Noncommutative Gauge Theory on the $q$-deformed Euclidean Plane
Frank Meyer\inst{1,2}
Harold Steinacker \inst{2}
Proceedings_Dubrovnik_Meyer.tex
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\begin{document}

\title*{Noncommutative Gauge Theory on the $q$-deformed Euclidean Plane}

\titlerunning{Noncommutative Gauge Theory on the $q$-deformed Euclidean Plane} 
% your contribution title if the original one is too long
\author{Frank Meyer\inst{1,2} \and
Harold Steinacker \inst{2}}
% Please full name, not just initials!
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% your contribution title if the original one is too long
\institute{Max-Planck-Institute for Physics (Werner-Heisenberg-Institut), 
F\"ohringer Ring 6, D-80805 M\"unchen, Germany
\texttt{meyerf@mppmu.mpg.de}
\and University of Munich, Theresienstr. 37, D-80333 M\"unchen, Germany 
\texttt{hsteinac@theorie.physik.uni-muenchen.de}  }

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\maketitle

\section{Introduction} \label{sec:intro}

The physical nature of space at very short distances is still not known.
Already Heisenberg proposed in a letter to Peierls \cite{letter_to_Peierls}
that spacetime is quantized below some scale, suggesting that 
this could help to resolve
the problem of infinities in quantum field theories. 
With this motivation, 
there has been a lot of work and progress in the 
formulation of quantum field theory on quantized or noncommutative spaces.
Noncommutativity is implemented by replacing a differentiable 
space-time manifold by an algebra of noncommutative coordinates 
\beq \label{eq:noncomm}
[x^i,x^j] = \theta^{ij}(x) \neq 0 \,.
\eeq
The simplest case is the so-called canonical quantum plane 
$\mathbb R^n_\theta$, 
where $\theta^{ij}$ is a constant tensor independent of $x$.
This is the space which is 
usually considered in the literature \cite{Douglas}.
%case has been subject of much research activity \cite{Douglas}. 
However, most of the rotational symmetry is lost on $\mathbb R^n_\theta$. 
On the other hand,
there exist quantum spaces which admit a generalized notion of symmetry,
being covariant under a quantum group. 
Not much is known about field theory on this type of spaces.

One of the simplest spaces with quantum group symmetry is 
the Euclidean quantum plane $\mathbb{R}_q^2$. It 
is covariant with respect to the 
$q$-deformed two-dimensional Euclidean group $E_q(2)$. 
We report here on our work \cite{paper}, proposing a formulation of gauge theory 
based on the natural algebraic structures on this spaces and 
using a suitable star product.


\section{The $E_q(2)$-Symmetric Plane}
The $E_q(2)$-Symmetric Plane 
is generated by the complex
coordinates $z,\bar{z}$ with the commutation relation
\beq\label{eq:comEq(2)}
z\bar{z}=q^2\bar{z}z \,.
\eeq
We consider formal power series in these variables $z,\bar{z}$
as functions on this space, 
\beq
\mathbb{R}_q^2:=\mathbb {R} \langle \langle z,\bar{z} \rangle
\rangle/(z\bar{z}-q^{2}\bar{z}z) \,. 
\eeq
Notice that the simple commutation relation (\ref{eq:comEq(2)}) 
are inconsistent with the
usual formulas for differentiation and integration. 
We should therefore first discuss the appropriate %concepts of
differential calculus and invariant integration.
Finally, to get physical predictions i.e. real numbers from 
 the abstract algebra, we also need either
 a representation of the noncommutative algebra, 
or a realization of the algebra using a star product.

\subsection{Covariant Differential Calculus}

It is natural to require that there exist deformed spaces of $k$-forms 
$\Omega^k_q$ which are covariant with respect to $E_q(2)$, 
%i.e. they are $E_q(2)$-modules, 
and that the exterior differential 
$d: \Omega^k_q \rightarrow \Omega^{k+1}_q$ 
satisfies the
usual Leibniz-rule as well as $d^2=0$. One can show that
there exists a unique 
covariant differential calculus with these properties \cite{Chaichian}:
\begin{equation}
\label{eq: comm-rel dz,dzbar with z,zbar}
\begin{array}{cccccccc}
zdz & = & q^{-2}dzz, &  &  & \overline{z}dz & = & q^{-2}dz\overline{z}\\
zd\overline{z} & = & q^{2}d\overline{z}z, &  &  & \overline{z}d\overline{z} & = & q^{2}d\overline{z}\, \overline{z} \\
\end{array}
\end{equation}
The following result is particularly useful for the construction of 
gauge field theories on $\mathbb{R}_q^2$:
\begin{lemma}
Consider the one-forms
\beq
\theta \equiv \theta^{z} := z^{-1}\overline{z}dz \quad , \quad \overline{\theta } \equiv \theta^{\overline{z}} := d\overline{z}z\overline{z}^{-1} \,
\eeq
and define 
\beq
\Theta :=\theta ^{i}\lambda _{i}, \quad \mbox{where} \quad
\lambda _{z} :=\frac{1}{1-q^{-2}}\overline{z}^{-1}\, , \, \lambda _{\overline{z}} := -\frac{1}{1-q^{-2}}z^{-1}.
\eeq 
Then for all functions \( f\in \mathbb {R}_{q}^{2} \) and one-forms $\alpha$,
the following holds:
\bea
[\theta ,f] &=& [\overline{\theta },f]=0,   \\
df &=& [\Theta ,f]=[\lambda _{i},f]\theta ^{i}
\label{Thetacomm} \\
d\alpha &=&\{\Theta ,\alpha \}
\label{Thetacomm-1}
\eea
denoting with $\{\cdot,\cdot\}$ the anti-commutator.
\end{lemma}


