\documentclass[prd,twocolumn,showpacs,amsmath,amssymb]{revtex4}
\usepackage{graphicx}

\begin{document}
\title{Unified field theory in four dimensions}
\author{Haigang Lu}
\email{lhg@chem.pku.edu.cn}
\affiliation{Quantum chemistry group, Department of chemistry,
Peking University, Beijing 100871, China}


\date{\today}

\begin{abstract}
To explain the simple integral ratio of $m_w/m_z$, an unified field 
theory in four dimensions was suggested, in which electromagnetic 
and strong interactions were put in gauge group $SU(4)$ and weak 
interaction and gravitation in $SL(4,{\bf R})$. To get the phenomenological 
mixing of weak
and electromagnetic interactions, decomposition of 4-dimensional 
Clifford algebra
and rotations around the origin were performed to representations of
$SU(4)$. The angle of rotations did be in agreement with the 
Weinberg angle in experiments and the simple integral ratio $15/17$ 
of $\cos \theta _w$ was verified. The geometric origin of this unified field 
theory was 3-dimensional projective geometry, in 
which a series of physical questions such as gauge group 
$SU(4) \otimes SL(4,{\bf R})$, concept of internal space, splitting and breaking
of gauge group, existence of dark matter, 
3 generations of fermions and so on, were solved geometrically.
\end{abstract}

\pacs{12.10.-g, 12.10.Kt, 02.40.Dr}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

Now, though the standard model(SM) of particle physics, based 
on the gauge symmetry $SU(2)_L \otimes U(1)_Y \otimes SU(3)_c$ has 
been in excellent 
agreement with observations, at least up to energies of order 
$100 GeV$ \cite{hepex}, the grand unified theory(GUT) of three 
interactions,
weak, electromagnetic and strong, is still an open question. 
These three interactions and gravitation could be described by 
gauge theories based on Riemann-Cartan geometry \cite{popov}, but 
we do not know the biggest gauge group and how to generate the
SM by breaking.

Weinberg and Salam independently established the electroweak 
theory based on $SU(2)_L \otimes U(1)_Y$ in 1967-68 \cite{wein, salam}; 
Georgi and Glashow 
proposed the simple GUT model based on $SU(5)$ in 1974 \cite{georgi}; 
there were a lot of different GUT models \cite{lang} subsequently, 
most of gauge 
groups of which included $SU(5)$ as a subgroup. The supersymmetric 
SM and supersymmetric GUT \cite{haber} were established soon after 
supersymmetry was discovered in 1970s. Superstring had become a 
powerful candidate of GUT and unified field theory since 1984 \cite{polch},
though there is no predict from the experiments up to present.
Superstring theory was based on 10- or 11-dimensional spacetime,
so that it
would generate many different 4-dimensional universes after compacting.

The SM and general relativity of gravitation were based on  
4-dimensional spacetime, so we would establish the unified field 
theory(UFT) in $4$ dimensions. The paper included two parts. The 
first part (\ref{phys}) discussed physical theory of UFT, 
which included the 
below. At first (in \ref{fact}), 
a fact in experiments was revealed that 
ratio of mass of weak gauge bosons was a simple integral 
ratio. Secondly (in \ref{sem}),
15 fermions were proposed in representations of
gauge group $SU(4)$, whose breaking led to charge quantization. 
Thirdly (in \ref{weakg}), the 
Weinberg angle in electroweak theory, whose cosine was integral 
ratio, was found by particular rotations of sub-representations of 
$SU(4)$, from which weak gauge group appeared and unification of 
weak interaction with gravitation was constructed. 
At last (in \ref{uft}), The 
UFT based on $SU(4) \otimes SL(4,{\bf R})$ was 
established, which unified all 
four interactions. The second part (\ref{geom}) 
discussed geometric basis 
of UFT, which included the below. 
At first (in \ref{geomphy}), we discussed the relation
of physics and geometry. 
Secondly (in \ref{3d}), it was argued that 
the space we exist in was 3-dimensional. 
Thirdly (in \ref{proj}), the fundamental 
elements and the biggest projective group 
$SL(4,{\bf R}) \otimes SL(4,{\bf R})$
in 3-dimensional projective geometry were given. 
Fourthly (in \ref{line}), it was 
not point geometry but line geometry which was endowed 
with the content of internal space in quantum and
particle physics for its 
completeness. There were two way to describe line geometry. 
At last (in \ref{split}), 
geometric principle of breaking of gauge group was discussed. 
A summary (\ref {sum})
was given at end of paper.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{Physical Theory}
\label{phys}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{A wonderful fact in experiments}
\label{fact}

