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\begin{document}

\draft
\title{Strong and Electroweak Interactions, 
and Their Unification with Noncommutative
Space-Time}
\author{Xiao-Gang He}
\address{Department of Physics, National Taiwan University,\\
Taipei, Taiwan 10764, R.O.C.}
\maketitle

\begin{abstract}
Quantum field theories based on noncommutative space-time (NCQFT) 
have been extensively studied
recently. However no NCQFT model, which can
uniquely describe the strong and electroweak interactions, 
has been constructed. This 
prevents consistent and systematic study of 
noncommutative space-time. In this work we construct a NCQFT model based
on the trinification gauge group $SU(3)_C\times SU(3)_L\times SU(3)_R$.  
A unique feature of this model, that 
all matter fields (fermions and Higgses) are assigned to
(anti-)fundamental representations of the factor $SU(3)$ groups,
allows us to construct a NCQFT model for strong and 
electroweak interactions and their unification without ambiguities. 
This model provides an example which allows
consistent and systematic study of
noncommutative space-time phenomenology. 
We also comment on some related issues regarding extensions to $E_6$ 
and $U(3)_C\times U(3)_L\times U(3)_R$ models.
\end{abstract}
\pacs{PACS numbers:22.20-z,11.15-q,12.10, 12.60-i
 }

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\pagestyle{plain} Noncommutative quantum field theory (NCQFT), based on
modification of the space-time commutation relations, provides an
alternative to the ordinary quantum field theory. A simple way to modify the
space-time properties is to change the usual space-time coordinate $x$ to
noncommutative coordinate $\hat X$ such that\cite{1}

\begin{eqnarray}
[ \hat{X}^{\mu },\hat{X}^{\nu }]=i\theta ^{\mu \nu },
\end{eqnarray}
where $\theta ^{\mu \nu }$ is a real anti-symmetric matrix. We will consider
the case where $\theta ^{\mu \nu }$ is a constant and commutes with $\hat{X}%
^{\mu }$.

NCQFT based on the above commutation relation can be easily studied using
the Weyl-Moyal correspondence replacing the product of two fields $A(\hat X)$
and $B(\hat X)$ with noncommutative coordinates by product of the same
fields but ordinary coordinate $x$ through the star ``*'' product,

\begin{eqnarray}
A(\hat X) B(\hat X) \to A(x)*B(x) = Exp[i {\frac{1}{2}} \theta^{\mu\nu}
\partial_{x,\mu} \partial_{y,\nu}] A(x) B(y)|_{x=y}.
\end{eqnarray}

Properties related to NCQFT have been studied extensively 
recently\cite{2,3,3a,4,5,6,7,8,9,10,11,12}.
NCQFT for a pure 
$U(1)$ group is easy to study. Related phenomenology have been
studied recently\cite{2}. But it is more complicated for non-abelian groups.
Due to the ``*'' product nature, there are fundamental differences between
ordinary and noncommutative gauge theories 
and cause many difficulties to 
construct a unique and consistent model for strong and electroweak 
interactions\cite{3,3a,4,5,6,7,8,9,10,11,12}.  
Some of the main problems are associated gauging the $SU(N)$ group and 
uniquely determining kinetic energy for a $U(1)$ group 
which is a factor group.
The construction of NCQFT model with 
standard $SU(3)_C\times SU(2)_L\times U(1)_Y$ gauge group 
therefore faces both problems above. 
So far no NCQFT model which can consistently describe strong and 
electroweak interactions has been constructed in the literature.
This fact prevents a consistent and systematic
phenomenological investigation to test
the hypothesis of noncommutative space-time.
 
In this work we construct a NCQFT model based
on the trinification gauge group $SU(3)_C\times SU(3)_L\times SU(3)_R$.  
In this model there is no $U(1)$
factor group in the theory, and  
all matter fields, fermions and Higgses, are assigned to
(anti-)fundamental representations of the factor $SU(3)$ groups.
These properties provide key elements to solve the above mentioned
problems.
We show that NCQFT model for strong and electroweak
interactions and their unification can be consistently constructed.
This model therefore provides an example which allows a consistent and 
systematic
study of noncommutative space-time phenomenology. 
We also comment on some
related issues regarding generalization of the model to $U(3)_C\times
U(3)_L\times U(3)_R$.

