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\begin{document}
\title{Hamilton-Jacobi quantization of a free particle on a $(n-1)$-sphere}
\author{Soon-Tae Hong}
\email{soonhong@ewha.ac.kr}
\affiliation{Department of Science
Education, Ewha Womans University, Seoul 120-750, Korea\\
Institut f\"ur Theoretische Physik, Universit\"at Heidelberg,
Philosophenweg 16, D-69120 Heidelberg, Germany}
\author{Klaus D. Rothe}
\email{K.Rothe@thphys.uni-heidelberg.de}
\affiliation{Institut
f\"ur Theoretische Physik, Universit\"at Heidelberg,
Philosophenweg 16, D-69120 Heidelberg, Germany}
\date{February 13, 2003}%
%\date{\today}%
\begin{abstract}
We study a free particle on a $(n-1)$-sphere in an extended phase
space, where the originally second-class Hamiltonian and
constraints are now in strong involution.  This allows for a
Hamilton-Jacobi formulation of the model, and enables us to
quantize the system canonically and to obtain the free particle
energy spectrum corresponding to that of a rigid rotator.
\end{abstract}
\pacs{02.30.Jr; 11.10.Ef; 11.10.Lm; 11.30.Na}
\keywords{Hamilton-Jacobi; nonlinear model; Dirac quantization;
spectrum} \maketitle

The quantization of constraint systems has been extensively
discussed in the literatures~\cite{dirac64,hong02pr}.  In
particular, the embedding of a second-class system into a
first-class one~\cite{bft}, where constraints are in strong
involution, has been of much interest, and has found a large
number of applications~\cite{hong02pr}. However, the importance
and usefulness of having the constraints in strong involution has,
in our view, not been sufficiently emphasized. To do this in terms
of simple
model is the main objective of the present paper.\\

\noindent (1) {\it Quantum mechanics for a free particle on a $(n-1)$-sphere}\\

\indent Consider the motion of a particle on a hypersphere
$S^{n-1}$, as described by the Lagrangian,
\begin{equation}
L_{0}=\frac{1}{2}\dot{q}_{a}\dot{q}_{a}+\lambda q_{a}\dot{q}_{a},
\label{lag}
\end{equation}
where $q_{a}$ $(a=1,2,...,n)$ are the coordinates parameterizing
the  $S^{n-1}$ manifold, and $\lambda$ is the Lagrange multiplier
implementing the second-class constraint $q_{a}\dot{q}_{a}\approx
0$ associated with the geometrical constraint $q_{a}q_{a}={\rm
constant}$.\footnote{It turns out to be convenient not to impose
the constraint in the form $\lambda (q_{a}q_{a}-1)$.}  From the
Lagrangian (\ref{lag}) we obtain for the canonical momenta
conjugate to the multiplier $\lambda$ and the coordinates $q_{a}$
\beq p_{\lambda}=0,~~~ p_{a}=\dot{q}_{a}+\lambda q_{a},
\label{momenta} \eeq and the corresponding canonical Hamiltonian
reads
\begin{equation}
H_{0}=\frac{1}{2}(p_{a}-\lambda q_{a})(p_{a}-\lambda q_{a}).
\label{canH}
\end{equation}
The usual Dirac algorithm is readily shown to lead to the pair of
second-class constraints $\Omega_{i}$ $(i=1,2)$: \beq
\Omega_{1}=p_{\lambda}\approx 0,~~~\Omega_{2}=q_{a}p_{a}-\lambda
q_{a}q_{a}\approx 0, \label{const22} \eeq satisfying the
constraint algebra with $\epsilon_{12}=-\epsilon_{21}=1$
\begin{equation}
\Delta_{kk^{\prime}}=\{\Omega_{k},\Omega_{k^{\prime}}\}
=\epsilon_{kk^{\prime}}q_{a}q_{a}. \label{delta}
\end{equation}

