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\begin{document}

%\title{ On the Schwinger realization of polynomial $su(2)$ algebra through the  $ {\cal A}_{
%\kappa } (1) $ Weyl-Heisenberg algebra }
%\author{ Won Sang Chung }
%\email{mimip4444@hanmail.net}
%\affiliation{Department of Physics and Research Institute of Natural Science, College of
%Natural Science, Gyeongsang National University, Jinju 660-701, Korea }
%\author{ Mohammed Daoud}
%\email{m_daoud@hotmail.com} \affiliation{Facult\'e des Sciences,
%D\'epartement de Physique, Agadir, Morocco\\
%and\\
%Abdus Salam International Centre for Theoretical Physics, Trieste,
%Italy}
%\date{\today}

%\begin{abstract}
%In this paper the Schwinger realization of polynomial algebra is constructed
%by using the non-linear deformation of Heisenberg algebra called $ A_{ \{
%\kappa \} } (1) $ algebra which is first proposed by by Daoud and Kibler
%[M.Daoud and M.Kibler , J.Phys.A\textbf{43} 115303 (2010) , \textbf{45}
%244036 (2012) ]
%\end{abstract}



\begin{abstract}


This paper deals with a Schwinger realization of polynomial $su(2)$ algebras. It
involves generalized  Weyl-Heisenberg algebras  $ A_{\kappa} (1)$ ($\kappa$ characterizes  the deviation from the usual
boson algebra). Hilberitian   as well as analytical representations are defined. To show the interest for
such algebras, we investigate the dynamical symmetry of the Kepler system in a
two-dimensional curved space and we discuss their relevance in the  super-symmetric
quantum mechanics.




%We discuss the Schwinger realization of $su(2)$ algebra by means of
%non linear Weyl-Heisenberg algebra  $ A_{\kappa} (1) $ where the
%real parameter $\kappa$ characterizes  the deviation from the usual
%boson algebra. We give the Hilbert representations of the obtained
% polynomial $su(2)$ algebra which admits two types of discrete
%representations according to the sign of the parameter $\kappa$. We
%also construct the corresponding coherent states which provides a
%Bargmann realization of the obtained polynomial $su(2)$ algebra. As
%illustrations, the dynamical symmetry of the Kepler system in a
%two-dimensional curved space is investigated and the relevance of
%non linear angular momentum in the context of super-symmetric
%quantum mechanics is discussed.

\end{abstract}









\maketitle

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%[-3mm] \hbox to\hsize{\hfill November 2011}\\
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\section{Introduction}
\noindent The generalized $su(2)$ algebras developed in the context of the theory of quantum algebras have
attracted a lot of interest in the literature. This  is especially by motivated their relevance in
different areas of physics as for instance exactly solvable quantum
potentials, statistical physics, Yang-Mills theory, field theory, two
dimensional integrable models, quantum optics, etc. [1-17]. Many
variants of generalized algebras were  investigated for different purposes. The generalized $su(2)$ algebras can be
defined in a unified scheme.  Indeed, n terms of  the raising and lowering generators
$J_{\pm}$ and the diagonal one $J_3$, the very well-known
linear $su(2)$ algebra is characterized by the structure relations
\begin{equation}\label{su2-gen}
[J_3 , J_+ ] = J_+ , ~~ [ J_3 , J_- ] = - J_-, ~~ [J_+ , J_- ] = f(
J_3 )
\end{equation}
where the  structure function characterizes the deviation from the linear $su(2)$ algebra. Clearly, many
different generalization schemes can be defined. of particular interest are polynomial $su(2)$ alegebras
in which $f(J_3 ) $ is a polynomial in $J_3$. For such  algebras, the Casimir operator
is given by
\begin{equation}
C = J_+ J_- + g( J_3 -1 ) = J_- J_+ + g ( J_3 )
\end{equation}
with the function $g(J_3)$ satisfies
\begin{equation}
g(J_3 ) - g( J_3 -1 ) = f( J_3 )
\end{equation}
Unitary irreducible representations were  for some specific forms of the polynomial structure
function $f( J_3 )$ (see for instance [14]). Coherent states associated with specific polynomial $su(2)$
algebras were defined [23-26]. Also bosonic realizations of such algebras
by means of generalized bosons were discussed in the literature [23-26]. They can classified  into two main categories. One, essentially
based on Jackson's $q$- calculus, uses the concept of $q$-deformed boson algebra $(q \in \mathbb{C})$ firstly proposed by Arik and Coon [28] and developed
further by Macfarlane [29] and Biedenharn [30] to provide bosonic and fermionic realizations of quantum algebras. The second category involves
new variants of generalized Weyl-Heisenberg
algebras [31-33]. The main characteristics of the
these algebras is the fact that are not related to ones defined in
the framework of quantum algebras and the $q$-calculus. Among them,
we shall  consider the single boson extended
Weyl-Heisenberg algebra $ {\cal A}_{\kappa}(1) $  [31]. It is generated by the operators $a^+$, $a^-$ and  $N$
satisfying the following commutation relations
\begin{equation}\label{Akappa1}
[ a^- , a^+ ] = 1 + 2 \kappa N, ~~~[N , \ad ] = \ad , ~~~ [N, a ] =-a,
\end{equation}
labeled by a real parameter
$\kappa$.  The usual boson algebra is recovered when $\kappa$ goes to zero.
In this context, the main of this work is as follows.
We shall introduce a one parameter family of nonlinear $su(2)$ algebras of type (\ref{su2-gen}). This is  realized \`a la
Schwinger in terms of two  commuting copies of generalized $ {\cal A}_{\kappa}(1) $  algebras (\ref{Akappa1}).
The representation spaces are defined using the Fock representation of $ {\cal A}_{\kappa}(1) $  algebras derived
in [31].  To provide physical illustration of such polynomial generalized bosonic realizations, we  give in the second section
the general procedure to derive the Casimir operator polynomial $su(2)$ algebras of an arbitrary order.
Section III is devoted to the Schwinger realization of polynomial $su(2)$ algebras and the associated discrete
representation spaces.  In section IV,  Fock-Bragmann  representations and  coherent states are constructed.
To exemplify the relevance of this family of nonlinear algebras,  some  physical models are discussed in Section V. Closing remarks
close this paper.




