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\begin{document}   
\title{Note On The Extended Noncommutativity Of Coordinates}
\author{\\Amina BOULAHOUAL$^1$  and  My. Brahim SEDRA $^{1,2}$\\
\small{Abdus Salam International Centre For Theoretical Physics ICTP,
Trieste, Italy}\\
\small {$1$ Laboratoire de Physique Th\'eorique et Appliqu\'ee LPTA,
 Facult\'e des Sciences}\\
\small {D\'epartement de Physique, B.P. 133, Kenitra, Morocco}\\
\small {$2$ UFR-HEP, Faculte des Sciences,  D\'epartement de Physique, Rabat,
Morocco}}

\maketitle
\hoffset=-1cm
\textwidth=11,5cm
                      
\vspace*{0.5cm}
\begin{abstract}
We present in this short note an idea about a possible extension of the standard 
noncommutative algebra to the formal differential operators framework. In this sense, 
we develop an analysis and derive an extended noncommutative algebra given by $[x_{a} , x_{b}]_{\star} = 
i(\theta + \chi)_{ab}$ where $\theta_{ab}$ is the standard noncommutative parameter 
and $\chi_{ab}(x)\equiv \chi^{\mu}_{ab}(x)\partial_{\mu} =\frac{1}{2}(x_a \theta^{\mu}\,_{b} - x_b \theta^{\mu} 
\, _{a})\partial_\mu$ is an antisymmetric non-constant vector-field shown to play the role of the extended
deformation parameter. This idea was motivated by the importance of noncommutative geometry 
framework in the current subject of D-brane and matrix theory physics.  
\end{abstract}
\newpage
\section {Introduction}
Recently there has been a revival interest in the noncommutativity of coordinates in string theory and 
D-brane physics[1-6]. This interest is known to concern also noncommutative quantum mechanics and noncommutative 
field theories [7, 8]. Before going into presenting the aim of our work, we will try in what follows to expose 
some of the results actually known in litterature.\\
The sharing property between all the above interesting areas of research is that the corresponding space exhibits 
the following structure

\begin{equation}
\begin{array}{lcr}
[x_{i}, x_{j}]_{\ast'}= i \theta_{ij}
\end{array}
\end{equation}
where $x_{i}$ are noncommuting coordinates which can describe also the space-time
coordinates operators and $\theta_{ij}$ is a constant antisymmetric tensor.
 \\
Quantum field theories living on this space are necessarily noncommutative field theories. Their formulation 
is simply obtained when the algebra (1) is realized in the space of fields (functions) by means of the Moyal bracket 
according to which the usual product of functions is replaced by the star-product as follows [9]

\begin{equation}
(f*g)(x) = f(x)e^{\frac{i}{2}
\theta^{ab}
\overleftarrow{\partial_a}\overrightarrow{\partial_b}} g(x) ,
\end{equation}
The link with string theory consist on the correspondence between the $\theta^{ij}$-constant
parameter and the constant antisymmetric two-form potential $B^{ij}$ on the brane as follows [1].

\begin{equation}
\theta^{ij} = (\frac{1}{B})^{ij}
,
\end{equation} 
such that in the presence of this $B$-field, the end points of an open string become noncommutative on the 
D-brane.\\
In this letter, we try to go beyound the standard noncommutative algebra (1) by presenting some computations 
leading to consider among other a non-constant antisymmetric $\widehat\theta$-parameter satisfying an 
extended noncommuative Heisenberg-type algebra given by
\begin{equation}
[x_{i}, x_{j}]_{\star}= i \widehat\theta_{ij}(x),
\end{equation}
where $\widehat\theta_{ij}(x)=(\theta +\chi)_{ij}$ and where $\chi_{ij}(x) = \chi_{ij}^{\mu}(x)\partial_{\mu}=\frac{1}{2}(x_a \theta^{\mu}\,_{b} - x_b \theta^{\mu} 
\, _{a})\partial_\mu$ describes
a non-constant vector-field deformation parameter. 
This is important since the obtained algebra (4) can be reduced to the standard noncommutative algebra (1) once one 
forget about the operatorial part of $\widehat\theta$ namely $\chi_{ij} = \chi_{ij}^{\mu}\partial_{\mu}$. This construction is also 
interesting as it may help to build a correspondence between the noncommutative geometry framework, based on the 
algebra (4), and the string theory with a non-constant B-field.