For the discussion to follow it will be of central importance to
convert this second-class constraints algebra into a strongly
involutive one, by suitably embedding our model into a larger
dimensional phase space.  Following the Batalin-Fradkin-Tyutin
scheme~\cite{bft,hong02pr}, we systematically achieve this by
introducing a pair of canonically conjugate auxiliary coordinates
$(\theta, p_{\theta})$  with the Poisson brackets
\begin{equation}
\{\theta, p_{\theta}\}=1. \label{phii}
\end{equation}
In this enlarged phase space one systematically constructs the
first-class constraints $\tilde{\Omega}_{i}$ as a power series in
these auxiliary coordinates, by requiring that they be in strong
involution $\{\tilde{\Omega}_{i},\tilde{\Omega}_{j}\}=0$, \beq
\tilde{\Omega}_{1}=\Omega_{1}+\theta,~~~
\tilde{\Omega}_{2}=\Omega_{2}-q_{a}q_{a}p_{\theta}.
\label{1stconst} \eeq Note that from here on all Poisson brackets
are understood to be taken with respect to the variables
$(q_{a},p_{a},\lambda,p_{\lambda},\theta,p_{\theta})$ of the
extended phase space.

We next construct the first-class coordinates $\tilde{{\cal F}}
=(\tilde{q}_{a},\tilde{p}_{a})$, corresponding to the original
coordinates defined by ${\cal F}=(q_{a},p_{a})$ in the extended
phase space. They are again obtained as a power series in the
auxiliary fields $(\theta,p_{\theta})$ by demanding that they be
in strong involution with the first-class constraints
(\ref{1stconst}), that is $\{\tilde{\Omega}_{i}, \tilde{{\cal
F}}\}=0$.  After some tedious algebra, we obtain for the
first-class coordinates
\begin{eqnarray}
\tilde{q}_{a}&=&q_{a}\left(\frac{q_{c}q_{c}+2\theta}{q_{c}q_{c}}\right)^{1/2}\nonumber \\
\tilde{p}_{a}&=&\left(p_{a}+2q_{a}\lambda\frac{\theta}{q_{c}q_{c}}
+2q_{a}p_{\theta}\frac{\theta}{q_{c}q_{c}}\right)
\left(\frac{q_{c}q_{c}}{q_{c}q_{c}+2\theta}\right)^{1/2}\nonumber\\
\tilde{\lambda}&=&\lambda+p_{\theta},~~~\tilde{p}_{\lambda}=p_{\lambda}+\theta.
\label{pitilde}
\end{eqnarray}
In terms of these coordinates the first-class Hamiltonian can be
written in the compact form
\begin{equation}
\tilde{H}_{0}=\frac{1}{2}(\tilde{p}_{a}-
\tilde{q}_{a}\tilde{\lambda})(\tilde{p}_{a}-
\tilde{q}_{a}\tilde{\lambda}). \label{htilde}
\end{equation}
It will be useful to note that this Hamiltonian is just the
Legendre transform of the first-class Lagrangian, which consists
of the Lagrangian (\ref{lag}) and a Wess-Zumino Lagrangian: \beq
\tilde{L}_{0}=\frac{1}{2}\dot{\tilde{q}}_{a}\dot{\tilde{q}}_{a}+\tilde{\lambda}
\tilde{q}_{a}\dot{\tilde{q}}_{a}=\frac{1}{2}\dot{q}_{a}\dot{q}_{a}+\lambda
q_{a}\dot{q}_{a}+
\frac{1}{q_{c}q_{c}}\dot{q}_{a}\dot{q}_{a}{\theta}
-\frac{1}{2(q_{c}q_{c})^{2}}\dot{\theta}^{2}.\label{wzlag} \eeq In
terms of the first-class coordinates (\ref{pitilde}), the strongly
involutive constraints (\ref{1stconst}) take the natural form \beq
\tilde{\Omega}_{1}=\tilde{p}_{\lambda},~~~
\tilde{\Omega}_{2}=\tilde{q}_{a}\tilde{p}_{a}-\tilde{\lambda}\tilde{q}_{a}\tilde{q}_{a},
\label{oott} \eeq which thus display manifest form invariance with
respect to the second-class constraints (\ref {const22}).  One
readily checks that one has the following Poisson brackets: \bea
\{\tilde{q}_{a},\tilde{p}_{b}\}&=&\delta_{ab},~~~~
\{\tilde{q}_{a},\tilde{q}_{b}\}=0,~~~~\{\tilde{p}_{a},\tilde{p}_{b}\}=0,\label{commst00}\\
\{\tilde{\lambda},\tilde{q}_{a}\}&=&-\tilde{q}_{a},~~~
\{\tilde{\lambda},\tilde{p}_{a}\}=\tilde{p}_{a},~~~
\{\tilde{p}_{\lambda},\tilde{q}_{a}\}=0,~~~\{\tilde{p}_{\lambda},\tilde{p}_{a}\}=0.
\label{commst01} \eea  It follows from these Poisson brackets that
the constraints (\ref{oott}) and the first-class Hamiltonian
(\ref{htilde}) are all in strong involution: \beq
\{\tilde{\Omega}_{i},\tilde{\Omega}_{j}\}=0,~~~\{\tilde{\Omega}_{i},\tilde{H}_{0}\}=0.\label{totoij}
\eeq This will simplify the discussion to follow.  Indeed, since
the constraints are now in strong involution, we can impose them
strongly. Solving $\tilde{\Omega}_{2}=0$ for $\tilde{\lambda}$,
and choosing $\tilde{q}_{a}\tilde{q}_{a}=1$ for convenience, we
may reduce the Hamiltonian (\ref{htilde}) to the form \beq
\tilde{H}_{0}=\frac{1}{2}[\tilde{p}_{a}-\tilde{q}_{a}(\tilde{q}_{b}\tilde{p}_{b})]
[\tilde{p}_{a}-\tilde{q}_{a}(\tilde{q}_{b}\tilde{p}_{b})].
\label{htilde2} \eeq  Effectively we have thus again only $2n$
independent degrees of freedom in the extended phase space to
describe the free particle motion on the $(n-1)$-sphere.