One of the keys in SM was Weinberg-Salam model of electromagnetic 
and weak interactions, in which the electroweak gauge group 
$SU(2)_L \otimes U(1)_Y$ broke 
to electromagnetic gauge group $U(1)_{em}$ spontaneously and three
weak gauge bosons acquired mass simultaneously. 
For $Z^0$ was eigenvector of mass matrix of $U(1)_Y$ and 
$SU(2)_L$ neutral gauge 
bosons, its mass was larger than $W^\pm$, i.e.,
\begin{equation*}
\frac{{m_w }}
{{m_z }} = \cos \theta _w,
\end{equation*}
in which $\theta _W$ was called Weinberg angle and determined by experiments. 
The Weinberg angle should be predicted in GUT and UFT.

From the last experimental data of mass of $Z^0$ and $W^\pm$ \cite{pdg},
a very simple integral ratio was found:
\begin{equation*}
\begin{gathered}
  m_z  = 91.1876 \pm 0.0021{\text{Gev,}} \hfill \\
  m_w  = 80.423 \pm 0.039{\text{Gev}}, \hfill \\
  \cos \theta _W^{{\text{Experiment}}}  = \frac{{m_w }}
{{m_z }} = 0.88195 \pm 0.00043, \hfill \\
  \cos \theta _W^{{\text{Theory}}}  = \frac{{4^2  - 1}}
{{4^2  + 1}} = \frac{{16 - 1}}
{{16 + 1}} = \frac{{15}}
{{17}} = 0.88235, \hfill \\
  {\text{Error}} \approx 0.045\% . \hfill \\ 
\end{gathered}
\end{equation*}
Noted that the error of $m_z$ was ignored in the 
calculation of error of $m_w/m_z$.

Note: $(8, 15, 17)$ is a Pythagorean number, that is, $a$, $b$ 
and $c$ are integers satisfying $a^2  + b^2  = c^2 $.
All of Pythagorean numbers are $(a = 2mn,{\text{ }}b
= n^2  - m^2 ,{\text{ }}c = n^2  + m^2)$
  and the group of 
integers here satisfies that $n=4$, $m=1$.

%%%%%%%%%%%%%%%%%%%%%%

\subsection{Unification of electromagnetic and
strong interactions in $SU(4)$}
\label{sem}

The biggest compact group in $4$ dimensions is $SU(4)$, which is smaller
than the simple GUT group $SU(5)$ of Georgi and Glashow, so it is 
impossible to 
include SM gauge group $SU(2)_L \otimes U(1)_Y \otimes SU(3)_C$ directly
and wholly. 
Notably and essentially, weak interaction was different one 
in contrast with electromagnetic and strong interactions \cite{perkins}.
\begin{enumerate}
\item Weak gauge bosons were massive, but electromagnetic and 
strong ones massless.
\item Some conservation laws were broken by weak interaction, but not 
by electromagnetic and strong interactions.
\end{enumerate}
It concluded that electromagnetic and strong interactions came from 
the same origin and weak interaction from another origin.
So we would unify strong and electromagnetic interactions in 
gauge theory of $SU(4)$
directly and $SU(4)$ could break to $SU(3)_C \otimes U(1)_{em}$
subsequently, and then charge quantization appeared.

The representations of $SU(4)$, \b{1}+\b{4}+\b{6}+\b{4}$^*$+\b{1}$^*$, 
formed a 4-dimensional Clifford algebra \cite{loun},
which was
\[
\begin{gathered}
  \left[ 0 \right] \\ 
  \left[ {\begin{array}{*{20}c}
   {l_1 } & {l_2 } & {l_3 } & {l_4 }  \\

 \end{array} } \right] \\ 
  \left[ {\begin{array}{*{20}c}
   0 & {l_1  \wedge l_2 } & {l_1  \wedge l_3 } & {l_1  \wedge l_4 }  \\
   {l_2  \wedge l_1 } & 0 & {l_2  \wedge l_3 } & {l_2  \wedge l_4 }  \\
   {l_3  \wedge l_1 } & {l_3  \wedge l_2 } & 0 & {l_3  \wedge l_4 }  \\
   {l_4  \wedge l_1 } & {l_4  \wedge l_2 } & {l_4  \wedge l_3 } & 0  \\

