---
abstract: 'Electroweak, gluons, and gravity fields arise as gauge fields from probabilities of physical events.'
author:
- |
    G. Quznetsov\
    gunn@mail.ru, gunn@chelcom.ru
date: 'August 17, 2007'
title: |
    All four forces without superstrings.\
    (Other game in town)
---

Introduction
============

As is obvious, the strings theory arrives at it’s logical finish [@Sch]. But it is possible that Nature is done more simply and more naturally than that. In this article I’m propose the deduction of the electroweak, gluons, and gravity forces from the representation of physical events’ probabilities by spinors as “other game in town” [@Sch1].

Electroweak fields
==================

Let $\left\langle \rho \left( \underline{x}\right) ,j_1\left( \underline{x}%
\right) ,j_2\left( \underline{x}\right) ,j_3\left( \underline{x}\right)
\right\rangle =\left\langle \rho \left(t,\mathbf{x}\right) ,\mathbf{j}%
\left( t,\mathbf{x}\right) \right\rangle $ be a probability density 3+1-vector of any physical event [@Q0].

Complex functions $\varphi _1\left( \underline{x}%
\right) $, $\varphi _2\left( \underline{x}\right) $, $\varphi _3\left( 
\underline{x}\right) $, $\varphi _4\left( \underline{x}\right) $ exist [@Q1] such that

$$\begin{aligned}
\rho &=&\sum_{s=1}^4\varphi _s^{*}\varphi _s\mbox{,}
\label{j} \\
\frac{j_{\alpha}}{\mathrm{c}} &=&-\sum_{k=1}^4\sum_{s=1}^4%
\varphi _s^{*}\beta _{s,k}^{\left[ \alpha \right] }\varphi _k  \nonumber\end{aligned}$$

for every such density vector. Here $\alpha \in \left\{ 1,2,3\right\} $ and $\beta ^{\left[ \alpha \right]
}$ - are diagonal elements of the light Clifford’s pentad [@Q2].

If

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

then [@Q4]

$$\frac 1{\mathrm{c}}\partial _t\varphi +\left( \mathrm{i}\Theta _0+\mathrm{i}%
\Upsilon _0\gamma ^{\left[ 5\right] }\right) \varphi =\left( 
\begin{array}{c}
\beta ^{\left[ 1\right] }\partial _1+\mathrm{i}\Theta _1\beta ^{\left[
1\right] }+\mathrm{i}\Upsilon _1\beta ^{\left[ 1\right] }\gamma ^{\left[
5\right] }+ \\ 
+\beta ^{\left[ 2\right] }\partial _2+\mathrm{i}\Theta _2\beta ^{\left[
2\right] }+\mathrm{i}\Upsilon _2\beta ^{\left[ 2\right] }\gamma ^{\left[
5\right] }+ \\ 
+\beta ^{\left[ 3\right] }\partial _3+\mathrm{i}\Theta _3\beta ^{\left[
3\right] }+\mathrm{i}\Upsilon _3\beta ^{\left[ 3\right] }\gamma ^{\left[
5\right] }+ \\ 
+\mathrm{i}M_0\gamma ^{\left[ 0\right] }+\mathrm{i}M_4\beta ^{\left[
4\right] }- \\ 
-\mathrm{i}M_{\zeta ,0}\gamma _\zeta ^{[0]}+\mathrm{i}M_{\zeta ,4}\zeta
^{[4]}- \\ 
-\mathrm{i}M_{\eta ,0}\gamma _\eta ^{[0]}-\mathrm{i}M_{\eta ,4}\eta ^{[4]}+
\\ 
+\mathrm{i}M_{\theta ,0}\gamma _\theta ^{[0]}+\mathrm{i}M_{\theta ,4}\theta
^{[4]}
\end{array}
\right) \varphi \mbox{.}  \label{ham0}$$

with real $\Theta _k$, $\Upsilon _k$, $M_0$, $M_4$, $M_{\zeta ,0}$, $%
M_{\zeta ,4}$, $M_{\eta ,0}$, $M_{\eta ,4}$, $M_{\theta ,0}$, $M_{\theta ,4}$ and $$\gamma ^{\left[ 5\right] }\stackrel{def}{=}\left[ 
\begin{array}{cc}
1_2 & 0_2 \\ 
0_2 & -1_2
\end{array}
\right] \mbox{,}  \label{g5}$$

here $\gamma ^{\left[ 0\right] }$, $\beta ^{\left[ 4\right] }$ are antidiagonal elements of the light Clifford’s pentad, and $\gamma _\zeta
^{[0]}$, $\zeta ^{[4]}$, $\gamma _\eta ^{[0]}$, $\eta ^{[4]}$, $\gamma
_\theta ^{[0]}$, $\theta ^{[4]}$ are antidiagonal elements of colored Clifford’s pentads [@Q3].

