# On a charged spinless point particle minimally coupled to a constant
magnetic field in a noncommutative plane

S. Hasibul Hassan Chowdhury<EMAIL_ADDRESS>Department of Mathematics
and Natural Sciences, BRAC University, 66 Mohakhali, Dhaka 1212, Bangladesh
Talal Ahmed Chowdhury<EMAIL_ADDRESS>Department of Physics, University of
Dhaka, Dhaka 1000, Bangladesh  Department of Physics and Astronomy, University
of Kansas, Lawrence, Kansas 66045, USA  The Abdus Salam International Centre
for Theoretical Physics, Strada Costiera 11, I-34151 Trieste, Italy

###### Abstract

In this paper, we provide a mathematically and physically consistent minimal
prescription for a charged spinless point particle coupled to a constant
magnetic field in a 2-dimensional noncommutative plane. It turns out to be a
gauge invariant prescription in contrast to the widely and carelessly used
naive minimal prescription in the context of 2-dimensional quantum mechanics
in a noncommutative plane. Besides, we explore the noncommutative U(1) gauge
theoretic structure of the underlying noncommutative system by explicitly
computing the 1-parameter family of Seiberg-Witten maps.

##  I Introduction

A quantum phase space (see [1]) is specified by an operator algebra. The
operator algebra that concerns us is generated by 4 self-adjoint operators
$\hat{X}^{r}$, $\hat{Y}^{r}$, $\hat{p}_{x}$, $\hat{p}_{y}$ and the identity
operator $\hat{\mathbb{I}}$ on $L^{2}(\mathbb{R}^{2},dx\;dy)$ obeying the
following set of commutation relations:

$[\hat{X}^{r},\hat{Y}^{r}]=i\vartheta\hat{\mathbb{I}},\hskip
36.135pt[\hat{X}^{r},\hat{p}_{x}]=[\hat{Y}^{r},\hat{p}_{y}]=i\hbar\hat{\mathbb{I}},$
(1.1)

the other commutators vanish identically. The pertaining operator algebra
represents a family of noncommutative 2-planes indexed by the real parameter
$r$. The second set of equalities in (1.1) refers to the quantum mechanical
structure in the underlying operator algebra while the first equality spells
out the noncommutative structure of the corresponding planes. Hence, equation
(1.1) points out that the operator algebra that we are dealing with studies
quantum mechanics in noncommutative spaces. Here, the real parameter $r$ does
not affect the underlying commutation relations and hence is called the gauge
parameter. Equation (1.1) yields a 1-parameter family of irreducible self-
adjoint representations of the universal enveloping algebra
$\mathcal{U}(\mathfrak{g}_{\hbox{\tiny{NC}}})$ of the Lie algebra
$\mathfrak{g}_{\hbox{\tiny{NC}}}$ of the 7-dimensional real nilpotent Lie
group $G_{\hbox{\tiny{NC}}}$ studied extensively in earlier papers [2, 3, 4,
5]. All the representations in the family are unitarily equivalent to each
other. This family of irreducible representations is labelled by the ordered
triple $(\hbar,\vartheta,0)$. In this sense, all the noncommutative 2-planes
in the family are equivalent or more precisely gauge equivalent. Observe that
the magnetic field pertaining to the noncommutativity of momenta coordinates
is taken to be zero in this case. This is the reason the last coordinate of
the unitary dual of $G_{\hbox{\tiny{NC}}}$ is taken to be zero in this case,
i.e., $(\hbar,\vartheta,0)$. One such representative of the family due to
$r=1$ is worked out in case III.3 of [3]. Compare the irreducible
representation of $G_{\hbox{\tiny{NC}}}$ appearing in case III.3 with the
1-parameter family of noncommutative planes in (2.2) by plugging in $r=1$.

Next, we minimally couple this 1 parameter ($r$) family of noncommutative
planes with an external uniform magnetic field $B$ using noncommutative U(1)
gauge fields (parametrized also by $r$) whose operatorial representation is
given by

$\hat{\mathbf{A}}^{r}=\left(\frac{-2(1-r)\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}}\hat{Y}^{r},\frac{2r\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}}\hat{X}^{r}\right).$ (1.2)

The familiar Landau and symmetric gauges correspond to the gauge parameter
values $r=1$ and $r=\frac{1}{2}$, respectively. We emphasise at this stage
that the unitary dual of $G_{\hbox{\tiny{NC}}}$ corresponding to the
1-parameter ($r$) family of noncommutative planes that concerns us is labeled
by $(\hbar,\vartheta,0)$. The magnetic field $B$ appearing in the operatorial
expression (1.2) of 1-parameter family of gauge fields is an external one,
i.e., it has nothing to do with intrinsic phase space noncommutativity. Here,
$e$ is the coupling parameter and we follow minimal coupling prescription ,
i.e., the respective covariant derivative operators are written as

$\hat{\Pi}^{r}_{i}=\hat{p}_{i}-e\hat{\mathbf{A}}^{r}_{i},$ (1.3)

where $i=x,y$. And the covariant derivative operators obey the following
commutation relation:

$[\hat{\Pi}^{r}_{x},\hat{\Pi}^{r}_{y}]=i\hbar B\hat{\mathbb{I}}.$ (1.4)

Again, the gauge parameter $r$ disappears from the expression of the
commutator between the covariant derivative operators along the $x$ and $y$
directions. It is important to note that

$[\hat{X}^{r},\hat{\Pi}^{r}_{x}]\neq i\hbar\hat{\mathbb{I}},\hskip
36.135pt[\hat{Y}^{r},\hat{\Pi}^{r}_{y}]\neq i\hbar\hat{\mathbb{I}},$ (1.5)

as is reflected in the last $2$ equations of (2.8). It is perfectly fine as
$\hat{\Pi}^{r}_{x}$ and $\hat{\Pi}^{r}_{y}$ obtained using (1.3) do not
represent generators of the Lie group $G_{\hbox{\tiny{NC}}}$ while
$\hat{p}_{x}$ and $\hat{p}_{y}$ indeed do and satisfy the governing
commutation relations (1.1) of 1-parameter family of noncommutative planes.
Gauge independence of the commutator in (1.4) ensures the gauge invariance of
the energy spectra of the Landau Hamiltonian as is proven both using operator
formalism and star-product formalism in section (III). There are instances in
the literature [23, 24, 25, 26, 27] where the authors write the operator
representing the gauge field as

$\hat{\mathbf{A}}^{r}=((r-1)B\hat{Y}^{r},rB\hat{X}^{r}),$ (1.6)

with the self-adjoint operators $\hat{X}^{r}$ and $\hat{Y}^{r}$ are as in
(2.2). Substitution of $\hat{\mathbf{A}}^{r}$ in (1.3) leads to

$[\hat{\Pi}^{r}_{x},\hat{\Pi}^{r}_{y}]=ie\hbar B\left[1-\frac{er(r-1)\vartheta
B}{\hbar}\right]\hat{\mathbb{I}}.$ (1.7)

In contrast to (1.4), presence of the gauge parameter $r$ in (1.7) contributes
gauge dependence to the energy spectra of the Landau Hamiltonian. Such minimal
prescription is termed as naive minimal prescription in [14].

Landau problem in a noncommutative plane can be looked upon as a
$U(1)_{\star}$ gauge theory. In fact, the relevant gauge field can be treated
as an ordered pair of formal power series in $\vartheta$ with coefficients in
$C^{\infty}(\mathbb{R}^{2})$. The set of such formal power series can be
endowed with the structure of a ring and is denoted by
$C^{\infty}(\mathbb{R}^{2})[[\vartheta]]$. Using the deformation quantization
technique, we deform the classical algebra $C^{\infty}(\mathbb{R}^{2})$ of
smooth functions on the configuration space $\mathbb{R}^{2}$ by introducing a
1-parameter family of noncommutative associative star-products $*^{r}$ (see
3.7) parametrized by the same gauge parameter $r$. The power series
representation of (1.2) is given by

$\mathbf{A}^{\hbox{\tiny{nc}}}\equiv(\mathbf{A}^{\hbox{\tiny{nc}}}_{x},\mathbf{A}^{\hbox{\tiny{nc}}}_{y})=\left(\frac{-2(1-r)\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}}y,\frac{2r\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}}x\right).$ (1.8)

Here, both the components of the ordered pair above belong to
$C^{\infty}(\mathbb{R}^{2})[[\vartheta]]$, i.e., each component above can be
expressed as a formal power series in $\vartheta$ with coefficients in
$C^{\infty}(\mathbb{R}^{2})$. The governing star-commutation relations of this
noncommutative ring of formal power series are all provided in (3.1,3.2,3.5).
All these star-commutation relations between the relevant observables are in
complete agreement with the operatorial commutation relations introduced in
(1.1). Let us take the liberty of sloppily calling a formal power series a
function. Whenever we refer to a function in
$(C^{\infty}(\mathbb{R}^{2}),*^{r})$, we essentially refer to a formal power
series in $\vartheta$ with coefficients lying in $C^{\infty}(\mathbb{R}^{2})$
with the product in the noncommutative ring given by $*^{r}$. At the expense
of being imprecise, we will benefit ourselves from obtaining the flavour of
the deformation of the commutative algebra (with respect to the pointwise
product) of classical observables in $C^{\infty}(\mathbb{R}^{2})$ by the
introduction of the 1-parameter family of star-products $*^{r}$ introduced in
(3.7).

