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\markboth{\small \em \quad Julius Wess \hfill}{\hfill\small \em Gauge theories on noncommutative spaces \quad}

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\begin{document}
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{\sl This article is dedicated to the memory of Michael S.\
Marinov,}
\rightline{\sl a great scientist and a close friend \qquad\qquad\quad\quad}
\medskip


\title{GAUGE THEORIES ON NONCOMMUTATIVE SPACES}
%{Talk given at the Workshop ``The quantum structure of spacetime
%and the geometric nature of fundamental interactions'',
%Berlin, October 2000}\\

\author{JULIUS WESS}
\address
{Sektion Physik der Ludwig-Maximilians-Universit\"at\\
Theresienstr. 37, D-80333 M\"unchen\\ and\\
Max-Planck-Institut f\"ur Physik\\
(Werner-Heisenberg-Institut)\\
F\"ohringer Ring 6, D-80805 M\"unchen\\
}
%\date{\vspace{3cm}
%This article is dedicated to the memory of Michael S.
%Marinov, a great scientist and a close friend.}


\maketitle

%\vspace{3cm}
\abstracts{A formalism is presented where gauge 
theories for nonabelian groups can be constructed
on a noncommutative algebra.}

\vspace{0.4cm}

\tableofcontents

\newpage

\section{Introduction}

One of our basic assumptions is that the laws of physics are based
on the 
space time structure of ${\mathbb R}^n$. The coordinates form
a differential manifold on which gauge
theories are built. These gauge theories describe all known 
physics from the theory of gravity to the standard model of 
elementary particle physics. All known experiments 
support this assumption, nevertheless we are obliged to ask if 
this will still be true at ever higher energies. This is 
motivated by the fact that the rules of
local quantum field theory that make
gauge theories in particle physics
so successful are not successful for
the theory of gravity. Quantum
gravity is plagued by too many infinities, which can be traced
to the short distance behaviour of the theory.

The question arises if at very short distances the theory can 
be changed, possibly by changing our notion 
of space-time. We recall that the structure of a differential 
manifold implies strong information on the topology of the space.
Points and their neighbourhoods are defined as well as 
the distance  between points. Is this really the true structure
of space-time at
very short distances? Or do we have to replace the concept
of a differential manifold by weaker assumptions?
A very weak assumption is that the coordinates form
an algebra\,\cite{JSSW} and that the rules of this algebra, addition
and multiplication, are the only relevant information to start
with. Thus in the chain coordinates -- differential manifold --
gauge theory we could try to replace differential manifold
by algebra and see how far we can develop such a theory.

This idea is not new.\cite{abc,DM} It was already expressed in a letter
by Heisenberg to Ehrenfest  in the year 1930. There noncommutative
coordinates that lead to an uncertainty relation for the
coordinates themselves were proposed. But Heisenberg could 
not formulate these ideas  mathematically. The idea,
however, propagated and led to a publication by Hartland 
S. Snyder in the year 1947. In this paper Snyder  discussed
quantized space time for a model with a Lie algebra 
structure for the coordinates. 
In a preceding letter to Pauli (1946) Snyder
discussed the interpretation of a theory with noncommutative
space-time  in a fully satisfactory way. This 
discussion was based on the rules of quantum mechanics,
treating the selfadjoint operators for space and time
as physical observables. In a letter to Bohr (1947), Pauli
mentioned the work of Snyder as a mathematically ingeneous
proposal which, however, seemed to Pauli a failure for
reasons of physics.

In the meantime our mathematical skill, also concerning 
algebraic properties, has seen a strong development, the
same is true for the art of designing new experiments at ever 
higher energies. It seems to be natural to try anew the 
noncommutative coordinates approach  and to see
if it still meets Pauli's objections. Finally, experiment
will have to decide, but at any energy reached at new 
accellerators the question on the causal structure of 
space-time has to be asked again, noncommutative 
space-time only being a particular version
of a modification. 

In this article I try to show that a gauge theory based
on an algebra can be constructed and that definite
physical consequences can be derived and tested experimentally. 
In this construction, coordinates more or
less lose their meaning as physical observables and only 
play a role as parameters in the formulaton of the theory.