\section {The idea}
Consider the noncommutative space defined by the relation (1) originated from the star product definition of two functions
$f$ and $g$ of an algebra $\mathcal A$ that is given by 
\begin{equation}
(f*'g)(x) = f(x)e^{\frac{i}{2}
\theta^{ab}
\overleftarrow{\partial_a}\overrightarrow{\partial_b}} g(x) ,
\end{equation}
where $\partial_a = \frac{\partial}{\partial {x^{a}}}$. The star-product in (5) is 
denoted by a prime for some reasons that we will explain later. With this star product, 
one can define the Moyal bracket as follows
\begin{equation}
[f(x),g(x)]_{\ast'} = f(x)*'g(x) - g(x)*'f(x).
\end{equation}
When the functions $f$ and $g$ coincide with the coordinates $x_{i}$ 
and $x_{j}$, we recover in a simple way (1). Actually, our idea starts from the observation 
that the derivatives $\partial _{a} = \frac{\partial}{\partial x^{a}}$ in the exponential (2) are 
differential operators which act in the following way:
\begin{equation}
   \partial _{a}  : \mathcal{A} \rightarrow \mathcal{A} 
,
\end{equation}
such that the prime derivative is given by 
\begin{equation}
\begin{array}{lcl}
\partial_{a}f &=& f'_{a}\\
\partial_{a}(fg) &=& f'_{a}g+fg'_{a}\\
\partial_{a}x_{i} &=& \delta_{ai}.
\end{array}
\end{equation}
Furthermore, for two given functions $f$ and $g$ of the algebra $\mathcal{A}$, the term $f*'g$ remains an element 
of $\mathcal{A}$.
So, the prime introduced in the definition of the $\ast$-product (5) is just to express the prime 
character of the derivative $\partial_a$ as shown in (7-8). \\

Looking for a possible generalization of the above analysis to the formal differential operators framework, 
we shall now introduce another kind of star-product, denoted by $\star$ and associated to an operatorial action of 
the derivative $\partial_a$. Before going into describing how does it works, let us first introduce the set 
$\Sigma^{(p,q)}$, $p \geq 0$ [10]. This is 
the algebra of local differential operators of arbitrary spins and positive degrees. The upper indices $(p,q)$ 
carried by $\Sigma$ are the lowest and the highest degrees. A particular example is given by $\Sigma^{(0,0)}$ which 
is nothing but the algera $\mathcal{A}$, the structure usually used in the standard $\ast$-product computations.
Furthermore, in terms of the spin quantum number $\Delta = s$, the space $\Sigma^{(p,q)}$ is given by

\begin{equation}
\Sigma^{(p,q)}{(\mathcal A)} = \oplus_{s\in N}\Sigma^{(p,q)}_{s}{(\mathcal A)}
.
\end{equation}
Typical elements of (9) are given for $(p,q) =(0,k)$ by 
\begin{equation}
{\mathcal L}^{(0,k)}_{s}= \sum_{m=0}^{k}\chi_{s-m}(x){\mathcal D}^{m}=\sum_{m=0}^{k}
\chi _{s-m}^{\mu_1...\mu_m}(x){\partial_{\mu_1}...\partial_{\mu_m}}
.
\end{equation}

Next, we assume that the derivative $\partial _{a}$ acts on the function $f$ as an operator
\begin{equation}
   \partial_{a}f(x) = f'_{a}(x)+ f(x)\partial_{a},
\end{equation}
a fact which means that our derivative should be defined as

\begin{equation}
   \partial_a   : \Sigma^{(0,i)} \rightarrow \Sigma^{(0,i+1)}.
\end{equation}
This way to define the derivatives is induced from the extended $\star$-product defined by

\begin{equation}
  (f \star g)(x) =f(x).g(x) + \frac{i}{2} \theta^{ij}(f'_i +f\partial_i)(g'_j +g\partial_j)+...
\end{equation}
This $\star$-product satisfy some non trivial relations that we summarize in section 3.\\
The major difference between the two different cases of $\ast$ and $\star$-products is that, for a given function $f$ of the algebra 
$\mathcal{A}$, the term $\partial_{a}f$ belongs on the first case to the algebra $\mathcal{A}$ while on the second case it is an 
element of the space $\Sigma^{(0,1)}$. In general $(\partial_1... \partial_n f)$ is an element of $\Sigma^{(0,n)}$ 
which is a particular set of the space of local differential operators denoted by ${\mathcal{O(A)}}$ and which we can 
realize as

\begin{equation}
{\mathcal{O(A)}} = \oplus_{0\leq p\leq q}\Sigma^{(p,q)}(\mathcal{A}),
\end{equation}