We obtain a function $\lambda^{\hbox{\tiny{nc}}}$ in
$C^{\infty}(\mathbb{R}^{2})$ (strictly speaking a formal power series in
$\vartheta$ with coefficients in $C^{\infty}(\mathbb{R}^{2})$) that produces
the desired $U(1)_{\star}$ phase factor
$U=e^{i\frac{\lambda^{\hbox{\tiny{nc}}}}{\hbar}}_{\star}$ of the underlying
noncommutative $U(1)$ gauge theoretic structure of the noncommutative Landau
problem. Here, $e^{x}_{\star}=\mathbb{I}+x+\frac{1}{2!}x*^{r}x+...$, with
$\mathbb{I}$ being the constant map in $C^{\infty}(\mathbb{R}^{2})$ that maps
everything to the constant real number $1$. We call this function
noncommutative gauge function which is given by the following expression:

$\lambda^{\hbox{\tiny{nc}}}=\lambda+3e\epsilon
r(r-1)\vartheta\frac{B^{2}xy}{\hbar}+\mathcal{O}(\vartheta^{2}),$ (1.9)

associated with an infinitesimal change $\epsilon=r^{\prime}-r$ of the gauge
parameter. Here, $\lambda=\epsilon Bxy$ is called the undeformed gauge
function. It is worth remarking at this stage that in the literature of
noncommutative gauge theory (see, for example, [11, 10]),
$\lambda^{\hbox{\tiny{nc}}}$ is referred to as gauge parameter. In contrast to
the existing literature, our gauge parameter $r$ is the one that parametrizes
not only the members of the same equivalence class $(\hbar,\vartheta,0)$
(representing the noncommutative 2-planes) of the unitary dual of
$G_{\hbox{\tiny{NC}}}$ (see 2.2) but also the noncommutative associative star-
products (see 3.7) $*^{r}$ that the deformed function space
$(C^{\infty}(\mathbb{R}^{2}),*^{r})$ over the noncommutative 2-planes are
equipped with. The same gauge parameter $r$ parametrizes an invertible family
of Seiberg-Witten like maps (see [11]) between the associated noncommutative
gauge field, field strength and their commutative counterparts that we explore
in detail in section IV.

The first purpose of the present paper is to give a consistent prescription
for a noncommutative plane to be minimally coupled with an external magnetic
field $B$111For earlier studies, please see [6, 7, 8]. By consistency we mean
the gauge invariance of the energy spectra of the underlying Landau
Hamiltonian. This mathematical and physical consistency is discussed using
both operator and star-product formalism. The next goal of the paper is to
address some unsettled issues in [9, 10]. The computations were carried out
there in symmetric gauge ($r=\frac{1}{2}$) and it was inferred there that
similar results should hold in other gauges also. For example, in [9] within
the symmetric gauge configuration, it was shown that although the particle’s
effective mass $m$ and the coupling $e$ change, the ratio
$\frac{e^{2}B^{2}}{m}$ remains unchanged. We verified this result for any
value of the gauge parameter $r$. The final goal of this paper is to find a
family of Seiberg-Witten maps [11] parametrized by $r\in\mathbb{R}$ relating
the underlying noncommutative gauge fields and magnetic field strength with
their commutative counterparts. We verify that the Moyal star-product [12, 13]
case (corresponding to $r=\frac{1}{2}$) reproduces the well-known Seiberg-
Witten map.

The paper is organized as follows. In section (II), we briefly discuss two
ways of approaching the problem of a charged spinless point particle of mass
$m$ placed in a noncommutative plane subjected to a constant magnetic field
$B$ (refer to section 6 of the excellent review article [10]). The first
approach (section 6.1 of [10]) involving a nonminimal prescription has been
addressed in [14]. Here, the uniform magnetic field $B$ refers to the one
originated from phase space noncommutativity and comes from the unitary dual
of the underlying kinematical symmetry group $G_{\hbox{\tiny{NC}}}$. The
second approach involves minimally coupling the particle of mass $m$ to an
external uniform magnetic field $B$. This external magnetic field $B$ is not
related to the unitary dual of $G_{\hbox{\tiny{NC}}}$. In section (III), we
discuss how one can obtain c-equivalent (in the sense of [15, 16, 17]) star
products $*^{r}$ parametrized by the gauge parameter $r$. The symmetric gauge
version ($r=\frac{1}{2}$) of the star products is the familiar Moyal product.
Later in this section, we show that the spectra of the Landau Hamiltonian is
independent of the gauge parameter $r$. Along the way, we prove that the ratio
$\frac{e^{2}B^{2}}{m}$ remains gauge invariant, i.e., independent of $r$ where
$e$ is the coupling through which the spinless point particle of mass $m$ is
coupled to the external magnetic field $B$. In section (IV), we obtain a
family of invertible maps parametrized by $r$ that relates the noncommutative
U(1) gauge fields and the relevant field strength with their commutative
counterparts. Again, $r=\frac{1}{2}$, here, yields the familiar Seiberg-Witten
map. Finally in section (V), we provide concluding remarks along with comments
on possible future research directions.

##  II Minimal prescription for quantum mechanics in noncommutative space

There are two equivalent ways of tackling the quantum mechanical problem of a
charged spinless point particle placed in a noncommutative plane under the
influence of a perpendicular constant magnetic field $B$ ([10]). In one
approach, one has the following set of commutation relations to start with

$\begin{split}[\hat{X}^{s},\hat{\Pi}^{r,s}_{x}]=[\hat{Y}^{s},\hat{\Pi}^{r,s}_{y}]=i\hbar\hat{\mathbb{I}},\;\;[\hat{\Pi}^{r,s}_{x},\hat{\Pi}^{r,s}_{y}]=i\hbar
B\hat{\mathbb{I}},\;\;[\hat{X}^{s},\hat{Y}^{s}]=i\vartheta\hat{\mathbb{I}}.\end{split}$
(2.1)

Here, $\hat{X}^{s}$, $\hat{Y}^{s}$, $\hat{\Pi}^{r,s}_{x}$ and
$\hat{\Pi}^{r,s}_{y}$ are some families of self-adjoint unbounded operators on
$L^{2}(\mathbb{R}^{2},dx\;dy)$ representing the non-central generators of a
nilpotent Lie group denoted by $G_{\hbox{\tiny{NC}}}$ ([3]). These families of
operators are indexed by $2$ parameters $r$ and $s$ in such a way (see
equation (2.8) of [14]) that they disappear from the right side of the
commutators given in (2.1). This is the reason the 2 parameters $r,s$ were
called gauge parameters in [14]. Certain values of the ordered pair
$(r,s)\in\mathbb{R}^{2}$, there, were seen to produce the symmetric and Landau
gauge representations of $G_{\hbox{\tiny{NC}}}$. Also, $\hat{\mathbb{I}}$ is
the identity operator on $L^{2}(\mathbb{R}^{2},dx\;dy)$. In this approach, the
magnetic field $B$ appears as one of the 3 parameters ($\hbar,\vartheta,B$)
labeling the unitary dual of $G_{\hbox{\tiny{NC}}}$. This approach has been
studied extensively in [14]. The 2 parameter ($r,s$) family of irreducible
representations of $G_{\hbox{\tiny{NC}}}$ in terms of the self-adjoint
operators $\hat{X}^{s}$, $\hat{Y}^{s}$, $\hat{\Pi}_{x}^{r,s}$,
$\hat{\Pi}_{y}^{r,s}$ satisfying the commutation relations 2.1 are easily seen
to be gauge equivalent, i.e., operators corresponding to distinct ordered
pairs $(r,s)\in\mathbb{R}^{2}$ and $(r^{\prime},s^{\prime})\in\mathbb{R}^{2}$
are unitarily equivalent.

In the other approach, one starts with a noncommutative plane characterized by
the following one parameter family of equivalent irreducible representations
of $\mathfrak{g}_{\hbox{\tiny{NC}}}$, the Lie algebra of
$G_{\hbox{\tiny{NC}}}$:

$\begin{split}\hat{X}^{r}&=\hat{x}+\frac{(r-1)\vartheta}{\hbar}\hat{p}_{y},\\\
\hat{Y}^{r}&=\hat{y}+\frac{r\vartheta}{\hbar}\hat{p}_{x},\\\
\hat{\Pi}_{x}&=\hat{p}_{x},\\\ \hat{\Pi}_{y}&=\hat{p}_{y}.\end{split}$ (2.2)

The ordered triple $(\hbar,\vartheta,0)\in\mathbb{R}^{2}$ in the unitary dual
of $G_{\hbox{\tiny{NC}}}$ determines the underlying equivalence class of
irreducible unitary representations of $G_{\hbox{\tiny{NC}}}$. Here, the
quantum mechanical position operators $\hat{x}$ and $\hat{y}$ act on a generic
element $\psi\in L^{2}(\mathbb{R}^{2},dx\;dy)$ in the following way:

$\begin{split}(\hat{x}\psi)(x,y)&=x\psi(x,y),\\\
(\hat{y}\psi)(x,y)&=y\psi(x,y),\end{split}$ (2.3)

while the quantum mechanical momenta operators $\hat{p}_{x}$ and $\hat{p}_{y}$
act as derivations on a generic element $\psi\in L^{2}(\mathbb{R}^{2},dx\;dy)$
in the following way:

$\begin{split}(\hat{p}_{x}\psi)(x,y)&=-i\hbar\frac{\partial\psi}{\partial
x}(x,y),\\\ (\hat{p}_{y}\psi)(x,y)&=-i\hbar\frac{\partial\psi}{\partial
y}(x,y).\end{split}$ (2.4)

The representation of $\mathcal{U}(\mathfrak{g}_{\hbox{\tiny{NC}}})$ given in
(2.2) obeys the following set of commutation relations:

$\begin{split}&[\hat{X}^{r},\hat{Y}^{r}]=i\vartheta\hat{\mathbb{I}}\;,\;[\hat{X}^{r},\hat{\Pi}_{x}]=i\hbar\hat{\mathbb{I}},\\\
&[\hat{Y}^{r},\hat{\Pi}_{y}]=i\hbar\hat{\mathbb{I}}\;,\;[\hat{\Pi}_{x},\hat{\Pi}_{y}]=0.\end{split}$
(2.5)