\section{The algebra}

Let me first exhibit the algebraic structure of ${\mathbb R}^n$
and then generalise to noncommutative coordinates. The 
coordinates $x^1 \dots x^n  \in {\mathbb R}^n$
are considered as elements of an associative algebra
over ${\mathbb C}$. The algebra, freely generated by these
elements, will be denoted by  ${\mathbb C}\left[[x^1,\dots,x^n]\right]$. 
The two brackets indicate that formal power series
are allowed in the algebra.

The elements of this  algebra are then subject to  relations
that make them commutative:
\be\label{1}
{\cal R:}\quad x^ix^j-x^jx^i=0\;.
\ee
These relations generate a two-sided ideal $I_R$; it consists
of all the elements of the algebra ${\mathbb C}\left[[x^1,\dots,x^n]\right]$
that can be obtained from 
the relation (\ref{1}) by multiplying (\ref{1}) from the left
and the right by all possible products of the coordinates.
We factor out this ideal and obtain the desired algebra:
\be\label{2}
{\cal A}_x=\frac{{\mathbb C}\left[[x^1,\dots,x^n]\right]}{I_{\cal R}}\;.
\ee
The elements of this algebra are the polynomials 
and the formal power
series in the 
commuting variables
$x^1, \dots,x^n \in {\mathbb R}$,
\bea\label{3}
f(x^1,\dots,x^n)&\in&{\cal A}_x\;,\\
f(x^1,\dots,x^n)&=&\sum^\infty_{r_i=0}f_{r_1\dots
  r_n}(x^1)^{r_1}\cdot\dots\cdot(x^n)^{r_n}\;.\nonumber
\eea
Multiplication in this algebra is the pointwise multiplication
of these functions.

This algebraic concept can be easily generalized to 
noncommutative coordinates. We consider algebras, freely
generated by elements $\hat{x}^1,\dots \hat{x}^n$, again
we call these elements coordinates, but now they are supposed to
satisfy relations that make them  noncommutative:
\be\label{4}
{\cal R}_{\hat{x},\hat{x}}:\quad [\hat{x}^i,\hat{x}^j]=i\theta^{ij}(\hat{x})\;.
\ee
Following L.~Landau, noncommutativity carries a hat. Again
the relations (\ref{4}) generate an ideal and we define our
algebra $\hat{{\cal A}}_{\hat{x}}$ as follows:
\bea\label{5}
{\cal A}_{\hat{x}}=\frac{{\mathbb C}
    [[\hat{x^1},\dots,\hat{x^n}]]}{I_{{\cal
      R}_{\hat{x},\hat{x}}}}\,,\qquad
\hat{f}\in&\hat{{\cal A}}_{\hat{x}}\,.
\eea

We impose one more condition on the algebra. The vectorspace
of the homogeneous polynomials of degree $m$, 
$\hat{V}^m_{\hat{x}}$ should
have the same dimension as $V^m_x$. Algebras of this type are
said to have the Poincare-Birkhoff-Witt property. In the 
following we shall consider such algebras only.

\section{The $*$ product}

The vectorspaces $V^m_x$ and $\hat{V}^m_{\hat{x}}$  are
finite-dimensional, thus they are isomorphic. To establish
an isomorphism we map a given basis of one space into a given
basis of the other space.
This then defines a vectorspace isomorphism between the 
vectorspaces $\hat{V}_{\hat{x}}$ and $V_x\,$.

We now change the algebra ${\cal A}_x$ to extend the above
vector space isomorphism to an algebra isomorphism. 
For this purpose we have to
change the multiplication law in ${\cal A}_x$. When we multiply two
elements in $\hat{{\cal A}}_{\hat{x}}$ we can compute from the 
multiplication law in $\hat{{\cal A}}_{\hat{x}}$ the coefficient 
function of the product in a given basis. 
We define the product in the vectorspace
$V_x$ to be the element with the same coefficient function
as it was  calculated in $\hat{{\cal A}}_{\hat{x}}\,$. This 
multiplication rule we call  $*$ (star) product and this defines 
the algebra  $*{\cal A}_x\,$. The algebras $\hat{{\cal A}}_{\hat{x}}$
and $*{\cal A}_x$ are isomorphic.

It is natural to use the elements of ${*\cal A}_x$ as objects
in physics. The pointwise product has to be replaced by the
$*$ product. In all the cases of interest the  $*$ product can be
expressed with the help of a differential operator. This
makes it possible to extend the $*$ product to functions without 
referring to power series expansion. Thus we treat the elements 
${\cal A}_x$ like ordinary fields but replace the pointwise
product by the $*$ product. This would be the starting point
of deformation quantization. As we have based the concept
on associative algebras, associativity of the $*$ product
is guaranteed.

