=}\left[ 
\begin{array}{cc}
-1_2 & 0_2 \\ 
0_2 & 1_2
\end{array}
\right] \mbox{, }\underline{\Gamma }^{\left[ 4\right] }\stackrel{Def}{=}%
\left[ 
\begin{array}{cc}
0_2 & 1_2 \\ 
1_2 & 0_2
\end{array}
\right] \mbox{.} 
\]



\section*{References}

\begin{thebibliography}{8}

%\section{}


\bibitem{Cp1}
M. Capek, {\it The Philosophical Impact oj Contemporary Physics}, 
D. Van Nostrand, Princeton, N.J. (1961)

\bibitem{Cp2} 
M. Capek, Particles or events. in {\it Physical Sciences and History of 
Physics}, Ed. R. S. Cohen and M. W. Wartorsky. Reidel, Boston, Mass. 
(1984)

\bibitem{W}   
E. C. Whipple jr. Nuovo Cimento A, {\bf 11}, 92 (1986)

\bibitem{J}   
J. Jeans, {\it The New Background of Science}, Macmillan, N. Y. (1933)


\bibitem{Jd3}  
Ph. Blanchar and A. Jadczyk {\it Preprint}  


\bibitem{Md}
for instance, E. Madelung, {\it Die Mathematischen Hilfsmittel 
des Physikers.} (Springer Verlag, 1957) p.29

\bibitem{Kn}
for instance, Gordon Kane, {\it Modern Elementary Particle
Physics.} (Addison-Wesley Publishing Company, Inc., 1987)


\bibitem{Pf}
for instance, Stephanie J. Morris, The Pythagorean Theorem,
The University of Georgia, Department of Mathematics Education J. Wilson,
EMT 669




\end{thebibliography}


\end{document}


