---
abstract: 'We derive the Holstein-Primakoff oscillator realization on the coadjoint orbits of the $SU(N+1)$ and $SU(1,N)$ group by treating the coadjoint orbits as a constrained system and performing the symplectic reduction. By using the action-angle variables transformations, we transform the original variables into Darboux variables. The Holstein-Primakoff expressions emerge after quantization in a canonical manner with a suitable normal ordering. The corresponding Dyson realizations are also obtained and some related issues are discussed.'
address:
- |
    Department of Physics\
    Sung Kyun Kwan University\
    Suwon 440-746, Korea
- |
    Department of Physics\
    Chonbuk National University\
    Chonju, 561-756, Korea
author:
- 'Phillial Oh[@poh]'
- 'Chaiho Rim[@rim]'
title: ' Holstein-Primakoff Realizations on Coadjoint Orbits'
---

Introduction
============

It is well known that the Holstein-Primakoff (HP) [@hols] and Dyson [@dyso] realizations of $su(2)$ algebra in terms of a single oscillator are very useful in describing the spin-density wave phenomena and many others in condensed matter physics [@kitt] and nuclear physics [@klei]. The HP realization also appears in the $q$-deformation [@ques] of the quantum algebras $su_q(2)$ and $su(1,1)_q$ [@bied] although $q$-deformation approach of of Jordan-Schwinger type [@bied1; @sun] is more conventional.

Since the HP and Dyson representations of $su(2)$ algebra can be interpreted as quantum mechanical operators on $S^2$, which is the coadjoint orbit of $SU(2)$ group, it is useful to consider them on the coadjoint orbits of an arbitrary group in extending to higher group [@okub]. So far, the generalization was performed mostly to minimal $CP(N)$ orbits [@rand] or Grassmanian manifold [@mead] which was largely based on the coherent state method [@klau].

In this letter, we discuss general representations of HP and Dyson oscillator realizations for the $su(N+1)$ and $su(1,N)$ algebras on the coadjoint orbits of $SU(N+1)$ and $SU(1,N)$ by treating the coadjoint orbits as a constrained classical system and by explicitly performing a symplectic reduction. Compared with non-linear realization method on coset space [@rand], this approach can have some advantage of exploiting the well-developed mathematical tool of symplectic reduction [@arno] which in our case deals mainly with quadratic constraints. The HP realization will emerge, if we transform the reduced system into canonical one by using the action-angle variable and then quantize it in a standard manner with the normal ordering prescription. Then, the Dyson realization will be obtained by shifting the square-root factor in HP realizations[@hech]. One of the merits of this coadjoint orbit approach is to provide a unified framework for finding explicit expressions for HP and Dyson realization in the compact and non-compact case. We will be mainly concerned with minimal and maximal orbits of $SU(N+1)$ and $SU(1,N)$ to make the presentation simple.

We start by briefly explaining our notation. Let us denote a column vector as a ket $\vert Z>=(Z^{(1)},Z^{(2)},\cdots,Z^{(N+1)})^T$ and introduce its bra $<\bar Z\vert= ( \bar Z_{(1)}, \bar Z_{(2)},\cdots , \bar Z_{(N+1)})$. Then, $<\bar Z\vert Z>=\sum_{i=1}^{N+1}\bar Z_{(i)}Z^{(i)}
\equiv \bar Z_{(i)}M^{ij}Z_{(j)}$. The raising and lowering are done with respect to the metric $M=\mbox{diag}(1,\epsilon,
\cdots,\epsilon)$. $\epsilon=1(-1)$ for $SN(N+1)(SU(1,N))$. Let us express the element $g$ of $SU(N)$ and $SU(1,N)$ by $N+1$ kets $(\vert Z_1>, \vert Z_2>,\cdots, 
\vert Z_{N+1}>)$ with $ \vert Z_p>=(Z_p^{(1)},Z_p^{(2)},\cdots,Z_p^{(N+1)})^T$. Then, $Mg^\dagger M$ is composed of $N+1$ bras $<\bar Z^p\vert$’s such that $<\bar Z^p\vert=
( \bar Z^p_{(1)}, \bar Z^p_{(2)},\cdots , \bar Z^p_{(N+1)})$. With the notation $< \bar Z_p\vert =< \bar Z^q\vert M_{qp}$, $Mg^\dagger Mg=I$ gives $$< \bar Z_p\vert Z_q>=M_{pq}, \quad \mbox{det}
(\vert Z_1>, \vert Z_2>,\cdots, \vert Z_{N+1}>) =1.
\label{cond}$$