\section{Gauge theory}

In this context it is possible to formulate a gauge theory.\cite{MSSW} We
start from a Lie algebra:
\be\label{6}
[T^a,T^b]=if^{ab}_cT^c\;.
\ee
In a usual gauge theory on commutative spaces the fields will
span a representation of this Lie algebra and they will transform
under the usual gauge transformation with Lie algebra valued
parameters:
\be\label{7}
\delta_{\alpha^0}\psi(x)=i\alpha^0(x)\psi(x)\;.
\ee
The commutator of two such transformations remains
Lie algebra valued.
\bea\label{8}
&&\hspace{-6mm}(\delta_{\alpha^0}\delta_{\beta^0}-\delta_{\beta^0}\delta_{\alpha^0})\psi
=  -(\beta^0\alpha^0-\alpha^0\beta^0)\psi
=  i(\alpha^0\times\beta^0)\psi=\delta_{\alpha^0\times\beta^0}\psi\;,\\[1mm]
&&\hspace{6mm} \alpha^0\times\beta^0  \equiv  \alpha^0_a\beta^0_bf^{ab}_cT^c.\nonumber
\eea


For a theory on non-commutative spaces we start with fields that 
are elements of ${*\cal A}_x$. Gauge transformations have to be defined 
with the $*$ product:
\be\label{9}
\delta_\alpha\psi(x)=i\alpha(x)*\psi(x)
\ee
The star product of functions is not commutative. The commutator
of two Lie algebra valued transformations does not reproduce
a Lie algebra valued parameter. Thus we shall assume that the
infinitesimal transformation parameters are enveloping algebra
valued:\,\cite{JSSW}
\be\label{10}
\alpha(x)=\alpha^0_a(x)T^a+\alpha^1_{ab}(x):T^aT^b:+\dots+\alpha^{n-1}_{a_1
\dots
  a_n}(x):T^{a_1}\cdots T^{a_n}:+\cdots
\ee
We have adopted the $::$ notation for a basis in the enveloping 
algebra of the Lie algebra. Completely symmetrized products could serve 
as a basis:
\bea\label{11}
:T^a:  =  T^a\,,\quad
:T^aT^b:  =  \frac{1}{2}(T^aT^b+T^bT^a)\,,\quad \mbox{ etc.}
\eea


The commutator of two transformations is certainly enveloping
algebra valued.
\be\label{12}
(\delta_\alpha\delta_\beta-\delta_\beta\delta_\alpha)\psi=
[\alpha\ds\beta]*\psi\;.
\ee
The disadvantage of this approach  is that 
infinitely many parameters $\alpha^n(x)$ have to be introduced.

It is a surprise that it is possible to define gauge transformations
where all the parameters $\alpha^n(x)$ depend on the finite set of 
parameters $\alpha^0(x)$ (Lie algebra valued) and in addition on the 
gauge potential $a(x)$ of a usual gauge theory and on their
derivatives. The gauge potential $a(x)$ has the usual transformation
properties:
\bea\label{13}
&&\delta a_i=\partial_i\alpha^0+i[\alpha^0,a_i]\;,\\
&&\delta a_{i,a}=\partial_i\alpha^0_a-\alpha^0_bf^{bc}_aa_{i,c}\;.\nonumber
\eea
We will call the new type of transformation parameters 
$\Lambda_{\alpha^0}(x)\,$. The new transformations are supposed
to close under a commutator into a transformation
characterized by $(\alpha^0\times\beta^0)\,$,
\bea\label{14}
\delta_{\alpha^0}\psi(x) & = & i\Lambda_{\alpha^0(x)}(x)*\psi(x)\,,\nonumber\\
(\delta_{\alpha^0}\delta_{\beta^0}-\delta_{\beta^0}\delta_{\alpha^0})\psi
& = & \delta_{\alpha^0\times\beta^0}\psi\,,\\
(\alpha^0\times\beta^0)_a & = & \alpha^0_b\beta^0_cf^{bc}_a\,.\nonumber
\eea
These equations 
define $\Lambda_{\alpha^0}(x)\,$. We shall see that
all  $\alpha_n(x)$ in (\ref{10}) can be defined in 
terms of $\alpha^0(x)$ and the
gauge potential $a(x)$. The transformation property 
(\ref{14}) then holds
as a consequence of (\ref{13}). The solution of this
problem, however, is not unique, this will be seen in
the following.