\section{ Two ${\cal A}_{\kappa}$-boson realization of $su(2)$ algebra}

The two generalized boson construction discussed in this section is a straightforward
generalization of the usual Schwinger
realization.
We consider two commuting copies of the generalized Weyl-Heisenberg algebra ${\cal A}_{\kappa}(1)$ (\ref{Akappa1}). Each one is  spanned by the three linear operators $
a_i^-$, $a_i^+$ and $N_i$ $(i = 1,2)$ satisfying the following relations
\begin{eqnarray}
[a_i^- , a_i^+] = I + 2 \kappa N_i \qquad [N_i , a_i^{\pm}] = \pm a_i^{\pm}
\qquad \left( a_i^- \right)^{\dagger} = a_i^+ \qquad N_i^{\dagger} = N_i.
\label{thealgebra}
\end{eqnarray}
 We denote the Fock space finite or infinite dimensional
\begin{eqnarray}
\mathcal{F}_{\kappa} = \mathcal{F}_{1} \otimes \mathcal{F}_{2} = \{ \vert
n_1 , n_2 \rangle, ~ n_1, n_2 ~~\mathrm{ranging} \}.
\end{eqnarray}
The actions of creation, annihilation and number operators on $\mathcal{F} $
are defined by
\begin{eqnarray}
& & a_i^+ \vert n_1 , n_2 \rangle = \sqrt{F(n_i+1)} ~ \vert n_1 + s^-_i ,
n_2 + s^+_i \rangle,  \nonumber \\
& & a_i^- \vert n_1 , n_2 \rangle = \sqrt{F(n_i)} ~ \vert n_1 - s^-_i , n_2
- s^+_i \rangle,  \label{action  n} \\
& & a_1^- \vert 0, n_2 \rangle = 0 \quad a_2^- \vert n_1, 0 \rangle = 0
\quad N_i \vert n_i \rangle = n_i ~\vert n_i \rangle  \nonumber
\end{eqnarray}
where the quantity $s^{\pm}_i$ is defined by
\[
s^{\pm}_i = \frac{1}{2} ( 1 \pm (-)^i).
\]
In (\label{action  n}), the positive function  is given by [31]
\begin{eqnarray}\label{F(n)}
F(n_i) = n_i [1 + \kappa (n_i - 1)] \qquad i = 1, 2.
\end{eqnarray}
It is quadratic in $n_i$ except the special case $\kappa = 0$ where it
reduces to a linear function. The Hilbertian representation of $\mathcal{A}_{\kappa}(1)$ was
investigated in [31]. Indeed, it has been shown that for $\kappa \geq 0$, the Fock
space $\mathcal{F}$ is infinite dimensional. It follows that for two generalized bosons of type (\ref{Akappa1}), the Fock state is
\[
\mathcal{F}_{\kappa \geq 0} \equiv \mathcal{F}_{+} = \{ \vert n_1 , n_2
\rangle, ~ n_1 \in \mathbf{N},~ n_2 \in \mathbf{N} \}.
\]
For $\kappa < 0$, the algebra $\mathcal{A}_{\kappa}(1)$ admits finite dimensional Hilbert space [31].  Indeed, it is simply verified
from (\ref{F(n)}) that for $\kappa < 0$, the function $F(n_i)$ is positive for the integers $n_i \leq d-1$  where $d$ stands for the integer part of $\frac{-1}{\kappa}$.
Thus, in this case, the Fock space  is finite dimensional
\[
\mathcal{F}_{\kappa < 0} \equiv \mathcal{F}_{-} = \{ \vert n_1 , n_2
\rangle, ~ n_1= 0, 1, \cdots, d-1; ~~n_2 =0, 1, \cdots, d-1 \},
\]
and the dimension is $d^2$.