The isospin charges on the coadjoint orbits are defined by $$Q^a=-2\mbox{Tr}(gxg^{-1}T^a).$$ where $x=i~\mbox{diag}(x_1,x_2,\cdots,x_{N+1})$ with $\sum_{i=1}^{N+1}x_i=0$. The $x_i$’s are real and $T^a$’s are the anti-hermitian generators of the group which satisfy the Lie algebra with real structure constant $f_{ab}^{\ \ c}$: $[T_a, T_b]=f_{ab}^{\ \ c}T_c$ and Tr$(T^aT^b)=-\frac{1}{2}\eta^{ab}$. By making use of the second equation of the Eq.(\[cond\]), $\vert Z_{N+1}>$ can be eliminated and subsequently we find that $Q^a$ can be expressed as [@oh955] $$Q^a=-2i\sum_{p=1}^{N}J_p<\bar Z^p\vert T^a\vert Z_p>.
\label{isosp1}$$ where $J_p=x_p-x_{N+1}=x_1+\cdots +2x_p+\cdots +x_{N}$.

Let us consider a classical system defined on the coadjoint orbit of $SU(N)$ described by a Lagrangian $$L=2\mbox{Tr}(xg^{-1}\dot g)-H(Q^a)=2
i\sum_{p=1}^{N+1}x_p <\bar Z^p\vert \frac{d}{dt}\vert Z_p>-H(Q^a).
\label{lagggg}$$ By using the second equation of the Eq.(\[cond\]) again, we find $$L=2i\sum_{p=1}^{N}J_p<\bar Z^p\vert \frac{d}{dt}\vert Z_p>-H(Q^a).$$ Note that there still exist the constraints $<\bar Z_p\vert Z_q>-M_{pq}=0\  (p,q=1,\cdots,N)$.

Using the symplectic structure of the above Lagrangian, one can show that the isospin charges satisfy the $su(N+1)$ and $su(1,N)$ algebras [@oh955]: $$\{Q^a, Q^b\}=f^{ab}_{\ \ c}Q^c.$$ HP realizations will be found if one finds a quantum mechanical expression of the above isospin charges in terms of canonical variables and so it is essential to bring the Lagrangian (\[lagggg\]) into a canonical form. We will achieve this by transforming the above system into action-angle variables. In passing, we mention that action-angle variables approach on the coadjoint orbits was also considered before [@alek] in the path integral quantization of the orbits in the compact case.

Minimal Orbits 
===============

Let us first apply the above formalism to minimal orbits, $CP(N)$ and its non-compact counter part. In this case, we have $x=i~\mbox{diag}(J,-J/N,\cdots,-J/N)$ and $J_1=J$, $J_2=\cdots=J_N=0$. In the compact case, $J$ is an integer for quantizable orbits. For non-compact case, $J$ depends on the various types of representations of non-compact groups [@pere]. With the notation $Z=(Z_0,Z_1,\cdots,Z_N)^T$ and introducing $\bar Z= ( \bar Z_0, \bar Z_1,\cdots , 
\bar Z_N)$, we find that the Lagrangian can be written as $$L_Z=i J(\bar Z M \dot Z -\dot{\bar Z}MZ)-H(Q^a)
\label{lag1}$$ with the constraint $\bar ZMZ=1$. Note that the notation in the above equation denotes the conventional matrix product rather that the abstract bracket inner product. In addition, the component is relabeled from $0$ to $N$ instead of $1$ to $N+1$. We mention that the above Lagrangian in the compact case was used in describing the internal degrees of freedom of non-Abelian Chern-Simons particles [@oh].

