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\begin{center}
{\Large{\bf Moving D$_2$-brane in the Noncommutative Spacetime}}

\vskip .5cm
{\large Davoud Kamani}
\vskip .1cm
 {\it Institute for Studies in Theoretical Physics and
Mathematics (IPM)
\\  P.O.Box: 19395-5531, Tehran, Iran}\\
{\sl e-mail: kamani@theory.ipm.ac.ir}
\\
\end{center}

\begin{abstract}

In this paper we study the noncommutativity of 
a moving membrane with background fields. The open string variables are 
analyzed. Some scaling limits are studied. 
The unitarity of the corresponding field 
theory is discussed. The conditions for equivalence of 
noncommutativity of the $T$-dual theory in the rest frame and
noncommutativity of the original theory in the moving frame are obtained. 
                                                                       
\end{abstract}
\vskip .5cm
Keywords: String theory; T-duality; Noncommutativity; D-brane.\\
PACS: 11.25.-w

\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Over the past years there have been attempts to explain 
noncommutativity on D-brane worldvolume
through the study of open strings in the presence of background fields
\cite{1,2}. From the DBI action, the effective action of open string theory
\cite{3}, it is known that on a D-brane no electric field can be stronger 
than a critical electric field \cite{4,5}, while the same is not true  
for magnetic fields. Also if a D$_p$-brane is put in a non-zero background 
electric field, it was found that one cannot define a low energy field 
theory limit \cite{4,6}.
Furthermore, it is known that Lorentz boosts act on 
electromagnetic backgrounds. This affects the noncommutativity 
parameter and effective open string metric.
According to these facts, the decoupling limit of D-branes with electric and 
magnetic backgrounds has been studied \cite{4,5}. 
Also by combining the spatial 
noncommutative SYM limit and Lorentz transformations, light-like
noncommutative SYM limit has been obtained \cite{7}.

We use the above facts to study the noncommutativity of a moving membrane
with electric and magnetic background fields. There are cases 
that the open string metric is proportional to the 
closed string metric. For the pure electric case we find decoupling limits,
which lead to the noncommutative theories. Note that some
field theories with space-time noncommutativity appear from studying 
string theory in the presence of a background electric field.
We shall see that the pure electric case is not unitary, while the 
pure magnetic case is unitary.

We obtain the effects of $T$-duality on the effective metric and 
noncommutativity parameter. Therefore, we find
speeds and background fields for the membrane such that the 
noncommutativity matrix of the $T$-dual theory is equivalent to the 
noncommutativity matrix of the original theory in the moving frame.

This paper is organized as follows. In section 2, we study the behavior 
of the effective variables of the open string
attached to a membrane, in terms  of the background fields and the 
speed of the membrane. In section 3, the unitarity of the 
corresponding field theory will be
discussed. In section 4, we study the $T$-duality of the theory and 
conditions for equivalence of two noncommutativity parameters.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Open string variables}
The analysis of Ref.\cite{2} leads to the definitions of 
the open string metric $G$, the noncommutativity parameter $\Theta$ and 
the effective open string coupling constant $G_s$
\bea
&~& G^{\mu \nu} = \bigg{(}(g+2 \pi \alpha' B)^{-1}g (g-2 \pi
\alpha' B)^{-1} \bigg{)}^{\mu \nu}\;, 
\nonumber\\
&~& G_{\mu \nu} = \bigg{(}(g-2 \pi \alpha' B)g^{-1} (g+2 \pi
\alpha' B) \bigg{)}_{\mu \nu}\;, 
\nonumber\\
&~& \Theta^{\mu \nu} = -(2 \pi \alpha')^2 \bigg{(}(g+2 \pi
\alpha' B)^{-1} B (g-2 \pi
\alpha' B)^{-1} \bigg{)}^{\mu \nu}\;,
\nonumber\\
&~& G_s = g_s \bigg{(} \frac{\det G}{\det
(g+2\pi \alpha'B)}\bigg{)}^{\frac{1}{2}} \;,
\eea
where $g_{\mu \nu}, B_{\mu \nu}$ and $g_s$ are closed string variables.