Now, one couples the nonrelativistic spinless point particle of mass $m$
situated in the noncommutative plane given by (2.2) to a constant magnetic
field $B$ minimally ($e$ being the coupling) through the following 1-parameter
family of representations:

$\begin{split}&\hat{X}^{r}=\hat{x}+\frac{(r-1)\vartheta}{\hbar}\hat{p}_{y},\\\
&\hat{Y}^{r}=\hat{y}+\frac{r\vartheta}{\hbar}\hat{p}_{x},\\\
&\hat{\Pi}^{r}_{x}=\frac{2(1-r)e\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\hat{y}+\left[1+\frac{2r(1-r)e\vartheta
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}}\right]\hat{p}_{x},\\\
&\hat{\Pi}^{r}_{y}=\frac{-2re\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\hat{x}+\left[1+\frac{2r(1-r)e\vartheta
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\right]\hat{p}_{y}.\end{split}$ (2.6)

Now the minimal coupling of the noncommutative plane given by (2.2) to the
electromagnetic field can be seen by a simple manipulation of (2.6):

$\begin{split}&\hat{X}^{r}=\hat{x}+\frac{(r-1)\vartheta}{\hbar}\hat{p}_{y},\\\
&\hat{Y}^{r}=\hat{y}+\frac{r\vartheta}{\hbar}\hat{p}_{x},\\\
&\hat{\Pi}^{r}_{x}=\hat{p}_{x}+\frac{2e(1-r)\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\left(\hat{y}+\frac{r\vartheta}{\hbar}\hat{p}_{x}\right),\\\
&\hat{\Pi}^{r}_{y}=\hat{p}_{y}-\frac{2re\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\left[\hat{x}+\frac{(r-1)\vartheta}{\hbar}\hat{p}_{y}\right].\end{split}$
(2.7)

The representation (2.7) satisfy the following set of commutation relations:

$\begin{split}&[\hat{X}^{r},\hat{Y}^{r}]=i\vartheta\hat{\mathbb{I}},\hskip
10.84006pt[\hat{\Pi}^{r}_{x},\hat{\Pi}^{r}_{y}]=ie\hbar B\hat{\mathbb{I}},\\\
&[\hat{X}^{r},\hat{\Pi}_{x}^{r}]=\left[1+\frac{2(1-r)e\vartheta
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\right]i\hbar\hat{\mathbb{I}},\\\
&[\hat{Y}^{r},\hat{\Pi}_{y}^{r}]=\left[1+\frac{2re\vartheta
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\right]i\hbar\hat{\mathbb{I}}.\end{split}$ (2.8)

Writing $\hat{\mathbf{\Pi}}^{r}\equiv(\hat{\Pi}^{r}_{x},\hat{\Pi}^{r}_{y})$
and $\hat{\mathbf{p}}\equiv(\hat{p}_{x},\hat{p}_{y})$, one obtains from the
expressions of kinematical momenta provided by (2.7) that

$\hat{\mathbf{\Pi}}^{r}=\hat{\mathbf{p}}-e\hat{\mathbf{A}}^{r},$ (2.9)

where the noncommutative U(1) gauge fields are given by

$\hat{\mathbf{A}}^{r}=\left(\frac{-2(1-r)\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}}\hat{Y}^{r},\frac{2r\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}}\hat{X}^{r}\right).$ (2.10)

The familiar symmetric and Landau gauges correspond to the following values of
$r$:

$r_{\hbox{\tiny{sym}}}=\frac{1}{2},\;\hbox{and}\;r_{\hbox{\tiny{Lan}}}=1.$

The gauge field expressions can be read off using (2.10) as

$\begin{split}&\hat{\mathbf{A}}^{\frac{1}{2}}=\left(-\frac{\hbar
B}{\hbar+\sqrt{\hbar^{2}+e\hbar\vartheta B}}\hat{Y}^{\frac{1}{2}},\frac{\hbar
B}{\hbar+\sqrt{\hbar^{2}+e\hbar\vartheta B}}\hat{X}^{\frac{1}{2}}\right),\\\
&\hat{\mathbf{A}}^{1}=(0,B\hat{x}).\end{split}$ (2.11)

The representation (2.6) takes the following form in Landau gauge:

$\begin{split}\hat{X}^{1}=\hat{x},\;\;\hat{Y}^{1}=\hat{y}+\frac{\vartheta}{\hbar}\hat{p}_{x},\;\;\hat{\Pi}_{x}^{1}=\hat{p}_{x},\;\;\hat{\Pi}_{y}^{2}=-eB\hat{x}+\hat{p}_{y},\end{split}$
(2.12)

while in the symmetric gauge (2.6) looks as follow:

$\begin{split}&\hat{X}^{\frac{1}{2}}=\hat{x}-\frac{\vartheta}{2\hbar}\hat{p}_{y},\\\
&\hat{Y}^{\frac{1}{2}}=\hat{y}+\frac{\vartheta}{2\hbar}\hat{p}_{x},\\\
&\hat{\Pi}_{x}^{\frac{1}{2}}=\frac{e\hbar
B}{\hbar+\sqrt{\hbar^{2}+e\hbar\vartheta B}}\hat{y}+\left[1+\frac{e\vartheta
B}{2(\hbar+\sqrt{\hbar^{2}+e\hbar\vartheta B})}\right]\hat{p}_{x},\\\
&\hat{\Pi}_{y}^{\frac{1}{2}}=-\frac{e\hbar
B}{\hbar+\sqrt{\hbar^{2}+e\hbar\vartheta B}}\hat{x}+\left[1+\frac{e\vartheta
B}{2(\hbar+\sqrt{\hbar^{2}+e\hbar\vartheta B})}\right]\hat{p}_{y}.\end{split}$
(2.13)

######  Remark II.1.

It is important to note that the 1-parameter family of gauge fields provided
by (2.10) satisfies

$\partial_{x}\hat{\mathbf{A}}^{r}_{y}-\partial_{y}\hat{\mathbf{A}}^{r}_{x}-\frac{ie}{\hbar}[\hat{\mathbf{A}}^{r}_{x},\hat{\mathbf{A}}^{r}_{y}]=B\hat{\mathbb{I}}=:\hat{F}_{xy},$
(2.14)

where $\partial_{x}=\frac{\partial}{\partial x}$ and
$\partial_{y}=\frac{\partial}{\partial y}$ and $\hat{\mathbf{A}}^{r}_{x}$ and
$\hat{\mathbf{A}}^{r}_{y}$ are the components of the noncommutative vector
potential $\hat{\mathbf{A}}^{r}$ defined in (2.10). One also defines
$\partial_{i}\hat{\mathbf{A}}^{r}_{j}=[\partial_{i},\hat{\mathbf{A}}^{r}_{j}]$
for $i,j=x,y$. Here, the $c$-number operator $\hat{F}_{xy}$ stands for the
noncommutative field strength associated with the constant magnetic field $B$.

##  III Star product approach to quantum mechanics on a noncommutative space

We consider the function space to be $C^{\infty}(\mathbb{R}^{2})$. Now, we
define $4$ functions $x,y,\Pi_{x},\Pi_{y}\in C^{\infty}(\mathbb{R}^{2})$ by
means of each of their noncommutative $*$ product with some other
(sufficiently well-behaved) vector $\psi\in L^{2}(\mathbb{R}^{2},dx\;dy)$ as
follow:

$\begin{split}&x*^{r}\psi:=\hat{X}^{r}\psi=x\psi-i(r-1)\vartheta\frac{\partial\psi}{\partial
y},\\\ &y*^{r}\psi:=\hat{Y}^{r}\psi=y\psi-
ir\vartheta\frac{\partial\psi}{\partial x},\\\
&\Pi_{x}*^{r}\psi:=\hat{\Pi}^{r}_{x}\psi=\frac{2(1-r)e\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}y\psi-i\hbar\left[1+\frac{2r(1-r)e\vartheta
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\right]\frac{\partial\psi}{\partial x},\\\
&\Pi_{y}*^{r}\psi:=\hat{\Pi}^{r}_{y}\psi=\frac{-2re\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}x\psi-i\hbar\left[1+\frac{2r(1-r)e\vartheta
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\right]\frac{\partial\psi}{\partial y}.\end{split}$ (3.1)

The last two equations in (3.1) merit elaboration: the $2$ functions $\Pi_{x}$
and $\Pi_{y}$ are actually derived objects using the relations
$\Pi_{x}=p_{x}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{x}$ and
$\Pi_{y}=p_{y}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{y}$. The 2 smooth functions
$p_{x}$ and $p_{y}$ are given by the following star-products with sufficiently
well-behaved wave function $\psi\in L^{2}(\mathbb{R}^{2},dx\;dy)$:

$p_{x}*\psi=\hat{p}_{x}\psi=-i\hbar\frac{\partial\psi}{\partial
x},\;\;\;\;p_{y}*\psi=\hat{p}_{y}\psi=-i\hbar\frac{\partial\psi}{\partial y}.$
(3.2)

And, the vector valued function $\mathbf{A}^{\hbox{\tiny{nc}}}$ is given by

$\mathbf{A}^{\hbox{\tiny{nc}}}\equiv(\mathbf{A}^{\hbox{\tiny{nc}}}_{x},\mathbf{A}^{\hbox{\tiny{nc}}}_{y})=\left(\frac{-2(1-r)\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}}y,\frac{2r\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}}x\right).$ (3.3)

Using (3.2), (3.3), (3.1) and linearity of star product, one now easily
verifies that the following hold:

$\displaystyle\Pi_{x}*^{r}\psi=(p_{x}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{x})*^{r}\psi=p_{x}*\psi-e\mathbf{A}^{\hbox{\tiny{nc}}}_{x}*^{r}\psi,$
(3.4)
$\displaystyle\Pi_{y}*^{r}\psi=(p_{y}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{y})*^{r}\psi=p_{y}*\psi-e\mathbf{A}^{\hbox{\tiny{nc}}}_{y}*^{r}\psi.$