It is well known that the constraint can be solved explicitly in terms of the projective coordinates defined by $\xi_i=Z_i/Z_0(Z_0\neq 0, i=1,2,\cdots,N)$ with a real gauge condition: [@alek; @oh; @ito] $$\chi=\frac{1}{2}( Z^*_0 -Z_0) = 0\label{gauge1}.$$ Then, the solution to the constraint $\bar ZMZ=1$ is given by $${\bar Z}_0=Z_0=\frac{1}{\sqrt{1+\epsilon\vert \xi\vert^2}},\ \ \ \ \vert\xi\vert^2
=\sum_i\vert\xi_i\vert^2,
\label{sol1}$$ By substituting $Z_I=(Z_0,Z_0\xi_i)$ and Eq. (\[sol1\]) into Eq. (\[lag1\]), we obtain the following reduced Lagrangian: $$L_\xi=i J\epsilon \frac{{\bar \xi}\dot \xi-\dot{\bar \xi}\xi}
{1+\epsilon\mid\xi\mid^2}- H(Q^a).
\label{xilag}$$ The isospin charges of Eq. (\[isosp1\]) becomes $$Q^a=-i\frac{2J}{1+\epsilon\vert\xi\vert^2}
\left( T^a_{00}+T^a_{0i}\xi_i+\epsilon T^a_{i0}\bar\xi_i
+\epsilon T^a_{ij}\bar\xi_i\xi_j\right).
\label{defi}$$

To make contact with HP representations, we make the following action-angle transformation of variables [@oh945]: $$I_i=\frac{2\vert J\vert \vert\xi_i\vert^2}{1+\epsilon\vert\xi\vert^2}
\equiv \bar\alpha_i\alpha_i,
\label{darboux}$$ and the angle variables are given by the phases of the $\alpha_i$’s. Assuming a positive value for $J\epsilon$, we have the Lagrangian (\[xilag\]) given by $$L_\alpha=\frac{i}{2}({\bar \alpha}_i\dot \alpha_i-\dot{\bar \alpha_i}
\alpha_i) - H(Q^a).
\label{test}$$ The Poisson bracket is defined in a canonical way $$\{\alpha_i,\bar\alpha_j\}=i\delta_{ij}.
\label{canonical}$$ Note that $J$ is negative in the non-compact case. Otherwise, the role of $\alpha_i$ and $\bar\alpha_i$ would be interchanged in the canonical commutation relation (\[canonical\]). The isospin functions (\[defi\]) are expressed as follows: $$\begin{aligned}
Q^a&=&-i\left[(2J-\epsilon\vert\alpha\vert^2)T^a_{00}
+\epsilon\bar\alpha_iT^a_{ij}\alpha_j\right.\nonumber\\
&+&\left.\epsilon\sqrt{2J-\epsilon\vert\alpha\vert^2}\bar\alpha_iT^a_{i0}+
\sqrt{2J-\epsilon\vert\alpha\vert^2}T^a_{0i}\alpha_i\right].
\label{isosp}\end{aligned}$$