In general a D$_2$-brane parallel to the $X^1 X^2$-plane has the 
NS$\otimes$NS background $B$-field as the following matrix
\bea
B_{\mu \nu} = \left( \begin{array}{ccc}
0 & E & E'\\
-E & 0 & b\\
-E' & -b & 0
\end{array} \right)\;,
\eea
where $\mu , \nu =0,1,2$. 
We shall discuss pure magnetic and pure electric cases.
Let the closed string metric 
along the worldvolume directions of the membrane be
\bea
g_{\mu \nu} = \left( \begin{array}{ccc}
-g_0 & 0 & 0 \\
0 & g_1 & g'\\
0 & g' & g_2
\end{array} \right).
\eea

Note that according to the equations (1), the effective open string 
coupling $G_s$ does not
change under the Lorentz boosts. Because $g_s$ is the exponential of the 
scalar field dilation, and the ratio of two determinants 
also does not change.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pure magnetic field}

For pure magnetic field (i.e., $E=E'=0$), the noncommutativity matrix is
\bea
\Theta^{\mu \nu} = \left( \begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & \theta \\
0 & -\theta & 0
\end{array} \right)\;,
\eea
where the parameter $\theta$ is
\bea
&~& \theta = - \frac{(2\pi \alpha')^2 b}{g+(2\pi \alpha')^2 b^2}\;,
\nonumber\\
&~& g \equiv  g_1g_2 -g'^2 \;.
\eea
This relation implies that two different magnetic fields
$b_\pm = -\frac{1}{2\theta} \pm \frac{1}{2} \sqrt{\frac{1}{\theta^2}
-\frac{g}{\pi^2 \alpha'^2}}$ produce the same noncommutativity on the 
membrane. Since $g$ is positive, the function $\theta (b)$ 
has maximum $\theta_0$ (minimum $-\theta_0$) at $b=-b_0$, $(b=b_0)$
where
\bea
&~& b_0 = \frac{\sqrt{g}}{2\pi \alpha'}\;,
\nonumber\\
&~& \theta_0 = \frac{\pi \alpha'}{\sqrt{g}}\;.
\eea
The effective open string coupling is
\bea
G_s = g_s \sqrt{1+ \bigg{(}\frac{b}{b_0}\bigg{)}^2}\;.
\eea
Now we proceed to study the expressions for the various geometrical 
quantities in different frames, appropriate to different states of motion
of the brane.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Motion along the $X^1$-direction}

Now we perform  Lorentz transformations on the coordinates $X^0$ and $X^1$,
\bea
&~& X'^1 = \gamma (X^1 - v \sqrt{g_0}X^0) \;,
\nonumber\\
&~& X'^0 = \gamma (X^0 - \frac{v}{\sqrt{g_0}}X^1) \;,
\eea
where $\gamma = 1/\sqrt{1-v^2}$. The effect of these transformations on the
noncommutativity matrix is
\bea
\Theta'^{\mu \nu} = \gamma \theta \left( \begin{array}{ccc}
0 & 0 & -\frac{v}{\sqrt{g_0}} \\
0 & 0 & 1 \\
\frac{v}{\sqrt{g_0}} & -1 & 0
\end{array} \right)\;.
\eea
Therefore, in the moving frame the noncommutativity matrix has non-zero 
time-like element. Note that this time-like element does not change the
unitarity of the corresponding field theory. 
In the next section unitarity will be discussed.

The transformation of the open string metric is
\bea
G'_{\mu \nu} = \left( \begin{array}{ccc}
-\gamma^2 g_0(1-g_1 av^2) & -\gamma^2 v \sqrt{g_0}(1-g_1 a) & 
\gamma v \sqrt{g_0}g'a \\
-\gamma^2 v \sqrt{g_0}(1-g_1 a) & \gamma^2 (g_1 a-v^2) & \gamma g'a\\
\gamma v \sqrt{g_0}g'a & \gamma g'a & g_2 a
\end{array} \right)\;,
\eea
where
\bea
a=1+\frac{4\pi^2 \alpha'^2 b^2}{g}\;.
\eea
For $v=0$, this metric reduces to the open string metric for static
membrane with magnetic field. The resulted metric is like the closed 
string metric (3) that its spatial elements are scaled by the factor $a$.
Note that the equation (11) gives $a>1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Motion along the $X^3$-direction}