As we are interested in nonlinear extension of $su(2)$ using a Schwinger
procedure based on the generalized Weyl-Heisenberg algebra $\mathcal{A}%
_{\kappa}(1)$, we decompose the Fock spaces in angular momentum
subspaces. In this respect, we introduce the quantum numbers $j$ and
$m$ defined as
\[
j = \frac{1}{2} (n_1 + n_2) \qquad m= \frac{1}{2} (n_1 - n_2)
\]
and we set the following correspondence
\[
\vert n_1 , n_2 \rangle \equiv \vert \frac{n_1 + n_2}{2} , \frac{n_1 - n_2}{2%
} \rangle = \vert j , m\rangle
\]
It follows that one can write the Fock spaces $\mathcal{F}_{+} $ and $%
\mathcal{F}_{-} $ as
\[
\mathcal{F}_{+} = \bigoplus_{j=0}^{\infty}~~ \mathcal{F}_{j} =
\bigoplus_{j=0}^{\infty}~~ \{ \vert j , m \rangle , ~m = -j , -j + 1, \cdots
, j-1, +j\}
\]
and
\[
\mathcal{F}_{-} = \bigoplus_{j=0}^{d}~~ \mathcal{F}_{j} =
\bigoplus_{j=0}^{d-1} ~~\{ \vert j , m \rangle ,~ m = -j , -j + 1, \cdots ,
j-1, +j\}
\]
Clearly, for $\kappa < 0$, the Schwinger realization produces only
angular momentum with $j = 0,{\frac{ 1}{2}}, 1, \cdots d-1$ such
that $\mathrm{dim} \mathcal{F}_{-} = d^2 $. This constitutes the
main difference with the case where $\kappa$ is positive including
the case $\kappa = 0$ which gives the standard Schwinger realization
of $su(2)$ algebra where all possible values of $j$ can be
generated. In the Schwinger picture, the $su(2)$ operators are
realized as
\[
J_3 = \frac{1}{2} ( N_2 - N_1 ) , ~~~ J_0 = \frac{1}{2} ( N_2 + N_1 )
\]
\begin{equation}
J_+ =a_2^{\dagger} a_1 , ~~~ J_- = a_2 a_1^{\dagger},
\end{equation}
This gives
\begin{equation}
[ J_+ , J_- ] = 2J_3 \bigg( 1 - \kappa + 2\kappa J_0 (1 + \kappa J_0)\bigg) %
+ 4\kappa^2 J_3^2.
\end{equation}
and the Casimir operator
\begin{equation}
C = 2 J_- J_+ + 2J_3 ( J_3 + 1 ) \bigg( 1 - \kappa + 2\kappa J_0 (1 + \kappa
J_0)\bigg) + 2\kappa^2 J_3^2 ( J_3 +1 )^2
\end{equation}
To simplify the notations in deriving the coherent states in the
following subsection a suitable basis for the subspaces
$\mathcal{F}_{j} $ is introduced as follows. We set
\[
\vert j , m \rangle = \vert n \rangle \quad \mathrm{with} \quad n = j+m .
\]
In this basis, the actions of the generalized $su(2)$ algebra
rewrites
\[
J_- \vert n \rangle = \sqrt{f(n)} \vert n-1 \rangle
\]
\[
J_- \vert n \rangle = \sqrt{f(n+1)} \vert n+1 \rangle
\]
\[
J_3 \vert n \rangle = (n-j)\vert n+1 \rangle
\]
where
\[
f(n) = F(n)F(2j-n+1)
\]
in terms of the structure function of the generalized
Weyl-Heisenberg algebra $\mathcal{A}_{\kappa}(1)$ given by
(\ref{F(n)}). Notice that the function $f(N)$ is of order 4 in $N$
and the obtained non linear $su(2)$ algebra can be idenified with
polynomial Weyl-Heisenberg algebra of order 4 introduced in [32].

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Bargmann realization and Coherent states}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