With noncommutative space-time there are modifications for the fields compared 
with the ordinary ones. We
indicate the fields in NCQFT with a hat and the ordinary ones without hat. 
The definition of gauge
transformation $\hat \alpha$ of a gauge field $\hat A_\mu$ for a $SU(N)$ 
is similar to the ordinary one 
but with usual product replaced by the ``*''
product. For example

\begin{eqnarray}
\delta_\alpha \hat \phi = i \hat \alpha * \hat \phi,
\end{eqnarray}
where $\hat \phi$ is a fundamental representation of $SU(N)$. We use the
notation $\hat A_\mu = \hat A^a_\mu T^a$, $\hat \alpha = \alpha^a T^a$
with $T^a$ being the $SU(N)$ generator normalized as 
$Tr(T^aT^b) = \delta^{ab}/2$.

Due to the noncommutativity of the space-time, 
two consecutive local transformations $\hat{\alpha}$ and $\hat{\beta}$
of the type in the above,

\begin{eqnarray}
(\delta_\alpha \delta_\beta - \delta_\beta\delta_\alpha) = (\hat \alpha*\hat 
\beta - \hat \beta * \hat \alpha),
\end{eqnarray}
cannot be reduced to the matrix commutator of the generators of the Lie
algebra due to the noncommutativity of the space-time. They have to be in
the enveloping algebra

\begin{eqnarray}
\hat \alpha = \alpha + \alpha^1_{ab} :T^aT^b: + ... +
\alpha^{n-1}_{a_1...a_n}:T^{a_1}...T^{a_n}: +...
\end{eqnarray}
where $:T^{a_1}...T^{a_n}:$ is totally symmetric under exchange of $a_i$.
This poses a difficulty in constructing non-abelian gauge theories\cite
{3}.

Seiberg and Witten have shown\cite{4} that the fields defined in
noncommutative coordinate can be mapped on to the ordinary fields, the
Seiberg-Witten mapping. In Ref.\cite{5} it was shown that this mapping
actually can be applied to the ``*'' product with any gauge groups. It is
possible to study non-abelian gauge group theories. Using the above
enveloping algebra, one can obtain the noncommutative fields in terms of the
ordinary fields with corrections in powers of the noncommutative parameter, 
$\theta^{\mu\nu}$, order by order. To the first order in $\theta^{\mu\nu}$
noncommutative fields can be expressed as

\begin{eqnarray}
&&\hat \alpha = \alpha +{\frac{1}{4}} \theta^{\mu\nu}\{\partial_\mu \alpha, 
A_\nu\}
+ c \theta^{\mu\nu}[\partial_\mu \alpha, A_\nu],  \nonumber \\
&&A_\mu = -{\frac{1}{4}} \theta^{\alpha\beta}\{A_\alpha, \partial_\beta
A_\mu + F_{\beta\mu}\} + c \theta^{\alpha\beta}([A_\alpha,\partial_\mu
A_\beta] +i[A_\alpha A_\beta, A_\mu]),  \nonumber \\
&&\hat \phi = a\theta^{\mu\nu} F_{\mu\nu} \phi -{\frac{1}{2}}
\theta^{\mu\nu} A_\mu\partial_\nu \phi + i({\frac{1}{4}} + c)\theta^{\mu\nu}
A_\mu A_\nu\phi,  \label{ncf}
\end{eqnarray}
where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu -ig_N[A_\mu,
A_\nu]$. The term proportional to $a$ can be absorbed into the redefinition
of the matter field $\phi$. The parameter $c$ can not be removed by
redefinition of the gauge field. It must be a purely imaginary number from
the requirement that the gauge field be self conjugate.

Using the above noncommutative fields, the action $S$ of a $SU(N)$ NCQFT are
given by\cite{5,6},

\begin{eqnarray}
S &=&\int Ld^{4}x,  \nonumber \\
L &=&-{\frac{1}{2}}Tr(F_{\mu \nu }F^{\mu \nu })+{\frac{1}{4}}g_{N}\theta
^{\mu \nu }Tr(F_{\mu \nu }F_{\rho \sigma }F^{\rho \sigma }-4F_{\mu \rho
}F_{\nu \sigma }F^{\rho \sigma })  \nonumber \\
&+&\bar{\phi}(i\gamma ^{\mu }D_{\mu }-m)\phi -{\frac{1}{4}}\theta ^{\alpha
\beta }\bar{\phi}F_{\alpha \beta }(i\gamma ^{\mu }D_{\mu }-m)\phi -{\frac{1}{%
2}}\theta ^{\alpha \beta }\bar{\phi}\gamma ^{\mu }F_{\mu \alpha }iD_{\beta
}\phi ,  \label{ss}
\end{eqnarray}
where $D_\mu =\partial_{\mu }-ig_{N}T^{a}A_{\mu }^{a}$. We note that the
parameter $c$ does not appear in the Lagrangian. The Lagrangian is
uniquely determined to order $\theta $. We will therefore work with a simple
choice $c=0$ from now on. In the above, if $\phi $ is a chiral field, $m=0$.