By virtue of the commutation relations (2.8), the nonvanishing star-
commutation relations can be read off as

$\begin{split}&[x\\!\overset{\;\;*^{r}}{,}y]=i\vartheta\mathbb{I},\;\;[\Pi_{x}\\!\\!\overset{\;\;*^{r}}{,}\Pi_{y}]=ie\hbar
B\mathbb{I},\\\
&[x\\!\overset{\;\;*^{r}}{,}\Pi_{x}]=\left[1+\frac{2(1-r)e\vartheta
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\right]i\hbar\mathbb{I}=i\hbar\mathbb{I}-e[x\\!\overset{\;\;*^{r}}{,}\\!\mathbf{A}^{\hbox{\tiny{nc}}}_{x}],\\\
&[y\\!\overset{\;\;*^{r}}{,}\Pi_{y}]=\left[1+\frac{2re\vartheta
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\right]i\hbar\mathbb{I}=i\hbar\mathbb{I}-e[y\\!\overset{\;\;*^{r}}{,}\\!\mathbf{A}^{\hbox{\tiny{nc}}}_{y}].\end{split}$
(3.5)

Also, $\mathbb{I}$ is the constant map in $C^{\infty}(\mathbb{R}^{2})$ that
maps everything to the constant value $1$ and the star-commutator between 2
functions $F(x,y)$ and $G(x,y)$ in $C^{\infty}(\mathbb{R}^{2})$ is defined as

$[F(x,y)\\!\overset{\;\;*^{r}}{,}G(x,y)]=F(x,y)*^{r}\\!G(x,y)-G(x,y)*^{r}\\!F(x,y),$
(3.6)

where the gauge parameter $r$ dependent star-product between $F(x,y),G(x,y)\in
C^{\infty}(\mathbb{R}^{2})$ is given by

$F(x,y)*^{r}\\!G(x,y)=F(x,y)e^{-i(r-1)\vartheta\overset{\leftarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}-ir\vartheta\overset{\leftarrow}{\partial_{y}}\overset{\rightarrow}{\partial_{x}}}G(x,y).$
(3.7)

In fact, (3.7) tells us that $*:C^{\infty}(\mathbb{R}^{2})\times
C^{\infty}(\mathbb{R}^{2})\rightarrow
C^{\infty}(\mathbb{R}^{2})[[\vartheta]]$, where
$C^{\infty}(\mathbb{R}^{2})[[\vartheta]]$ is the ring of all formal power
series in the deformation parameter $\vartheta$ with coefficients in the ring
$C^{\infty}(\mathbb{R}^{2})$. In fact, $*$ can be extended to the ring
$C^{\infty}(\mathbb{R}^{2})[[\vartheta]]$ so that it becomes a map
$*:C^{\infty}(\mathbb{R}^{2})[[\vartheta]]\times
C^{\infty}(\mathbb{R}^{2})[[\vartheta]]\rightarrow
C^{\infty}(\mathbb{R}^{2})[[\vartheta]]$ [15]. One can now go on to show that
$2$ star products $*^{r}$ and $*^{r^{\prime}}$ given by (3.7) are gauge
equivalent (or C-equivalent in the sense of [15]), i.e., there exists an
invertible operator $T$ on $C^{\infty}(\mathbb{R}^{2})$ such that

$T(F*^{r}G)=T(F)*^{r^{\prime}}T(G).$ (3.8)

Let us now observe that
$T=e^{i(r-r^{\prime})\vartheta\overset{\rightarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}}$
is the required invertible operator which satisfies (3.8). Indeed, one has

$\displaystyle T(F*^{r}G)$
$\displaystyle=e^{i(r-r^{\prime})\vartheta\overset{\rightarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}}[Fe^{-i(r-1)\vartheta\overset{\leftarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}-ir\vartheta\overset{\leftarrow}{\partial_{y}}\overset{\rightarrow}{\partial_{x}}}G]$
(3.9)
$\displaystyle=Fe^{i(r-r^{\prime})\vartheta\overset{\leftarrow}{\partial_{x}}\overset{\leftarrow}{\partial_{y}}+i(r-r^{\prime})\vartheta\overset{\leftarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}+i(r-r^{\prime})\vartheta\overset{\leftarrow}{\partial_{y}}\overset{\rightarrow}{\partial_{x}}+i(r-r^{\prime})\vartheta\overset{\rightarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}}e^{-i(r-1)\vartheta\overset{\leftarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}-ir\vartheta\overset{\leftarrow}{\partial_{y}}\overset{\rightarrow}{\partial_{x}}}G$
$\displaystyle=Fe^{i(r-r^{\prime})\vartheta\overset{\leftarrow}{\partial_{x}}\overset{\leftarrow}{\partial_{y}}+i(r-r^{\prime})\vartheta\overset{\leftarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}+i(r-r^{\prime})\vartheta\overset{\leftarrow}{\partial_{y}}\overset{\rightarrow}{\partial_{x}}+i(r-r^{\prime})\vartheta\overset{\rightarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}-i(r-1)\vartheta\overset{\leftarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}-ir\vartheta\overset{\leftarrow}{\partial_{y}}\overset{\rightarrow}{\partial_{x}}}G$
$\displaystyle=Fe^{i(r-r^{\prime})\vartheta\overset{\leftarrow}{\partial_{x}}\overset{\leftarrow}{\partial_{y}}-i(r^{\prime}-1)\vartheta\overset{\leftarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}-ir^{\prime}\vartheta\overset{\leftarrow}{\partial_{y}}\overset{\rightarrow}{\partial_{x}}+i(r-r^{\prime})\vartheta\overset{\rightarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}}G.$

On the other hand,

$\displaystyle T(F)*^{r^{\prime}}T(G)$
$\displaystyle=[e^{i(r-r^{\prime})\vartheta\overset{\rightarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}}F]e^{i(r^{\prime}-1)\vartheta\overset{\leftarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}-ir^{\prime}\vartheta\overset{\leftarrow}{\partial_{y}}\overset{\rightarrow}{\partial_{x}}}[e^{i(r-r^{\prime})\vartheta\overset{\rightarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}}G]$
(3.10)
$\displaystyle=Fe^{i(r-r^{\prime})\vartheta\overset{\leftarrow}{\partial_{x}}\overset{\leftarrow}{\partial_{y}}-i(r^{\prime}-1)\vartheta\overset{\leftarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}-ir^{\prime}\vartheta\overset{\leftarrow}{\partial_{y}}\overset{\rightarrow}{\partial_{x}}+i(r-r^{\prime})\vartheta\overset{\rightarrow}{\partial_{x}}\overset{\rightarrow}{\partial_{y}}}G.$

From (3.9) and (3.10) follows (3.8).

######  Remark III.1.

It is important to note that one can reproduce the familiar Moyal product by
taking $r=\frac{1}{2}$ in the general definition of 1-parameter family of
star-products given by (3.7). In other words, Moyal star-product is the
symmetric gauge version of the star product (3.7).

The associativity of the star product follows from the general definition of
it presented in (3.7). Now let us consider the Hamiltonian for the
2-dimensional Landau problem: a charged spinless point particle of mass $m$
and charge $e$ placed in a constant magnetic field. Using star product the
Landau Hamiltonian is the following smooth function on $\mathbb{R}^{2}$:

$H=\frac{1}{2m}(p_{x}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{x})*^{r}(p_{x}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{x})+\frac{1}{2m}(p_{y}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{y})*^{r}(p_{y}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{y}).$
(3.11)

Hence, the pertinent Schrodinger equation for a sufficiently well-behaved
function
$\psi\in L^{2}(\mathbb{R}^{2},dx\;dy)$ reads

$H*^{r}\psi=E\psi.$ (3.12)

Let us first see that

$\begin{split}(p_{x}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{x})*^{r}\psi&=(\hat{p}_{x}-e\hat{\mathbf{A}}^{r}_{x})\psi\\\
&=\left[\hat{p}_{x}+\frac{2e(1-r)\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}}\hat{Y}^{r}\right]\psi\\\
&=\left[1+\frac{2r(1-r)e\vartheta
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\right]\left[\hat{p}_{x}-\frac{e}{1+\frac{2r(1-r)e\vartheta
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}}\mathbf{A}^{\hbox{\tiny{nc}}}_{x}\right]\psi.\end{split}$ (3.13)

Let us denote

$\bar{\Lambda}(r)=1+\frac{2r(1-r)e\vartheta
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}}.$ (3.14)

Then (3.13) reads

$(p_{x}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{x})*^{r}\psi=\bar{\Lambda}(r)\left(\hat{p}_{x}-\frac{e}{\bar{\Lambda}(r)}\mathbf{A}^{\hbox{\tiny{nc}}}_{x}\right)\psi,$
(3.15)

which leads to the following

$\begin{split}\frac{1}{2m}(p_{x}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{x})*^{r}(p_{x}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{x})*^{r}\psi&=\frac{[\bar{\Lambda}(r)]^{2}}{2m}\left(\hat{p}_{x}-\frac{e}{\bar{\Lambda}(r)}\mathbf{A}^{\hbox{\tiny{nc}}}_{x}\right)^{2}\psi,\\\
&=\frac{1}{2m_{*}}(\hat{p}_{x}-e_{*}\mathbf{A}^{\hbox{\tiny{nc}}}_{x})^{2}\psi,\end{split}$
(3.16)

where the reduced mass and reduced charge, denoted by $m_{*}$ and $e_{*}$,
respectively, are given by

$m_{*}=\frac{m}{[\bar{\Lambda}(r)]^{2}}\hskip
36.135pte_{*}=\frac{e}{\bar{\Lambda}(r)}.$ (3.17)

One can similarly go on to find

$\frac{1}{2m}(p_{y}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{y})*^{r}(p_{y}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{y})*^{r}\psi=\frac{1}{2m_{*}}(\hat{p}_{y}-e_{*}\mathbf{A}^{\hbox{\tiny{nc}}}_{y})^{2}\psi,$
(3.18)

where $m_{*}$ and $e_{*}$ are as given by (3.17).