On the other hand, from (\ref{pitilde}) one finds that these $2n$
independent degrees of freedom satisfy the Poisson algebra
(\ref{commst00}) which is evidently in accordance with the
first-class Lagrangian (\ref{wzlag}).  Hence in the formulation of
the Hamiltonian (\ref{htilde2}) we can treat this free particle
motion as that of an unconstrained system.  This will prove very
useful in what
follows.\\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
(2) {\it Hamilton-Jacobi quantization}\\

\indent If our Lagrangian would describe an unconstrained system,
we would have only one Hamilton-Jacobi (HJ) equation for the
Hamilton principal function $S$, as given by a partial
differential equation (PDE) of the form \beq
H^{\pr}_{0}:=\frac{\pa S}{\pa t }+H_{0}\left(t,
q_{a},\lambda,\frac{\pa S}{\pa q_{a}},\frac{\pa S}{\pa
\lambda}\right)=0. \label{hj0}\eeq If $H_{0}$ does not depend
explicitly on $t$, we may then seek a solution of the form
$S=\alpha t+W$, with $\alpha={\rm constant}$. In the case of our
model, the primary constraint $p_{\lambda}=0$ leads to an
additional PDE of the type \beq H^{\pr}_{1}:=\frac{\pa S}{\pa
\lambda}+H_{1}\left(t, q_{a},\lambda,\frac{\pa S}{\pa
q_{a}},\frac{\pa S}{\pa \lambda}\right)=0, \label{hj1} \eeq where
$H_{1}$ vanishes for the case in question, but has been kept in
(\ref{hj1}) in order to streamline the notation.

Collecting $H_{0}^{\pr}$ and $H_{1}^{\pr}$ into the ``vector"
$H_{\alpha}^{\pr}$ $(\alpha =0,1)$, the minimal requirement for
this system of equations to have a solution is that the
$H_{\alpha}^{\pr}$ be in strong involution:
$\{H_{\alpha}^{\pr},H_{\beta}^{\pr}\}=0$. Since
$\{H_{0}^{\pr},H_{1}^{\pr}\}\neq 0$, this is not the case. We are
thus forced to consider the first-class formulation of the model
in the extended phase space discussed above.  In this formulation
(\ref{hj0}) and (\ref{hj1}) are replaced by \beq
\tilde{H}^{\pr}_{0}=\tilde{p}_{0}+\tilde{H}_{0}=0,~~~
\tilde{H}^{\pr}_{1}=\tilde{p}_{\lambda}+\tilde{H}_{1}=0,
\label{hj} \eeq where again $\tilde{H}_{1}=0$, and the following
identification \beq \tilde{p}_{0}=\frac{\pa S}{\pa
t},~~~\tilde{p}_{a}=\frac{\pa S}{\pa
\tilde{q}_{a}},~~~\tilde{p}_{\lambda}=\frac{\pa
S}{\pa\tilde{\lambda}}\eeq are understood.