The quantum mechanical operator realizations are obtained after quantizing the above operators by replacing the Poisson bracket (\[canonical\]) with Dirac bracket $\alpha_i\longrightarrow a^\dagger_i,~\bar\alpha_i\longrightarrow
a_i$, and perform the normal ordering of the resulting operators by putting the creation operators $a^\dagger$ to the left of annihilation operators $a$. Following the above procedure, we get the following HP realization: $$\begin{aligned}
\hat Q_{\mbox{hp}}^a&=&-i\left[(2J-\epsilon a^\dagger\cdot a)T^a_{00}
+\epsilon T^a_{ij}a^\dagger_j a _i\right.\nonumber\\
&+&\left.\epsilon\sqrt{2J-\epsilon a^\dagger\cdot a}T^a_{i0} a_i+
 a^\dagger_i T^a_{0i}
 \sqrt{2J-\epsilon  a^\dagger\cdot a}\right].
\label{isospin}\end{aligned}$$ If we shift the square root in front of $T^a_{i0} a_i$ to the back of $a^\dagger_i T^a_{0i}$, we get the following generalized Dyson realization: $$\begin{aligned}
\hat Q_{\mbox{d}}^a&=&-i\left\{\epsilon
[ T^a_{i0}+ T^a_{ij}a^\dagger_j-
T^a_{00}a^\dagger_i-T^a_{0j}a^\dagger_ja^\dagger_i]a_i\right.\nonumber\\
&+&\left.2J T^a_{00}+2J T^a_{0i}a^\dagger_i\right\}.
\label{bargmannfock}\end{aligned}$$ It is easy to check that the above realization satisfies the algebras for both the compact and non-compact cases. For the compact case with $\epsilon=1$, the above expression was obtained as an holomorphic differential operator acting on coherent state [@oh955]. Note that shifting the square root makes $\hat Q^a$’s and $\hat Q^{a\dagger}$’s not manifestly conjugate to each other in the Dyson case. To make the representation unitary, the inner product should be defined [@hech] with respect to the Liouville measure, while the Bargmann measure is used for the HP case.

We note that a similar expression in the compact case appeared in the study of the generalized spin system [@rand]. Our result reduces to it after a trivial rescaling of the variables and choosing a specific representation. Our phase space is in the canonical form (see the Eq. (\[canonical\])) and the result holds for arbitrary representation of the group. In addition, Eq. (\[isospin\]) also covers the non-compact case.

To put the above expressions into a more familiar form, we consider $SU(N+1)$ case. It is convenient to use a representation in which the the ladder operators, $E^a_{pq}\equiv E^{\alpha\beta}_{pq} ~(\alpha,\beta=1,
\cdots, N; p,q=0,1,\cdots N)$, are given as follows: $$E^{0\alpha}_{pq}=-\delta^0_p\delta^\alpha_q,~~ 
E^{\alpha 0}_{pq}=-\delta^\alpha_p\delta^0_q,~~
E^{\alpha\beta}_{pq}=\delta^\alpha_p\delta^\beta_q~ (p\neq q) ,
\label{hp1}$$ and the Cartan subalgebra is given by $N$ diagonal matrices which are denoted by $H^m\equiv T^{m^2+2m},~ m=1,2,\cdots N$ and expressed as follows: $$H^m_{pq}=(\sum_{k=0}^{m-1}\delta_{pk}\delta_{qk}
-m\delta_{pm}\delta_{qm})/\sqrt{2m(m+1)}.
\label{hp2}$$ Then, we obtain the following HP realization from the Eq. (\[isospin\]): $$\begin{aligned}
\hat Q^{0i}&=&a_i^\dagger\sqrt{2J-a^\dagger\cdot a},
~~\hat Q^{i0}=\sqrt{2J-a^\dagger\cdot a}a_i=\hat Q^{0i\dagger}
\nonumber\\
\hat Q^{ij}&=&-a_j^\dagger a_i\label{hp4} ~~(i\neq j)\\
\hat Q^m&=&
\frac{1}{\sqrt{2m(m+1)}}\left((m+1)a^\dagger_m a_m
+\sum_{k=m+1}^Na^\dagger_k a_k -2J\right).\nonumber\end{aligned}$$ The corresponding Dyson realizations of $SU(N+1)$ is given by [@rowe]: $$\begin{aligned}
\hat Q^{0i}&=&a_i^\dagger(2J-a^\dagger\cdot a),
~~\hat Q^{i0}=a_i
\nonumber\\
\hat Q^{ij}&=&-a_j^\dagger a_i~~(i\neq j)\label{dy3}\\
\hat Q^m&=&
\frac{1}{\sqrt{2m(m+1)}}\left((m+1)a^\dagger_m a_m
+\sum_{k=m+1}^Na^\dagger_k a_k -2J\right).\nonumber\end{aligned}$$