 \end{array} } \right] \\ 
  \left[ {\begin{array}{*{20}c}
   {l_2  \wedge l_3  \wedge l_4 } & {l_3  \wedge l_4  \wedge l_1 } 
   & {l_4  \wedge l_1  \wedge l_2 } & {l_1  \wedge l_2  \wedge l_3 }  \\

 \end{array} } \right] \\ 
  \left[ {l_1  \wedge l_2  \wedge l_3  \wedge l_4 } \right]. \\ 
\end{gathered} 
\]
(By the way, the representations of fermions of $SU(5)$ 
given by Georgi and Glashow, \b{5}$^*$+\b{10},
did not form an algebra and was not close.) 
15 fermions of first generation and vacuum
could be filled in the above close
16-dimensional representations of $SU(4)$ 
following Georgi and Glashow:
\[
\begin{gathered}
  \left[ 0 \right] \\ 
  \left[ {\begin{array}{*{20}c}
   {d^1 } & {d^2 } & {d^3 } & {e^ +  }  \\

 \end{array} } \right] \\ 
  \left[ {\begin{array}{*{20}c}
   0 & {u_3^c } & { - u_2^c } & { - u^1 }  \\
   { - u_3^c } & 0 & {u_1^c } & { - u^2 }  \\
   {u_2^c } & { - u_1^c } & 0 & { - u^3 }  \\
   {u^1 } & {u^2 } & {u^3 } & 0  \\

 \end{array} } \right]{\text{ }} \\ 
  \left[ {\begin{array}{*{20}c}
   {e^ -  } & {d_{\text{3}}^c } & {d_2^c } & {d_{\text{1}}^c }  \\

 \end{array} } \right] \\ 
  \left[ {\nu _e } \right] \\ 
\end{gathered} 
\]
By the mixing of vacuum $[0]$ and neutrino $[\nu _e]$, the 
right-hand anti-neutrino and left-hand neutrino were generated, i.e.
$[0] + [\nu ] \to [\nu ]_L  + [\tilde \nu ]_R $.
Then the above representations became to
\[
\begin{gathered}
  \left[ {\tilde \nu _{\text{e}} } \right]_R  \\ 
  \left[ {\begin{array}{*{20}c}
   {d^1 } & {d^2 } & {d^3 } & {e^ +  }  \\

 \end{array} } \right] \\ 
  \left[ {\begin{array}{*{20}c}
   0 & {u_3^c } & { - u_2^c } & { - u^1 }  \\
   { - u_3^c } & 0 & {u_1^c } & { - u^2 }  \\
   {u_2^c } & { - u_1^c } & 0 & { - u^3 }  \\
   {u^1 } & {u^2 } & {u^3 } & 0  \\

 \end{array} } \right] \\ 
  \left[ {\begin{array}{*{20}c}
   {e^ -  } & {d_3^c } & {d_2^c } & {d_1^c }  \\

 \end{array} } \right] \\ 
  \left[ {\nu _e } \right]_L  \\ 
\end{gathered} 
\]
in which the charge of fermions and strong interaction were 
satisfied entirely.

%%%%%%%%%%%%%%%%%%%%%

\subsection{Weak interaction and gravitation}
\label{weakg}

As the gauge group $SU(4)$ was broken to $SU(3)_C \otimes U(1)_{em}$,
the 4-dimensional Clifford algebra was split to two 3-dimensional 
Clifford subalgebras:
\[
\begin{array}{*{20}c}
   {\left\{ {\left[ {\tilde \nu _e } \right]_R {\text{  }}\left[
   {\begin{array}{*{20}c}
   {d^1 }  \\
   {d^2 }  \\
   {d^3 }  \\

 \end{array} } \right]{\text{  }}\left[ {\begin{array}{*{20}c}
   {u_1^c }  \\
   {u_2^c }  \\
   {u_3^c }  \\

 \end{array} } \right]{\text{  }}\left[ {e^ -  } \right]} \right\}}  \\
    \oplus   \\
   {\left\{ {\left[ {e^ +  } \right]{\text{  }}\left[ {\begin{array}{*{20}c}
   {u^1 }  \\
   {u^2 }  \\
   {u^3 }  \\

 \end{array} } \right]{\text{  }}\left[ {\begin{array}{*{20}c}
   {d_1^c }  \\
   {d_2^c }  \\
   {d_3^c }  \\

 \end{array} } \right]{\text{  }}\left[ {\nu _e } \right]_L } \right\}}  \\

 \end{array},
\]
from which the representations of weak group $SU(2)$ were
generated one-to-one.