According to Bargmann prescription, the coherent states provides the
analytical realizations of a given algebra. For the nonlinear algebra $su(2)$
discussed above, for a representation $j$, the coherent states are of the
form
\begin{equation}
\vert z \rangle = \sum_{n = 0}^{2j} a_n z^n \vert n \rangle
\end{equation}
where $z$ is a complex variable, $n = j+m$ and the $a_n$ coefficients to be
determined. In the Bargmann realization any vector is realized as follows
\begin{equation}
\vert n \rangle \longrightarrow a_{n} z^{n} \equiv \langle \bar z\vert n
\rangle
\end{equation}
and the operator $J^-$ is assumed acts as a derivation according to
\begin{equation}
J_-\longrightarrow \frac{d}{dz}.
\end{equation}
Thus, using the expression of the action of $J_-$ on the Fock space and the
Bargamnn correspondence, it is simple to see that the coefficients $a_n$
satisfy the recursion relation
\begin{equation}
n a_{n}= \sqrt{f(n)}a_{n-1}
\end{equation}
where
\[
f(n) = F(2j-n+1) F(n).
\]
This yields
\begin{equation}
a_{n}= a_0 \frac{\sqrt{f(n)!}}{n!}
\end{equation}
where $f(n)!= f(1) f(2) \cdots f(n)$ with $f(0)! = 1$. The coefficient $a_o$
is obtained from the normalization condition of the coherent states . As
result, one gets
\begin{equation}
\vert z \rangle = \mathcal{N}^{-1} \sum_{n=0}^{2j} \frac{\sqrt{f(n)!}}{n!}
z^n \vert n \rangle
\end{equation}
where the $\mathcal{N}$ normalization factor is given by the sum
\begin{equation}
|\mathcal{N}|^2 = \sum_{n = 0}^{2j} \frac{f(n)!}{(n!)^2} \vert z \vert^{2n}.
\end{equation}
%The explicit expression of the factorial $f(n)!$ is
%$$ f(n)! = $$
In this realization, the operators $J_3$ and $J_+$ act,
respectively, as follows
\begin{equation}
N \longrightarrow {z} \frac{d}{d{z}} - j \qquad J_+\longrightarrow {z} %
\bigg( 1 + \kappa{z} \frac{d}{d{z}}\bigg) \bigg( 2j - {z} \frac{d}{d{z}}%
\bigg) \bigg( 1 + 2j\kappa - \kappa {z} \frac{d}{d{z}}\bigg).
\end{equation}

Any state $\vert \Psi \rangle$
\[
\vert \Psi \rangle = \sum_{n=0}^{2j} \Psi_{n}\vert n \rangle
\]
corresponds to an analytical function given by the following correspondence
\begin{equation}  \label{psi(z)}
\vert \Psi \rangle \longrightarrow \Psi(z) = \mathcal{N} \langle \bar z
\vert \Psi\rangle = \sum_{n=0}^{2j} \Psi_{n} f_{n}(z)
\end{equation}
where the monomials $f_{n}(z)$ are the analytical functions associated with
the vectors $n$
\begin{equation}
f_{n} (z) = \mathcal{N} \langle \bar z \vert 2n \rangle = \frac{\sqrt{f(n)!}
}{n!} z ^{n}.
\end{equation}
It easy to check
\[
J_3 f_{n}(z) = ( j- n) f_{n}(z)
\]
\[
J_+ f_{n}(z) = \sqrt{f(n+1)} f_{n+1}(z)
\]
\[
J_- f_{n}(z) = \sqrt{f(n)} f_{n-1}(z)
\]

It is simple to verify that, in the limiting case $\kappa
\rightarrow 0$, one recovers the $su(2)$ coherent states and the
standard Bargamnn realization based on spin coherent states.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The ${\cal A}_{\{\kappa\}}$ algebra}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We start with the generalized Weyl-Heisenberg algebra on $\mathbb{C}$ spanned by
the linear operators $a^-$ (annihilation operator), $a^+$ (creation operator)
and $N$ (number operator) satisfying the commutation relations
	\begin{eqnarray}
	[a^- , a^+] =  G(N) \qquad
	[N, a^-] = - a^- \qquad
	[N, a^+] = + a^+ 	
	\label{algebre}
	\end{eqnarray}
and the Hermitian conjugation conditions
	\begin{eqnarray}
	a^+ = (a^-)^{\dagger} \qquad N = N^{\dagger}.
	\label{hermiticity conditions}
	\end{eqnarray}
The $G$ function in (\ref{algebre}) is such that
	\begin{eqnarray}
	G(N) = \big(G(N)\big)^{\dagger}.
	\label{hermiticity of G}
	\end{eqnarray}
Of course, the case $G(N) = I$, where $I$ is the identity operator, corresponds
to the usual Weyl-Heisenberg algebra or harmonic oscillator algebra. Various realizations
of $G$ are known in the literature
\cite{Carballo, newdaokib, daoud-kibler1, daoud-kibler2,
Daskaloyannis, Fernandez, Gavrilik, Katriel, Man'ko, Quesne}. In
the present paper, we shall be concerned with a class of polynomial Weyl-Heisenberg algebras
characterized by
	\begin{eqnarray}
	G(N) = F(N+I) - F(N)
	\label{G as difference of two F}
	\end{eqnarray}
with the $F$ function defined by
	\begin{eqnarray}
	F(N) = N [ I + \kappa_1(N-I)]
	         [ I + \kappa_2(N-I)] \cdots
	         [ I + \kappa_r(N-I)]
	\label{generalformofF}
	\end{eqnarray}
where the $\kappa_i$'s ($i = 1, 2, \cdots , r$) are real parameters (for instance, see \cite{Gavrilik}). We note
${\cal A}_{\{\kappa\}}$, with $\{\kappa\} \equiv \{\kappa_1, \kappa_2, \cdots, \kappa_r\}$, the
generalized Weyl-Heisenberg algebra (or generalized oscillator algebra) defined via
(\ref{algebre})-(\ref{generalformofF}).