If $M_{\zeta ,0}=0$, $M_{\zeta ,4}=0$, $M_{\eta ,0}=0$, $M_{\eta ,4}=0$, $%
M_{\theta ,0}=0$, $M_{\theta ,4}=0$ then the Dirac lepton moving equation is derived from (\[ham0\]):

$$\left( \mathrm{i}\frac 1{\mathrm{c}}\partial _t-\Theta _0-\Upsilon _0\gamma
^{\left[ 5\right] }\right) \varphi =\sum_{k=1}^3\left( \beta ^{\left[
k\right] }\left( \mathrm{i}\partial _k-\Theta _k-\Upsilon _k\gamma ^{\left[
5\right] }\right) -m\gamma \right) \varphi  \label{eq3}$$

with $m=\sqrt{M_0^2+M_4^2}$ and $\gamma =\left( \frac{M_0}{\sqrt{M_0^2+M_4^2}%
}\gamma ^{\left[ 0\right] }+\frac{M_4}{\sqrt{M_0^2+M_4^2}}\beta ^{\left[
4\right] }\right) $.

Let $x_4$, $x_5$ be some real variables such that

$$-\frac {\pi\mathrm{c}}{\mathrm{h}}\leq x_5\leq \frac {\pi\mathrm{c}}{\mathrm{%
h}},-\frac {\pi\mathrm{c}} {\mathrm{h}}\leq x_4\leq \frac {\pi\mathrm{c}}{%
\mathrm{h}}\mbox{.}$$

and let

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

In this case $\widetilde{\varphi }$ obeys to the following moving equation:

$$\left( \sum_{s=0}^3\beta ^{\left[ s\right] }\left( \mathrm{i}\partial
_s-\Theta _s-\Upsilon _s\gamma ^{\left[ 5\right] }\right) -\gamma ^{\left[
0\right] }\mathrm{i}\partial _5-\beta ^{\left[ 4\right] }\mathrm{i}\partial
_4\right) \widetilde{\varphi }=0$$

(here $\beta ^{\left[ 0\right] }=-1$).

This equation can be transformated to the following form [@Q5]:

$$\left( \sum_{s=0}^3\beta ^{\left[ s\right] }\left( \mathrm{i}\partial
_s+F_s+0.5g_1YB_s\right) -\gamma ^{\left[ 0\right] }\mathrm{i}\partial
_5-\beta ^{\left[ 4\right] }\mathrm{i}\partial _4\right) \widetilde{\varphi }%
=0  \label{eq1}$$

with real $F_s$, $B_s$, real positive $g_1$, and with the charge matrix[^1] :

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

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

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

then equation (\[eq1\]) is invariant for the following transformations:

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

Therefore, $B_\mu $ are components of the Standard Model gauge field $B$.

Further $\Im _{\mathbf{J}}$ be [@DVB], [@AV] the space spanned of the following basis [@Q7]:

$$\mathbf{J}\stackrel{def}{=}\left\langle 
\begin{array}{c}
\frac{\mathrm{h}}{2\pi \mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{%
\mathrm{c}}\left( s_0x_4\right) \right) \epsilon _1,\frac{\mathrm{h}}{2\pi 
\mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{\mathrm{c}}\left(
s_0x_4\right) \right) \epsilon _2, \\ 
\frac{\mathrm{h}}{2\pi \mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{%
\mathrm{c}}\left( s_0x_4\right) \right) \epsilon _3,\frac{\mathrm{h}}{2\pi 
\mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{\mathrm{c}}\left(
s_0x_4\right) \right) \epsilon _4, \\ 
\frac{\mathrm{h}}{2\pi \mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{%
\mathrm{c}}\left( n_0x_5\right) \right) \epsilon _1,\frac{\mathrm{h}}{2\pi 
\mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{\mathrm{c}}\left(
n_0x_5\right) \right) \epsilon _2, \\ 
\frac{\mathrm{h}}{2\pi \mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{%
\mathrm{c}}\left( n_0x_5\right) \right) \epsilon _3,\frac{\mathrm{h}}{2\pi 
\mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{\mathrm{c}}\left(
n_0x_5\right) \right) \epsilon _4
\end{array}
\right\rangle  \label{TJ}$$

with some integer numbers $s_0$ and $n_0$ and with

$$\epsilon _1=\left[ 
\begin{array}{c}
1 \\ 
0 \\ 
0 \\ 
0
\end{array}
\right] ,.\epsilon _2=\left[ 
\begin{array}{c}
0 \\ 
1 \\ 
0 \\ 
0
\end{array}
\right] ,.\epsilon _3=\left[ 
\begin{array}{c}
0 \\ 
0 \\ 
1 \\ 
0
\end{array}
\right] ,.\epsilon _4=\left[ 
\begin{array}{c}
0 \\ 
0 \\ 
0 \\ 
1
\end{array}
\right] \mbox{.}$$