The orders of Clifford algebras formed Pascal triangle.
The Pascal triangle of orders of $0-4$ dimensional Clifford algebras,
decomposition of 4-dimensional Cilfford algebra to two 3-dimensional
subalgebras, and angle between the two 3-dimensional subalgebras were
shown in figure \ref{fang},
\begin{figure}
\includegraphics[width=200pt]{fang}
\caption{
  \label{fang}   
  Pascal triangle, decomposition of 4-dimensional Clifford algebra,
  and angle between two subalgebras.
  }
\end{figure}
from which we noted that the representations of $SU(2)$
for weak interaction could be not generated one-to-one directly, and it 
should be obtained by rotating the subalgebras around the center for 
half of the angle $\theta$ in figure \ref{fang} synchronously.
Then, we found that
${\tan}\frac{\theta }
{2} = \frac{1}
{4}{\text{  }} \Rightarrow {\text{ }}
\left\{ 
{\begin{array}{*{20}ccl}
  {\cos}\theta &{{ = }}&15/17, \\
  {\sin}\theta &{{ = }}&8/17, \\
  {\sin}^{{2}} \theta & = &0.22145. \\

 \end{array} } \right.$

Our result $\sin^2 \theta = 0.22145$ was in agreement 
with the experimental data 
$\sin ^2 \theta _w  = 0.232$  (from $Z^0$) and
$0.223$ (from $W^\pm$) \cite{erler}. For ${m_w}/
{m_z } = \cos \theta _w $, the prediction of our theory was in
agreement with the experiments exactly. The simple integral ratio 
of mass of $W^\pm$ and $Z^0$,
\[\frac{m_w}{m_z} =\frac{15}{17} = \frac {4^2-1}{4^2+1},\]
implied affinities between weak interaction and 4-dimensional spacetime.

Now, weak gauge group would be given. The basis of Einstein 
gravitation was Riemann metric $g_{\mu \nu}$, a symmetric
$4 \times 4$ matrix. The biggest group in 4-dimensional space is 
$SL(4,{\bf R})$, which could decompose into $SO(4) \otimes _s g_{\mu \nu}$.
$SO(4) \cong SU(2)_a \otimes SU(2)_b$ just is the weak gauge 
group, which is related as below:
\[
\begin{gathered}
  \left( {\begin{array}{*{20}c}
   0 & {x_1 } & {x_2 } & {x_4 }  \\
   { - x_1 } & 0 & {x_3 } & {x_5 }  \\
   { - x_2 } & { - x_3 } & 0 & {x_6 }  \\
   { - x_4 } & { - x_5 } & { - x_6 } & 0  \\

 \end{array} } \right) \hfill \\
  =\frac{{\text{1}}}
{{\text{2}}}\left( {\begin{array}{*{20}c}
   {\text{0}} & {\sigma _1^a  + \sigma _1^b } & {\sigma _2^a  + 
   \sigma _2^b } & {\sigma _3^a  - \sigma _3^b }  \\
   { - (\sigma _1^a  + \sigma _1^b )} & {\text{0}} & {\sigma _3^a 
   + \sigma _3^b } & {\sigma _2^a  - \sigma _2^b }  \\
   { - (\sigma _2^a  + \sigma _2^b )} & { - (\sigma _3^a  + 
   \sigma _3^b )} & {\text{0}} & {\sigma _1^a  - \sigma _1^b }  \\
   { - (\sigma _3^a  - \sigma _3^b )} & { - (\sigma _2^a  - 
   \sigma _2^b )} & { - (\sigma _1^a  - \sigma _1^b )} & {\text{0}}  \\

 \end{array} } \right) \hfill \\ 
\end{gathered} 
\]
in which $\sigma _i^{a,b}$ are coefficients of Pauli matrix,
\[
\sigma _1  = \left( {\begin{array}{*{20}c}
   0 & 1  \\
   1 & 0  \\

 \end{array} } \right),{\text{ }}\sigma _2  = \left( {\begin{array}{*{20}c}
   0 & { - i}  \\
   i & 0  \\

 \end{array} } \right),{\text{ }}\sigma _3  = \left( {\begin{array}{*{20}c}
   1 & 0  \\
   0 & { - 1}  \\