One finally arrives at the following:

$\begin{split}&\frac{1}{2m}(p_{x}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{x})*^{r}(p_{x}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{x})*^{r}\psi+\frac{1}{2m}(p_{y}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{y})*^{r}(p_{y}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{y})*^{r}\psi\\\
&=\frac{1}{2m_{*}}[(\hat{p}_{x}-e_{*}\mathbf{A}^{\hbox{\tiny{nc}}}_{x})^{2}+(\hat{p}_{y}-e_{*}\mathbf{A}^{\hbox{\tiny{nc}}}_{y})^{2}]\psi=:\hat{H}^{r}_{*}\psi.\end{split}$
(3.19)

Here, $\hat{H}^{r}_{*}$ with the expressions for the reduced charge $e_{*}$
and mass $m_{*}$ as given in (3.17) is called the deformed Hamiltonian of the
noncommutative Landau problem.

Now, one finds out from (3.3) that

$\partial_{x}\mathbf{A}^{\hbox{\tiny{nc}}}_{y}-\partial_{y}\mathbf{A}^{\hbox{\tiny{nc}}}_{x}=\frac{2\hbar
B}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\mathbb{I}=:\bar{B}(r,\vartheta)\mathbb{I}.$ (3.20)

Gathering (3.20) with (3.3) and the first equation of (3.5), one verifies that
the star product version of (2.14) given by the following holds

$\partial_{x}\mathbf{A}^{\hbox{\tiny{nc}}}_{y}-\partial_{y}\mathbf{A}^{\hbox{\tiny{nc}}}_{x}-\frac{ie}{\hbar}[\mathbf{A}^{\hbox{\tiny{nc}}}_{x}\\!\\!\overset{\;\;*^{r}}{,}\\!\mathbf{A}^{\hbox{\tiny{nc}}}_{y}]=B\mathbb{I}.$
(3.21)

Using the 1-dimensional harmonic oscillator technique of solving the canonical
Landau problem (page 21 in [10]), one arrives at the following eigenvalues of
the deformed Hamiltonian $H_{*}$ as given in (3.19) (we choose the speed of
light $c=1$):

$E_{n}=\hbar\frac{|e_{*}\bar{B}(r,\vartheta)|}{m_{*}}\left(n+\frac{1}{2}\right)\hskip
36.135pt\hbox{with}\;\;n=0,1,2,...$ (3.22)

By using (3.14), (3.17) and (3.20), one finds that the eigenvalues given in
(3.22) are indeed gauge invariant, i.e., independent of the gauge parameter
$r$:

$\begin{split}E_{n}&=\hbar\frac{|e\bar{\Lambda}(r)\bar{B}(r,\vartheta)|}{m}\left(n+\frac{1}{2}\right),\\\
&=\hbar\frac{|eB|}{m}\left(n+\frac{1}{2}\right).\end{split}$ (3.23)

Each of the eigenstates associated with the eigenvalues above are infinitely
degenerate as expected. The eigenvalues are also found to be independent of
the noncommutativity parameter $\vartheta$.

Using (3.20) along with (3.14), one finds that

$\displaystyle\bar{\Lambda}(r)\bar{B}(r,\vartheta)$
$\displaystyle=\frac{2\hbar B[\hbar-2r(r-1)e\vartheta
B+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}]}{[\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}]^{2}},$ (3.24)
$\displaystyle=\frac{2\hbar B[\hbar-2r(r-1)e\vartheta
B+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}]}{[2\hbar^{2}-4r(r-1)e\hbar\vartheta
B+2\hbar\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B]}},$ $\displaystyle=B.$

(3.24) is shown to hold true [9, 10] in the symmetric gauge only, i.e., for
$r=\frac{1}{2}$. Also, if we manipulate the expression for
$\bar{B}(r,\vartheta)$ (see (3.20)), one finds out

$B=\left[1-\frac{r(r-1)e\vartheta\bar{B}}{\hbar}\right]\bar{B},$ (3.25)

so that combining (3.25) with (3.24), one obtains an expression for
$\bar{\Lambda}(r)$ using the effective magnetic field $\bar{B}(r,\vartheta)$
in a given gauge:

$\bar{\Lambda}(r)=1-\frac{r(r-1)e\vartheta\bar{B}}{\hbar},$ (3.26)

the symmetric gauge value of which agrees with [9, 10]:

$\bar{\Lambda}\left(\frac{1}{2}\right)=1+\frac{e\vartheta\bar{B}}{4\hbar}.$
(3.27)

Assuming $r=\frac{1}{2}$, on the other hand, in the expression for
$\bar{B}(r,\vartheta)$ in (3.20), one obtains

$\begin{split}\bar{B}\left(\frac{1}{2},\vartheta\right)&=\frac{2\hbar
B}{\hbar+\sqrt{\hbar^{2}+e\hbar\vartheta B}},\\\
&=\frac{2}{e\vartheta}(\sqrt{\hbar^{2}+e\hbar\vartheta B}-\hbar),\end{split}$
(3.28)

which again agrees with the effective magnetic field expression for the
symmetric gauge given in [9, 10, 18].

Also, the term $\frac{e^{2}B^{2}}{m}$ is inferred to be gauge invariant in
[9]. Indeed, we find here that

$\frac{e_{*}^{2}B^{2}}{m_{*}}=\frac{\frac{e^{2}}{[\bar{\Lambda}(r)]^{2}}B^{2}}{\frac{m}{[\bar{\Lambda}(r)]^{2}}}=\frac{e^{2}B^{2}}{m}.$
(3.29)

Although we have seen explicitly that the spectra of the Landau Hamiltonian in
the context of noncommutative space turn out to be gauge invariant by means of
(3.23), let us try to investigate this gauge invariance group theoretically
and study its relation with the star product approach introduced earlier in
this section.

From the earlier discussion, we have seen that there exists an invertible
operator $T$ on $C^{\infty}(\mathbb{R}^{2})$ that satisfies

$T(H)*^{r^{\prime}}T(\psi)=T(H*^{r}\psi)=ET(\psi),$ (3.30)

where $H$ is the Hamiltonian function (3.11) in $C^{\infty}(\mathbb{R}^{2})$
for the noncommutative Landau problem and $\psi\in
L^{2}(\mathbb{R}^{2},dx\;dy)$ is a sufficiently well-behaved wavefunction. If
one writes $T(H)=\widetilde{H}$ and $T(\psi)=\widetilde{\psi}$, then (3.30)
reads

$\widetilde{H}*^{r^{\prime}}\widetilde{\psi}=E\widetilde{\psi}.$ (3.31)

In view of (3.12) and (3.31), one finds that under gauge transformation
Schrodinger equation transforms covariantly so that the underlying spectra $E$
remain invariant.

Let us now denote by $\hat{H}_{*}^{r}$ and
$\hat{\widetilde{H}}_{*}^{r^{\prime}}$, the operators representing the smooth
functions $H$ and $\widetilde{H}$ in $C^{\infty}(\mathbb{R}^{2})$,
respectively. One can refer to (3.19) for an explicit expression of this
operator. In order to obtain $\hat{\widetilde{H}}_{*}^{r^{\prime}}$, one just
replaces $r$ with $r^{\prime}$ throughout in (3.19). The one-parameter family
of Hamiltonians $\hat{H}^{r}$ is expressed using the one parameter family of
equivalent irreducible self-adjoint representations of
$\mathfrak{g}_{\hbox{\tiny{NC}}}$ due to a fixed ordered pair
$(\hbar,\vartheta)$ (2.2). Physically all these gauge equivalent
representations (parametrized by $r$) represent the same noncommutative plane.
Hence the unbounded self-adjoint operator $\hat{H}_{*}^{r}$ and
$\hat{\widetilde{H}}_{*}^{r^{\prime}}$ are related by

$\hat{\widetilde{H}}^{r^{\prime}}_{*}=\hat{U}\hat{H}_{*}^{r}\hat{U}^{-1},$
(3.32)

with $\hat{U}$ being a unitary operator on $L^{2}(\mathbb{R}^{2},dx\;dy)$. In
operatorial language, (3.12) translates to

$\hat{H}_{*}^{r}\psi=E\psi,$ (3.33)

leading to

$\hat{U}\hat{H}_{*}^{r}\psi=E\hat{U}\psi,$ (3.34)

so that one obtains

$\hat{U}\hat{H}_{*}^{r}\hat{U}^{-1}\hat{U}\psi=E\hat{U}\psi.$

In other words, the eigenvalue of the gauge transformed Hamiltonian
$\hat{\widetilde{H}}^{r^{\prime}}$ remains the same, i.e., $E$ corresponding
to the eigenvector $\widetilde{\psi}=\hat{U}\psi$:

$\hat{\widetilde{H}}^{r^{\prime}}\\!\\!\\!\widetilde{\psi}=E\widetilde{\psi}.$
(3.35)

The star-product version of (3.35) is of course (3.31). If $U$ is an
invertiable smooth function in $C^{\infty}(\mathbb{R}^{2})$ that is
represented by the unitary operator $\hat{U}$ appearing in (3.32), then the
star-product version of (3.32) reads

$T(H)=\widetilde{H}=U*^{r}H*^{r}U^{-1}.$ (3.36)

At the wavefunction level, one also has

$T(\psi)=\widetilde{\psi}=U*^{r}\psi,$ (3.37)

so that in terms of the invertible smooth function $U\in
C^{\infty}(\mathbb{R}^{2})$, (3.31)now reads

$U*^{r}H*^{r}U^{-1}*^{r^{\prime}}U*^{r}\psi=E(U*^{r}\psi).$ (3.38)

Since $U^{-1}*^{r^{\prime}}U=\mathbb{I}$, the map in
$C^{\infty}(\mathbb{R}^{2})$ that maps everything to 1, one finds that the
following holds

$U*^{r}H*^{r}\psi=E(U*^{r}\psi),$ (3.39)

which is indeed the star-product version of (3.34).