According to the Lorentz transformations
\bea
&~& X'^3 = \gamma (X^3 - v\sqrt{g_0} X^0) \;,
\nonumber\\
&~& X'^0 = \gamma (X^0 - \frac{v}{\sqrt{g_0}}X^3) \;,
\eea
the noncommutativity matrix $\Theta^{\mu \nu}$ does not change. 
The open string metric $G_{\mu \nu}$ transforms to
\bea
G'_{\mu \nu} = \left( \begin{array}{ccc}
-\gamma^2 g_0 & 0 & 0 \\
0 & g_1 a & g' a \\
0 & g' a & g_2 a
\end{array} \right)\;.
\eea
For special speeds there is $\gamma^2 = a$, or equivalently
\bea
v = \pm \frac{2\pi \alpha' b}{\sqrt{g+(2\pi \alpha'b)^2}}\;.
\eea
Since $g$ is positive, these speeds are not greater than the speed of light.
For the speeds (14), the transformed open string metric is 
\bea
G'_{\mu \nu} = a g_{\mu \nu}\;,
\eea
that is, in the moving membrane with the speeds (14),
the open string metric is scaled by the 
closed string metric. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pure electric field}

Now consider an electric field along the $X^1$-direction i.e., $E' = b=0$.
The noncommutativity matrix is
\bea
\Theta^{\mu \nu} = T(E) \left( \begin{array}{ccc}
0 & g_2 & -g' \\
-g_2 & 0  & 0 \\
g' & 0 & 0 
\end{array} \right)\;,
\eea
where the function $T(E)$ is
\bea
T(E) = \frac{1}{g_2}\frac{E}{E_0^2- E^2}\;,
\eea
and the electric field $E_0$ is
\bea
E_0 = \frac{1}{2\pi \alpha'}\sqrt{\frac{g_0 g}{g_2}}\;.
\eea
The electric field $E$ produces spacetime noncommutativity.
Since all elements of the noncommutativity matrix are proportional to 
the function $T(E)$, this factor shows the strength of the noncommutativity.
The effective open string coupling is
\bea
G_s = g_s \sqrt{1- \bigg{(}\frac{E}{E_0}\bigg{)}^2}\;\;.
\eea
According to this root factor there is
$-E_0 \leq E \leq E_0$. Therefore, the function
$T(E)$ is one to one i.e., each noncommutativity strength corresponds only to 
a one value of the electric field. Near the electric field $E_0$, open 
strings do not interact and the strength of the noncommutativity is infinite.

Although the effective theory of the open string is singular and
ill-defined at the critical electric field $E_0$, it is possible to define a
space-time noncommutative theory by taking an appropriate scaling limit.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Motion along the $X^1$-direction}

The Lorentz transformations (8) give the noncommutativity matrix as
\bea
\Theta'^{\mu \nu} = T(E) \left( \begin{array}{ccc}
0 & g_2 & -\gamma g' \\
-g_2 & 0  & \gamma v g' \sqrt{g_0} \\
\gamma g' & -\gamma v g' \sqrt{g_0} & 0 
\end{array} \right)\;,
\eea
and the open string metric as
\bea
&~& G'_{00} = -\gamma^2 g_0 \bigg{[} 1-g_1 v^2 -\bigg{(}1-\frac{g}{g_2}v^2
\bigg{)}\frac{E^2}{E_0^2}\bigg{]}\;,
\nonumber\\
&~& G'_{01} = -\gamma^2 v \sqrt{g_0} \bigg{[} 1-g_1 -\bigg{(}1-\frac{g}{g_2}
\bigg{)}\frac{E^2}{E_0^2}\bigg{]}\;,
\nonumber\\
&~& G'_{02} = \gamma v g' \sqrt{g_0} \;,
\nonumber\\
&~& G'_{11} = \gamma^2 \bigg{[} g_1- v^2 -\bigg{(}\frac{g}{g_2}-v^2
\bigg{)}\frac{E^2}{E_0^2}\bigg{]}\;,
\nonumber\\
&~& G'_{12} = \gamma g' \;,
\nonumber\\
&~& G'_{22} = g_2\;.
\eea
Near the electric field $E=E_0$, when there is $g' \rightarrow 0$, 
for every value of the speed $v$, 
all elements of the transformed metric $G'_{\mu \nu}$,
except $G'_{22}$, go to zero
\bea
G'_{\mu \nu} = 0 \;\;\;\; {\rm except}\; G'_{22}\;, {\rm when} 
\;\;g' \rightarrow 0\;\;{\rm and}
\;\;E \rightarrow E_0\;\;.
\eea

