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\title{Quantization 
via Star Products}
\author{Takayuki HORI\footnote{E-mail: hori@main.teikyo-u.ac.jp}and ~Takao KOIKAWA\footnote{E-mail: koikawa@otsuma.ac.jp}}
\vspace{1cm}
\address{
	$^{*)}$ Department of Economics, Teikyo University,\\
	Hachioji 192-0395,~Janan\\
        $^{\dagger)}$ School of Social Information Studies,\\
        Otsuma Women's University\\
        Tama 206-0035,~Japan\\ 
}
\vspace{1cm}

We study quantization via star products. We pursue a quantization scheme, where 
a quantum theory is described entirely in terms of the function space without any 
reference to operators in the Hilbert space. For consistency of the theory, the associativity law plays the essential role in excluding the unwanted solutions to the stargen-value equation. This is exhibited in the $D$-dimensional harmonic oscillator explicitly. As the by-products, the interplay between the Laguerre polynomials and the creation and annihilation functions with the star product is demonstrated.  

\newpage
  
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\vspace{.5cm}

\section{Introduction}

In 1949 Moyal discovered an interesting description of the quantum mechanics\cite{moya}, 
called Moyal quantization, in which the Wigner function\cite{Wig} is used as the 
phasespace distribution function similar to that used in the statistical mechanics. 
The transformation function connecting the distribution functions at different times 
satisfies a differential equation similar to the classical equations of 
motion, where the Poisson bracket is replaced by the  Moyal bracket.
The complexity of the Moyal bracket, however, has  prevented physicists from 
practical use of this scheme, and it has not attracted their wide interests with some 
exceptions (see, {\it e.g.}, \cite{CUZ} and references therein).

 Recently, a renewed interest to the Moyal theory is provoked, partially due to 
the invention of the non-commutative geometry which may be regarded as the proper 
framework of  some string theories(see \cite{Mia}, for a review).
Another motivation for the Moyal theory rises in its idea that 
the theory is thought  to be  constructed entirely in terms of functions on the 
phasespace, not referring to 
the Hilbert space of state vectors and operators acting to them, which would enable one to 
understand the connection between the classical and quantum theories in a more perspective manner.
Precisely speaking, however,  
the Moyal theory implicitly relies on the notions of the ordinary quantum mechanics, {\it e.g.},
 on the notion of state vectors and operators. 
The  fundamental equation of Moyal theory derived from the ordinary quantum mechanics 
  has solutions  which cannot be
properly interpreted in the ordinary quantum mechanics, 
and general rule to exclude such solutions has not been known so far.
An example is the solution corresponding to the state with negative energy 
in a non-relativistic theory.



In the present paper we propose  an abstract framework for the Moyal quantization, which is 
completely free from the concept of state vectors and operators acting to them.
Our scheme is based on 
the so-called star product and on a function space of the canonical variables, which are 
defined so that the star product satisfies some reasonable axioms.
We call the scheme  {\it star quantization} in order to distinguish it from the procedure 
presented by Moyal.
The crucial axiom by which the solutions with negative eigenvalues are excluded is the 
associative law of the 
star product.
In the abstract framework using these axioms 
the procedure of obtaining all physical predictions is completely determined, once the
concrete definition of the star product satisfying the axioms is given.
Furthermore the classical theory and the quantum theory are, in a sense, 
defined on the same footing, and 
the difference of them are attributed only to the difference of their specific star 
products.



Applying the scheme to the $D$-dimensional harmonic oscillator, we find that the axiom of the 
associativity of the star product excludes the presence of negative energy, 
and the general solutions of 
our scheme include just the same contents as the ordinary quantum mechanics.
This strongly suggests the equivalence between the ordinary quantum mechanics and the theory 
with proper choice of the star product in the abstract scheme  of this paper.


This paper is organized as follows.
In Section 2 six axioms are presented for the general framework 
for quantization. In Section 3 two choices of the star products are made, 
which correspond to the classical and the quantum mechanics, respectively.
In Section 4 solutions to the stargen-value equation are 
obtained in analytic and algebraic methods, and the role of the 
star product is discussed.
Section 5 is devoted for summary and discussions. 



 \vspace{.5cm}


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\section{General Framework for Quantization}


In this section we construct an abstract framework for physical theory, based on the Moyal quantization.
Some of its ingredients  are derived by the Moyal method from the ordinary quantum mechanics, 
and  others are not.