 \end{array} } \right).
 \]
Figure \ref{rotate1} showed SO(4) how to act on the 
weak representations of fermions. 
We noted that the representations of 
fermions induced matrix of $SO(4)$ to be rotated 45 degree 
around the center.
\begin{figure}
\includegraphics[width=200pt]{rotate1}
\caption{
  \label{rotate1}   
  The way SO(4) acting on weak representations of fermions,
and $\sigma _i^+= \sigma _i^a+ \sigma _i^b$, 
$\sigma _i^-= \sigma _i^a- \sigma _i^b$(i = 1,2,3).
  }
\end{figure}
Two $SU(2)$s both are weak gauge groups, so there are
two degenerate $Z^0$ and two pairs of degenerate $W^\pm$.
The weak interaction and gravitation were unified in $SL(4, {\bf R})$, 
so that these two interactions have the same origin, and 
the weak gauge bosons should be massive. As strong and 
electromagnetic gauge group $SU(4)$ would act on weak and 
gravitation gauge group $SL(4,{\bf R})$, then weak gauge bosons could 
have charge. We do not know the mechanisms of generating 
mass and charge yet.

%%%%%%%%%%%%%%%%%%%%

\subsection{Unified field theory}
\label{uft}

All four interactions have been unified in $SU(4) \otimes 
SL(4,{\bf R})$ gauge theory in 4-dimensional space. 
For acquiring quantitative information, we need metric 
so that $SL(4,{\bf R})$ split across to
gravitation $g _{\mu \nu}$ and weak interaction 
$SO(4) \cong [SU(2)]^2$, i.e. $SL(4,{\bf R})\rightarrow  SO(4)\otimes _s
g_{\mu\nu}$. The splitting of gravitation and weak 
interaction induced the internal space acted on by $SU(4)$,
split to $3$ and $1$ dimensions., i.e. $4\rightarrow 3+1$
( see \ref{split}), 
so that $SU(4)$ was split to strong $SU(3)_C$
and electromagnetic $U(1)_{em}$ interactions, i.e. 
$SU(4)\rightarrow SU(3)_C\otimes U(1)_{em}$, from which the 
charge quantization appeared. The electroweak theory was a 
phenomenological unification. Only in the unification of four 
interactions, the basic concepts such as charge and mass
could be understand correctly.

Here, we gave the broken processes in this unification:
\[
\begin{array}{ccl}
   SU(4) & \to & SU(3)_C  \otimes U(1)_{em}  \\
    \otimes & \uparrow & \\
   SL(4,{\bf R}) & \to &  [SO(4)_w  \cong [SU(2)_w ]^2 ] \otimes _s
   g_{\mu \nu }.   \\

 \end{array} 
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Geometric basis}
  \label{geom}

%%%%%%%%%%%%%%%%%%%%

\subsection{Geometrization of physics}
\label{geomphy}

It is well known that there are deep relations between physics and 
geometry. Geometry is mathematical basis of physical theory and 
serves as a fundamental tool in study of physics, so that the 
geometrization of physics is always pursued. In quantum physics, 
there is no progress for geometrization. For example, wave function, 
the most basic concept in quantum physics, did not have any geometric 
explain before. 
In \cite{faus}, Fauser summarized projective relativity,
in which ``quantum physics is incidence physics'' was concluded.
In particle physics, geometrization was concentrated on 
gauge theory based on Riemann-Cartan geometry.
Gauge theory just is fibre bundle, in which gauge field corresponds 
to connection and field strength to curvature. The above fibres are 
homogeneous spaces such as Lie groups, Euclid spaces and so on. 
In past, the importance of fibre space, 
in particular Klein homogenous geometry as a 
fundamental scene of spacetime, was 
neglected. 

In order to lay the complete geometric foundation for modern physics,
we would discuss the important station of Klein homogeneous spaces.


%%%%%%%%%%%%%%%%%%%%%%

\subsection{Space is 3-dimensional.}
\label{3d}

At beginning of 20 century, Poincar\'e discussed the space 
in which we exist. He resulted that the fundamental and single 
property of our space in topology was that it was 3-dimensional 
continuum \cite{poin}.
The recent experimental \cite{hoyl} results showed that the square 
inverse law of Newton gravitation was satisfied in high 
precision as to $200 \mu m$, which verified that the space 
was 3-dimensional. 