##  IV On 1-parameter family of Seiberg-Witten maps for noncommutative U(1)
gauge theory

N. Seiberg and E. Witten were the first to provide a set of formulae in [11]
establishing the relationship between the noncommutative quantities (gauge
fields, field strength etc.) and their commutative counterparts. A variant of
the magnetic field $B$ is used to this end in [10] (see equation (65) at p.
20) which we call commutative Seiberg-Witten field strength denoted by
$\mathfrak{B}$. This gauge dependent quantity in terms of the physical
magnetic field $B$ reads

$\mathfrak{B}=\frac{\hbar B}{\hbar-4r(r-1)e\vartheta B}.$ (4.1)

Note that the symmetric gauge (corresponding to $r=\frac{1}{2}$) version of
(4.1) matches with the one provided in [10] (equation (65) at p.20). In terms
of $\mathfrak{B}$, the commutative Seiberg-Witten gauge fields are defined as
(in [10] expressed in symmetric gauge, i.e., for $r=\frac{1}{2}$)

$\mathbf{A}^{r}\equiv(\mathbf{A}_{x},\mathbf{A}_{y})=((r-1)\mathfrak{B}y,r\mathfrak{B}x).$
(4.2)

We will be using $\mathbf{A}$ instead of $\mathbf{A}^{r}$ for the sake of
notational clarity to denote the redefined commutative gauge field, i.e., the
1-parameter family of commutative Seiberg-Witten gauge fields.

The 1-parameter family of noncommutative Seiberg-Witten gauge fields are
precisely the same as the ones introduced in (3.3):

$\mathbf{A}^{\hbox{\tiny{nc}}}\equiv(\mathbf{A}^{\hbox{\tiny{nc}}}_{x},\mathbf{A}^{\hbox{\tiny{nc}}}_{y})=((r-1)\bar{B}y,r\bar{B}x),$
(4.3)

where $\bar{B}$ in terms of the commutative Seiberg-Witten field strength
$\mathfrak{B}$ is given by:

$\bar{B}(r,\vartheta)=\frac{\hbar}{2r(1-r)e\vartheta}\left[\frac{1}{\sqrt{1+\frac{4r(r-1)e\vartheta\mathfrak{B}}{\hbar}}}-1\right],$
(4.4)

The noncommutative gauge function $\lambda^{\hbox{\tiny{nc}}}$ can then be
obtained using the infinitesimal variation $\epsilon=r^{\prime}-r$ of the
gauge parameter associated with the noncommutative gauge fields given in (1.8
or 4.3):

$\delta\mathbf{A}^{\hbox{\tiny{nc}}}_{j}=\partial_{j}\lambda^{\hbox{\tiny{nc}}}+\frac{ie}{\hbar}[\lambda^{\hbox{\tiny{nc}}}\\!\overset{\;\;*^{r}}{,}\mathbf{A}^{\hbox{\tiny{nc}}}_{j}],\\\
$ (4.5)

where $j=x,y$. A straightforward but lengthy computation then leads to the
following expression relating the noncommutative gauge function with its
commutative counterpart:

$\lambda^{\hbox{\tiny{nc}}}=\lambda+3e\epsilon
r(r-1)\vartheta\frac{B^{2}xy}{\hbar}+\mathcal{O}(\vartheta^{2}),$ (4.6)

where the undeformed gauge function $\lambda$ is defined as $\lambda=\epsilon
Bxy$. The detailed derivation of (4.6) is provided in the Appendix. Now, a
sufficiently well-behaved wavefunction $\psi\in C^{\infty}(\mathbb{R}^{2})$
transforms under the underlying $U(1)_{\star}$ gauge transformation as

$\psi\rightarrow U_{\lambda^{\hbox{\tiny{nc}}}}*^{r}\psi,$ (4.7)

where
$U_{\lambda^{\hbox{\tiny{nc}}}}=e^{\frac{ie}{\hbar}\lambda^{\hbox{\tiny{nc}}}}_{\star}$.
Here, $e^{x}_{\star}$ denotes star-exponential, i.e., in the power series
expansion of $e^{x}$, all the products are taken to be star-products. Using
(4.5), one verifies that the star-exponent
$\frac{e\lambda^{\hbox{\tiny{nc}}}}{\hbar}$ is indeed dimensionless. Under
$U(1)_{\star}$ gauge transformation the noncommutative gauge fields transform
as

$\mathbf{A}_{j}^{\hbox{\tiny{nc}}}\rightarrow
U_{\lambda^{\hbox{\tiny{nc}}}}*^{r}\mathbf{A}^{\hbox{\tiny{nc}}}_{j}*^{r}U_{\lambda^{\hbox{\tiny{nc}}}}^{-1}+\frac{i\hbar}{e}U_{\lambda^{\hbox{\tiny{nc}}}}*^{r}\partial_{j}U^{-1}_{\lambda^{\hbox{\tiny{nc}}}},$
(4.8)

with $j=x,y$. Indeed, one verifies that (4.8) is in agreement with the
infinitesimal gauge transformation given in (4.5) due to an infinitesimal
variation $\epsilon=r^{\prime}-r$ of the gauge parameter. Also, note that the
inverse of $U_{\lambda^{\hbox{\tiny{nc}}}}$ is given by
$U_{\lambda^{\hbox{\tiny{nc}}}}^{-1}=e^{-\frac{ie}{\hbar}\lambda^{\hbox{\tiny{nc}}}}_{\star}$.

Of course, (4.3) satisfies,

$[p_{x}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{x}\\!\overset{\;\;*^{r}}{,}p_{y}-e\mathbf{A}^{\hbox{\tiny{nc}}}_{y}]=ie\hbar
B\mathbb{I}.$ (4.9)

Now, one has

$\displaystyle\mathbf{A}^{\hbox{\tiny{nc}}}_{x}$
$\displaystyle=(r-1)\frac{\hbar}{2r(1-r)e\vartheta}\left[\frac{1}{\sqrt{1+\frac{4r(r-1)e\vartheta\mathfrak{B}}{\hbar}}}-1\right]y,$
(4.10)
$\displaystyle=\frac{\hbar}{2re\vartheta}\left[1-\frac{1}{\sqrt{1+\frac{4r(r-1)e\vartheta\mathfrak{B}}{\hbar}}}\right]y,$
$\displaystyle=\frac{\hbar}{2er\vartheta}\left[1-\left\\{1+\frac{4r(r-1)e\vartheta\mathfrak{B}}{\hbar}\right\\}^{-\frac{1}{2}}\right]y,$
$\displaystyle=\frac{\hbar
y}{2er\vartheta}\left[\frac{2r(r-1)e\vartheta\mathfrak{B}}{\hbar}-\frac{6r^{2}(r-1)^{2}e^{2}\vartheta^{2}\mathfrak{B}^{2}}{\hbar^{2}}+\cdots\right],$
$\displaystyle=(r-1)\mathfrak{B}y-\frac{3r(r-1)^{2}e\vartheta\mathfrak{B}^{2}y}{\hbar}+\mathcal{O}(\vartheta^{2}),$
$\displaystyle=\mathbf{A}_{x}-\frac{2er}{\hbar}\theta^{xy}\mathbf{A}_{x}\partial_{y}\mathbf{A}_{x}-\frac{e(r-1)}{\hbar}\theta^{xy}\mathbf{A}_{x}\partial_{x}\mathbf{A}_{y}+\mathcal{O}(\vartheta^{2}),$
$\displaystyle=\mathbf{A}_{x}-\frac{e}{\hbar}\theta^{xy}\mathbf{A}_{x}\left[(3r-1)\partial_{y}\mathbf{A}_{x}+(1-r)F_{yx}\right]+\mathcal{O}(\vartheta^{2}),$

where $\theta^{xy}=-\theta^{yx}=\vartheta$, are the entries of the real
$2\times 2$ antisymmetric matrix $\theta$ given by

$\theta=\begin{bmatrix}0&\vartheta\\\ -\vartheta&0\end{bmatrix},$ (4.11)

and

$F_{xy}=\partial_{x}\mathbf{A}_{y}-\partial_{y}\mathbf{A}_{x}=\mathfrak{B}\mathbb{I}=-F_{yx},$
(4.12)

are the entries of the following $2\times 2$ antisymmetric real matrix $F$
representing the commutative Seiberg-Witten field strength:

$F=\begin{bmatrix}0&\partial_{x}\mathbf{A}_{y}-\partial_{y}\mathbf{A}_{x}\\\
\partial_{y}\mathbf{A}_{x}-\partial_{x}\mathbf{A}_{y}&0\end{bmatrix}.$ (4.13)

Also, from (4.3) and (4.4), it follows that

$\displaystyle\mathbf{A}^{\hbox{\tiny{nc}}}_{y}$
$\displaystyle=r\frac{\hbar}{2r(1-r)e\vartheta}\left[\frac{1}{\sqrt{1+\frac{4r(r-1)e\vartheta\mathfrak{B}}{\hbar}}}-1\right]x,$
(4.14)
$\displaystyle=\frac{\hbar}{2(1-r)e\vartheta}\left[\left\\{1+\frac{4r(r-1)e\vartheta\mathfrak{B}}{\hbar}\right\\}^{-\frac{1}{2}}-1\right]x,$
$\displaystyle=\frac{\hbar
x}{2(1-r)e\vartheta}\left[-\frac{2r(r-1)e\vartheta\mathfrak{B}}{\hbar}+\frac{6r^{2}(r-1)^{2}e^{2}\vartheta^{2}\mathfrak{B}^{2}}{\hbar^{2}}-\cdots\right],$
$\displaystyle=r\mathfrak{B}x+\frac{3r^{2}(1-r)e\vartheta\mathfrak{B}^{2}x}{\hbar}+\mathcal{O}(\vartheta^{2}),$
$\displaystyle=\mathbf{A}_{y}+\frac{2e(r-1)}{\hbar}\theta^{yx}\mathbf{A}_{y}\partial_{x}\mathbf{A}_{y}+\frac{er}{\hbar}\theta^{yx}\mathbf{A}_{y}\partial_{y}\mathbf{A}_{x}+\mathcal{O}(\vartheta^{2}),$
$\displaystyle=\mathbf{A}_{y}-\frac{e}{\hbar}\theta^{yx}\mathbf{A}_{y}\left[(2-3r)\partial_{x}\mathbf{A}_{y}+rF_{xy}\right]+\mathcal{O}(\vartheta^{2}).$