 
Let us consider an abstract  dynamical system described by real coordinates, $q_i, (i=1,...,N)$, 
and their canonical momenta, $p_i$.
We make the following assumptions:\\
(1) There exist a function space, ${\cal S}$, of complex valued and integrable functions of $(q, p)$, 
and a  map, $\star :{\cal S}\times{\cal S} \rightarrow {\cal S}$, satisfying the distributive and the 
associative laws and the Hermiticity property, 
$(f_1\star f_2)^* = f^*_2\star f^*_1$ for $f_i \in {\cal S}$, 
and the property $1\star f = f\star 1 = f$ for $f \in {\cal S}$. 
We postulate that the pair $(\star, {\cal S})$ of operator
 (we refer to star product hereafter) and the function space, having the above properties, is given, and 
physical states and observables are described by functions in ${\cal S}$.\\
(2) There exists a set of mutually commuting observables, ${\cal O}=\{O_1, O_2,..., O_r\}$, in the
sense of the star product. Here $r$ may or may not be finite.
 

We  can define the eigenvalue problem associated with the star product as in the 
theory of vector space.
For  functions, $g^i \in {\cal S}$, the function, $f_{\{\lambda\}\{\lambda'\}}\in{\cal S}$, satisfying
\begin{eqnarray}
          g^i\star f_{\{\lambda\}\{\lambda'\}} &=& \l^if_{{\{\lambda\}\{\lambda'\}}},\\
          f_{\{\lambda\}\{\lambda'\}}\star g^i &=& \lambda'^if_{{\{\lambda\}\{\lambda'\}}},
\end{eqnarray}
is called the phasespace eigenfunction\cite{moya}  of $g^i$, 
and $\{\lambda\}$($\{\lambda'\}$)  the set of left(right) eigen-values.
The above equations are sometimes called the stargenvalue equation with respect to $g^i$. 
For  observables, $O^i$, and an observable or a state function, $\phi\in{\cal S}$,
let us define
\begin{eqnarray}
    {\phi}^{(O)}_{\{\lambda\}\{\lambda'\}} := \int \frac{dqdp}{(2\pi\kappa)^N}~{\phi}(q, p)\star O_{\{\lambda'\}\{\lambda\}}(q, p), \label{mtrix0}
\end{eqnarray}
where $O_{\{\lambda'\}\{\lambda\}}$ is the eigenfunction of $O^i$. We  call it  matrix element of $\phi$ with respect to the observables, $O^i$. Note the orders of subscripts in the above equation.
In terms of these quantities we postulate the following assumptions concerning  observations.


\noindent (3) An ideal observation of a set of observables, $O_i$, results in  specific  set of values, 
and we assume it to be one of left-right eigenvalues of $O_i$:
\begin{eqnarray}
  O_i\star O_{\{\lambda\}\{\lambda\}} = O_{\{\lambda\}\{\lambda\}}\star O_i = \l_iO_{\{\lambda\}\{\lambda\}}.
\end{eqnarray}
(4) If the state is described by $\Psi$, the probability that one gets a set of values, $\l$, as a result of the observation of observables, $O^i\in{\cal S}$,  is
\begin{eqnarray}
        P(\Psi; O; \{\lambda\}) = \Psi^{(O)}_{\{\lambda\}\{\lambda\}}, \label{prob}
\end{eqnarray}
where r.h.s. is defined by (\ref{mtrix0}). We assume that the state function, $\Psi(q, p)$, and the 
eigenfunction, $O_{\{\lambda\}\{\lambda\}}\in{\cal S}$, are normalized as
\begin{eqnarray}
          \int \frac{dqdp}{(2\pi\kappa)^N}~O_{\{\lambda\}\{\lambda'\}}(q, p) &=& \delta_{\{\lambda\}\{\lambda'\}} ,    \label{norm1} \\  
          \int \frac{dqdp}{(2\pi\kappa)^N}~{\Psi}(q, p) &=& 1, \label{norm2}
\end{eqnarray}    
where r.h.s. of (\ref{norm1}) is the Kronecker delta or the delta function in the discrete or 
continuous spectra, respectively.\\
(5) After an ideal observation the state reduces to the one for which the probability of getting the same 
result by a succeeding observation of the same observable is unity.

The last assumption (5) may be seen as  slightly different from the usual one in the ordinary quantum mechanics where the state reduces, after the observation, to the corresponding eigenstate of the observable. 
The definition  of the {\it reduction of state} in (5), 
however, turns out to be more appropriate for the abstract setting of the theory including 
classical and quantum mechanics on the same footing, and it is in fact equivalent to that in the 
case of the ordinary quantum mechanics (see  next section).

For a non-ideal observation in which the observed value lies in 
a certain range, 
the probability is the sum of r.h.s. of eq.(\ref{prob}) over that range.
Thanks to the axiom of Hermiticity, the probability (\ref{prob}) is a real number.
Furthermore one can make it positive definite by a proper choice of the star product (see section 3).
  