As a consequence of the $a$ dependence of $\Lambda_{\alpha}^0$
we have to transform $\Lambda_{\alpha}^0$ under the second
variation in the commutator. This changes equation (\ref{12})
and this is the reason why the new approach works.

\section{Constant $\theta$}

To illustrate this approach we restrict it to the
algebra where $\theta^{\mu\nu}$ is a constant. In this case
we obtain in a fully symmetrized basis the following $*$ product:\,\cite{Weyl}
\bea\label{15}
(f*g)(x)&=&e^{\frac{i}{2}\frac{\partial}{\partial
    x^i}\theta^{ij}\frac{\partial}{\partial
    y^j}}f(x)g(y)\Big|_{y\Rightarrow x}\\
&=&\int d^ny\,\,\delta^n(x-y)e^{\frac{i}{2}\frac{\partial}{\partial
    x^i}\theta^{ij}\frac{\partial}{\partial y^j}}f(x)g(y).\nonumber
\eea
We  expand in $\theta$, 
\be\label{16}
\Lambda_{\alpha^0}=\alpha^0_aT^a+\theta^{ij}\Lambda^1_{\alpha^0,ij}+\cdots\,.
\ee
The $*$ product has to be expanded as well.
Finally we expand the defining equation for $\Lambda_{\alpha^0}$,
\bea\label{161}
(\delta_{\alpha^0}\delta_{\beta^0}-\delta_{\beta^0}\delta_{\alpha^0})\psi
& = &
i(\delta_{\alpha^0}\Lambda_{\beta^0}-\delta_{\beta^0}\Lambda_{\alpha^0})*\psi
+ [\Lambda_{\alpha^0}\ds\Lambda_{\beta_0}]*\psi\,,\\
& = &
\delta_{\alpha^0\times\beta^0}\psi=i\Lambda_{\alpha^0\times\beta^0}*\psi\,.\nonumber 
\eea 
The zeroeth order in the expansion of (\ref{161}) defines $\alpha^0$
as Lie algebra valued. 

In first order we obtain 
\bea\label{17}
\theta^{ij} \big( (\delta_{\alpha^0}\Lambda^1_{\beta^0,ij}&-&\delta_{\beta^0}
\Lambda^1_{\alpha^0,ij})-i([\alpha^0,\Lambda^1_{\beta^0,ij}]\\
& - & [\beta^0,\Lambda^1_{\alpha^0,ij}]) \big)
 + \frac{1}{2}\partial_i\alpha^0_a\partial_j\beta^0_b
:T^aT^b: = \theta^{ij}\Lambda^1_{\alpha^0\times\beta^0,ij}\,.\nonumber
\eea
A closer look shows that this  is an
inhomogeneous linear equation for $\Lambda^1$. The inhomogeneous
term is known, it contains $\alpha^0$ and $\beta^0$ only. A particular 
solution of (\ref{17}) is
\be\label{18}
\theta^{ij}\Lambda^1_{\alpha^0,ij}=\frac{1}{2}\theta^{ij}(\partial_i\alpha^0_a)a_{j,b} :T^aT^b:\;.
\ee
Any solution of the homogeneous part of 
(\ref{17}) can be added to (\ref{18}).

We can proceed order by order in $\theta$, the structure of the
equations will always be the same. It will be an inhomogeneous
linear equation, the homogeneous remains 
the same, the inhomogeneous part will contain 
known quantities only. 
This way we obtain $\Lambda_{\alpha^0}$
in a $\theta$ expansion,
\be\label{19}
\Lambda_{\alpha^0}=\alpha^0_aT^a+\frac{1}{2}\theta^{ij}(\partial_i\alpha_a^0)a_{j,b}:T^aT^b:+\cdots\,.
\ee
Such a construction of the transformation parameter first occured  in the 
context of the Seiberg-Witten map.\cite{JSW1}

\section{Covariant coordinates}

In a usual gauge theory we would procede with the definition of 
covariant derivatives. Derivatives, however, are not a natural concept for
algebras. It is more natural to introduce covariant coordinates. Based on
such a concept gauge theories can be developed as well. 