Farther $U$ be any linear transformation of space $\Im _{\mathbf{J}}$ such that for every $\widetilde{\varphi }$: if $\widetilde{\varphi }\in \Im _{%
\mathbf{J}}$ then:

$$\begin{array}{c}
\left( U\widetilde{\varphi }\right) ^{\dagger }\left( U\widetilde{\varphi }%
\right) =\rho \mbox{,} \\ 
\left( U\widetilde{\varphi }\right) ^{\dagger }\beta ^{\left[ s\right]
}\left( U\widetilde{\varphi }\right) =-\frac{j_{,s}}{\mathrm{c}}
\end{array}
\label{un1}$$

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

Matrix $U$ factorized as the following [@Q8]:

$$U=\exp \left( \mathrm{i}\varsigma \right) \widetilde{U}U^{\left( -\right)
}U^{\left( +\right) }$$

with real $\varsigma $ and with

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

and

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

with real $a$, $b$, $c$, $q$, $u$, $v$, $k$, $s$.

Matrix $U^{\left( +\right) }$ refers to antiparticles [@Q9]. And transformation $U^{\left( -\right) }$ reduces equation (\[eq1\]) to the following shape [@Q10]:

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

with real positive $g_2$ and with

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

with real $W_{0,\mu }$, $W_{1,\mu }$ and $W_{2,\mu }$ .

Equation (\[hW\]) is invariant under the following transformation [@Q12]:

$$\begin{aligned}
\ &&\varphi \rightarrow \ \varphi ^{\prime }=U\varphi \mbox{,}  \nonumber \\
\ &&x_4\rightarrow x_4^{\prime }=\left( \ell _{\circ }+\ell _{*}\right)
ax_4+\left( \ell _{\circ }-\ell _{*}\right) \sqrt{1-a^2}x_5\mbox{,} 
\nonumber \\
\ &&x_5\rightarrow x_5^{\prime }=\left( \ell _{\circ }+\ell _{*}\right)
ax_5-\left( \ell _{\circ }-\ell _{*}\right) \sqrt{1-a^2}x_4\mbox{,}
\label{tt2} \\
\ &&x_\mu \rightarrow x_\mu ^{\prime }=x_\mu \mbox{,
for }\mu \in \left\{ 0,1,2,3\right\} \mbox{,} \nonumber\\
\ &&B_\mu \rightarrow B_\mu ^{\prime }=B_\mu \mbox{,}\nonumber \\
\ &&W_\mu \rightarrow W_\mu ^{\prime }=UW_\mu U^{\dagger }-\frac{2\mathrm{i}%
}{g_2}\left( \partial _\mu U\right) U^{\dagger }\nonumber\end{aligned}$$

with [@Q11]

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

Hence $W_\mu $ behaves as components of the weak field $W$ of Standard Model.

Field $W_{0,\mu }$ obeys the following equation [@Q13]:

$$\left( -\frac 1{\mathrm{c}^2}\partial _t^2+\sum_{s=1}^3\partial _s^2\right)
W_{0,\mu }=g_2^2\left( \widetilde{W}_0^2-\widetilde{W}_1^2-\widetilde{W}_2^2-%
\widetilde{W}_3^2\right) W_{0,\mu }+\Lambda  \label{eq2}$$

with

$$\widetilde{W}_\nu =\left[ 
\begin{array}{c}
W_{0,\nu } \\ 
W_{1,\nu } \\ 
W_{2,\nu }
\end{array}
\right]$$

and $\Lambda $ is the action of other components of field $W$ on $W_{0,\mu }$.

Equation (\[eq2\]) looks like to the Klein-Gordon equation of field $%
W_{0,\mu }$ with mass

$$m=\frac{\mathrm{h}}{\mathrm{c}}g_2\sqrt{\widetilde{W}_0^2-\sum_{s=1}^3%
\widetilde{W}_s^2}  \label{z10}$$

and with additional terms of the $W_{0,\mu }$ interactions with others components of $\widetilde{W}$. Fields $W_{1,\mu }$ and $W_{2,\mu }$ have similar equations.