To obtain a theory which can describe the strong and electroweak
interactions such as the Standard Model (SM), there are additional
difficulties because the charge quantization problem of a $U(1)$ group. Due
to the noncommuting nature of the ``*'' product, even with a $U(1)$ gauge
theory the charges of matter fields in the theory are fixed to only three
possible values, 1, 0, -1\cite{3}. Therefore the $U(1)_Y$ part of the SM can
not be simply implemented in noncommutative space-time. However, this
difficulty can be overcome, again, with the use of the Seiberg-Witten
mapping\cite{4} .

To solve the $U(1)$ charge quantization problem, one associates each charge $%
gq^{(n)}$ of the nth matter field a gauge field $\hat A_\mu^{(n)}$\cite{7}.
In the commutative limit, $\theta^{\mu\nu} \to 0$, $\hat A_\mu^{(n)}$
becomes the single gauge field $A_\mu$ of the ordinary commuting space-time $%
U(1)$ gauge theory. But at non-zero orders in $\theta^{\mu\nu}$, $\hat A%
^{(n)}_\mu$ receives corrections\cite{7}. 
%\begin{eqnarray}
%\hat A^{(n)}_\xi
%= A_\xi + {gq^{(n)}\over 4} \theta^{\mu\nu}\{A_\nu, \partial_\mu
%A_\xi \} + {gq^{(n)}\over 4} \theta^{\mu\nu} \{F_{\mu\xi}, A_\nu\} +
%O(\theta^2).
%\end{eqnarray}
In doing so, the kinetic energy of the gauge boson will, however, be
affected. Depending on how the kinetic energy is defined (weight over
different field strength of $\hat A_\mu^{(n)}$), the resulting kinetic
energy will be different even though the proper normalization to obtain the
correct kinetic energy in the commutative limit is imposed\cite{7}.

In the SM there are six different matter field multiplets for each
generation, i.e. $u_R$, $d_R$, $(u,\;d)_L$, $e_R$, $(\nu,\;e)_L$ and $%
(H^0,\;H^-)$, a priori one can choose a different $g_i$ for each of them.
After identifying three combinations with the usual $g_3$, $g_2$ and $g_1$
couplings for the SM gauge groups, there is still a freedom to choose
different gauge boson self interaction couplings at non-zero orders in $%
\theta^{\mu\nu}$. This leads to ambiguities in self interactions of gauge
bosons when non-zero order terms in $\theta^{\mu\nu}$ are included\cite{7}.
This problem needs to be resolved. A way to solve this problem is to have a
theory without the use of $U(1)$ factor group.

There are many groups without $U(1)$ factor 
group which contain the SM gauge group
and may be used to describe the strong and electroweak interactions. However
not all of them can be easily extended to a full NCQFT using the formulation
described above because in some of these models the matter fields are not
all assigned in fundamental representations. For example one can easily
obtain unique gauge boson self interactions in $SU(5)$ theory\cite{8}. But
the matter fields requires non-fundamental representations 10 which causes
additional complications\cite{9}. To obtain a consistent NCQFT, one needs to
find a theory which uses fundamental representations for matter fields. To
this end we propose to use the trinification gauge group\cite{13}, $%
SU(3)_C\times SU(3)_L\times SU(3)_R$ with a $Z_3$ symmetry.

This theory leads to unification of strong and electroweak interactions. An
important feature of this theory is that the matter and Higgs fields are
assigned to (anti-)fundamental representations of the factor $SU(3)$ groups
and therefore the formalism described earlier can be readily used.

In the trinification model, the gauge fields are in the adjoint
representation,

\begin{eqnarray}
24= A^C + A^L + A^R = (8,1,1) + (1,8,1) + (1,1,8),
\end{eqnarray}
which contains 24 gauge bosons. $A^C$ contains the color gluon bosons, a
linear combination of component fields of $A^L$ and $A^R$ forms the $U(1)_Y$
gauge boson, and $A^L$ contains the $SU(2)_L$ gauge bosons. The rest are
integrally charged heavy gauge bosons which do not mediate proton decays.