\subsection{Invariant Integral}
In order to define an invariant action, we need an integral on 
\( \mathbb {R}_{q}^{2} \) which is invariant
under quantum group transformations. 
This means that
\begin{equation}
\label{eq: invariance condition for the q-integral}
\int ^{q}f(z,\overline{z})\triangleleft X=\varepsilon (X)\int ^{q}f(z,\overline{z})
\end{equation}
for all \( f \in \mathbb{R}^2_q\) and \( X\in U_{q}(e(2)) \). 
Here \( U_{q}(e(2)) \) is the $q$-deformed universal enveloping Lie algebra 
of the two-dimensional Euclidean
group, $\triangleleft$ denotes the right action of \( U_{q}(e(2)) \) on \(\mathbb{R}^2_q\) and 
$\varepsilon (X)$ is the counit. Now any function 
in \(\mathbb{R}^2_q\)  can be decomposed as 
\(f(z,\overline{z})=\sum _{m\in \mathbb {Z}}z^{m}f_{m}(z\overline{z}). \)
It can be shown that (\ref{eq: invariance condition for the q-integral}) 
is satisfied for the following discrete
quantum traces \cite{Koelink}
\begin{equation} \label{inv_int}
\int^{q,(r_0)}f(z,\overline{z}):=r^{2}_{0}(q^{2}-1)\sum ^{\infty
}_{k=-\infty }q^{2k}f_0(q^{2k}r_{0}^{2}),
\end{equation}
where $r_0 \in \mathbb{R}$ labels the irreducible representations of \(\mathbb{R}^2_q\). 
The most general invariant integral is given by
superpositions of these integrals,
\begin{equation}
\label{de: Eq(2) invariant q-integral 1}
\int^{q} f(z,\overline{z}) = \int_1^q dr_0 \mu(r_0) \int^{q,(r_0)}f_0(z\overline{z}) 
\end{equation}
with arbitrary  "weight" function $\mu(r) >0$. 
It is quite  remarkable and 
useful that for the special choice $\mu(r_0)=\frac 1{r_{0}(q^{2}-1)}$, 
one recovers 
the usual  Riemannian integral, which is therefore
also invariant under  \( U_{q}(e(2)) \) \cite{paper}. 
%This is generally not the case for other quantum groups. 