Now, we only supposed that 3-dimensional
space was the basis of existence of matter. 
Space and time have same degrees in (special or general) 
relativity so that evolvement of time has no influence to 
dimension of space, and only changes the mode of existence in 
space. That is to say, 3-dimensional space would not 
change following time. Below the energy of Planck scale, 
we would think that dimension of space could not increase with 
the energy. All in all, 3-dimensional space must be the 
basis of existence of matter so that it is the basis of 
gauge bosons, fermions and dark matter.

%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Projective geometry in $3$ dimensions}
\label{proj}

Cayley said \cite{cayley}:`` Projective geometry is all geometry.''
F. Klein classed the homogeneous geometry beginning from 
projective geometry in the famous Erlangen Program \cite{Klein,Klein1}. 
The projective geometry in $3$ dimension was needed because 
the space we exist in was 3-dimensional in topology, 
and the biggest
homogeneous geometry was projective geometry.

From the Klein's point of view, geometry studies 
the geometric invariant figures and relations under 
one transformation group. Projective geometry studies 
the invariant figures and relations under 
projective transformation group \cite{sam}. A point would be
described by homogeneous coordinate in projective 
geometry. For example a points in n-dimensional
projective geometry is 
described by $n+1$ parameters. By this means, all 
points including infinite point is in equal degree,
which agrees with the fundamental claim of special or
general relativity: there is
no special point in universe.

Because there are $n+1$ parameters for coordinate of a point, 
the projective collineation group is $SL(n+1, {\bf R})
=GL(n+1, {\bf R})/{\bf R}$, 
a $(n+1) \times (n+1)$ special matrix,whose dimension is $(n+1)^2-1$. 

Another transformation in projective geometry is duality. 
The points and $(n-1)$-dimensional superplanes are 
represented by the same number of parameters in projective geometry,
relation between the lines and $(n-2)$-dimensional superplanes 
is similar with above, and so on. The transformations 
between points and $(n-1)$-dimensional elements, between 
lines and $(n-2)$-dimensional elements and so on, 
are duality. For point space of projective geometry, 
dualistic transformations is not a group, 
but it does be a group for combination of point space 
and $(n-1)$-dimensional superplane space. The dual group is 
$SL(n+1,{\bf R})$, a $(n+1)\times (n+1)$ special matrix, whose 
dimensions is $(n+1)^2-1$ also. In projective geometry 
the dual group and collineation group
forms the biggest group called
projective transformation group.

In 3-dimensional projective geometry, the fundamental 
elements is point, whose coordinate is $(x_0, x_1, x_2, x_3)$,
plane, whose coordinate is $(S_0, S_1, S_2, S_3)$, and line. 
For two points give a line or two planes intersect on a line,
coordinate of line could be represented by two points or two planes:
\[
\left( {\begin{array}{*{20}c}
   {x_0 } \hfill & {x_1 } \hfill & {x_2 } \hfill & {x_3 } \hfill  \\
   {y_0 } \hfill & {y_1 } \hfill & {y_2 } \hfill & {y_3 } \hfill  \\

 \end{array} } \right)
 \]
or 
\[
\left( {\begin{array}{*{20}c}
   {S_0 } \hfill & {S_1 } \hfill & {S_2 } \hfill & {S_3 } \hfill  \\
   {T_0 } \hfill & {T_1 } \hfill & {T_2 } \hfill & {T_3 } \hfill  \\

 \end{array} } \right).
 \]
The collineation group in 3-dimensional projective geometry 
is $SL(4,{\bf R})$ and dual group also is $SL(4,{\bf R})$, but the duality 
does not form a group in point space or plane space. 
In combination of point and plane space, duality forms a group. 
The duality transforms a line to another line, so duality 
forms group in line space. The biggest transformation 
group is $SL(4,{\bf R}) \otimes SL(4,{\bf R})$ in point-plane space or 
line space.

The generators of $SL(4,{\bf R})$ and $SU(4)$ have the similar 
commutative relations, they could translate each other 
according to the mechanism of Kosyakov's 
``spontaneous symmetry deformation'' \cite{kosy}, which indicated that 
$SL(4,{\bf R})$ could deform to $SU(4)$ at ``high temperature''.
Then our group $SL(4,{\bf R}) \otimes SL(4,{\bf R})$
could deform to $SU(4)\otimes SL(4,{\bf R})$,
which just is the gauge group of UFT.