The $F(N)$ polynomial of order $r+1$ with respect to $N$ can be developed as
	\begin{eqnarray}
	F(N) = N \sum_{i=0}^{r} s_i (N-I)^i
	\label{development of F(N)}
	\end{eqnarray}
in terms of the coefficients (totally symmetric under permutation group $S_r$)
	\begin{eqnarray}
	s_0 = 1 \qquad s_i = \sum_{j_1 < j_2 < \cdots < j_i}  \kappa_{j_1} \kappa_{j_2} \cdots \kappa_{j_i} \quad
	(i = 1, 2, \cdots , r)
	\label{vieta formula}
	\end{eqnarray}
where the indices $j_1, j_2, \cdots , j_i$ take the values $1, 2, \cdots , r$. Then, the $G(N)$ operator can be written
	\begin{eqnarray}
	G(N) = I + \sum_{i = 1}^r s_i \bigg[ (N+I)N^i - N(N-I)^i \bigg]
	\label{development of G(N)}
	\end{eqnarray}
which clearly indicates that ${\cal A}_{\{\kappa\}}$ with $\{\kappa\} \equiv \{0, 0, \cdots, 0\}$	coincides with
the usual Weyl-Heisenberg algebra.

The ${\cal A}_{\{\kappa\}}$ $r$-parameter algebra covers the cases of (i) the extended
harmonic oscillator algebra \cite{Quesne}, (ii) the fractional oscillator
algebra \cite{newdaokib}, and (iii) the $W_k$ algebra introduced in the
context of fractional supersymmetric quantum mechanics of order $k$
\cite{daoud-kibler1, Kibler}. As a particular case, algebra ${\cal A}_{\{\kappa\}}$
with $\kappa_1 = \kappa$ and $r = 1$ is nothing but the ${\cal A}_{\kappa}$
algebra worked out in \cite{daoud-kibler2} and corresponding to
	\begin{eqnarray}
	G(N) = I + 2\kappa N.
	\label{G(N) de Akappa}
	\end{eqnarray}
Algebra ${\cal A}_{\kappa}$ defined by (\ref{algebre}), (\ref{hermiticity conditions}) and (\ref{G(N) de Akappa}) turns
out to be of particular interest when dealing with dynamical symmetries of some exactly solvable quantum systems. More
precisely, ${\cal A}_{\kappa = 0}$ corresponds to the usual oscillator system while ${\cal A}_{\kappa < 0}$ and
${\cal A}_{\kappa > 0}$ are relevant to the Morse and P\"oschl-Teller systems, respectively
\cite{daoud-kibler1, daoud-kibler2}. Note also that the ${\cal A}_{\kappa}$ one-parameter algebra provides a unified
scheme to deal with the $su(2)$ algebra (for $\kappa < 0$), the $su(1,1)$ algebra (for $\kappa > 0$), and the usual
Weyl-Heisenberg algebra (for $\kappa = 0$) \cite{Atakishiyev, daoud-kibler2}. More generally, the ${\cal A}_{ \{ \kappa \} }$
algebra can be viewed as a special class of the polynomial extensions of $su(2)$ and $su(1,1)$ discussed in \cite{Bonatsos}
and \cite{Shreecharan}, respectively.











%\section*{Conclusion}

%In this paper we discussed the structure and representation of the $q=-1$ oscillator algebra.
%This algebra was shown to be different from the ordinary fermion algebra.
%We investigated the basic thermodynamic property by using the $q=-1$ oscillator algebra.
%We used the $q=-1$ oscillator algebra to realize the $\s$ algebra and obtained the eigenvector
%of the position operator represented in terms of the generators of the $\s$ algebra.
%Finally, we used $sl_q(2)$ algebra to discuss the rotational spectra for molecule
%and nuclei. By taking the limit $q \rightarrow -1 $, we investigated the rotational spectra related to the $\s$ algebra which depends on the value of $j$.

\section{Concluding remarks}

%In this paper we discussed the unitary representation of Higgs algebra.
%We introduced the linear ( or quadratic ) deformation of Heisenberg algebra
%to construct the single mode  Holstein-Primakoff realization of Higgs algebra.
%We also proposed the q-extension of Higgs algebra and its Holstein-Primakoff realization by
%using a new kind of q-deformed oscillator algebra. Finally, we discussed the relations between our
% work and some physical models such as Kepler problem in a two-dimensional curved space and SUSY quantum mechanics.

%There exists several types of realizations such as Dyson realization and
%Villain realization. We think that our method will be applied to construction
%of several types of realizations. We think that this work and its related topics will become clear in the near future.



In this paper, we discussed the realization of polynomial $su(2)$
algebra by generalizing the standard Schwinger realization. In this
generalized realization, we used two generalized boson characterized
by the  extended Weyl-Heisenberg algebra ${\cal A}_{\kappa}(1)$
[31]. Other kinds of extended algebras were recently introduced in
the literature and in this respect, we believe that the procedure
developed in this work can be easily adapted to generate other
variants of polynomial $su(2)$ algebras. In the same spirit, the
Bargmann realization discussed here may be naturally applied to
construct the associated coherent states.
