Now, consider the $\star$-product definition for the coordinates $x_a$ and $x_b$, we obtain by using the above 
analysis 
\begin{equation}
\begin{array}{lcl}
x_{a} \star x_{b} &=& \sum\limits_{\alpha = 0}^{\infty}(x_a \star x_b)_{\alpha} \\
&=& x_{a}x_{b}+ \sum\limits_{{\alpha} = 1}^{\infty} \frac{1}{\alpha!}(\frac{i}{2})^{\alpha}\prod\limits_{i=1}^{\alpha}
\theta^{\mu_i \nu_i}(\prod\limits_{j=1}^{\alpha}\partial_{\mu_{j}}x_a)(\prod\limits_{k=1}^{\alpha}\partial_{\nu_k}x_b)
\end{array}
\end{equation}
and explicitly we have, 
\begin{equation}
\begin{array}{lcl}
(x_a \star x_b)_{\alpha} = \frac{1}{\alpha!}(\frac{i}{2})^{\alpha}&\Bigg\{&x_{a}x_{b}\prod\limits_{i=1}^{\alpha}
(\theta^{\mu_i \nu_i}\partial_{\mu_i}\partial_{\nu_i})\\

&+& \sum\limits_{i=1}^{\alpha}\Big\{\theta_{ab}\prod\limits_{k\ne i}(\theta^{\mu_{k} \nu_{k}}\partial_{\mu_k}\partial_{\nu_k})\\ 

&+& x_a {\theta^{\mu_{i}}\,_{b}}\prod\limits_{j\ne i}(\theta^{\mu_{j} \nu_{j}}
\partial_{\nu_j})\prod\limits_k \partial_{\mu_k}\\

&+&(x_a \theta^{\nu_{i}}\,_{b}
 
+ x_b \theta^{\nu_a} \, _{a})\prod\limits_{j\ne i}(\theta^{\mu_{j} \nu_{j}}\partial_{\mu_j})\prod\limits_{k}\partial_{\nu_k}\\ 

&+&\sum\limits_{j\ne i}\theta^{\nu_{i}}\,_{a}\theta^{\nu_{j}}\,_{b}\prod\limits_{(k\ne i , k\ne j)}(\theta^{\mu_{k} \nu_{k}}
\partial_{\mu_k})\prod_l \partial_{\nu_l} \\


&+&\sum\limits_{j\ne i}\theta_{a}\,^{\nu_i}\theta^{\mu_j}\,_{b}(\prod\limits_{l\ne j}\partial_{\nu_l})
(\prod\limits_{(k\ne i, k\ne j)}\theta^{\mu_k \nu_k})(\prod\limits_{m\ne i}\partial_{\mu _m})   \Big\}\Bigg\}

\end{array}
\end{equation}
This result is obtained by using the derived recurrence formula (30).\\
Furthermore, using the antisymmetry property of $\theta$, we can easilly check that (16) can be more simplified. 
Concrete examples are given by $x_{a}x_{b}\prod\limits_{i=1}^{\alpha}
(\theta^{\mu_i \nu_i}\partial_{\mu_i}\partial_{\nu_i})$ which is a vanishing term.
Also $\theta_{ab}\prod\limits_{k\ne i}(\theta^{\mu_{k} \nu_{k}}
\partial_{\mu_k}\partial_{\nu_k})$ as well as $x_a {\theta^{\mu_{i}}\,_{b}}\prod\limits_{j\ne i}(\theta^{\mu_{j} \nu_{j}}
\partial_{\nu_j})\prod\limits_k \partial_{\mu_k}$ and $(x_a \theta^{\nu_{i}}\,_{b}
+ x_b \theta^{\nu_a} \, _{a})\prod\limits_{j\ne i}(\theta^{\mu_{j} \nu_{j}}\partial_{\mu_j})\prod\limits_{k}
\partial_{\nu_k}$ are terms which contribute only for the value $\alpha =1$.\\