To avoid this singularity, we can do the following 
scaling limit. For $E \rightarrow E_0$, $g' \rightarrow 0$ and finite 
$g_2$, we should have
\bea
\gamma g' = \kappa\;,\;\;\;\;\gamma^2(1-\frac{E^2}{E_0^2}) = \rho\;,
\eea
where $\kappa$ and $\rho$ are finite constants. We conclude that the
boost velocity approaches to the speed of light, $v \rightarrow 1$.
Therefore, the metric (21) becomes
\bea
G'_{\mu \nu} = \left( \begin{array}{ccc}
-\rho g_0(1-g_1) & -\rho {\sqrt g_0}(1-g_1) & \kappa{\sqrt g_0} \\
-\rho {\sqrt g_0}(1-g_1) & \rho (g_1-1)  & \kappa  \\
\kappa{\sqrt g_0} & \kappa  & g_2 
\end{array} \right)\;.
\eea
All elements of this metric are finite. To restore interactions of open
strings, the string coupling $g_s$ can be scaled to infinity i.e., 
$g_s \sim \gamma$. This leads to a finite $G_s$. The noncommutativity
parameter $\Theta'^{\mu\nu}$ near the critical field
$E_0$ also should be finite. Therefore,
we scale $\alpha'$ as $\alpha' = \mu (1- \frac{E^2}{E_0^2})$,
where $\mu$ is finite. In other words, after scaling we have
\bea
\Theta'^{\mu \nu} = \frac{2\pi \mu}{\sqrt{g_0g_2g}}
\left( \begin{array}{ccc}
0 & g_2 & -\kappa \\
-g_2 & 0 & \kappa \sqrt{g_0} \\
\kappa & -\kappa \sqrt{g_0} & 0 
\end{array} \right)\;.
\eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Motion along the $X^3$-direction}

According to the Lorentz transformations (12) we obtain 
$\Theta' = \gamma \Theta$, where $\Theta$ is given by the equation (16). 
The open string metric becomes
\bea
G'_{\mu \nu} = \left( \begin{array}{ccc}
-\gamma^2 g_0 (1- \frac{E^2}{E_0^2}) & 0 & 0 \\
0 & g_1 (1-\frac{g}{g_1 g_2}\frac{E^2}{E_0^2})  & g'  \\
0 & g'  & g_2 
\end{array} \right)\;.
\eea
Again in the limit $E \rightarrow E_0 $ but arbitrary speed $v$ and $g'$, 
we should introduce a scaling limit. Let the elements of the closed string 
metric $g_0$, $g'$ and $g_2$ be finite. In the limit
$E \rightarrow E_0$, we can put $\gamma^2 (1-\frac{E^2}{E_0^2}) = \rho$,
which gives, 
\bea
G'_{\mu \nu} = \left( \begin{array}{ccc}
-\rho g_0 & 0 & 0 \\
0 & \frac{g'}{g_2} & g'  \\
0 & g'  & g_2 
\end{array} \right)\;.
\eea
In this limit, the speed $v$ approaches to the speed of light i.e., 
$v \rightarrow 1$.

To have a finite noncommutativity parameter, the parameter
$\alpha'$ should go to zero like 
$\alpha' = \beta(1- \frac{E^2}{E_0^2})^{3/2}$, where $\beta$ is another 
finite constant. Therefore, we obtain 
\bea
\Theta'^{\mu \nu} = 2\pi \beta \sqrt{\frac{\rho}{g_0g_2g}}
\left( \begin{array}{ccc}
0 & g_2 & -g' \\
-g_2 & 0 & 0 \\
g' & 0 & 0 
\end{array} \right)\;.
\eea

For $v=0$, the metrics (21) and (26) reduce to the open string metric for
the static membrane with electric field.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Unitarity}
Space-like noncommutative quantum field theories are always unitary \cite{8}.
In Ref.\cite{9} it was shown that quantum theories with light-like 
noncommutativity also are unitary. The unitarity of quantum field theories 
requires that the inner product $p \circ p$ be never negative \cite{8,9},
where $p$ is an external momentum and 
\bea
p \circ p=-p_\mu \Theta^{\mu \rho} G_{\rho \sigma}\Theta^{\sigma \nu} 
p_\nu\;.
\eea
Since all indices are contracted, this quantity is frame independent. 
Therefore, we study it in the rest frame.