Maximal Orbits 
===============

Now, let us turn to the maximal orbits, flag manifold of the group. Here, in order to make the presentation simple, we will restrict to the $SU(3)$ and $SU(1,2)$ case. Extension to higher group is straightforward. Let us choose the element $x$ as $x=i \mbox{diag}(x_1,x_2,-(x_1+x_2))$. Then, $J_1=2x_1+x_2, J_2=x_1+2x_2$. We require $x_1\neq x_2, 
J_1\neq 0, J_2\neq 0$. Introduce again $Z_i=(Z_{i0},Z_{i1},Z_{i2})^T ~(i=1,2)$ and $\bar Z_i= ( \bar Z_{i0}, 
\bar Z_{i1},\bar Z_{i2})$, we find $$L=i\sum_{i=1,2}J_iM^{ii}
(\bar Z_iM \dot Z_i -\dot{\bar Z_i}MZ_i)-H(Q^a).
\label{lag2}$$ The constraints are given by $$\bar  Z_i M Z_j=M_{ij}
\label{constraint}$$

To solve the constraints, we again choose the real gauge conditions: $$\bar Z_{10}=Z_{10}~(\neq 0),~~ \bar Z_{22}=Z_{22} ~(\neq 0).$$ Defining the projective coordinates $z_i=Z_{1i}/Z_{10},~ \xi_\alpha=Z_{2\alpha}/Z_{20}$ $(i=1,2;\alpha=0,1)$, the above constraints can be solved as $$Z_{10}=\frac{1}{\sqrt{1+\epsilon\vert z\vert^2}},~~~
Z_{22}=\frac{1}{\sqrt{1+\epsilon\vert \xi_0\vert^2
+\vert \xi_1\vert^2}},
\label{sol2}$$ and the remaining constraints become $\xi_0=-\epsilon(\xi_1\bar z_1+\bar z_2)$. To compare with the known results of symplectic structure on the maximal orbit [@pick], we redefine the variables by $\bar\xi_1\rightarrow -z_3, \bar\xi_0\rightarrow -z_4$ with the remaining constraint given by $z_4=\epsilon (z_2-z_1z_3)$. Then the canonical one form of the Lagrangian (\[lag2\]) is given by $\theta=i(\partial-\bar\partial)W$, where $W$ is given by $$W=\log (1+\epsilon\vert z_1\vert^2+\epsilon\vert z_2\vert^2)^m
(1+\vert z_3\vert^2+\epsilon\vert z_2-z_1z_3\vert^2)^n
\label{symp}$$ with $J_1=m$ and$J_2=-n$. In the compact case with $\epsilon=1$ and $m,n=$ integers, the above expression precisely reduces to the form given in Ref. [@pick]. For non-compact case, they need not be integers. From here on, we will use interchangeably use the variables $z_1,z_2,z_3,z_4$ or $z_1,z_2, w_0\equiv z_4, w_1\equiv z_3$. With our new notation, the isospin function $Q^a$’s of the Eq.(\[isosp1\]) becomes $$\begin{aligned}
Q^a&=&-i\frac{2m}{1+\epsilon\vert z\vert^2}
\left( T^a_{00}+T^a_{0i}z_i+\epsilon T^a_{i0}\bar z_i
+\epsilon T^a_{ij}\bar z_iz_j\right)\nonumber\\
&-&i\frac{2n}{1+\epsilon\vert w_0\vert^2
+\vert w_1\vert^2}
\left(-\epsilon T^a_{22}+\epsilon T^a_{2\alpha}\bar w_\alpha+
 T^a_{\alpha 2}\zeta_\alpha
-  T^a_{\alpha\beta}\bar w_\beta\zeta_\alpha\right)
\label{defii}\end{aligned}$$ with $\zeta_\alpha=(w_0, \epsilon w_1)$.