It is
obvious that coordinates do not commute with
gauge transformations, it is also natural to introduce covariant 
coordinates in analogy to covariant derivatives:
\bea\label{20}
X^i=x^i+A^i(x)\,,\\
\delta_{\alpha^0}X^i*\psi=i\Lambda_{\alpha^0}*X^i*\psi\,.\nonumber
\eea
This leads to the following transformation law for the 
gauge potential:
\be\label{22}
\delta A^i=-i[x^i\ds\Lambda_{\alpha^0}]+i[\Lambda_{\alpha^0}\ds A^i]\,.
\ee
To satisfy such a transformation law we have again to
assume that $A(x)$ is enveloping algebra valued. In general,
this would imply infinitely many gauge fields. For the restricted 
gauge transformations $\Lambda_{\alpha^0}$ it is possible to construct a gauge
potential that depends on the Lie algebra valued potential
$a(x)$ and its derivatives only. The transformation
law (\ref{13}) for $a(x)$ will imply the transformation law
for $A(x)\,$. 
This is the main achievement of the Seiberg-Witten map, see Eq.~(\ref{28}).

The construction of gauge fields that transform to tensorial
follows the usual concept as we know it from covariant
derivatives. An obvious definition is
\be\label{23}
X^\mu X^\nu -X^\nu X^\mu -i\theta^{\mu\nu}(X)=\hat{F}^{\mu\nu}\,.
\ee
It is chosen in such a way that $\tilde{F}^{\mu\nu}$  vanishes for
a vanishing gauge potential $A^\mu\,$.

The tensorial transformation law of  $\tilde{F}^{\mu\nu}$ 
follows directly from (\ref{22}),
\be\label{24}
\delta_\alpha\tilde{F}^{\mu\nu}=[\Delta_{\alpha 0}*\tilde{F}^{\mu\nu}]\,.
\ee
It should be noted, however, that the trace
in the
representation space of the Lie algebra of a tensor is not an invariant 
because the star product is not commutative. 

\section{The integral}

An invariant 
action can be constructed only if the integral has its trace
property,
\be\label{25}
\int f*g=\int g*f\;.
\ee
Integration is not a natural concept in an algebra. It is
supposed to be a linear map from ${\cal A}_{\hat{x}}$ into
${\mathbb C}$,
\bea\label{26}
\int:\quad{\mathcal A}_{\hat{x}}\to{\mathbb C}\;,\qquad
\int(c_1\hat{f}+c_2\hat{g})=c_1\int\hat{f}+c_2\int\hat{g}\;.
\eea
In addition the trace property is required:
\be\label{27}
\int\hat{f}\hat{g}=\int\hat{g}\hat{f}.
\ee
This is equivalent to (\ref{25}).

\section{Gauge theory for constant $\theta$}

For constant $\theta$ the
usual integral in $x$-space will have the trace property.
This can be shown by a direct calculation.

Let us have a look at this formalism for constant
$\theta^{\mu\nu}\,$.

\noindent
The Seiberg-Witten map:
\bea\label{28}
A^i(x) & = & \theta^{ij}V_j\;,\nonumber\\
V_j(x) & = &
a_{j,a}T^a-\frac{1}{2}\,\theta^{ln}a_{l,a}(\partial_na_{j,b}+F_{nj,b}
:T^aT^b:+\cdots\,,\\
F_{nj,b} & = &
\partial_na_{j,b}-\partial_ja_{n,b}+f^{cd}_ba_{n,c}a_{j,d}\,.\nonumber 
\eea
\noindent
The field strength:
\be\label{29}
\tilde{F}_{ij}=F_{ij,a}T^a+\theta^{ln}(F_{il,a}F_{jn,l}-\frac{1}{2}
a_{l,a}(2\partial_nF_{ij,b}+a_{n,c}F_{ij,d}f^{cd}_e)):T^aT^b:+\cdots\,.
\ee
\noindent
The Lagrangian:
\be\label{30}
L=\frac{1}{4}\mbox{Tr}\,F_{ij}*F^{ij}\,.
\ee
\noindent
The invariant action:
\bea\label{301}
W  =  \frac{1}{4}\int\mbox{Tr}F_{ij}*F^{ij}
 =  \frac{1}{4}\int\mbox{Tr}F_{ij}F^{ij}\,.\nonumber
\eea


New coupling terms arise, $\theta^{\mu\nu}$ appears as a
coupling constant, it is a Lorentz tensor and the interaction
term breaks Lorentz invariance. This was to be expected 
because the defining relation (\ref{1}) already breaks Lorentz 
invariance.