It now remains to compare the noncommutative Seiberg-Witten field strength
with its commutative counterpart. The noncommutative Seiberg-Witten field
strength $F^{\hbox{\tiny{nc}}}_{xy}$ is given by

$F^{\hbox{\tiny{nc}}}_{xy}=\partial_{x}\mathbf{A}^{\hbox{\tiny{nc}}}_{y}-\partial_{y}\mathbf{A}^{\hbox{\tiny{nc}}}_{x}-\frac{ie}{\hbar}[\mathbf{A}^{\hbox{\tiny{nc}}}_{x}\\!\\!\overset{\;\;*^{r}}{,}\mathbf{A}^{\hbox{\tiny{nc}}}_{y}]=B\mathbb{I},$
(4.15)

with $\mathbb{I}$ being the real valued constant map in
$C^{\infty}(\mathbb{R}^{2})$ that maps everything to $1$. Here, we used (3.21)
to arrive at (4.15).

Now, using (3.25) in (4.15), one obtains

$\displaystyle F^{\hbox{\tiny{nc}}}_{xy}$
$\displaystyle=\bar{B}\left[1-\frac{r(r-1)e\vartheta\bar{B}}{\hbar}\right]\mathbb{I},$
(4.16)
$\displaystyle=\frac{\hbar}{2r(1-r)e\vartheta}\left[\frac{1}{\sqrt{1+\frac{4r(r-1)e\vartheta\mathfrak{B}}{\hbar}}}-1\right]\left[1+\frac{1}{2}\left\\{\frac{1}{\sqrt{1+\frac{4r(r-1)e\vartheta\mathfrak{B}}{\hbar}}}-1\right\\}\right]\mathbb{I},$

upon using (4.4). Let

$\frac{4r(r-1)e\vartheta\mathfrak{B}}{\hbar}=t$ (4.17)

in (4.16) and simplify.

$\displaystyle F^{\hbox{\tiny{nc}}}_{xy}$
$\displaystyle=\frac{2\mathfrak{B}}{t}\left[1-\frac{1}{\sqrt{1+t}}\right]\left[1+\frac{1}{2}\left(\frac{1}{\sqrt{1+t}}-1\right)\right]\mathbb{I},$
(4.18)
$\displaystyle=\frac{\mathfrak{B}}{t}\left[1-\frac{1}{\sqrt{1+t}}\right]\left[1+\frac{1}{\sqrt{1+t}}\right]\mathbb{I},$
$\displaystyle=\left(\frac{\mathfrak{B}}{1+t}\right)\mathbb{I}.$

Substituting back the value of $t$ from (4.17) in (4.18), one finally obtains

$\displaystyle F^{\hbox{\tiny{nc}}}_{xy}$
$\displaystyle=\mathfrak{B}\left[1-\frac{4r(1-r)e\vartheta\mathfrak{B}}{\hbar}\right]^{-1}\mathbb{I},$
(4.19)
$\displaystyle=\mathfrak{B}\mathbb{I}+\frac{4r(1-r)e\vartheta\mathfrak{B}^{2}}{\hbar}\mathbb{I}+\mathcal{O}(\vartheta^{2}),$
$\displaystyle=F_{xy}+\frac{4er(1-r)}{\hbar}\theta^{yx}F_{xy}F_{yx}+\mathcal{O}(\vartheta^{2}).$

Here, $F_{xy}=-F_{yx}$ are as given in (4.12). Equations (4.10), (4.14) and
(4.19) tell us how noncommutative U(1) gauge field and Field strength are
related with their commutative counterparts in the first order of the
deformation parameter $\vartheta$ in a chosen gauge parametrized by a real
value of $r$. These equations take the following familiar forms in the
symmetric gauge configuration corresponding to $r=\frac{1}{2}$:

$\begin{split}\mathbf{A}^{\hbox{\tiny{nc}}}_{x}&=\mathbf{A}_{x}-\frac{e}{2\hbar}\theta^{xy}\mathbf{A}_{x}(\partial_{y}\mathbf{A}_{x}+F_{yx})+\mathcal{O}(\vartheta^{2}),\\\
\mathbf{A}^{\hbox{\tiny{nc}}}_{y}&=\mathbf{A}_{y}-\frac{e}{2\hbar}\theta^{yx}\mathbf{A}_{y}(\partial_{x}\mathbf{A}_{y}+F_{xy})+\mathcal{O}(\vartheta^{2}),\\\
F^{\hbox{\tiny{nc}}}_{xy}&=F_{xy}+\frac{e}{\hbar}\theta^{yx}F_{xy}F_{yx}+\mathcal{O}(\vartheta^{2}).\end{split}$
(4.20)

We call the 1-parameter family of maps given by (4.14), (4.10) and (4.19), the
Seiberg-Witten maps.

##  V Conclusion and Future Perspectives

In this paper, we have undertaken noncommutative U(1) gauge theoretic study of
a spinless point particle placed in a noncommutative plane minimally coupled
to an external uniform magnetic field. In the process, we have seen how the
gauge parameter $r$ gives rise to $c$-equivalent star products $*^{r}$ in the
sense of [15]. We also have obtained an induced family of invertible
(parametrized by the same gauge parameter $r$) maps that relate the
noncommutative U(1) gauge field and the corresponding field strength with
their commutative counterparts. These invertible maps for $r=\frac{1}{2}$
agrees with the so-called Seiberg-Witten map constructed in [11].

Geometrically, one has a 1-dimensional $\mathcal{A}$-module
$\mathbb{M}=\mathbb{C}\times\mathcal{A}$ where the noncommutative $*$-algebra
is $(C^{\infty}(\mathbb{R}^{2}),*^{r})$. Connections on $\mathbb{M}$ are then
maps $\nabla:\hbox{Der}(\mathcal{A})\times\mathbb{M}\rightarrow\mathbb{M}$
satisfying some nice properties (see appendix of [19]). There is a more
interesting construction of deformation of principal G bundle
$\hbox{pr}:\mathcal{P}\circlearrowleft G\rightarrow\mathcal{M}$ [20, 21] where
$\mathcal{M}$ is a smooth manifold. According to the construction presented in
[22], if one has a $\star$ product defined on $\mathcal{M}$, i.e.,
$\star:C^{\infty}(\mathcal{M})[[\lambda]]\times
C^{\infty}(\mathcal{M})[[\lambda]]\rightarrow
C^{\infty}(\mathcal{M})[[\lambda]]$ with $\lambda$ being the deformation
parameter, then one has an induced right $\star$-module structure $\bullet$ on
$C^{\infty}(\mathcal{P})[[\lambda]]$ by

$F\bullet f=F\hbox{pr}^{*}f+\sum_{n=1}^{\infty}\lambda^{r}\varrho_{n}(F,f),$
(5.1)

with $\varrho_{n}:C^{\infty}(\mathcal{P})\times C^{\infty}(M)\rightarrow
C^{\infty}(\mathcal{P})$ is a bidifferential operator along the map pr for all
$r\geq 1$ such that one also has the $G$-equivariance

$g^{*}(F\bullet f)=g^{*}F\bullet f.$ (5.2)

Here, $F\in C^{\infty}(\mathcal{P})$, $f\in C^{\infty}(\mathcal{M})$ and $g\in
G$. Also, $\hbox{pr}^{*}:C^{\infty}(\mathcal{M})\rightarrow
C^{\infty}(\mathcal{P})$ and the right action of the group $G$ on
$\mathcal{P}$ induces a map $g^{*}:C^{\infty}(\mathcal{P})\rightarrow
C^{\infty}(\mathcal{P})$.

We look forward to studying this construction in the present context where,
$\mathcal{M}=\mathbb{R}^{2}$, $\lambda=\vartheta$, $G=U(1)$ and $\star=*^{r}$.
We would like to define an equivalence class of $*^{r}$-module structures
$\bullet^{r}$ in the sense of [22] and study its resulting algebraic
properties in a future publication.

Furthermore, the Landau problem is formulated for the particle moving in a
two-dimensional field. However, it can be extended to the higher dimensional
space as shown, for example, in [28, 29]. Besides, the magnetic field can be
non-uniform, as pointed out in [10]. It will be interesting to explore the
noncommutative generalization of the higher dimensional case and the presence
of a non-uniform magnetic field which we have set for a future publication.

### Acknowledgement

We acknowledge fruitful discussions with Tibra Ali that helped us understand
the simple underlying gauge theoretic structures. TAC would also like to thank
the High Energy Physics Theory group of the Department of Physics and
Astronomy at the University of Kansas for the hospitality and support. SHHC
gratefully acknowledges his six year old daughter Nefertiti’s graciousness in
allowing him to use her notebook to do important calculations.