\section{Star Product Approach}
\subsection{The Star Product}
The noncommutative algebra $\mathbb{R}^2_q$ can be realized
on the algebra of commutative functions on
$\mathbb{R}^2$ using a new, noncommutative product, 
called star product. Let us denote the commutative variables on 
$\mathbb{R}^2$ by greek letters $\zeta, \bar{\zeta}$ to distinguish
them from the generators $z,\bar{z}$, and let $q=:e^h$. Then
a hermitian star product for $\mathbb{R}^2_q$ is given by
\begin{equation}
\label{de: q-symm star product}
f\star g:=\mu \circ e^{h(\zeta \partial_{\zeta}\otimes 
\overline{\zeta}\partial_{\overline{\zeta}}-\overline{\zeta}\partial_{\overline{\zeta}}\otimes \zeta\partial
_{\zeta})}(f\otimes g)=fg+h\zeta \overline{\zeta}(\partial_{\zeta} f \partial_{\overline{\zeta}} g -
\partial_{\overline{\zeta}} f \partial_{\zeta} g) +
\mathcal{O}(h^2) \, .
\end{equation}
\subsection{Noncommutative Gauge Transformations}
The formalism of covariant coordinates was established in 
\cite{Madore} for an arbitrary Poisson structure. 
This leads to problems in the semi-classical limit\footnote{To obtain in the classical limit the classical gauge
field $a_i$ we have to invert $\theta^{ij}$. This is only well-defined if $\theta$ is invertible, and even then
it spoils the covariant transformation property whenever $\theta$ is not constant. To maintain covariance one
has to "invert $\theta$ covariantly" as done in \cite{Diploma}, leading to complicated expressions.}.
Therefore we propose the 
following approach, taking advantage of the
frame $\theta, \bar{\theta}$ which commutes with all functions 
and the generator $\Theta$ of the exterior differential. 
We define infinitesimal gauge transformations of a matter 
field as 
\beq
\delta \psi = i \Lambda \star \psi , \qquad \delta \zeta^i=0 \,.
\eeq
Let us introduce the "covariant derivative" (or covariant one-form) as 
\beq
D:=\Theta -i A \,.
\eeq  
Then requiring that $D\psi$ transforms covariantly, i.e.
\beq
\delta D\star \psi \stackrel{!}{=}i\Lambda \star D \star \psi
\eeq
leads to the following gauge transformation property for the gauge field $A$:
\beq
\delta A=[\Theta \stackrel{\star}{,}\Lambda]+i[\Lambda \stackrel{\star}{,}A]=d\Lambda +i[\Lambda
\stackrel{\star}{,}A] \,.
\eeq
We define the field strength as the two-form 
\beq
F:=D \wedge_q D
\eeq
where $\wedge_q$ the star-wedge \cite{paper}. Then
\beq
\delta F = i[\Lambda \stackrel{\star}{,}F].
\eeq
As a two-form, the field strength can be written as 
$F=f\theta \wedge_q \bar{\theta}$. Since the frame $\theta,
\bar{\theta}$ commutes with functions, $f$ transforms covariantly as well:
\beq \label{gaugetrafo_f}
\delta f= i[\Lambda\stackrel{\star}{,}f] \,.
\eeq
We note that all transformations have the correct 
classical limit as $q \rightarrow 1$.

\subsection{Seiberg-Witten Map}
The Seiberg-Witten map \cite{SW} allows 
to express the noncommutative gauge fields in terms of
the commutative ones. Hence the noncommutative theory can be 
interpreted as a deformation of the
commutative theory. Its physical predictions can be explicitly obtained
by expanding in the deformation parameter $h$, and the commutative theory is
reproduced in the limit $h\rightarrow0$.
The Seiberg-Witten map is based on the following requirement:
\begin{itemize}
\item The \emph{consistency condition:}
\begin{equation}
\label{eq:consistency}
\begin{array}{cccc}
 & (\delta _{\alpha }\delta _{\beta }-\delta _{\beta }\delta _{\alpha })\Psi  & = & \delta _{-i[\alpha ,\beta ]}\Psi \\
\Leftrightarrow  & i\delta _{\alpha }\Lambda _{\beta }-i\delta _{\beta }\Lambda _{\alpha }+[\Lambda _{\alpha }\stackrel{\star}{,}\Lambda _{\beta }] & = & i\Lambda _{-i[\alpha ,\beta ]},
\end{array}
\end{equation}
\item Noncommutative gauge transformations are related to commutative ones:
\begin{eqnarray}
A_{i}[a_{i}]+\delta_{\L} A_{i}[a_{i}] & = & A_{i}[a_{i}+\delta_{\a} a_{i}]\label{eq:SWequ} \\
\Psi [\psi ,a_{i}]+\delta_{\L} \Psi [\psi ,a_{i}] & = & \Psi [\psi +\delta_{\a} \psi ,a_{i}+\delta_{\a} a_{i}]\, .\nonumber 
\end{eqnarray}
\end{itemize}
Solving these conditions in our case, we obtain  the following
expression for the field strength expanded in powers of $h$:
\begin{eqnarray}
f &=& F^0_{12}+h\big \{F^0_{12}+
\theta^{12}(F^{0}_{12}F^0_{12}-a_{\zeta}\partial_{\overline{\zeta}}F^0_{12}+a_{\overline{\zeta}}\partial_{\zeta}F^0_{12})
+\partial_{\zeta}\theta^{12}(a_{\zeta}\partial_{\overline{\zeta}}a_{\overline{\zeta}}
 \label{SW_f} \\ 
& &
+a_{\overline{\zeta}}\partial_{\overline{\zeta}}a_{\zeta}+2a_{\overline{\zeta}}\partial_{\zeta}a_{\overline{\zeta}})+\partial_{\overline{\zeta}}\theta^{12}(a_{\zeta}\partial_{\zeta}a_{\overline{\zeta}}+a_{\overline{\zeta}}\partial_{\zeta}a_{\zeta}+2a_{\zeta}\partial_{\overline{\zeta}}a_{\zeta})
\big \}+\mathcal{O}(h^2) \, ,
\nonumber
\end{eqnarray}
where $a_{\zeta_i}$ is the classical gauge field and $F^0_{ij}=\partial_{\zeta_i}a_j-\partial_{\zeta_j}a_i$ is
the classical field strength. 