On the other hand, performing straightforward but lenghty computations, we find the 
following noncommutative extended $\star$-algebra
\begin{equation}
[x_{a} , x_{b}]_{\star} = i\widehat{\theta}_{ab}(x)
,
\end{equation}
where the only non-vanishing term among a long mathematical series is given by
\begin{equation}
\widehat{\theta}_{ab}=\theta_{ab}+\frac{1}{2}(x_a \theta^{\mu}\,_{b} - x_b \theta^{\mu} \, _{a})\partial_\mu.
\end{equation}
Later on, we will denote the vector-field apearing on the rhs of (18) simply by 
\begin{equation}
\chi_{ab}\equiv \chi^{\mu}_{ab}(x)\partial_{\mu} =\frac{1}{2}(x_a \theta^{\mu}\,_{b} - x_b \theta^{\mu} \, _{a})
\partial_\mu, 
\end{equation}
In this way, $\chi$ is interpreted as a deformation parameter term such that the algebra (17) becomes
\begin{equation}
[x_{a} , x_{b}]_{\star} =i\Big(\theta + \chi\Big)_{ab}.
\end{equation}
 
Note by the way that the long series we obtained for the $\widehat\theta$-parameter, before simplifying to (18), is 
given by
\begin{equation}
\widehat{\theta}_{ab}(x)=\sum\limits_{\alpha =1}^{\infty}\widehat{\theta}_{ab}^{\alpha},
\end{equation}
with
\begin{equation}
\widehat{\theta}_{ab}^{\alpha}= \frac{1}{\alpha!}\frac{i^{\alpha -1}}{2^{\alpha}}\sum_{i=1}^{\alpha}
\{2\theta_{ab}\prod_{k\ne i}(\theta^{\mu_{k} \nu_{k}}\partial_{\mu_k}\partial_{\nu_k})+(x_a \theta^{\mu_i}\,_{b}-
x_b \theta^{\mu_i}\,_{a})\prod_{j\ne i}(\theta^{\mu_{j} \nu_{j}}\partial_{\nu_j})\prod_{k=1}^{\alpha}\partial_{\mu_k}\}.
\end{equation}
 

\section{Some Useful Formulas}
{\bf U.1} let $c$ and $c'$ be constant numbers, we have
\begin{equation}
\begin{array}{lcl}
c\star c' =c.c'
\end{array}
\end{equation}
{\bf U.2} For every function $f(x)$ on the algebra $\Sigma^{(0,0)}\equiv \mathcal A$, we can show by using explicit 
computations that
\begin{equation}
\begin{array}{lcl}
f(x)\star c = f.c + \sum \limits_{\alpha=1}^{\infty}\frac{c}{\alpha!}(\frac{i}{2})^{\alpha}
\prod_{i=1}^{\alpha}\theta^{\mu_{i} \nu_{i}}f_{\mu_{1}...\mu_{\alpha}}^{(\alpha)}\prod_{j=1}^
{\alpha}\partial_{\nu_{j}}
\end{array}
\end{equation}
where for example $f_{a_1}^{(1)}$ is the prime derivative with respect to $\partial_{a_{1}}$\\
{\bf U.3} We have also
\begin{equation}
\begin{array}{lcl}
c\star f(x)= c.f(x)
\end{array}
\end{equation}
{\bf U.4} Combining (24-25) we find for the particular case $f=x$ 
\begin{equation}
\begin{array}{lcl}
[x_{\mu} , c]_{\star} = \frac{i}{2} c.\theta_{\mu}\, ^{\nu}\partial_{\nu}
\end{array}
\end{equation}
{\bf U.5} On the other hand,
\begin{equation}
\begin{array}{lcl}
\partial_{a}\star \partial_{b}=\partial_{a}.\partial_{b}
\end{array}
\end{equation}
{\bf U.6} Now, using (25), we can easily show that
\begin{equation}
\begin{array}{lcl}
\partial_{a}\star f(x) &=& \partial_{a}.f(x)\\
&=& f'_{a}(x)+f\partial_a
\end{array}
\end{equation}
{\bf U.7} In general we have 
\begin{equation}
\begin{array}{lcl}
\prod_{j=1}^
{\alpha}\partial_{\nu_{j}}\star f(x) = \prod_{j=1}^
{\alpha}\partial_{\nu_{j}} . f(x)
\end{array}
\end{equation}
{\bf U.8}
\begin{equation}
\partial_{\mu_1}...\partial_{\mu_n}x_a = x_a \partial_{\mu_1}...\partial_{\mu_n} + \sum\limits_{i=1}^{n}\delta_{a \mu_i}
\partial_{\mu_1}...\check{\partial}_{\mu_i}...\partial_{\mu_n},
\end{equation}
or equivalently
\begin{equation}
(\prod\limits_{i=1}^{n}\partial_{\mu_i})x_a = x_a (\prod\limits_{i=1}^{n}\partial_{\mu_i})+
\sum\limits_{i=1}^{n}\delta_{a \mu_i}
(\prod\limits_{j\ne i}^{n}\partial_{\mu_j}).
\end{equation}