These new terms in the Lagrangian will give rise to
new interactions. Due to the breaking of Lorentz invariance
interaction terms will   occur that are forbidden in a
Lorentz invariant theory. A good example is the
$Z^0\rightarrow \gamma\gamma$ decay. From (\ref{301})
we find the following interaction terms that contribute
to this decay if the gauge theory is based on the
standard model:\,\cite{BZW}
\bea\label{3011}
{\mathcal L}_{Z_{\gamma\gamma}} \!\!\!& = &\!\!\! 
{g' \over \sqrt{(g^2+g'^2)^3} } 
        \left( \alpha g^2 + \beta (g'^2-2g^2) \right) \theta^{kl} \\
\!\!\!& \times &\!\!\!
        \Big( 2(-\partial_iZ_k+\partial_kZ_i) \partial_j A_l (\partial^i A^j -
        \partial^j A^i)\nonumber\\
\!\!\!& + &\!\!\! (\partial_i A_k \partial_j A_l +\partial_k A_i \partial_l A_j
        - 2 \partial_k A_i \partial_j A_l)(-\partial^i Z^j + \partial^j Z^i )
        \nonumber\\
\!\!\!& + &\!\!\! (-2\partial_k Z_i \partial_l A_j + 2\partial_j Z_l \partial_k A_i
        + 2\partial_i Z_j \partial_k A_l + \partial_k Z_l \partial_i A_j)
        (\partial^i A^j - \partial^j A^i) \Big)\,.\nonumber
\eea
This expression is gauge invariant under the usual
Lie algebra valued gauge transformation. It 
contributes to the branching ratio of the $Z^0$ decay.

We still have to learn how the gauge potential couples to the
matter fields. This will be done via covariant derivatives,
\bea\label{302}
&&{\mathcal D}_i*\psi=(\partial_i-iV_i)*\psi\;,\\
&&\delta_{\alpha^0}{\mathcal D}_i*\psi=
i\Lambda_{\alpha^0}*{\mathcal D}_i*\psi\;.
\nonumber
\eea 

\section{Derivatives}

First we have to define derivatives.  In general, the
star product will depend on the coordinates,
when we differentiate it the coordinate dependence
of the $*$ product will contribute as well. Nevertheless,
we demand a Leibniz rule of the type
\be\label{303}
\partial_\mu * (f*g) = (\partial_\mu f) * g + 
{\mathcal O}^\nu_\mu(f) * \partial_\nu g\,.
\ee
From the associativity of the $*$ product follows
that $O_{\mu}^{\nu}(f)$ has to be an algebra
homomorphism.

It is easier to define derivatives for $\hat {\mathcal A}_{\hat x}\,$.
A general
procedure was outlined in Ref.\,\citebk{WZ}. We first extend the
algebra by algebraic elements $\hat \partial$ and consider the
algebra ${\mathbb C}\left[ [ \hat{x}^1,\dots,\hat{x}^n,
\dots, \hat{\pat}^1, \dots, \hat{\pat}^n ] \right]$. 
This algebra has to be divided by the ideal 
$I_{\hat x, \hat x}$
as before. Then we have to construct a derivative, based on
a Leibniz rule that is a map in ${\mathbb C}\left[[\hat{x}^1,\dots,\hat{x}^n,
\dots, \hat{\pat}^1, \dots, \hat{\pat}^n]\right]/I_{\hat{x},\hat{x}}\,$.
This leads to consistency relations for the Leibniz rule
The Leibniz rule can now be interpreted
as a relation and the respective ideals  can be 
constructed and factored out. Finally this has to be
supplemented by $\hat{\pat}, \hat{\pat}$ relations.
We treat these relations as usual and after dividing
by the respective ideal we arrive at an algebra
that we call $\hat {\mathcal A}_{\hat x,\hat \pat}$.

In more detail 
the generalized Leibniz rule is supposed to have the form:
\be\label{31}
\hat{\pat}_i(\hat{f}\hat{g})=(\hat{\pat}_i\hat{f})\hat{g}+O_i^l(\hat{f})\hat{\pat}_l\hat{g}\,.
\ee
From the law of associativity in  $\hat {\mathcal A}_{\hat x}$
follows that the
map $0$ has to be an algebra homomorphism 
\be\label{32}
O^i_j(\hat{f}\hat{g})=O^i_l(\hat{f})O^l_j(\hat{g})\,.
\ee
If we define the Leibniz rule on the linear
coordinates we can
generalize it to all elements.