Following the HJ scheme~\cite{pi98,HKPR}, one obtains from
(\ref{hj}), the familiar Hamiltonian equations of motion for a
constrained system~\cite{dirac64}, \beq
d\tilde{q}_{\un{a}}=\frac{\pa \tilde{H}^{\prime}_{\alpha}} {\pa
\tilde{p}_{\un{a}}}d\tilde{t}_{\alpha},~~~
d\tilde{p}_{\un{a}}=-\frac{\pa \tilde{H}^{\prime}_{\alpha}} {\pa
\tilde{q}_{\un{a}}}d\tilde{t}_{\alpha}, \label{hjem0} \eeq with
$\tilde{t}_{\alpha}=(t,\tilde{\lambda})$ and
$\tilde{q}_{\un{a}}=(\tilde{q}_{a},\tilde{\lambda})$.  Explicitly
\bea
&&d\tilde{q}_{a}=(\tilde{p}_{a}-\tilde{q}_{a}\tilde{\lambda})dt,~~~
d\tilde{p}_{a}=(\tilde{p}_{a}-\tilde{q}_{a}\tilde{\lambda})
\tilde{\lambda}dt\label{eom1}\\
&&d\tilde{\lambda}=d\tilde{\lambda},~~~~~~~~~~~~~~~~~
d\tilde{p}_{\lambda}=0. \label{eom2} \eea Since the algebra of the
first-class constraints (\ref{oott}) is strongly involutive, we
may implement the constraint $\tilde{\Omega}_{2}=0$ strongly.
Solving equation (\ref{oott}) for $\tilde{\lambda}$ and
substituting into (\ref{eom1}), we have now from (\ref{hj}) the HJ
equation \beq \tilde{H}^{\pr}_{0}:=\frac{\pa S}{\pa t
}+\tilde{H}_{0}\left(\tilde{q}_{a},\frac{\pa S}{\pa
\tilde{q}_{a}}\right)=0,\label{hj2}\eeq where $\tilde{H}_{0}$ is
given by (\ref{htilde2}). Note that due to the commutator
relations (\ref{commst00}) we can now treat this free particle
system on the $(n-1)$-sphere as an unconstrained system with the
Hamiltonian (\ref{htilde2}).  The equations of motion (\ref{eom2})
for $(\tilde{\lambda},\tilde{p}_{\lambda})$ have thus become
irrelevant to our system.

Inserting the first-class constraint $\tilde{\Omega}_{2}=0$,
together with $\tilde{q}_{a}\tilde{q}_{a}=1$, into (\ref{eom1}),
we can rewrite the equations of motion for the independent
variables $(\tilde{q}_{a},\tilde{p}_{a})$ as follows \beq
d\tilde{q}_{a}=(\tilde{p}_{a}-\tilde{q}_{a}(\tilde{q}_{b}\tilde{p}_{b}))dt,~~~~~~
d\tilde{p}_{a}=(\tilde{p}_{a}-\tilde{q}_{a}(\tilde{q}_{b}\tilde{p}_{b}))
(\tilde{q}_{c}\tilde{p}_{c})dt. \label{eom3}\eeq For the
differential of the Hamilton principal function these equations
imply \beq dS=\frac{\pa S}{\pa t}dt +\frac{\pa S}{\pa
\tilde{q}_{a}}d\tilde{q}_{a}
=\tilde{p}_{0}dt+\tilde{p}_{a}d\tilde{q}_{a}
=(-\tilde{H}_{0}+\tilde{p}_{a}\dot{\tilde{q}}_{a})dt,
\label{sddetails} \eeq which can be integrated in time to yield
\beq S=\int {\rm d}t~\tilde{L}_{0}. \label{action0fin} \eeq Hence
$S$ can be identified with the action associated with the
first-class Lagrangian $\tilde{L}_{0}$ on the space of solutions
of the equations of motion \beq
\ddot{\tilde{q}}_{a}+\tilde{q}_{a}(\dot{\tilde{q}}_{b}\dot{\tilde{q}}_{b})=0,
\label{eomlam} \eeq following from (\ref{eom3}).