\section{Concluding Remarks}

Following this construction, some remarks are in order:\\ \\
{\bf R.1} A first important remark concerning the obtained algebra (17), is that it does not close as a standard 
algebra. This property is easily observed since the extended Moyal bracket of $x_a$ and $x_b$; which are coordinates elements 
of $\Sigma_{-1}^{(0, 0)}= {\mathcal A}_{-1}^{(0, 0)}$; gives $\widehat{\theta}_{ab}(x)=\theta_{ab}+\frac{1}{2}
(x_a \theta^{\mu}\,_{b} - x_b \theta^{\mu} \, _{a})\partial_\mu$ which is an element of $\Sigma_{-2}^{(0, 1)}$. 
However, if we forget about the vector field term $\chi _{ab}^{\mu}\partial_{\mu}$ in $\widehat{\theta}_{ab}(x)$, we 
recover the standard noncommutative structure (1) which is a closed algebra.\\ \\
{\bf R.2} Related to {R.1}, we can also check that the associativity with respect to the operatorial $\star$-product 
operation is not satisfied. As an example consider 
\begin{equation}
\begin{array}{lcl}
{(f\star 1)\star 1} &=& {\mathcal D}(x)\star 1,\\
{f\star (1\star 1)} &=& {\mathcal D}(x)
\end{array}
\end{equation}
where the differential operator ${\mathcal D}(x)$ is just the result of $f \star 1$. Then we can easilly check that 
${\mathcal D}(x) \ne {\mathcal D}(x) \star 1$ as shown in the formulas (24-25).\\ \\
{\bf R.3} Concerning the mentionned properties (R.1-2), the problem of closure of the derived algebra (17) 
can be also explained by using the analogy with the non-linear Zamolodchikov $W_3$-algebra which exhibits a similar property. Namely 
the non-closure of the algebra due to the presence of the spin-4 term in the commutation relations of $W_3$ currents. 
For a review see [11].\\ \\
{\bf R.4} The noncommutative extended parameter $\widehat\theta_{ab} = \Big(\chi + \theta\Big)_{ab}$ is not a 
constant object contrary to $\theta_{ab}$ and thus the associated algebra (17) is not a trivial structure as it 
corresponds to a noncommutative deformation of the standard algebra (1) by the vector fields $\chi_{ab}$.\\ \\
{\bf R.5} Using the derived relation (26), we can easilly show that the non constant deformation parameter $\chi_{ab}$ 
is given by $\chi_{ab} \equiv i\big \{ x_b.(x_a \star 1) - x_a.(x_b \star 1) \big \}$.\\ \\ 
{\bf R.6} $\theta_{ab}$ as well as the antisymmetric tensor $\chi_{ab}(x)$ are objects of conformal weights $\Delta =-2$, 
since $\Delta(\partial_{\mu})= -\Delta(x)=1$.\\ \\ 
{\bf R.7} From the mathematical point of view, $\widehat{\theta}_{ab}^{(\alpha)}$ given in (21-22) are general objetcs 
which belong to the subspaces
\begin{equation}
\Sigma_{-2}^{(2\alpha -2, 2\alpha -1)},
\end{equation}
and $\widehat{\theta}_{ab}$ given in (18) is nothing but the first contribution for $\alpha =1$ and consequently is 
an object of $\Sigma_{-2}^{(0, 1)}$.\\ \\
{\bf R.8} We easily obtain the standard noncommutative algebra (1) from (17) just by considering the following 
quotient space
\begin{equation}
\Sigma_{-2}^{(0, 1)} \Big/ \Sigma_{-2}^{(1,1)}
\end{equation}
which consist simply on forgeting about vector fields $\chi^{\mu}\partial_{\mu}$.\\ \\

\vspace*{3cm}  {\bf Acknowledgements} \\

The authors would like to thank the Abdus Salam International Centre for Theoretical Physics and the
considerable help of the High Energy Section where this work was done. Amina Boulahoual is especially grateful 
to Prof. Seif Randjbar-Daemi for scientific help. We would like also to thank S. Majid and M.Seikh-Jabbari 
for valuable discussions. This work was done withing the framework of the associate ship scheme of the Abdus Salam ICTP.

\newpage                                 
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