Let us again consider the action-angle variable transformations $$\alpha_i=\sqrt{2m}\frac{z_i}{\sqrt{1+
\epsilon\vert z\vert^2}},
~~\beta_\alpha=\sqrt{2n}\frac{w_\alpha}{\sqrt{1+
\epsilon\vert w_0\vert^2
+\vert w_1\vert^2}}
\label{darboux2}$$ which renders the Lagrangian (\[lag2\]) into a canonical form $$L=\frac{i}{2}\left[ \epsilon({\bar \alpha}\dot \alpha-\dot{\bar \alpha}\alpha)
+\epsilon ({\bar \beta}m\dot \beta-\dot{\bar \beta}m\beta)
\right]- H(Q^a)
\label{can}$$ with $m$ given by $m=\mbox{diag}(1,\epsilon)$. We also have the isospin functions given by $$\begin{aligned}
Q^a&=&-i\left[(2m-\epsilon\vert\alpha\vert^2)T^a_{00}
+\epsilon\bar\alpha_iT^a_{ij}\alpha_j+
\epsilon\sqrt{2m-\epsilon\vert\alpha\vert^2}\bar\alpha_iT^a_{i0}+
\sqrt{2m-\epsilon\vert\alpha\vert^2}T^a_{0i}\alpha_i
\right]\nonumber\\
&-&i\left[-\epsilon(2n-\epsilon\vert\gamma\vert^2)T^a_{22}
-\gamma_\alpha T^a_{\alpha\beta}\bar\beta_\beta
+\sqrt{2n-\epsilon\vert\gamma\vert^2}
\gamma_\alpha T^a_{\alpha 2}+
\epsilon\sqrt{2n-\epsilon\vert\gamma\vert^2}T^a_{2\alpha}
\bar\beta_\alpha\right].
\label{isosp2}\end{aligned}$$ with $\gamma_\alpha=(\beta_0,\epsilon\beta_1)$. The quantum mechanical operator realizations are obtained after going through the same steps as in the minimal case.

To deal with the remaining constraint $z_4=z_2-z_1z_3$, we will restrict to the compact case for convenience. One is tempted to substitute this constraint directly into the Eq.(\[isosp2\]) and then quantize the system. However, this would change the canonical structure of the Eq.(\[can\]) in a very complicated manner. Another way to carry out the analysis is to impose the constraint on the quantum state. The constraint in terms of $\alpha_1,
\alpha_2, \alpha_3\equiv \beta_1, \alpha_4\equiv \beta_0$ is given by $$\Phi_{\mbox{hp}}=\alpha_4\sqrt{l_1}-\alpha_2\sqrt{l_2}+\alpha_1\alpha_3=0$$ where $l_1=2m-\vert\alpha_1\vert^2-\vert\alpha_2\vert^2,~
l_2=2n-\vert\alpha_3\vert^2-\vert\alpha_4\vert^2$. One can easily check that the constraints are second class.