Next, we introduce a linear operator $X_{0}$ corresponding to the
equations of motion (\ref{eom3}) as follows \beq
X_{0}f=\{f,\tilde{H}^{\pr}_{0}\}=[\tilde{p}_{a}-\tilde{q}_{a}(\tilde{q}_{b}
\tilde{p}_{b})]\left(\frac{\pa f }{\pa
\tilde{q}_{a}}+\tilde{q}_{b}\tilde{p}_{b}\frac{\pa f} {\pa
\tilde{p}_{a}}\right). \label{x0op}\eeq From (\ref{commst00}) we
see that $\tilde{q}_{a}$ and $\tilde{p}_{a}$ are canonically
conjugate variables, so that on quantum level \beq
\left[\tilde{q}_{a},\tilde{q}_{b}\right]=0,~~~\left[\tilde{p}_{a},\tilde{p}_{b}\right]=0,~~~
\left[\tilde{q}_{a},\tilde{p}_{b}\right]=i\hbar\delta_{ab}.
\label{commst3} \eeq  In the Schr\"odinger representation, we thus
have for the quantum commutators, \beq
\tilde{p}_{a}=-i\hbar\frac{\pa}{\pa \tilde{q}_{a}},~~~
\tilde{q}_{a}=i\hbar\frac{\pa}{\pa \tilde{p}_{a}}.
\label{op-pi}\eeq Inserting these operators into (\ref{x0op}), we
arrive at \beq X_{0}f=\frac{i}{\hbar}[\tilde{p}_{a}-\tilde{q}_{a}
(\tilde{q}_{b}\tilde{p}_{b})][\tilde{p}_{a}-\tilde{q}_{a}
(\tilde{q}_{b}\tilde{p}_{b})]f=\frac{2i}{\hbar}\tilde{H}_{0}f,
\eeq which shows that in the case of a free particle on the
$(n-1)$-sphere we have only one quantum operator, namely the
Hamiltonian.

Following the symmetrization procedure of ref.~\cite{lee81}
together with ({\ref{htilde2}) and (\ref{op-pi}), we obtain for
the Hamilton quantum operator of a free particle on the
$(n-1)$-sphere, \bea
\tilde{H}_{0}&=&:\frac{1}{2}\left(-i\hbar\frac{\pa}{\pa
\tilde{q}_{a}} +i\hbar\tilde{q}_{a}\tilde{q}_{c}\frac{\pa}{\pa
\tilde{q}_{c}}\right)\left(-i\hbar\frac{\pa}{\pa \tilde{q}_{a}}
+i\hbar\tilde{q}_{a}\tilde{q}_{c}\frac{\pa}{\pa \tilde{q}_{c}}\right):\nonumber\\
&=&\frac{1}{2}\hbar^{2}\left[-\frac{\pa^{2}}{\pa \tilde{q}_{a}\pa
\tilde{q}_{a}}+(n-1)\tilde{q}_{a}\frac{\pa}{\pa
\tilde{q}_{a}}+\tilde{q}_{a}\tilde{q}_{b}\frac{\pa}{\pa
\tilde{q}_{a}}\frac{\pa}{\pa \tilde{q}_{b}}+\frac{(n+1)(n-3)}{4}
\right]. \label{op-htilde} \eea Note that the quantum Hamiltonian
(\ref{op-htilde}) has only terms of order $\hbar^{2}$, so that one
has rotational energy contributions of order $\hbar^{2}$, without
any vibrational modes of order $\hbar$. In fact, the starting
Lagrangian (\ref{lag}) does not possess any vibrational degrees of
freedom, since it involves only the kinetic term describing the
motions of the particle residing on the $S^{n-1}$ manifold.