Each generation of fermions is assigned to a 27,

\begin{eqnarray}
\psi &=& \psi^{LR} + \psi^{RC} + \psi^{CL} = (1,3,\bar 3) + (\bar 3,1,3) +
(3,\bar 3,1),  \nonumber \\
\psi^{LR} &=& \left ( 
\begin{array}{lll}
E^0 & E^- & e^- \\ 
E^+ & \bar E^0 & \nu \\ 
e^+ & N_1 & N_2
\end{array}
\right ),\;\; \psi^{RC} = \left ( 
\begin{array}{lll}
\bar u_1 & \bar u_2 & \bar u_3 \\ 
\bar d_1 & \bar d_2 & \bar d_3 \\ 
\bar B_2 & \bar B_2 & \bar B_2
\end{array}
\right ),\;\; \psi^{CL} = \left ( 
\begin{array}{lll}
u_1 & d_1 & B_1 \\ 
u_2 & d_2 & B_2 \\ 
u_3 & d_3 & B_3
\end{array}
\right ).
\end{eqnarray}
In the above we have written the fermions in left-handed chiral fields. The $%
B$ field is a heavy particle.

The Higgs fields which break the trinification to the SM gauge group are
also assigned to 27 representations. In order to have correct mass patterns,
at least two 27 Higgs representations are needed\cite{13}. We indicate them
by

\begin{eqnarray}
\phi_i = \phi_i^{LR} + \phi_i^{RC} + \phi_i^{CL} = (1,3,\bar 3)_i + (\bar 3,
1,3)_i + (3, \bar 3,1)_i.
\end{eqnarray}

The $Z_3$ symmetry operates in the following way. If $(C,L,R)$ is a
representation under the $SU(3)_C\times SU(3)_L\times SU(3)_R$, the effect
of $Z_3$ is to symmetrize it to

\begin{eqnarray}
Z_3(C,L,R) = (C,L,R) + (R,C,L) + (L,R,C).
\end{eqnarray}
Requirement of the Lagrangian to be invariant under $Z_3$ relates the gauge
couplings $g^{C,L,R}$ of the gauge groups to be equal, $g^C=g^L=g^R=g^U$, at
a scale which is the unification scale of the model.

The vacuum expectation values of the Higgs scalars break the symmetry to $%
SU(3)_C\times SU(2)_L \times U(1)_Y$ have the following form

\begin{eqnarray}
<\phi^{LR}_1> = \left ( 
\begin{array}{lll}
\hat 0 & 0 & 0 \\ 
0 & \hat 0 & 0 \\ 
0 & 0 & v_1
\end{array}
\right ),\;\; <\phi^{LR}_2> = \left ( 
\begin{array}{lll}
\hat 0 & 0 & 0 \\ 
0 & \hat 0 & \hat 0 \\ 
0 & v_2 & \hat 0
\end{array}
\right ).
\end{eqnarray}
Non-zero values of $v_{1,2}$ break the symmetry to the SM group. The scale
of $v_{1,2}$ are at the unification scale which is above $10^{15}$ GeV. At
this stage 12 of the gauge bosons, and $B$, $E_i$ and $N_i$ particles
receive masses. They are therefore very heavy. The entries indicated by $%
\hat 0$ can develop VEV's of order $m_W$. These VEV's break the SM group to 
$SU(3)_C\times U(1)_{em}$ and provide masses for the ordinary quarks and
leptons.

It is clear that the trinification model provides an easy framework for
building a consistent NCQFT model for strong and electroweak interactions
because only adjoint and (anti-)fundamental representations of the factor
groups are present. To the first order in $\theta^{\mu\nu}$, the
noncommutative gauge fields are the same form for the gauge fields as in eq.
(\ref{ncf}). The fermion and Higgs fields are in the same representation and
are all (anti-)fundamental representations $\phi$ of the type $(3,\bar 3)$
under the subgroups $SU(3)\times SU(3)$. The noncommutative fields expressed
in the ordinary fields are given by

\begin{eqnarray}
\hat \phi = \phi - {\frac{1}{2}} \theta^{\mu\nu} ( A_\mu \partial_\nu \phi - 
{\frac{i}{2}} A_\mu A_\nu \phi + \partial_\nu \phi A^{\prime}_\mu +{\frac{i}{%
2}}\phi A^{\prime}_\nu A^{\prime}_\mu),  \label{nncf}
\end{eqnarray}
where $A_\mu$ and $A^{\prime}_\mu$ are the gauge fields of the first $SU(3)$
and the second $SU(3)$ gauge groups, respectively.