Using the expression (\[isosp2\]) and canonically quantizing the system, we obtain the following HP realizations in the standard notation of the generators $E$’s, the Eqs. (\[hp1\]) and (\[hp2\]): $$\begin{aligned}
\hat Q^{1+i2}&=&a_1^\dagger\sqrt{\hat l_1}-a^\dagger_4a_3,~
\hat Q^{1-i2}=\sqrt{\hat l_1}a_1-a^\dagger_3a_4
\nonumber\\
\hat Q^3&=&a_1^\dagger a_1+\frac{1}{2}(a_2^\dagger a_2-
a_3^\dagger a_3+a_4^\dagger a_4)-m
\nonumber\\
\hat Q^{4+i5}&=&a_2^\dagger\sqrt{\hat l_1}+a^\dagger_4\sqrt{\hat l_2},~
\hat Q^{4-i5}=\sqrt{\hat l_1}a_2+\sqrt{\hat l_2}a_4\label{hpp}\\
\hat Q^{6+i7}&=&-a_2^\dagger a_1-a^\dagger_3\sqrt{\hat l_2},~
\hat Q^{6-i7}=-a_1^\dagger a_2-\sqrt{\hat l_2}a_3\nonumber\\
\hat Q^8&=&\frac{\sqrt{3}}{2}(a_2^\dagger a_2+
a_3^\dagger a_3+a_4^\dagger a_4)-\frac{1}{\sqrt{3}}m-\frac{2}{\sqrt{3}}n
\nonumber\end{aligned}$$ where $\hat l_1=2m-a_1^\dagger a_1-a_2^\dagger a_2,
~ \hat l_2=2n-a_3^\dagger a_3-a_4^\dagger a_4$. Since the constraints are second class, only half of the constraints is imposed on the physical state $$\hat\Phi_{\mbox{hp}}\vert\mbox{phys}>
=(a^\dagger_4\sqrt{\hat l_1}-
a^\dagger_2\sqrt{\hat l_2}+a^\dagger_1a^\dagger_3)\vert\mbox{phys}>=0.
\label{coo}$$ The physical states are labeled by $(m,n)$ and can be obtained by successive applications of $m$-times of $a^\dagger_1, a^\dagger_2$ combined and $n$-times of $a^\dagger_3, a^\dagger_4$ combined to the vacuum state. The above condition (\[coo\]) will give some restrictions on $m$ and $n$. The result will determine irreducible representations of the $SU(3)$ group according to the Borel-Weil-Bott theorem[@kiri]. The detailed analysis on the relations between the Eqs. (\[hpp\]) and (\[coo\]) and the irreducible representations is not of concern here and will be reported elsewhere.

By shifting the square root, we again get the corresponding constrained Dyson realizations: $$\begin{aligned}
\hat Q^{1+i2}&=&a_1^\dagger {\hat l_1}-a^\dagger_4a_3,~
\hat Q^{1-i2}=a_1-a^\dagger_3a_4
\nonumber\\
\hat Q^3&=&a_1^\dagger a_1+\frac{1}{2}(a_2^\dagger a_2-
a_3^\dagger a_3+a_4^\dagger a_4)-m
\nonumber\\
\hat Q^{4+i5}&=&a_2^\dagger{\hat l_1}+a^\dagger_4{\hat l_2},~
\hat Q^{4-i5}=a_2+a_4\label{diy}\\
\hat Q^{6+i7}&=&-a_2^\dagger a_1-a^\dagger_3{\hat l_2},~
\hat Q^{6-i7}=-a_1^\dagger a_2-a_3
\nonumber\\
\hat Q^8&=&\frac{\sqrt{3}}{2}(a_2^\dagger a_2+
a_3^\dagger a_3+a_4^\dagger a_4)-\frac{1}{\sqrt{3}}m-\frac{2}{\sqrt{3}}n
\nonumber\end{aligned}$$ And we can infer that the constraint changes into $$\hat\Phi_{\mbox{d}}\vert \mbox{phys}>=
(a^\dagger_4-a^\dagger_2+a^\dagger_1a^\dagger_3)\vert \mbox{phys}>=0.
\label{dyyyy}$$