%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Line geometry}
\label{line}

In modern physics, it is not complete for describing 
an event with point only, that is to say, a point has no 
enough information for completely describing an object 
we studied, and the information about energy and momentum must 
be appended. This corresponds with the case in projective geometry: 
point space is not complete under projective 
transformation group, and the combination of point and 
plane space is complete. We discovered that the line 
space alone is complete under projective transformation group,
which shows that the information of line is complete. So 
the line corresponds with the wave function in quantum 
physics for completeness. 
Therefore the geometrization of wave function is that
wave function is the line of 3-dimensional projective geometry and 
can describe events completely. Point is 
intersection of lines and plane is combination of lines.

Now, we will give the way to describe line in
3-dimensional projective geometry \cite{pott}.

The coordinate of line could be described by coordinates 
of two points or two planes, which is a $2 \times 4$ matrix. 
To obtain vector representations of line, Pl\"ucker coordinate or 
Klein coordinate was used.

Let $\delta _{ij}  = \left| {xy} \right|_{ij}  
= \left| {\begin{array}{*{20}c}
   {x_i } & {x_j }  \\
   {y_i } & {y_j }  \\

 \end{array} } \right|{\text{  }}(i,j = 0,1,2,3),$
in which $x_i$ and $y_i$ are coordinates of points or planes. Then 
$\delta _{ij}  =  - \delta _{ji} $.
Let
\[
\begin{array}{*{20}c}
   {p_1  = \delta _{01} ,} \hfill & {p_2  = \delta _{02} ,} 
   \hfill & {p_3  = \delta _{03} ,} \hfill  \\
   {p_4  = \delta _{23} ,} \hfill & {p_5  = \delta _{31} ,}
   \hfill & {p_6  = \delta _{12}, } \hfill  \\

 \end{array} 
 \]
and six numbers $p_i$ , which are not equal to zero all, 
are called homogeneous coordinates of line $p$ 
relative to reference system, i.e. Pl\"ucker coordinate.

The ratios of six numbers are not independent entirely, 
so the degree of freedom of line is $4$ in 3-dimensional geometry. 
They must satisfy the fundamental homogeneous quadric 
identity called Pl\"ucker identity:
\[
\Omega _{pp}  \equiv p_1 p_4  + p_2 p_5  + p_3 p_6  = 0.
\]

Line space of 3-dimensional projective geometry corresponds 
with 4-dimensional quadric supersurface $Q_2^4$ in 
5-dimensional projective geometry. After transformation
\[
p_1  = x_1  + ix_4 ,{\text{ }}p_4  = x_1  - ix_4,
{\text {and so on}},
\]
the Klein coordinate of line appears, in which the 
Pl\"ucker identity becomes
\[
\sum\limits_{i = 1}^6 {x_i^2 }  = 0.
\]
The quadric supersurface $Q_2^4$ is Grassmann geometry $G(2,4)$, 
whose dimension is $4$.
Using $[l_1, l_2, l_3, l_4]$ to represent a point of $G(2,4)$,
the basic representation space of fermions is 4-dimensional ,
and the formations 
of fermions shows as \ref{sem}. We noted that the point dimension 
of one line dimension is $[3/4]$ because the point space is 
3-dimensional and line space is 4-dimensional. 
Then dimension of fundamental fermions is $[3/4]$,
and of two orders $[3/2]$, and of three orders $[9/4]$.
For electron is representation of three order and positron is basic one, 
the apparent dimension of electron and positron in 
Lagrangian is $([3/4]+[9/4])/2= [3/2]$, which ia in agreement
with the results of quantum field theory.

Then we concluded that line space was internal space of 
wave function, which was 4-dimensional.

%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Splitting and breaking of projective group}
\label{split}

We can not give any quantitative result such as length 
and angle so far because there is no metric in projective 
geometry. According to Klein's point of view in Erlangen 
Program, in order to describe our world further, the metric 
must be endowed with projective geometry, which implies 
that the quadric surfaces in projective geometry must be 
determined.

Duality transforms points to planes and vice versa. 
Two dualities transform points to points, and planes to planes. 
When the same points must be transformed to the same planes under duality, 
3-dimensional projective space is split to quadric surfaces 
and null systems, and the dual group $SL(4,{\bf R})$ is split 
to symmetric part $S_{ij}$ and antisymmetric part $A_{ij}$
simultaneously, in which the symmetric part corresponds to
metric tensor $g_{\mu\nu}$ of gravitation and antisymmetric part,
compact subgroup of $SL(4,{\bf R})$, corresponds to group 
$SO(4) \cong [SU(2)]^2$ of weak interaction. 