In the $^*\!{\mathcal A}_x$ version of the algebra the Leibniz rule
takes the form (\ref{31}).
This rule can be found as follows: $\hat{\pat}$ introduces
a map on the basis of $\hat{\mathcal A}_{\hat x}$, this map defines a map in 
$^*\!{\mathcal A}_x\,$. This map has finally to be expressed with
ordinary $x$-derivatives. This then leads to (\ref{31}).

For constant $\theta^{\mu\nu}$ where the $*$ product does not
depend on $x$ the $\pat *$ derivatives are just the ordinary
$x$-derivatives.

Covariant derivatives are then defined as usual:
\bea\label{33}
{\mathcal D}_i*\psi=(\partial_i-iV_i)*\psi\,,\\
\delta_{\alpha^0}{\mathcal D}_i*\psi=i\Lambda_{\alpha^0}*{\mathcal D}_i*\psi\,.
\nonumber
\eea
The vector potential has to be enveloping algebra valued. Again,
it can be expressed in terms of $a_\mu$ by a Seiberg-Witten map.
Therefore we expect that $A_\mu$ and $V_\mu$ are related.

For constant $\theta^{\mu\nu}$ we find
\be\label{34}
A^i(x) =  \theta^{ij}V_j\\,.
\ee
Covariant derivatives exist for $\theta^{\mu\nu}=0$.
From (\ref{34}) follows that $A^\mu$ vanishes in this
case, coordinates are already covariant.



\section{Gauge couplings to matter fields}



The matter field $\psi$ that transforms like 
\be\label{35}
\delta_{\alpha^0}\psi(x)  =  i\Lambda_{\alpha^0(x)}(x)*\psi(x)
\ee
can be expressed in terms of a field $\psi^0$ that 
transforms with a Lie algebra valued parameter and
the Lie algebra valued vector potential $a$. The 
transformation property (\ref{35}) will be a
consequence of (\ref{7}) and (\ref{13}).

For constant $\theta$ we find
\be\label{37}
\psi = \psi^0 - {1\over 2}\,\theta^{\mu\nu}a^l_\mu T^l \partial_\nu \psi^0 + \dots\;.
\ee
This now leads to the Lagrangian
\bea
&&\!\!\!\!\!\!\int \bar{\psi} * \left( \gamma^\mu D_\mu *  - m \right) \psi 
d^4x  
   =   
   \int \bar{\psi^0} \left( \gamma^\mu D_\mu - m \right) \psi^0 d^4x
\label{38}\\
&&\!\!\!\!\!\!  -{1\over 4}\,\theta^{\mu\lambda}\! \int\bar{\psi^0} F^0_{\mu\lambda} 
    \left( \gamma^\mu D_\mu    - m \right) \psi^0  d^4x  
  - \frac{1}{4}\,\theta^{\sigma\lambda}\! \int \bar{\psi^0} \gamma^\mu F^0_{\mu\sigma} 
    D_\lambda \psi^0 d^4 x + \cdots\,. 
\nonumber
\eea
The fields $\psi^0$ and ${F^{\mu\nu}}^0$ transform
like the usual gauge fields with a Lie algebra
valued parameter, ${F^{\mu\nu}}^0$ is just
the usual field strength of a gauge theory.
Accordingly, $D_{\mu}\psi^0$ is the usual
covariant derivative with the field $a_{\mu}$
as a gauge potential.

%\vspace{-2mm}
 
\section{Conclusion}

Such a theory 
based on noncommutative coordinates
should only be relevant for a region
with very high energy density, thus for very short 
distances, i.e. well inside the confinement range.
For larger distances we know that physics is 
described very well with commuting coordinates. $\theta^{\mu\nu}(x)$
will be a complicated function, we treat this function in a 
power series expansion and start with constant 
$\theta^{\mu\nu}$. This has a chance to be relevant
for processes that take place at very short distances
where the constant $\theta^{\mu\nu}$ might be dominant.
The higher order contribution on the expansion become
relevant at distances where the process has already
occured. Such a process will not be
sensitive to the functional behaviour of  $\theta^{\mu\nu}(x)$
and the constant $\theta^{\mu\nu}$ approximation might be a good 
approximation.\cite{BZW} To find such a process demands physical
intuition. 

%\vspace{-3mm}

\section*{References}

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\end{thebibliography}



\end{document}