##  VI Appendix

Here we present the derivation of the expression (see 4.6) of the
noncommutative gauge function $\lambda^{\hbox{\tiny{nc}}}$ in detail. Using
(3.3), one obtains

$\displaystyle\delta\mathbf{A}^{\hbox{\tiny{nc}}}_{x}$
$\displaystyle=\mathbf{A}^{\hbox{\tiny{nc}}\prime}_{x}-\mathbf{A}^{\hbox{\tiny{nc}}}_{x}$
(6.1) $\displaystyle=\frac{-2(1-r^{\prime})\hbar
By}{\hbar+\sqrt{\hbar^{2}-4r^{\prime}(r^{\prime}-1)e\hbar\vartheta
B}}+\frac{2(1-r)\hbar By}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}},$
$\displaystyle=-2(1-r^{\prime})By\left[1+\sqrt{1-\frac{4r^{\prime}(r^{\prime}-1)e\vartheta
B}{\hbar}}\right]^{-1}+2(1-r)By\left[1+\sqrt{1-\frac{4r(r-1)e\vartheta
B}{\hbar}}\right]^{-1},$
$\displaystyle=\frac{\hbar}{2r^{\prime}e\vartheta}\left[1-\sqrt{1-\frac{4r^{\prime}(r^{\prime}-1)e\vartheta
B}{\hbar}}\right]-\frac{\hbar}{2re\vartheta}\left[1-\sqrt{1-\frac{4r(r-1)e\vartheta
B}{\hbar}}\right],$
$\displaystyle=\frac{\hbar}{2r^{\prime}e\vartheta}\left[1-\left\\{1-\frac{2r^{\prime}(r^{\prime}-1)e\vartheta
B}{\hbar}-\frac{2r^{\prime
2}(r^{\prime}-1)^{2}e^{2}\vartheta^{2}B^{2}}{\hbar^{2}}\right\\}\right]y,$
$\displaystyle\hskip
7.22743pt-\frac{\hbar}{2re\vartheta}\left[1-\left\\{1-\frac{2r(r-1)e\vartheta
B}{\hbar}-\frac{2r^{2}(r-1)^{2}e^{2}\vartheta^{2}B^{2}}{\hbar^{2}}\right\\}\right]y+\mathcal{O}(\vartheta^{2}),$
$\displaystyle=(r^{\prime}-1)By+r^{\prime}(r^{\prime}-1)^{2}\frac{e\vartheta
B^{2}y}{\hbar}-(r-1)By-\frac{r(r-1)^{2}e\vartheta
B^{2}y}{\hbar}+\mathcal{O}(\vartheta^{2}),$
$\displaystyle=(r^{\prime}-r)By+\frac{e\vartheta
B^{2}y}{\hbar}[r^{\prime}(r^{\prime}-1)^{2}-r(r-1)^{2}]+\mathcal{O}(\vartheta^{2}).$

Similarly, using (3.3), one can proceed to obtain the expression for the
infinitesimal variation of the $y$-component of the noncommutative gauge
field:

$\displaystyle\delta\mathbf{A}^{\hbox{\tiny{nc}}}_{y}=(r^{\prime}-r)Bx+\frac{e\vartheta
B^{2}x}{\hbar}[r^{\prime
2}(r^{\prime}-1)-r^{2}(r-1)]+\mathcal{O}(\vartheta^{2}).$ (6.2)

One, now takes the infinitesimal variation
$\delta\mathbf{A}^{\hbox{\tiny{nc}}}_{x}$ of the $x$-component of the
noncommutative gauge field from (6.1) and expand it in
$\epsilon=r^{\prime}-r$:

$\displaystyle\delta\mathbf{A}^{\hbox{\tiny{nc}}}_{x}$
$\displaystyle=By\epsilon+\frac{B^{2}e\vartheta
y}{\hbar}[(r+\epsilon-1)^{2}(r+\epsilon)-(r-1)^{2}r]+\mathcal{O}(\vartheta^{2}),$
(6.3) $\displaystyle=By\epsilon+\frac{B^{2}e\vartheta
y}{\hbar}[r(r-1)^{2}+r\epsilon^{2}+2r(r-1)\epsilon+\epsilon(r-1)^{2}+\epsilon^{3}+2(r-1)\epsilon^{2}-(r-1)^{2}r]+\mathcal{O}(\vartheta^{2}),$
$\displaystyle=By\epsilon+\frac{B^{2}e\vartheta
y}{\hbar}(r\epsilon^{2}+2r^{2}\epsilon-2r\epsilon+\epsilon
r^{2}-2r\epsilon+\epsilon+\epsilon^{3}+2r\epsilon^{2}-2\epsilon^{2})+\mathcal{O}(\vartheta^{2}),$
$\displaystyle=By\epsilon+\frac{B^{2}ye\vartheta\epsilon}{\hbar}(3r^{2}-4r+1)+\mathcal{O}(\epsilon^{2})+\mathcal{O}(\vartheta^{2}).$

Since $\epsilon=r^{\prime}-r$ is an infinitesimal, $\mathcal{O}(\epsilon^{2})$
can be neglected and (6.3) reduces to

$\delta\mathbf{A}^{\hbox{\tiny{nc}}}_{x}=By\epsilon+\frac{B^{2}ye\vartheta\epsilon}{\hbar}(3r^{2}-4r+1)+\mathcal{O}(\vartheta^{2}).$
(6.4)

Similarly, for the y-component of the noncommutative gauge field, (6.2) can be
expanded in $\epsilon=r^{\prime}-r$. Here, again one retains only the terms
linear in the infinitesimal $\epsilon$ to obtain,

$\delta\mathbf{A}^{\hbox{\tiny{nc}}}_{y}=Bx\epsilon+\frac{B^{2}xe\vartheta\epsilon}{\hbar}(3r^{2}-2r)+\mathcal{O}(\vartheta^{2}).$
(6.5)

But according to (4.5), the infinitesimal variation of the $x$ component of
the noncommutative gauge field is given by

$\delta\mathbf{A}^{\hbox{\tiny{nc}}}_{x}=\partial_{x}\lambda^{\hbox{\tiny{nc}}}+\frac{ie}{\hbar}[\lambda^{\hbox{\tiny{nc}}}\\!\overset{\;\;*^{r}}{,}\mathbf{A}^{\hbox{\tiny{nc}}}_{x}].$
(6.6)

Now, write

$\lambda^{\hbox{\tiny{nc}}}=\widetilde{B}\epsilon xy,$ (6.7)

and substitute in (6.6). Use (3.3) and the definition of the 1-parameter
family of star-products (3.7) together with (6.7) in (6.6) to obtain

$\displaystyle\delta\mathbf{A}^{\hbox{\tiny{nc}}}_{x}$
$\displaystyle=\widetilde{B}\epsilon
y+\frac{ie}{\hbar}\left[\widetilde{B}\epsilon
xy\\!\overset{\;\;*^{r}}{,}\frac{2(r-1)\hbar
By}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}}\right],$ (6.8)
$\displaystyle=\widetilde{B}\epsilon xy+\frac{ie\vartheta}{\hbar}\times
i\epsilon(r-1)\widetilde{B}\times\frac{2\hbar
By}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta B}},$
$\displaystyle=\widetilde{B}\left[1+\frac{2e(1-r)B\vartheta}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\right]\epsilon y.$

Now comparing (6.8) with (6.4), one obtains,

$\displaystyle\widetilde{B}\left[1+\frac{2e(1-r)B\vartheta}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\right]=B+\frac{eB^{2}\vartheta}{\hbar}(r-1)(3r-1)+\mathcal{O}(\vartheta^{2}).$
(6.9)

Rearrangement of the terms in (6.9) leads to

$\displaystyle\widetilde{B}$
$\displaystyle=\left[B+\frac{eB^{2}\vartheta}{\hbar}(r-1)(3r-1)\right]\left[1+\frac{2e(1-r)B\vartheta}{\hbar+\sqrt{\hbar^{2}-4r(r-1)e\hbar\vartheta
B}}\right]^{-1},$ (6.10)
$\displaystyle=\left[B+\frac{eB^{2}\vartheta}{\hbar}(r-1)(3r-1)\right]\left[1+\frac{2e(1-r)B\vartheta}{\hbar}\left\\{1+\sqrt{1-\frac{4r(r-1)e\vartheta
B}{\hbar}}\right\\}^{-1}\right]^{-1},$
$\displaystyle=\left[B+\frac{eB^{2}\vartheta}{\hbar}(r-1)(3r-1)\right]\left[1+\frac{2e(1-r)B\vartheta}{\hbar}\left\\{\frac{1}{2}+\frac{4er(r-1)\vartheta
B}{8\hbar}+...\right\\}\right]^{-1},$
$\displaystyle=B-e(1-r)\frac{B^{2}\vartheta}{\hbar}+\frac{eB^{2}\vartheta}{\hbar}(r-1)(3r-1)+\mathcal{O}(\vartheta^{2}),$
$\displaystyle=B+3er(r-1)\frac{B^{2}\vartheta}{\hbar}+\mathcal{O}(\vartheta^{2}),$

so that one finally obtains the following expression for the noncommutative
gauge function $\lambda^{\hbox{\tiny{nc}}}$:

$\lambda^{\hbox{\tiny{nc}}}=B\epsilon xy+3e\epsilon
r(r-1)\vartheta\frac{B^{2}xy}{\hbar}+\mathcal{O}(\vartheta^{2}).$ (6.11)

We leave it to the reader to verify that the above expression (6.11) of
noncommutative gauge function indeed satisfies the defining equation of the
infinitesimal variation of the y-component of the noncommutative gauge field
(see 4.5):

$\delta\mathbf{A}^{\hbox{\tiny{nc}}}_{y}=\partial_{y}\lambda^{\hbox{\tiny{nc}}}+\frac{ie}{\hbar}[\lambda^{\hbox{\tiny{nc}}}\\!\overset{\;\;*^{r}}{,}\mathbf{A}^{\hbox{\tiny{nc}}}_{y}],$
(6.12)

with $\mathbf{A}^{\hbox{\tiny{nc}}}_{y}$ is as in (3.3) and
$\delta\mathbf{A}^{\hbox{\tiny{nc}}}_{y}$ is as in (6.5).

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