%%%%%%%%%%%%%%%%%% References
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{References}

[1] M. Lakshmanan and K. Eswaran, J. Phys. A: Math. Gen. \textbf{
8}, 1658 (1975).

[2] P. Higgs, J. Phys. A: Math. Gen. \textbf{12}, 309 (1979).

[3] E. Sklyanin, Funct. Anal. Appl. \textbf{16}, 263 ( 1982).

[4] T. Curtright and Z. Zachos, Phys. Lett. B \textbf{ 243}, 237 (
1990).

[5] A. Polychronakos, Mod. Phys. Lett. A \textbf{ 5 }, 2325 ( 1990).

[6] M . Rocek, Phys. Lett. B \textbf{ 255}, 554 ( 1991).

[7] O. Gal'bert, Ya. I. Granovskii and A. Zhedanov, Phys. Lett. A
 \textbf{ 153}, 177 (1991).

[8] Ya. I. Granovskii , A. Zhedanov and I. Lutzenko, J. Phys. A:
Math. Gen.
 \textbf{ 24}, 3887 (1991).

[9] K. Schoutens, A. Servin and P. van Nieuwenhuizen , Phys. Lett. B  \textbf{ 255%
}, 549 (1991).

[10] A. Zhedanov, Mod. Phys. Lett. A \textbf{7}, 507 (1992).

[11] D. Bonatsos, C. Daskaloyannis and K. Kokkotas, Phys. Rev. A
\textbf{48}, 3407 ( 1993).

[12] C. Quesne, Phys. Lett. A \textbf{193}, 245 (1994).

[13] P. Letourneau and L. Vinet, Ann. Phys. \textbf{243}, 144
(1995).

[14] B. Abdesselam, J. Beckers, A. Chakrabarti and N. Debergh, J.
Phys. A: Math. Gen. \textbf{ 29}, 3075 (1996).

[15] J. de Boer, F. Harmsze and T. Tijn, Phys. Rep. \textbf{272},
139 ( 1996).

[16] D. Fernandez and V.Hussin , J. Phys. A: Math. Gen. \textbf{
32}, 3603 ( 1999).

[17] V. Karassiov, J. Sov. Laser Res. \textbf{13}, 188 ( 1992).

[18] V. Man'ko, G. Marmo, E. Sudarshan and F. Zaccaria, Phys. Scr.
\textbf{55}, 528 (1997).

[19] B. Roy and P. Roy, Quantum Semiclass. Opt. \textbf{1}, 341 (
1999).

[20] C. Quesne, Phys. Lett. A \textbf{272}, 313 (2000).

[21] J. Van der Jeugt and R. Jagannathan, J. Math. Phys.
\textbf{36}, 4507 (1995).

[22] G. Junker and P. Roy, Phys. Lett. A \textbf{257}, 113 (1999).

[23] V. Sunilkumar, B. Bambah, R. Jagannathan, P. Panigrah and V.
Srinivasan, J. Opt. B {\bf 2}, 126 (2000).

[24] V. Sunilkumar, B. Bambah, R. Jagannathan,  \textit{Three
dimensional quadratic algebras: Some realizations and
representations}, arXiv:math-ph/0108027 (2001).

[25] V. Sunilkumar, B. Bambah, R. Jagannathan,  \textit{Quadratic
algebras: Three mode bosonic realizations and applications},
arXiv:quant-ph/0010017 (2000).

[26] V. Sunilkumar,  \textit{Aspects of Polynomial algebras and their
physical applications}, Ph.D thesis at University of Hyderabad, India,
arXiv:math-ph/0203047 (2002).

[27] F. Jackson, Mess. Math. \textbf{38 }, 57 (1909).

[28] M. Arik and D. Coon, J. Math. Phys. {\bf 17}, 524 (1976).

[29] A. J. Macfarlane, . Phys. A: Math. Gen. {\bf  22}, 4581 (1989).

[30] L. Biedenharn, J. Phys. A: Math. Gen. {\bf 22}, L873 (1990).

[31] M. Daoud and M. Kibler , J. Phys. A: Math. Theor. {\bf 43},
115303 (2010) .

[32] M. Daoud and M. Kibler, J. Phys. A: Math. Theor. {\bf 45},
244036 (2012) .

[33] M. Daoud and M. Kibler , J. Math. Phys. {\bf 52}, 081201 (2011)
.

[34] P. Abraham and H. Moses, Phys. Rev. A \textbf{22}, 1333 (1980).

[35] B. Mielnik , J. Math. Phys.  \textbf{25}, 3387 (1984).

[36] M. Nieto, Phys. Lett. B \textbf{145}, 208 (1984).

[37] M. Luban and D. Pursey, Phys. Rev. D  \textbf{ 33}, 431 (1986).

[38] P. Roy and R. Toychoudhury, Z. Phys. C \textbf{ 31}, 111 (
1986).

[39] G. Darboux , Comptes Rendus Acad. Sci. (Paris)  \textbf{94 } ,
1456 (1982).