Now, the physical state described by the state function, $\Psi\in{\cal S}$, which  satisfies the condition
\begin{eqnarray}
       \Psi\star\Psi = \Psi, \label{pure}
\end{eqnarray}
is called a pure state.  A linear combination  of state functions describing pure states is another state function.
Since the probability (\ref{prob}) is linear with respect to the state function, the linear combination does {\it not} respect the  `quantum mechanical' coherence, hence   
we call it an incoherent mixture of these states.


Next let us consider the time development of the state 
function. For that purpose the crucial quantity is 
the Moyal bracket of two function, $A, B\in{\cal S}$, 
defined by
\begin{eqnarray}
       \{A, B\}_M := \frac{1}{i\hbar}(A\star B - B\star A).
\end{eqnarray}
The time development of our abstract system is
determined by the following assumption:

\noindent (6) The physical state function,$\Psi$, satisfies
\begin{eqnarray}
      \frac{\partial}{\partial t} \Psi = -\{\Psi, H\}_M, \label{SM}
\end{eqnarray}
where $H\in{\cal S}$ is called the Hamiltonian of the system.
We call  (\ref{SM}) as the  {Schr\"{o}dinger}-Moyal equation.
The observables are assumed to be time independent.


The above assumptions complete the construction of the star 
quantization.
The general procedure  obtaining the physical predictions 
is as follows: First  one must define a suitable 
star product operation and the function space on which 
it acts, which satisfy the axioms (1)-(6).
This fixes the framework of the physics which may be 
classical or quantum mechanics or other possible 
unknown dynamics if exists.
Next  the physical system is defined by introducing 
a Hamiltonian which determines the time evolution of 
state functions.
An observation of an observable determines the 
initial state function through the axiom (5).
Solving the {Schr\"{o}dinger}-Moyal equation, one gets the state 
function at a later time. The probability of 
getting a specific value of an observable at that time is obtained 
by Eq.(\ref{prob}).




Finally let us mention another way of defining the 
time evolution.
The definition (\ref{SM}) corresponds to the  {Schr\"{o}dinger} picture of 
the ordinary quantum mechanics.
One can also define the time evolution  corresponding to the Heisenberg
representation where the state functions are time
independent, and the observables satisfy
\begin{eqnarray}
      \frac{\partial}{\partial t}O  = \{O, H\}_M, \label{HM}
\end{eqnarray}
which we call   the Heisenberg-Moyal equation.
In this paper we use the definition corresponding to the
{Schr\"{o}dinger} representation.



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\section{Choices of Star Products}

We must choose the star product satisfying the axioms of the 
previous section.
In this section we present two choices corresponding to  well-known 
physical schemes, {\it i.e.}, classical and quantum mechanics.\vspace{.5cm}

\noindent (1) Classical mechanics.\\
The star product is defined by
\begin{eqnarray}
    A\star B = AB + \frac{i\hbar}{2}\{A, B\}_P,  \label{SP_C}
\end{eqnarray}
where the bracket in r.h.s. is the Poisson bracket.
The Planck constant appearing in (\ref{SP_C}) has no observable effect as is shown below. 
In this choice of the star product the Moyal bracket coincides with the Poisson bracket.
The observables corresponding to the coordinates and the momenta are 
\begin{eqnarray}
         X(q, p) = q, \quad P(q, p) = p,  
\end{eqnarray}
respectively.
The left-right eigenstates of these observables are
\begin{eqnarray}
     X_{QQ}(q, p) = \delta(q - Q),\quad  P_{PP}(q, p) = \delta(p - P),\label{EF0}
\end{eqnarray}
respectively.
The probability of getting the value $Q_0$ when one   
observes  the observable $X$ on the state described by 
the state function
\begin{eqnarray}
     \Psi(q, p) = \delta(q - Q)\delta(p - P),  \label{SF1}
\end{eqnarray}
is calculated by (\ref{prob}), with the result
\begin{eqnarray}
    P(X;\Psi;Q_0) = \delta(Q - Q_0).
\end{eqnarray}
Similarly the probability of getting the value $P_0$ 
of  $P$ on the same state is
\begin{eqnarray}
    P(P;\Psi;P_0) = \delta(P - P_0).
\end{eqnarray}
Thus we conclude by the axiom (5) that the reduced state after the 
above observation is described by the state function (\ref{SF1}), and 
 by {\it neither} one of eigenfunctions in eq.(\ref{EF0}).           
Note that the two observables, $X$ and $P$, have simultaneous 
observed values with unit probability, though they do {\it not} commute 
in the sense of the star product.


The state function at a later time is obtained by solving the 
{Schr\"{o}dinger}-Moyal equation (\ref{SM}) with the initial condition (\ref{SF1}).
The result is
\begin{eqnarray}
       \Psi(q, p, t) = \delta(q - Q(t))\delta(p - P(t)),   \label{SF2}
\end{eqnarray}
where $Q(t), P(t)$ are solutions of the canonical equations of motion
\begin{eqnarray}
       \dot{Q} &=& \{Q, H\}_P,\\
       \dot{P} &=& \{P, H\}_P,
\end{eqnarray}
with the initial condition $Q(0)=Q, P(0)=P$.
This is verified by using $F(x)\delta'(x - y) = F(y)\delta'(x - y) - F'(y)\delta(x - y)$.