\subsection{The Action}

To define a gauge-invariant action, 
we need an integral which is cyclic with respect to the star product,
since the field strength transforms in
the adjoint. The invariant integrals (\ref{inv_int}) 
do not have this property, which can be restored by introducing
a measure function $\mu$. It can be shown that for the measure function
$\mu:=\frac{1}{\zeta\bar{\zeta}}$, we can even drop the 
star under the integral: 
\beq
\int d\zeta d\bar{\zeta} \mu f\star g = \int d\zeta d\bar{\zeta} \mu fg = \int d\zeta d\bar{\zeta} \mu g\star f 
\eeq
Putting all this together, we can now define a gauge invariant action:
\beq
S:=\frac{1}{2}\int d\zeta d\bar{\zeta} \mu f\star f \,.
\eeq
Gauge invariance is guaranteed because of (\ref{gaugetrafo_f}). Moreover, the classical action for abelian
gauge field theory is reproduced in the classical limit $h\rightarrow 0$ because of (\ref{SW_f}).

Another possibility is to replace 
the measure function $\mu$ by a scalar ``Higgs'' 
field $\varphi$, which transforms
such that it restores the gauge invariance of the action.
One can in fact find a suitable potential for $\phi$ which admits 
a solution $\langle \phi \rangle = \mu = \frac{1}{\zeta\bar{\zeta}}$, 
leading to a $E_q(2)$-invariant action through spontaneous symmetry breaking.
%hence reconciling gauge invariance with a $E_q(2)$-invariant action.




% BibTeX users please use
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%
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\begin{thebibliography}{99.}
%
% and use \bibitem to create references.
%
% Use the following syntax and markup for your references!!!
%
% Monographs
\bibitem{letter_to_Peierls} W. Pauli: \textit{Scientific Correspondence}, Vol. II (Springer 1985) p 15.
\bibitem{Douglas} M. R. Douglas and N. A. Nekrasov: Rev. Mod. Phys. \textbf{73}, 977 (2001).
% Contributed Works
\bibitem{paper} F. Meyer and H. Steinacker: Int.J.Mod.Phys. \textbf{A19},
3349-3376 (2004).
% Journal
\bibitem{Chaichian} M.Chaichian and A. P. Demichev: Phys. Lett. \textbf{B320}, 273 (1994).
\bibitem{Koelink} H. T. Koelink: Duke Math. J. \textbf{76}, 483 (1994).
\bibitem{Madore} J. Madore, S. Schraml, P. Schupp and J. Wess: Eur. Phys. J. \textbf{C16}, 161 (2000).
\bibitem{Diploma} F. Meyer: \textit{Models of Gauge Field Theory on Noncommutative Spaces}, Diploma-Thesis, Univ.
of Munich, Chair Prof. J. Wess (2003), arXiv: hep-th/0308186.
\bibitem{SW} N. Seiberg and E. Witten: JHEP \textbf{09}, 032 (1999).



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hep-th0411091.extracted_bib
\begin{thebibliography}{99.}
%
% and use \bibitem to create references.
%
% Use the following syntax and markup for your references!!!
%
% Monographs
\bibitem{letter_to_Peierls} W. Pauli: \textit{Scientific Correspondence}, Vol. II (Springer 1985) p 15.
\bibitem{Douglas} M. R. Douglas and N. A. Nekrasov: Rev. Mod. Phys. \textbf{73}, 977 (2001).
% Contributed Works
\bibitem{paper} F. Meyer and H. Steinacker: Int.J.Mod.Phys. \textbf{A19},
3349-3376 (2004).
% Journal
\bibitem{Chaichian} M.Chaichian and A. P. Demichev: Phys. Lett. \textbf{B320}, 273 (1994).
\bibitem{Koelink} H. T. Koelink: Duke Math. J. \textbf{76}, 483 (1994).
\bibitem{Madore} J. Madore, S. Schraml, P. Schupp and J. Wess: Eur. Phys. J. \textbf{C16}, 161 (2000).
\bibitem{Diploma} F. Meyer: \textit{Models of Gauge Field Theory on Noncommutative Spaces}, Diploma-Thesis, Univ.
of Munich, Chair Prof. J. Wess (2003), arXiv: hep-th/0308186.
\bibitem{SW} N. Seiberg and E. Witten: JHEP \textbf{09}, 032 (1999).



\end{thebibliography}