\end{document}







\section{  Polynomial $su(2)$ algebra and Casimir operator}

To begin, we shall describe the procedure to derive  provide the
general algorithm to derive the Casimir operator for a polynomial $su(2)$ algebra. In particular, we consider the sitiation
where the structure function in the commutations relations (\ref{su2-gen}) is a polynom of order $r$ in the generator  $J_3$
\begin{equation}
f( J_3 ) = \sum_{n=0}^r C_n J_3^n
\end{equation}
%The polynomial
%algebra of degree $r$ is defined by the commutation relation
%\[
%[J_3 , J_+ ] = J_+ , ~~ [ J_3 , J_- ] = - J_-
%\]
%\begin{equation}
%[J_+ , J_- ] = \sum_{n=0}^r C_n J_3^n ,
%\end{equation}
where $C_n $'s are structure constants. Clearly, this algebra covers
Weyl-Heisenberg algebra, $su (2) $ algebra, quadratic algebra and
cubic algebra for $ r =0, 1, 2, 3 $, respectively. Now let us
compute the Casimir operator for the algebra (6). First let us set
\begin{equation}
x^n = \sum_{k=0}^n \beta^n_k (x)_k ,
\end{equation}
where
\begin{equation}
(x)_k = x ( x +1) (x+1) \cdots ( x + k-1) , ~~~ (x)_0 =1
\end{equation}
The recurrence relation is given by
\begin{equation}
\beta_1^{n+1} = - \beta_1^n
\end{equation}
\begin{equation}
\beta_k^{n+1} = \beta_{k-1}^n - k \beta_k^n , ~~~ ( k=2, 3, \cdots, n )
\end{equation}
\begin{equation}
\beta_{n+1}^{n+1} = \beta_{n}^{n}
\end{equation}
Solving the eq.(9) and eq.(11) yields
\begin{equation}
\beta_1^n = (-1)^{n+1} , ~~~~\beta_n^n = 1
\end{equation}
The first few values of $\beta_k^n $'s become
\[
\beta_{0}^{0} =1
\]
\[
\beta^{1}_{0} =0, ~~ \beta_{1}^{1} =1
\]
\[
\beta^{2}_{0} =0, ~~ \beta^{2}_{1} =-1, ~~~\beta_{2}^{2} =1
\]
\[
\beta^{3}_{0} =0, ~~ \beta^{3}_{1} =1, ~~~\beta^{3}_{2} =-3,
~~~\beta_{3}^{3} =1
\]
Let us assume that Casimir operator takes the following form
\begin{equation}
C = \sum_{n=0}^r C_n f_n (J_3 ),
\end{equation}
Then, we have
\begin{equation}
f_n ( J_3) - f_n ( J_3 -1 ) = J_3^n
\end{equation}
Using the formula
\begin{equation}
(x)_n - ( x-1)_n = n ( x)_{n-1},
\end{equation}
we get
\begin{equation}
C = \sum_{n=0}^r C_n \sum_{k=0}^n \alpha_k^n (J_3 )_{n+1-k} ,
\end{equation}
where
\begin{equation}
\alpha_{n-k}^n = \frac{ \beta_k^n }{k+1}
\end{equation}
The few functions $f_n (x) $'s become
\[
f_0 (x) = (x)_1
\]
\[
f_1 (x) = \frac{1}{2} (x)_2
\]
\[
f_2 (x) = \frac{1}{3} [ (x)_3 - \frac{3}{2} (x)_2 ]
\]
\[
f_3 (x) = \frac{1}{4} [ (x)_4 - 4 (x)_3 + 2 (x)_2 ]
\]
The general method, developed in this section, to derive the Casimir
operator for a polynomial algebra $su(2)$, of an arbitrary order
$r$, is useful in determining the Casimir operator for the algebra
obtained in the Schwinger scheme using the generalized
Weyl-Heisensebeg ${\cal A}_{\kappa}(1)$. This constitutes the main
of the following section.



\section{Applications}

In this section we discuss some physical applications of the
polynomial $su(2)$ algebra realized here above. More precisely,we
investigate the Kepler problem in the two dimensional curved space
[2] by using the Holstein-Primakoff realization of the Higgs
algebra. We also discuss the relation between our work and super
symmetric (SUSY) quantum mechanics.

\subsection{Kepler problem in two dimensional curved space}

Kepler hamiltonian in the two dimensional curved space is given by
\begin{equation}
H = \frac{1}{2} ( \pi_i \pi_i + \lambda L_z^2 ) - \frac{ \mu}{ r },
\end{equation}
where $\lambda $ is the curvature of the two dimensional curved space, $\mu$
is a constant , $L_z$ is a two-dimensional rotation operator, and $\pi_i $
are components of the momentum $\vec{ \pi } $ defined by
\begin{equation}
\pi_i = p_i - \frac {\lambda}{2} \{ x_i , ( \vec x \cdot \vec p ) \}, ~~~
i=1,2
\end{equation}
where $\{ A ,B \} = AB + BA $ and $p_i = - \frac{ \partial}{\partial x_i } $%