The ”mass” (\[z10\]) is invariant [@Q14] under the Lorentz transformations

$$\widetilde{W}_0^{\prime }\stackrel{def}{=}\frac{\widetilde{W}_0-\frac v{%
\mathrm{c}}\widetilde{W}_k}{\sqrt{1-\left( \frac v{\mathrm{c}}\right) ^2}}%
\mbox{, }\widetilde{W}_k^{\prime }\stackrel{def}{=}\frac{\widetilde{W}%
_k-\frac v{\mathrm{c}}\widetilde{W}_0}{\sqrt{1-\left( \frac v{\mathrm{c}%
}\right) ^2}}\mbox{, }\widetilde{W}_k^{\prime }\stackrel{def}{=}\widetilde{W}%
_k\mbox{, if }s\neq k \mbox{,}$$

is invariant under the turns of the $\left\langle \widetilde{W}_1,\widetilde{%
W}_2,\widetilde{W}_3\right\rangle $ space

$$\left\{ 
\begin{array}{c}
\widetilde{W}_r^{\prime }=\widetilde{W}_r\cos \lambda -\widetilde{W}_s\sin
\lambda \mbox{.} \\ 
\widetilde{W}_s^{\prime }=\widetilde{W}_r\sin \lambda +\widetilde{W}_s\cos
\lambda \mbox{;}
\end{array}
\right|$$

and invariant under a global weak isospin transformation $U$:

$$W_\nu \rightarrow W_\nu ^{\prime }=U^{\left( -\right) }W_\nu U^{\left(
-\right) \dagger }\mbox{,}$$

but is not invariant for a local transformation $U^{\left( -\right) }$. But local transformations for $W_{0,\mu }$, $W_{1,\mu }$ and $W_{2,\mu }$ is insignificant since all three particles are very short-lived.

That is the form (\[z10\]) varies in space, but locally acts like a mass - i.e. it does not allow to particles of this field to behave as a massless ones.

If

$$\begin{array}{c}
Z_\mu \stackrel{def}{=}\left( W_{0,\mu }\cos \alpha -B_\mu \sin \alpha
\right) \mbox{,} \\ 
A_\mu \stackrel{def}{=}\left( B_\mu \cos \alpha +W_{0,\mu }\sin \alpha
\right)
\end{array}$$

with

$$\alpha \stackrel{def}{=}\arctan \frac{g_1}{g_2}$$

then masses of $Z$ and $W$ fulfill to the following ratio [@Q15]:

$$m_Z=\frac{m_W}{\cos \alpha }\mbox{.}$$

If

$$\mathrm{e}\stackrel{def}{=}\frac{g_1g_2}{\sqrt{g_1^2+g_2^2}}\mbox{,}$$ and

$$\widehat{Z}_\mu \stackrel{def}{=}Z_\mu \frac 1{\sqrt{g_2^2+g_1^2}}\left[ 
\begin{array}{cccc}
\left( g_2^2+g_1^2\right) 1_2 & 0_2 & 0_2 & 0_2 \\ 
0_2 & 2g_1^21_2 & 0_2 & 0_2 \\ 
0_2 & 0_2 & \left( g_2^2-g_1^2\right) 1_2 & 0_2 \\ 
0_2 & 0_2 & 0_2 & 2g_1^21_2
\end{array}
\right] \mbox{,}$$

$$\widehat{W}_\mu \stackrel{def}{=}g_2\left[ 
\begin{array}{cccc}
0_2 & 0_2 & \left( W_{1,\mu }-\mathrm{i}W_{2,\mu }\right) 1_2 & 0_2 \\ 
0_2 & 0_2 & 0_2 & 0_2 \\ 
\left( W_{1,\mu }+\mathrm{i}W_{2,\mu }\right) 1_2 & 0_2 & 0_2 & 0_2 \\ 
0_2 & 0_2 & 0_2 & 0_2\cdot 1_2
\end{array}
\right] \mbox{,}$$

$$\widehat{A}_\mu \stackrel{def}{=}A_\mu \left[ 
\begin{array}{cccc}
0_2 & 0_2 & 0_2 & 0_2 \\ 
0_2 & 1_2 & 0_2 & 0_2 \\ 
0_2 & 0_2 & 1_2 & 0_2 \\ 
0_2 & 0_2 & 0_2 & 1_2
\end{array}
\right] \mbox{.}$$

then equation (\[hW\]) has got the following form [@Q16]:

$$\left( \sum_{\mu =0}^3\beta ^{\left[ \mu \right] }\mathrm{i}\left( \partial
_\mu +\mathrm{i}e\widehat{A}_\mu -\mathrm{i}0.5\left( \widehat{Z}_\mu +%
\widehat{W}_\mu \right) \right) +\gamma ^{\left[ 0\right] }\mathrm{i}%
\partial _5+\beta ^{\left[ 4\right] }\mathrm{i}\partial _4\right) \widetilde{%
\varphi }=0.  \label{AZW}$$

Here the vector field $A_\mu $ is* the electromagnetic potential*, and $\left( \widehat{Z}_\mu +\widehat{W}_\mu \right) $ is *the weak potential*.