The first order in $\theta^{\mu\nu}$ corrections to the Lagrangian $\tilde L$
for gauge and fermion kinetic energy terms are given by

\begin{eqnarray}
\tilde L &=& [{\frac{1}{4}}g^C \theta^{\mu\nu}Tr(F^C_{\mu\nu} F^C_{\alpha
\beta} F^{C\alpha\beta} -4F^C_{\alpha \mu} F^C_{\beta\nu} F^{C\alpha\beta})
+ (C\to L) + (C\to R)]  \nonumber \\
&-&[{\frac{i}{4}} \theta^{\mu\nu}Tr(\bar \psi^{LR} F^L_{\mu\nu}\gamma^\alpha
D_\alpha \psi^{LR} +F^R_{\mu\nu}\bar \psi^{LR}\gamma^\alpha D_\alpha
\psi^{LR}  \nonumber \\
&+&2\bar \psi^{LR} F^L_{\alpha\mu}\gamma^\alpha D_\nu \psi^{LR}
+2F^R_{\alpha\mu}\bar \psi^{LR}\gamma^\alpha D_\nu \psi^{LR}) + (LR \to LC)
+ (LR \to CR)],  \label{int}
\end{eqnarray}
where $D_\mu \psi^{LR} = \partial_\mu \psi^{LR} - ig_L A^L_\mu \psi^{LR} + i
g_R\phi^{LR} A^R_\mu$.

The above Lagrangian uniquely determine interactions due to noncommutative
space-time correction to the first order in $\theta ^{\mu \nu }$ without the
problems pointed out earlier. 
We emphasis that although the resulting theory at low energies appears
to have $U(1)$ factor group(s), the corresponding gauge self-interactions
are fixed because of the choice of trinification group which dictates
how gauge bosons interacte.
From the above Lagrangian one can easily study
new interactions due to noncommutative space-time and test the model by
experimental data. For illustration, we present the neutral gauge boson self
interactions and its interactions with the SM fermions. Expanding the above
Lagrangian we obtain

\begin{eqnarray}
L_{int} &=&{\frac{1}{4}}g^{C}\theta ^{\mu \nu }Tr(G_{\mu \nu }G_{\alpha
\beta }G^{\alpha \beta }-4G_{\alpha \mu }G_{\beta \nu }G^{\alpha \beta }) 
\nonumber \\
&+&{\frac{1}{16}}\theta ^{\mu \nu }g_{Y}[c_{W}({\frac{7}{15}}
c_{W}^{2}+s_{W}^{2})(F_{\mu \nu }F_{\alpha \beta }F^{\alpha \beta
}-4F_{\alpha \mu }F_{\beta \nu }F^{\alpha \beta })  \nonumber \\
&-&s_{W}({\frac{7}{15}}s_{W}^{2}+c_{W}^{2})(Z_{\mu \nu }Z_{\alpha \beta
}Z^{\alpha \beta }-4Z_{\alpha \mu }Z_{\beta \nu }Z^{\alpha \beta }) 
\nonumber \\
&+&c_{W}(c_{W}^{2}-{\frac{23}{15}}s_{W}^{2})(F_{\mu \nu }Z_{\alpha \beta
}Z^{\alpha \beta }+2Z_{\mu \nu }Z_{\alpha \beta }F^{\alpha \beta
}-4(Z_{\alpha \mu }Z_{\beta \nu }F^{\alpha \beta }+2F_{\alpha \mu }Z_{\beta
\nu }Z^{\alpha \beta }))  \nonumber \\
&-&s_{W}(s_{W}^{2}-{\frac{23}{15}}c_{W}^{2})(Z_{\mu \nu }F_{\alpha \beta
}F^{\alpha \beta }+2F_{\mu \nu }Z_{\alpha \beta }F^{\alpha \beta
}-4(F_{\alpha \mu }F_{\beta \nu }Z^{\alpha \beta }+2Z_{\alpha \mu }F_{\beta
\nu }F^{\alpha \beta }))]
\end{eqnarray}
where $c_{w}=\cos \theta _{W}$, $s_{W}=\sin \theta _{W}$. $G_{\mu \nu }$, $%
F_{\mu \nu }$, $Z_{\mu \nu }=\partial _{\mu }Z_{\nu }- \partial_{\nu }Z_{\mu
}$ are the field strengths for the gluon, photon and Z particles,
respectively. Note that the above interactions are obtained at the
unification scale where $g_{Y}=\sqrt{3/5}g^{U}$ and $\sin ^{2}\theta
_{W}=3/8 $, and $g^{C}=g^{L}=g^{R}=g^{U}$.