We now define the Casimir operator $\tilde{J}^{2}$ in terms of the
$(n-1)$-sphere Laplacian~\cite{vil68} \beq
\tilde{J}^{2}=\hbar^{2}\left[-\frac{\pa^{2}}{\pa \tilde{q}_{a}\pa
\tilde{q}_{a}}+(n-1)\tilde{q}_{a}\frac{\pa}{\pa
\tilde{q}_{a}}+\tilde{q}_{a}\tilde{q}_{b}\frac{\pa}{\pa
\tilde{q}_{a}}\frac{\pa}{\pa \tilde{q}_{b}}\right], \label{jop}
\eeq whose eigenvalue spectrum is given in terms of the
corresponding angular quantum number $l$ $(l={\rm integers})$ and
the dimension of the sphere as follows \beq
\tilde{J}^{2}|l\rangle=\hbar^{2}l(l+n-2)|l\rangle. \eeq We thus
have for the Hamiltonian operator of a free particle on the
$(n-1)$-sphere \beq
\tilde{H}_{0}=\frac{1}{2}\left[\tilde{J}^{2}+\frac{\hbar^{2}(n+1)(n-3)}{4}\right],
\label{spectrum} \eeq to yield the eigenvalue equation
$\tilde{H}_{0}|l\rangle=E_{l}|l\rangle$ with the spectrum given by
\beq
E_{l}=\frac{\hbar^{2}}{2}\left[l(l+n-2)+\frac{(n+1)(n-3)}{4}\right].\label{eigene}
\eeq Note that the Hamiltonian operator (\ref{spectrum}) is that
of a rigid rotator, and the energy eigenvalues (\ref{eigene})
involve global shifts depending on the dimension of the sphere,
which has not been included in the
references~\cite{neves00,podo28}.

Finally, we return to the first-class Hamiltonian (\ref{htilde}),
where the variable $\tilde{\lambda}$ has not yet been eliminated
using the constraint $\tilde{\Omega}_{2}=0$. Corresponding to
$\tilde{H}^{\pr}_{0}$ and $\tilde{H}^{\pr}_{1}$ in (\ref{hj}), we
introduce linear operators $X_{\alpha}$ $(\alpha=0,1)$ in the
extended phase space
$(\tilde{q}_{\un{a}},\tilde{p}_{\un{a}})=(\tilde{q}_{a},\tilde{\lambda},
\tilde{p}_{a},\tilde{p}_{\lambda})$, defined by \beq
X_{\alpha}f=\{f,\tilde{H}^{\pr}_{\alpha}\}=\frac{\pa f }{\pa
\tilde{q}_{\un{a}}}\frac{\pa\tilde{H}^{\pr}_{\alpha}}{\pa
\tilde{p}_{\un{a}}}-\frac{\pa\tilde{H}^{\pr}_{\alpha}}{\pa
\tilde{q}_{\un{a}}}\frac{\pa
f}{\pa\tilde{p}_{\un{a}}},\label{x0op2}\eeq where $f$ is an
arbitrary freedom of the phase space variables.  One readily
checks that $X_{0}$ and $X_{1}$ are given by \beq
X_{0}=(\tilde{p}_{a}-\tilde{q}_{a}\tilde{\lambda})\left(\frac{\pa
}{\pa \tilde{q}_{a}}+\tilde{\lambda}\frac{\pa } {\pa
\tilde{p}_{a}}\right),~~~X_{1}=\frac{\pa } {\pa \tilde{\lambda}}
\label{xxxx} \eeq and satisfy the closed Lie algebra \beq
\left[X_{0},X_{1}\right]=0, \label{lie} \eeq in agreement with the
general relation
$\left[X_{0},X_{1}\right]f=\{\{\tilde{H}^{\pr}_{0},
\tilde{H}^{\pr}_{1}\},f\}$~\cite{pi98} and (\ref{totoij}).
Moreover, one can easily check that, corresponding to the degrees
of freedom of $X_{\alpha}$, there exist $\tilde{H}^{\pr}_{\alpha}$
satisfying
$\dot{\tilde{H}}^{\pr}_{0}=\dot{\tilde{H}}^{\pr}_{1}=0$. Note that
the equations of motion in this formulation read \beq
\ddot{\tilde{q}}_{a}+\tilde{q}_{a}\dot{\tilde{\lambda}}=0.\eeq
Using the constraint (\ref{oott}), we can eliminate
$\dot{\tilde{\lambda}}$, to put this equation into the form
(\ref{eomlam}).

In conclusion, it can be said that the Batalin-Fradkin-Tyutin
embedding of the second-class constrained system into a
first-class one has played a central role in our problem for
obtaining the energy spectrum, as well as a Hamilton-Jacobi
formulation of our model.\\

STH would like to thank the warm hospitality of the Institut f\"ur
Theoretische Physik at the Universit\"at Heidelberg during his
stay, and to acknowledge financial support in part from the Korea
Science and Engineering Foundation Grant R01-2000-00015. We also
would like to thank Heinz Rothe for helpful discussions.

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\end{document}