Gluons fields
=============

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 equation (\[eq3\]) is invariant under the following turning transformations [@Q17]:

$$\begin{array}{c}
\widetilde{\varphi }\rightarrow \widetilde{\varphi }^{\prime }=U_{1,2}\left(
\alpha \right) \widetilde{\varphi }\mbox{,} \\ 
x_0\rightarrow x_0^{\prime }=x_0\mbox{,} \\ 
x_1\rightarrow x_1^{\prime }=x_1\cos \alpha -x_2\sin \alpha \mbox{,} \\ 
x_2\rightarrow x_2^{\prime }=x_1\sin \alpha +x_2\cos \alpha \mbox{,} \\ 
x_3\rightarrow x_3^{\prime }=x_3\mbox{,} \\ 
\Theta _0\rightarrow \Theta _0^{\prime }=\Theta _0\mbox{,} \\ 
\Theta _1\rightarrow \Theta _1^{\prime }=\Theta _1\cos \alpha -\Theta _2\sin
\alpha \mbox{,} \\ 
\Theta _2\rightarrow \Theta _2^{\prime }=\Theta _1\sin \alpha +\Theta _2\cos
\alpha \mbox{,} \\ 
\Theta _3\rightarrow \Theta _3^{\prime }=\Theta _3\mbox{,} \\ 
\Upsilon _0\rightarrow \Upsilon _0^{\prime }=\Upsilon _0\mbox{,} \\ 
\Upsilon _1\rightarrow \Upsilon _1^{\prime }=\Upsilon _1\cos \alpha
-\Upsilon _2\sin \alpha \mbox{,} \\ 
\Upsilon _2\rightarrow \Upsilon _2^{\prime }=\Upsilon _1\sin \alpha
+\Upsilon _2\cos \alpha \mbox{,} \\ 
\Upsilon _3\rightarrow \Upsilon _3^{\prime }=\Upsilon _3\mbox{,} \\ 
M_0\rightarrow M_0^{\prime }=M_0\mbox{,} \\ 
M_4\rightarrow M_4^{\prime }=M_4\mbox{:}
\end{array}
\label{T12}$$

and is invariant under all other turnings of Cartesian system [@Q100].

But under such rotation the mass members of colored pentads of equation (\[ham0\]) are interfused [@Q18]:

$$\begin{aligned}
M_{\zeta ,0}^{\prime }\cos \alpha +M_{\eta ,0}^{\prime }\sin \alpha
&=&M_{\zeta ,0}\mbox{,} \\
M_{\zeta ,4}^{\prime }\cos \alpha +M_{\eta ,4}^{\prime }\sin \alpha
&=&M_{\zeta ,4}\mbox{,} \\
M_{\eta ,0}^{\prime }\cos \alpha -M_{\zeta ,0}^{\prime }\sin \alpha
&=&M_{\eta ,0}\mbox{.} \\
M_{\eta ,4}^{\prime }\cos \alpha -M_{\zeta ,4}^{\prime }\sin \alpha
&=&M_{\eta ,4}\mbox{,} \\
M_{\theta ,0}^{\prime } &=&M_{\theta ,0}\mbox{,} \\
M_{\theta ,4}^{\prime } &=&M_{\theta ,4}\mbox{.}\end{aligned}$$

The mass members of colored pentads are interfused under other Cartesian rotations, too.

Therefore the chromatic triplet elements can not be separated in space. These elements must be *confined* into identical place (*confinement*).

Each chromatic pentad contains two mass elements. Hence, every family contains two sorts of the chromatic particles of tree colors. These particles are quarks.

The hamiltonian of form:

$$\begin{array}{c}
\widehat{H}_{clr}\stackrel{def}{=}\sum_{s=1}^3\beta ^{\left[ s\right]
}\left( \mathrm{i}\partial _s+F_s\right) + \\ 
+M_{\zeta ,0}\gamma _\zeta ^{[0]}-M_{\zeta ,4}\zeta ^{[4]}+M_{\eta ,0}\gamma
_\eta ^{[0]}+M_{\eta ,4}\eta ^{[4]}-M_{\theta ,0}\gamma _\theta
^{[0]}-M_{\theta ,4}\theta ^{[4]}\mbox{.}
\end{array}
\label{Hclr}$$

can be formulated as the following:

$$\widehat{H}_{clr}=\sum_{s=1}^33\lfloor \widehat{\beta }^{\left[ s\right]
}\left( \mathrm{i}\partial _s+F_s\right) +3\lfloor \widehat{M}\mbox{,}
\label{Hcl}$$

here $\lfloor $ is the left bracket of the product K-matricies designation [@Q19], $\widehat{M}$ is a K-matrix such that:

$$\widehat{M}\stackrel{def}{=}\left\| 
\begin{array}{ccc}
\left( M_{\zeta ,0}-\widehat{\mathrm{i}}M_{\zeta ,4}\right) \gamma _\zeta
^{[0]} & 0_4 & 0_4 \\ 
0_4 & \left( M_{\eta ,0}+\widehat{\mathrm{i}}M_{\eta ,4}\right) \gamma _\eta
^{[0]} & 0_4 \\ 
0_4 & 0_4 & \left( -M_{\theta ,0}-\widehat{\mathrm{i}}M_{\theta ,4}\right)
\gamma _\theta ^{[0]}
\end{array}
\right\|  \label{Mx}$$

with

$$\widehat{\mathrm{i}}\stackrel{def}{=}\mathrm{i}\gamma ^{\left[ 5\right]
}=\left[ 
\begin{array}{cccc}
\mathrm{i} & 0 & 0 & 0 \\ 
0 & \mathrm{i} & 0 & 0 \\ 
0 & 0 & -\mathrm{i} & 0 \\ 
0 & 0 & 0 & -\mathrm{i}
\end{array}
\right] \mbox{,}$$

$$\widehat{\beta }^{\left[ 1\right] }\stackrel{def}{=}-\left\| 
\begin{array}{ccc}
\zeta ^{[1]} & 0_4 & 0_4 \\ 
0_4 & \eta ^{[1]} & 0_4 \\ 
0_4 & 0_4 & \theta ^{[1]}
\end{array}
\right\| \mbox{,}$$

$$\widehat{\beta }^{\left[ 2\right] }\stackrel{def}{=}\left\| 
\begin{array}{ccc}
0_4 & \gamma _\zeta ^{[0]}\eta ^{[2]}\gamma _\eta ^{[0]} & 0_4 \\ 
0_4 & 0_4 & \gamma _\eta ^{[0]}\theta ^{[2]}\gamma _\theta ^{[0]} \\ 
\gamma _\theta ^{[0]}\zeta ^{[2]}\gamma _\zeta ^{[0]} & 0_4 & 0_4
\end{array}
\right\| \mbox{,}$$

$$\widehat{\beta }^{\left[ 3\right] }\stackrel{def}{=}\left\| 
\begin{array}{ccc}
0_4 & 0_4 & \gamma _\zeta ^{[0]}\theta ^{[3]}\gamma _\theta ^{[0]} \\ 
\gamma _\eta ^{[0]}\zeta ^{[3]}\gamma _\zeta ^{[0]} & 0_4 & 0_4 \\ 
0_4 & \gamma _\theta ^{[0]}\eta ^{[3]}\gamma _\eta ^{[0]} & 0_4
\end{array}
\right\| \mbox{.}$$

Equation

$$\widehat{H}_{clr}\widetilde{\varphi }=\sum_{s=1}^33\lfloor \widehat{\beta }%
^{\left[ s\right] }\left( \mathrm{i}\partial _s+F_s\right) \widetilde{%
\varphi }\rceil +3\lfloor \widehat{M}\widetilde{\varphi }\rceil \mbox{.}
\label{Hclr2}$$

is invariant [@Q20] under the following transformation:

$$\begin{aligned}
\widetilde{\varphi } &\rightarrow &\widetilde{\varphi }^{\prime }\stackrel{%
def}{=}\lfloor U\widetilde{\varphi }\rceil \mbox{,}  \label{TG} \\
\widehat{M} &\rightarrow &\widehat{M}^{\prime }\stackrel{def}{=}\lfloor U%
\widehat{M}U^{\dagger }\rceil \mbox{,}  \nonumber \\
\widehat{\beta }^{\left[ s\right] } &\rightarrow &\widehat{\beta }^{\left[
s\right] \prime }\stackrel{def}{=}\lceil U\widehat{\beta }^{\left[ s\right]
}U^{\dagger }\rceil \mbox{,}  \nonumber \\
F_s &\rightarrow &F_s^{\prime }\stackrel{def}{=}\left( F_s-\lceil \left( 
\mathrm{i}\partial _sU\right) U^{\dagger }\rceil \right)  \nonumber\end{aligned}$$

with an unitary 3$\times $3 K-matrix $U$.