From the above we see that the triple neutral gauge boson interactions are
uniquely determined unlike the case with SM gauge group studied in Ref. \cite
{7}. These interactions are also different from those predicted by $SU(5)$
model\cite{8}. This can be used to test the model\cite{10}.

The fermion-gauge boson interactions can readily be obtained by expanding
eq. (\ref{int}). The Yukawa coupling terms and Higgs potential terms can
also be obtained using results in eqs. (\ref{ncf}) and (\ref{nncf}). The
full details will be presented elsewhere.

We have constructed a NCQFT unification model of strong and electroweak
interactions based on $SU(3)_C\times SU(3)_L\times SU(3)_R\times Z_3$ group.
In this model all interactions are determined. New gauge boson self- and
fermion-gauge interactions are predicted. If the noncommutative scale turns
out to be low, the model can be tested experimentally. This model
provides an example which can consistently describe the strong and
electroweak interactions and their unification, and allows a systematic
investigation of the hypothesis of noncommutative space-time. 

Before closing we would like to make 
two comments on some possible extensions of the
model discussed here. One of them concerns about $E_6$ extension of the
model.
The $SU(3)_C\times SU(3)_L \times SU(3)_R$ group can be embedded into
$E_6$ group. One therefore can try to construct a NCQFT
based on $E_6$. 
With this group the gauge bosons are in the $78$ adjoint 
representation and the fermions and Higgses are in the $27$ 
fundamental representations\cite{e6}.  
A NCQFT model can be constructed following 
the precedures discussed earlier. This model is
very similar to the trinification model with the advantage that no additional
$Z_3$ symmetry is needed. There are however differences and complications. 
In addition to the
gauge bosons in the trinification model, there are also $54$ colored gauge 
bosons. These particles mediate proton decays. Therefore they 
have to be made heavy. To achieve this 
more Higgs representations will have to be introduced which complicate the
thoery\cite{e6}. 
The trinification model is simpler in terms of particle contents.

The other comment concerns another approach to construct
trinification model with noncommutative space-time without the use of
Seiberg-Witten mapping adopted in Ref.\cite{11}. In this approach one first
constructs fields in $U(N)$ product groups and then break the symmetry
spontaneously to $SU(N)$ product group. For the trinification model, one can
extend the group to $U(3)_C\times U(3)_L\times U(3)_R$. The gauge field
representation is

\[
27=(9,1,1)+(1,9,1)+(1,1,9). 
\]
In Ref.\cite{11} the symmetry breaking of $U(N)$ to $SU(N)$ is assumed to be
achieved by non-zero VEV's of representations $S_{i}$ which transform as
singlets under the $SU(N)$ but with non-zero charge for $U(1)$ subgroup of $%
U(N)$. Following Ref. \cite{11} we introduce three $S_{i}$ fields for each
of the factor $U(N)$ group. The VEV's of these fields break the group to the
trinification group discussed earlier producing three heavy gauge bosons.

The matter and Higgs fields are again assigned as 27 representations and are
fundamental and anti-fundamental representations of two of the factor
groups. This allows easy formulation of a gauge theory in noncommutative
space-time because that only fundamental, adjoint and singlets are allowed,
and also that a field can not transform non-trivially under more than two
factor groups\cite{12}. The difference between this model and the model
discussed earlier is that the fields in the earlier model are the ordinary
fields in commuting space-time while the fields in this model are not. More
detailed discussions on the physical meaning of the spontaneous symmetry
breaking model will be presented elsewhere.

This work was supported in part by National Science Council under grants NSC
89-2112-M-002-058 and NSC 89-2112-M-002-065, and in part by the Ministry of
Education Academic Excellence Project 89-N-FA01-1-4-3. I would like to thank
hospitalities provided by the Institute of Theoretical Sciences at the
University of Oregon and the Institute of Theoretical Physics of the
Academia Sinica in Beijing where part of this work was done.

Note Added: Another consistent 
noncommutative grand unified model based on $SO(10)$
has been constructed by P. Aschieri et al. , three months 
after this work.


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