. Then, the dynamical algebra is constructed with the two components of the
Runge-Lenz vectors $R_{\pm }$ and diagonal component of the angular momentum
$L_z $ ;
\[
[ L_z , R_{\pm }] = \pm R_{\pm}
\]
\[
[ R_+, R_-] = 4 \lambda L_z^3 + ( - 4 H + \lambda /2 ) L_z ,
\]
\begin{equation}
\{ R_+ , R_- \} = 2 \mu^2 + ( 2 H - \lambda L_z^2 ) ( 2 L_z^2 + \frac{1}{2}
) - 2 \lambda L_z^2
\end{equation}
The Runge-Lenz vectors are defined by
\begin{equation}
R_{\pm } = R_1 \pm i R_2
\end{equation}
and
\begin{equation}
R_i = \frac{1}{2} \{ L_z , \epsilon_{ij} \pi_j \} + \mu \frac{ x_i} {r},
\end{equation}
where $\epsilon_{ij} $ is a two-dimensional Levi-Civita symbol. Comparing
the eq.(36) with Higgs algebra (22), we get
\begin{equation}
2\bigg( 1 - \kappa + 2\kappa J_0 (1 + \kappa J_0)\bigg)= - 4 H + \lambda /2
, ~~~ \kappa^2 = \lambda
\end{equation}
From the eq.(23) and the eq.(36), we have
\begin{equation}
H= 2 ( C - \mu^2 )
\end{equation}
Using the eq.(23) and the eq.(40), we have
\begin{equation}
E_{jm} = \langle jm| H |jm\rangle = \frac{ - 2 \mu^2 + \kappa^2 j ( j+1) ( 1
+ 4 j ( j+1)) }{ 1 + 8j (j+1) }
\end{equation}
reflecting that the parameter $\kappa$ can reveal some perturbation
effects induced by the space curvature. Also, it must be noticed the
simple relation between the Hamiltonian and the Casimir operator of
the polynomial $su(2)$ algebra given by the equation (40).



%For Case A and Case C, we have $ - j \le m \le j $, so we have $(2j+1)$ energy eigenvalues.
%For Case B, we have $ j-(n_1 -1 ) \le m \le j $, so we have $ n_-$ eigenvalues.
%For Case D, we have $ \frac{3}{2} \le m \le j $, so we have $ j-1/2$ energy eigenvalues.




\subsection{ SUSY Quantum Mechanics}

The SUSY quantum mechanics provides us a powerful tool for finding pairs of
iso-spectral hamiltonians [34-37]. The Darboux method is based on the
existence of an operator $A$ and its adjoint $A^{\dagger} $ which act as
transformation operators between a pair of self-adjoint hamiltonian $H_{\pm}$
[38,39]. Obviously, $H_{\pm}$ are essentially iso-spectral. For the
single-mode, the algebra (18) can be written as

?????????????????????????
????????????????????????



Let us set
\begin{equation}
H_+ = A A^{\dagger} = N + \epsilon
\end{equation}
Then, the algebra (42) can be written as
\begin{equation}
[ a , \ad ] = 1 + 2 \kappa( H_+ - \epsilon) , ~~~[H_+ , \ad ] = \ad , ~~~
[H_+, a ] =-a
\end{equation}
If we consider the SUSY transformation [22]
\begin{equation}
B = A^{\dagger}a A , ~~~~ B^{\dagger} = A^{\dagger}a^+ A
\end{equation}
We have the following algebra
\begin{equation}
[ B, B^{\dagger} ] = ( 1 - 2 \epsilon + 2 \kappa \epsilon^2 ) H_- + ( 3 - 6
\kappa \epsilon ) H_-^2 + 4 \kappa H_-^3,
\end{equation}
where we used the relation $H_+ A = A H_- $ and the fact that $H_- $
is a SUSY partner of $H_+ $. Setting
\begin{equation}
B \leftrightarrow \frac{1}{\sqrt{\kappa} }J_- , ~~~ B^{\dagger}
\leftrightarrow \frac{1}{\sqrt{\kappa} }J_+, ~~H_- = J_3, ~~~~ \epsilon =
\frac{1 }{2\kappa}
\end{equation}
one obtains
\begin{equation}
[J_+ , J_- ] = ( \kappa -1 ) J_3 + 4 \kappa^2 J_3^3
\end{equation}
Comparing the equation (48) with the equation (22), we have
\begin{equation}
4 \kappa^2 J_0^2 + 4 \kappa J_0 + 3 ( 1 - \kappa) =0
\end{equation}
From the condition that $J_0 $ has a real value, we have $\kappa \ge
2/3 $. And the solution of the equation (49) is given by
\begin{equation}
J_0 = \frac{ -1 \pm \sqrt{ 3 \kappa -2 } }{ 2 \kappa }
\end{equation}
We note that for the special value $\kappa =2/3$, we have $J_0 =
-4/3 $. It is clear that the  transformation (45) provides us with a
single boson realization of a the non linear $su(2)$ algebra
characterized by the structure relation (48). This shows that,
beside the Schwinger approach,  the super-symmetric tools are also
useful in generating non linear angular momentum algebra.