There:

$$\lfloor \left( \mathrm{i}\partial _sU\right) U^{\dagger }\rceil =\mathrm{i}%
\widehat{\mathrm{i}}\sum_{r=1}^8\lambda _rg\frac 12\gamma ^{\left[ 5\right]
}G_s^r$$

with $G_s^r$ are real, $g$ is a real positive and:

$$U=\exp \left( \widehat{\mathrm{i}}\sum_{r=1}^8\lambda _r\alpha _r\right)$$

with

$$\lambda _1\stackrel{def}{=}\left\| 
\begin{array}{ccc}
0_4 & 1_4 & 0_4 \\ 
1_4 & 0_4 & 0_4 \\ 
0_4 & 0_4 & 0_4
\end{array}
\right\| \mbox{, }\lambda _2\stackrel{def}{=}\left\| 
\begin{array}{ccc}
0_4 & -\widehat{\mathrm{i}} & 0_4 \\ 
\widehat{\mathrm{i}} & 0_4 & 0_4 \\ 
0_4 & 0_4 & 0_4
\end{array}
\right\| \mbox{,}\lambda _3\stackrel{def}{=}\left\| 
\begin{array}{ccc}
1_4 & 0_4 & 0_4 \\ 
0_4 & -1_4 & 0_4 \\ 
0_4 & 0_4 & 0_4
\end{array}
\right\| \mbox{,}$$

$$\lambda _4\stackrel{def}{=}\left\| 
\begin{array}{ccc}
0_4 & 0_4 & 1_4 \\ 
0_4 & 0_4 & 0_4 \\ 
1_4 & 0_4 & 0_4
\end{array}
\right\| \mbox{, }\lambda _5\stackrel{def}{=}\left\| 
\begin{array}{ccc}
0_4 & 0_4 & -\widehat{\mathrm{i}} \\ 
0_4 & 0_4 & 0_4 \\ 
\widehat{\mathrm{i}} & 0_4 & 0_4
\end{array}
\right\| \mbox{, }\lambda _6\stackrel{def}{=}\left\| 
\begin{array}{ccc}
0_4 & 0_4 & 0_4 \\ 
0_4 & 0_4 & 1_4 \\ 
0_4 & 1_4 & 0_4
\end{array}
\right\| \mbox{,}$$

$$\lambda _7\stackrel{def}{=}\left\| 
\begin{array}{ccc}
0_4 & 0_4 & 0_4 \\ 
0_4 & 0_4 & -\widehat{\mathrm{i}} \\ 
0_4 & \widehat{\mathrm{i}} & 0_4
\end{array}
\right\| \mbox{, }\lambda _8\stackrel{def}{=}\frac 1{\sqrt{3}}\left\| 
\begin{array}{ccc}
1_4 & 0_4 & 0_4 \\ 
0_4 & 1_4 & 0_4 \\ 
0_4 & 0_4 & -2\cdot 1_4
\end{array}
\right\| \mbox{.}$$

Here $G_s^r$ are *the gluon fields*.

Gravity fields
==============

Equation (\[ham0\]) can be reformed as the following [@Q21]:

$$\left( 
\begin{array}{c}
\beta ^{\left[ 0\right] }\left( \partial _0+\mathrm{i}\Theta _0+\mathrm{i}%
\Upsilon _0\gamma ^{\left[ 5\right] }-\mathrm{i}M_0\gamma ^{\left[ 0\right]
}-\mathrm{i}M_4\beta ^{\left[ 4\right] }\right)  \\ 
+\beta ^{\left[ 1\right] }\left( \partial _1+\mathrm{i}\Theta _1+\mathrm{i}%
\Upsilon _1\gamma ^{\left[ 5\right] }+M_{\zeta ,0}\beta ^{\left[ 4\right]
}-M_{\zeta ,4}\gamma ^{\left[ 0\right] }\right)  \\ 
+\beta ^{\left[ 2\right] }\left( \partial _2+\mathrm{i}\Theta _2+\mathrm{i}%
\Upsilon _2\gamma ^{\left[ 5\right] }+M_{\eta ,0}\beta ^{\left[ 4\right]
}+M_{\eta ,4}\gamma ^{\left[ 0\right] }\right)  \\ 
+\beta ^{\left[ 3\right] }\left( \partial _3+\mathrm{i}\Theta _3+\mathrm{i}%
\Upsilon _3\gamma ^{\left[ 5\right] }-M_{\theta ,0}\beta ^{\left[ 4\right]
}-M_{\theta ,4}\gamma ^{\left[ 0\right] }\right) 
\end{array}
\right) \varphi =0\mbox{.}$$

Therefore the leptons mass members $M_0$, $M_4$ and the quarks mass members $%
M_{\zeta ,0}$, $M_{\zeta ,4}$, $M_{\eta ,0}$, $M_{\eta ,4}$, $M_{\theta ,0}$, $M_{\theta ,4}$ form a field

$$\mathcal{G}\left( M_0,M_4,M_{\zeta ,0},M_{\zeta ,4},M_{\eta ,0},M_{\eta
,4},M_{\theta ,0},M_{\theta ,4}\right)$$

which acts as other gauge fields $\Theta $ and $\Upsilon \gamma ^{\left[
5\right] }$. Since $\mathcal{G}$ is defined by values of masses then this field is *the gravity field*.