For the magnetic case the inner product is
\bea
p\circ p = \frac{(2\pi \alpha')^4 b^2}{g(g+(2\pi \alpha')^2 b^2)}\bigg{(}
\frac{g}{g_1}p_1^2 + g_1 (p_2 - \frac{g'}{g_1}p_1)^2 \bigg{)}\;,
\eea
where $p_0$, $p_1$ and $p_2$ are components of the external momentum $p$.
Since $g_1$ and $g$ are positive numbers, 
the unitarity constraint is satisfied.

For the electric case the inner product is
\bea
p\circ p= \bigg{(} \frac{E}{g_2 E_0} \bigg{)}^2 \frac{1}{E_0^2 - E^2}
\bigg{(} g_2 g p_0^2 - g_0 (g_2 p_1 - g' p_2)^2 \bigg{)}\;.
\eea
The condition $p\circ p \geq 0$ should hold for all available values of the 
momentum components $p_0$, $p_1$ and $p_2$. 
The third factor can be negative therefore, the 
theory with pure electric field is not unitary.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$T$-duality and equivalence of noncommutativities}
Now consider $T$-duality of the theory. In the case of toroidal
compactification, when $d$-spatial coordinates are compactified on torus
$T^d$, the $T$-duality group is $O(d,d; {\bf Z})$ \cite{10}. Assume that a
D$_p$-brane is wrapped on torus $T^p$. Under the action of a particular 
element of $O(p,p; {\bf Z})$ $T$-duality group i.e., 
\bea
T = \left( \begin{array}{cc}
{\bf 0} & {\bf 1}_{p \times p} \\
{\bf 1}_{p \times p}  & {\bf 0} 
\end{array} \right)\;,
\eea
the background fields have the transformations \cite{11}
\bea
(g+2\pi \alpha' B) \rightarrow ( {\tilde g}+ 2\pi
\alpha' {\tilde B})=(g+2\pi \alpha' B)^{-1}\;.
\eea
According to the equations (1) and (33) we obtain
\bea
&~& G^{\mu\nu} = {\tilde g}^{\mu\nu}\;,
\nonumber\\
&~& \Theta^{\mu\nu} = (2 \pi \alpha' )^2 {\tilde B}^{\mu\nu}\;,
\eea
that is, the open string metric and the noncommutativity parameter
appear as the background fields of the $T$-dual theory
of string theory. One can show that the 
effects of $T$-duality transformations on the 
open string metric $G^{\mu\nu}$ and 
on the noncommutativity parameter $\Theta^{\mu\nu}$, are as the followings
\bea
&~& {\tilde G}^{\mu\nu} = g^{\mu\nu}\;,
\nonumber\\
&~& {\tilde \Theta}^{\mu\nu} = (2\pi \alpha')^2 B^{\mu\nu}\;.
\eea
Therefore, the background fields of the string theory appear as the effective
metric and the noncommutativity parameter of the 
effective theory of the $T$-dual theory \cite{12}.

Now we find the background fields and the speed of the membrane such that the
noncommutativity parameter in the moving frame be equivalent to the 
noncommutativity parameter of the $T$-dual theory in the rest frame i.e.,
\bea
\Theta'^{\mu\nu} = - \lambda {\tilde \Theta}^{\mu\nu}\;,
\eea
where $\lambda$ is a positive constant. This equation means that, $T$-duality
acts as Lorentz transformations and vice-versa. In 
this case two noncommutativities (i.e., boosted and
$T$-dual) are equivalent. In other words, noncommutativity parameter
in the moving frame is proportional to the background field 
$B^{\mu \nu}$ in the rest frame. 
The equations (35) and (36) give the following table for various values
of $g'$, $v$ and $B_{\mu\nu}$