Let us compare the above formula with the other Dyson realization which can be obtained by the method of geometric quantization in the holomorphic coherent state approach [@oh1]: $$\begin{aligned}
\hat Q^1&=&-\frac{1}{2}\left[(z_1^2-1)\frac{\partial}{\partial z_1}
+z_1z_2\frac{\partial}{\partial z_2}
+(z_2-z_1z_3)\frac{\partial}{\partial z_3}-2m z_1\right]
\nonumber\\
\hat Q^2&=&-\frac{i}{2}\left[-(z_1^2+1)\frac{\partial}{\partial z_1}
-z_1z_2\frac{\partial}{\partial z_2}
-(z_2-z_1z_3)\frac{\partial}{\partial z_3}+2m z_1\right]
\nonumber\\
\hat Q^3&=&z_1\frac{\partial}{\partial z_1}
+\frac{z_2}{2}\frac{\partial}{\partial z_2}
-\frac{z_3}{2}\frac{\partial}{\partial z_3}-m
\nonumber\\
\hat Q^4&=&-\frac{1}{2}\left[z_1z_2\frac{\partial}{\partial z_1}
+(z_2^2-1)\frac{\partial}{\partial z_2}
+z_3(z_2-z_1z_3)\frac{\partial}{\partial z_3}-2m z_2-2n(z_2-z_1z_3)\right]
\nonumber\\
\hat Q^5&=&-\frac{i}{2}\left[-z_1z_2\frac{\partial}{\partial z_1}
-(z_2^2+1)\frac{\partial}{\partial z_2}
-z_3(z_2-z_1z_3)\frac{\partial}{\partial z_3}+2m z_2+2n(z_2-z_1z_3)\right]
\label{dyy}\\
\hat Q^6&=&\frac{1}{2}\left[-z_2\frac{\partial}{\partial z_1}
-z_1\frac{\partial}{\partial z_2}
+(z_3^2-1)\frac{\partial}{\partial z_3}-2nz_3\right]
\nonumber\\
\hat Q^7&=&\frac{i}{2}\left[z_2\frac{\partial}{\partial z_1}
-z_1\frac{\partial}{\partial z_2}
-(z_3^2+1)\frac{\partial}{\partial z_3}+2nz_3\right]
\nonumber\\
\hat Q^8&=&\frac{\sqrt{3}}{2}(z_2\frac{\partial}{\partial z_2}
+z_3\frac{\partial}{\partial z_3})-\frac{1}{\sqrt{3}}m
-\frac{2}{\sqrt{3}}n.
\nonumber\end{aligned}$$ We find that the expression (\[diy\]) reduces to the above one after naively using the Fock-Bargmann representation $a_i\rightarrow \partial/\partial z_i, a_i^\dagger\rightarrow
z_i$ with the substitution $z_4=z_2-z_1z_3$ and acting on the physical states annihilated by $a_4, \frac{\partial}
{\partial z_4}\vert\mbox{phys}>=0$. However, the relation between the two approach must be investigated further: the Eqs. (\[diy\]) and (\[dyyyy\]) which correspond to the process of reduction after quantization, in general, does not give the same result as the case of quantization after reduction, Eq. (\[dyy\]).

Conclusion
==========

We studied the HP oscillator realization on the coadjoint orbits of the $SU(N+1)$ and $SU(1,N)$ group by considering the symplectic reduction of these group and by using the action-angle variables transformations. The HP expressions were obtained after canonical quantization with a suitable normal ordering. In the minimal case, the constraints can be solved explicitly but in the maximal case, some of the constraints were imposed directly on the physical states. The corresponding Dyson realizations were also obtained.

It would be straightforward to extend the above formalism to other coadjoint orbits. Especially, it would be interesting to apply it in studying the generalized spin system, ferromagnet or antiferromagnet system on the flag manifold [@rand] and the Hermitian symmetric space [@ford; @ohpark]. Finally, the $q$-deformation of the Eqs. (\[hpp\]) and (\[diy\]) poses another interesting problem. Details will appear elsewhere.

We would like to thank J. Klauder and V. Manko for useful conversations. This work is supported by the KOSEF through the CTP at SNU and the project number (96-1400-04-01-3, 96-0702-04-01-3), and by the Ministry of Education through the RIBS (BSRI/96-1419,96-2434).

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