Conclusion
==========

Thus, all four forces arise from the representation of physical events’ probabilities by spinors. And all physics events are interpreted by these four forces. Superstrings are not necessary.

A researcher obtains from Nature only probabilities of physical events. But these probabilities give only the known by this time particles (leptons, quarks) and the known by this time gauge fields (electroweak, gluons, gravity) [@Q]. Therefore, if anybody shall assert that he found a super or he found a Higgs then this assertion is false because all these events[^2] can be interpreted by leptons, quarks, and by electroweak, gluons, gravity fields.

[99]{} Bert Schroer, String theory and the crisis in particle physics,\
arXiv:physics/0603112v4 \[physics.soc-ph\] 5 Oct 2006

Idem, p.29

Gunn Quznetsov, *Probabilistic Treatment of Gauge Theories*, in series Contemporary Fundamental Physics, ed. V. Dvoeglazov, Nova Sci. Publ. Inc., NY (2007), p. 38; G. Quznetsov, *Logical Foundation of Theoretical Physics*, Nova Sci. Publ. Inc., NY, (2006), p. 80

Gunn Quznetsov, *Probabilistic Treatment of Gauge Theories*, in series Contemporary Fundamental Physics, ed. V. Dvoeglazov, Nova Sci. Publ. Inc., NY (2007), p.39-41

Idem., p.38 (formula (2.1))

Idem., p.61 (formula (2.27))

Idem, p.39 (formulas 2.2, 2.3, 2.5, 2.7, 2.9)

Idem, p.62

Idem, p.64 (formula 2.34)

Dvoeglazov, V. V., Additional Equations Derived from the Ryder Postulates in the (1/2,0)+(0,1/2) Representation of the Lorentz Group. hep-th/9906083. *Int. J. Theor. Phys.* **37** (1998) 1909. *Helv. Phys. Acta* **70** (1997) 677. *Fizika B* **6** (1997) 75; *Int. J. Theor. Phys.* **34** (1995) 2467.* Nuovo Cimento* **108** A (1995) 1467. *Nuovo Cimento* **111** B (1996) 483. *Int. J. Theor. Phys.* **36** (1997) 635.

Ahluwalia, D. V., (j,0)+(0,j) Covariant spinors and causal propagators based on Weinberg formalism. nucl-th/9905047. *Int. J. Mod. Phys.* *A* **11** (1996) 1855.

Gunn Quznetsov, *Probabilistic Treatment of Gauge Theories*, in series Contemporary Fundamental Physics, ed. V. Dvoeglazov, Nova Sci. Publ. Inc., NY (2007), p.95

Idem, p.96-98

Idem, p.91-94

Idem, p.129

Idem, p.121

Idem, p.98

Idem, p.127

Idem, p.127-128

Idem, p.129

Idem, p.131

Idem, p.137

Idem, p.136

Idem, p.138

Idem, p.140-146

Idem, p.150-152

Idem, p.155

Gunn Quznetsov, *Probabilistic Treatment of Gauge Theories*, in series Contemporary Fundamental Physics, ed. V. Dvoeglazov, Nova Sci. Publ. Inc., NY (2007); G. Quznetsov, *Logical Foundation of Theoretical Physics*, Nova Sci. Publ. Inc., NY, (2006)

[^1]: I denote $n\times n$ matrices:

    $$1_n\stackrel{def}{=}\left[ 
    \begin{array}{cccc}
    1 & 0 & \cdots & 0 \\ 
    0 & 1 & \cdots & 0 \\ 
    \cdots & \cdots & \cdots & \cdots \\ 
    0 & 0 & \cdots & 1
    \end{array}
    \right] ,0_n\stackrel{def}{=}\left[ 
    \begin{array}{cccc}
    0 & 0 & \cdots & 0 \\ 
    0 & 0 & \cdots & 0 \\ 
    \cdots & \cdots & \cdots & \cdots \\ 
    0 & 0 & \cdots & 0
    \end{array}
    \right] \mbox{.}$$

[^2]: and a dark energy and a dark matter, may be, too