\begin{table}[ht]
\vspace{0.3cm}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$ $ & ${\rm magnetic\;and }\;v_1$&$ {\rm magnetic\;and}\;v_3$& 
${\rm electric\;and}\;v_1$& ${\rm electric\;and}\;v_3$\\
\hline
$g'$&$g'$&$g'$&$0\;\;\;\;\;\bigg{|}\;\;\;\;g'$&$g'$\\
\hline
$v$&$0$&$v$&$v\;\;\;\;\bigg{|}\;\;\;\;0$&$v$\\
\hline
$B_{\mu\nu}$&$ b=\pm \frac{1}{2\pi \alpha'}\sqrt{(\frac{1}{\lambda}-1)g}$
&$b=\pm \frac{1}{2\pi \alpha'}\sqrt{(\frac{1}{\lambda}-1)g}$&
$E=\pm E_0 \sqrt{1-\frac{1}{\lambda}}$&
$E=\pm E_0 \sqrt{1-\frac{\gamma}{\lambda}}$\\
\hline
$G_s$&$\frac{g_s}{\sqrt{\lambda}}$&$\frac{g_s}{\sqrt{\lambda}}$&
$\frac{g_s}{\sqrt{\lambda }}$&
$g_s \sqrt{\frac{\gamma}{\lambda }}$\\
\hline
\end{tabular}
\end{center}
where $v_i$ shows the membrane motion along the $X^i$-direction.
\end{table}

When the equation (36) holds, to find the corresponding
$G'_{\mu \nu}$, use the equations (10), (13), (21) and (26) and
the values of $g'$, $v$, $b$ and $E$ of this table. 
For the first and the second columns of 
the table, the spatial elements of the
metric $G'_{\mu \nu}$ are proportional to $g_{\mu \nu}$ with the scale factor
$\frac{1}{\lambda}$. For the special case that is given by the 
equation (15) we have $\lambda=\frac{1}{\gamma^2}$. According to 
the equation(35), the equation (15) can be written as 
$G'_{\mu \nu} = \frac{1}{\lambda} {\tilde G}_{\mu \nu}$. That is,
on the open string metric, 
Lorentz transformation also acts as $T$-duality.

Noncommutativity matrix in the moving frame is $\Theta'^{\mu \nu}=
-(2\pi \alpha')^2 \lambda B^{\mu \nu } $.
For the magnetic case ${\Theta'}^{\mu \nu }$ is similar to the matrix (4),
that $\theta$ should be replaced with the factor
$\mp 2\pi \alpha' \sqrt{\frac{\lambda -\lambda^2}{g}}$. 
This factor is the strength of the noncommutativity, and for
$\lambda = \frac{1}{2}$ has extremum $\mp \frac{\pi \alpha'}{\sqrt{g}}$.
For the electric case $\Theta'^{\mu \nu}$ is similar to the matrix
(16) in which the function $T(E)$ should be replaced with the factor
$\pm \frac{1}{g_2 E_0} \sqrt{\lambda^2 - \eta \lambda}$. For the motion
along the $X^1$ and $X^3$ directions $\eta$ is 1 and $\gamma$ respectively.

Note that the equation $\Theta^{\mu\nu} = 
- \lambda ({\tilde \Theta})'^{\mu\nu}$ for the 
membrane with magnetic field, produces the results of the equation (36).
The matrix $({\tilde \Theta})'^{\mu\nu}$ is noncommutativity parameter
of the $T$-dual theory in the moving frame.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
For the pure magnetic field on the membrane, there are two values of this 
field that produce the same noncommutativity on the membrane. For the special
speeds perpendicular to the brane, the open string metric is proportional
to the closed string metric. Also the corresponding field 
theory is always unitary.

For the pure electric case, the strength of 
the noncommutativity in terms of the electric field is
one to one. This is due to the fact that the effective open string coupling 
is real. When electric field on the membrane approaches to its critical 
value, we obtained some noncommutative theories from decoupling limits. 
Note that the corresponding field
theory with pure electric field is not unitary.

We saw that the background fields of string theory ($T$-dual of string 
theory), are effective metric and noncommutativity parameter of 
the effective $T$-dual theory (the effective theory of string theory). 
Therefore, for the special background fields and speeds 
of the membrane, the noncommutativity of the $T$-dual theory in the
rest frame appears like the noncommutativity of the original
theory in the moving frame. More precisely, the open string metric
of the original theory in the moving frame is equivalent to that one
of the $T$-dual theory in the rest frame.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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