The splitting above also make the line space $Q_2^4$ be split. 
A quadric surface of 3-dimensional projective geometry 
corresponds a quadric curve of 4-dimensional Klein line space, 
and a null symmetry to a linear complex, a 3-dimensional 
supersurface. Then the 4-dimensional Klein line space is split 
to $(1+3)$-dimensional space so that the collineation group 
$SU(4)$ is broken to $SU(3)$ and $U(1)$. In particle physics 
a series of results of transmutation of groups and 
line space above mean that splitting to gravitation and weak 
interaction of dual group induces the collineation group 
to be broken to strong and electromagnetic interactions.

The null system corresponds with 3-dimensional
subspace of strong interaction, in which there is only one quadric 
curve wholly. 
1-dimensional subspace of strong interaction
corresponds with representation of free quark,
but there is no quadric 
curve similar to electromagnetic interaction for it.
That is to say, there is no metric for 
free quark.Then the free quarks are not metrizable,
so that they are confined in nature.
It indicates that ``quark confinement'' is connected with 
the metrizable property of line space.

Different generations of fermions have different masses only. 
In order to classify generations of fermions, 
the different metric tensors must be classified, 
i.e. the different quadric surface in 3-dimensional 
projective geometry must be classified in  real 
field. According to ranks and signatures of metric 
tensors, there are $4$ classes and $8$ subclasses:

Rank=4, $\left\{ {\begin{array}{*{20}c}
  {x_1^2  + x_2^2  + x_3^2  + x_0^2  = 0} \\
  {x_1^2  + x_2^2  + x_3^2  - x_0^2  = 0} \\
  {x_1^2  + x_2^2  - x_3^2  - x_0^2  = 0} \\

 \end{array} } \right.$

Rank=3, $\left\{ {\begin{array}{*{20}c}
  {x_1^2  + x_2^2  + x_0^2  = 0} \\
  {x_1^2  + x_2^2  - x_0^2  = 0} \\

 \end{array} } \right.$

Rank=2, $\left\{ {\begin{array}{*{20}c}
  {x_1^2  + x_0^2  = 0} \\
  {x_1^2  - x_0^2  = 0} \\

 \end{array} } \right.$

Rank=1, $x_0^2  = 0$

It shows that there are $4$ classes of metric tensors, 
in which metric tensors of rank=$4, 3,2$ correspond to the 
first, second and third generation of fermions respectively, 
and the space of rank=$1$ is the whole 3-dimensional 
space and should correspond to vacuum. In a single generation, 
there are three or two subclasses. For rank=$4$ of the first 
generation, $x_1^2  + x_2^2  + x_3^2  - x_0^2  = 0$
 corresponds with Minkowski spacetime,  
 $x_1^2  + x_2^2  + x_3^2  + x_0^2  = 0$
should correspond with imaginary elliptical spacetime of dark matters, 
and  $x_1^2  + x_2^2  - x_3^2  - x_0^2  = 0$
 should be hyperbolic space of something we still do not 
 know so far. 
 For rank=$3$ of the second generation,
 $x_1^2  + x_2^2  - x_0^2  = 0$
 corresponds with cylindrical spacetime similar to 
 Minkowski spacetime, and 
$x_1^2  + x_2^2  + x_0^2  = 0$
should correspond with imaginary cylindrical spacetime of dark matter.
 It is similar to rank=$2$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Summary}
\label{sum}

In this paper, we got the biggest gauge group $SU(4)\otimes 
SL(4,{\bf R})$
in 4-dimensional field theory, in which the gauge group of 
standard model and gravitation were all included. In order to
describe our 
object quantitatively, metric tensor was needed for dual 
group and the gravitation and weak gauge group appeared, 
which made 4-dimensional basic representation of 
fermions split to ($1+3$)-dimensional one so that the collineation 
group was broken to gauge groups of strong and electromagnetic
interactions. The mixing of weak and electromagnetic interactions 
produced the Weinberg angle, whose value from our theory was in agreement
with the experimental data. This model not only included all 
known interactions and matters, but also gave logical geometric 
explains for a lot of physical concepts and phenomena such as 
concept of internal space, splitting and breaking of gauge group, 
quark confinement,
existence of dark matter, 3 generations of fermions and so on.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



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