Thus we can say that the Moyal quantized theory with the star product (\ref{SP_C}) is 
equivalent to the classical mechanics, with unit probability.\vspace{.5cm}




\noindent (2) Quantum mechanics.\\
The star product is defined by
\begin{eqnarray}
   A\star B(q, p) &=& e^{\frac{i\hbar}{2}\sum_{i}\left(\frac{\partial}{\partial q_i}\frac{\partial}{\partial {p'_i}} - \frac{\partial}{\partial {q'_i}}\frac{\partial}{\partial p_i}\right)}A(q,p)B(q',p')\biggm|_{q=q', p=p'},
\end{eqnarray}
which was introduced \cite{moya}\cite{groe} in connection with the time evolution of 
the Wigner function.
The basic function space is a set of well-behaved functions of the phasespace, 
and it should be consistent with, among others, the associative law.

First we review the {\it formal} equivalence of the Moyal scheme and the ordinary 
quantum mechanics, then discuss a subtle point concerning  the associativity of the
star product.
In the ordinary quantum mechanics a state is described  by a vector in Hilbert space, and an
observable is represented by an operator acting to the state vector.
A function of the phasespace is uniquely determined by an operator through the Wigner-Weyl 
correspondence defined by 
\begin{eqnarray}
       A_W(q, p) &=& \int du~e^{-\frac{i}{\hbar}pu}\left<q+\frac{u}{2}\Biggm|\hat{A}\Biggm|q-\frac{u}{2}\right>,
\end{eqnarray}
where the state vectors are time-independent eigenvectors of coordinate operator, $\hat{q}$. 
Conversely, the matrix element of the operator is expressed in terms of the function as
\begin{eqnarray}
       \langle q|\hat{A}|q'\rangle &=& \int\frac{dp}{(2\pi\hbar)^N}~e^{\frac{i}{\hbar}p(q - q')}A_W\left(\frac{q + q'}{2}, p\right).
\end{eqnarray}
In this paper we restrict ourselves to  the case without constraints. (See \cite{HKM} 
for a preliminary work on the Moyal quantization in constraint system.)

It is an easy task to show
\begin{eqnarray}
         (\hat{A}\hat{B})_W = A_W\star B_W,  \label{AstarB}
\end{eqnarray}
by which various relations between the quantities in the ordinary quantum mechanics and 
the Moyal scheme are obtained. For example, the phasespace eigenfunction of 
an observable $A$ is given by
\begin{eqnarray}
         A_{nm} = (|n\rangle\langle m|)_W,
\end{eqnarray}
where $|n\rangle$ is the eigenvector of $\hat{A}$ with eigenvalue $n$.
The state function $\Psi(q, p)$ is given in terms of the state vector $|\psi\rangle$ as
\begin{eqnarray}
    \Psi(q, p) = (|\psi\rangle\langle\psi|)_W.
\end{eqnarray}


Expressing the matrix element (\ref{mtrix0}) in terms of the operator and the state vector, we have
\begin{eqnarray}
        B^{(A)}_{nm} &=& \int \frac{dqdp}{(2\pi\hbar)^N}~B(q, p)\star A_{mn}(q, p) \nonumber  \\
                     &=& \int \frac{dqdp}{(2\pi\hbar)^N}~(\hat{B}|m\rangle\langle n|)_W    \nonumber  \\
                     &=& \int \frac{dqdpdu}{(2\pi\hbar)^N}~~e^{-\frac{i}{\hbar}pu}\langle q+u/2|\hat{B}|m\rangle\langle n|q - u/2\rangle    \nonumber  \\
                     &=& \langle n|\hat{B}|m\rangle.
\end{eqnarray}
Thus it coincides with the matrix element of the ordinary quantum mechanics.\\



The probability of getting a value $\l$ by the observation of the observable $\hat{A}$ in the
state $|\psi\rangle$ is calculated by (\ref{prob}), and we get
\begin{eqnarray}
        P(\Psi; A; \l) &=& \int\frac{dqdp}{(2\pi\hbar)^N}~\Psi(q, p)\star A_{\lambda\lambda}(q, p) \nonumber  \\
                      &=& \int\frac{dqdp}{(2\pi\hbar)^N}~(|\psi\rangle\langle\psi|\lambda\rangle\langle\l|)_W \nonumber  \\
                      &=& \int\frac{dqdpdu}{(2\pi\hbar)^N}~e^{-\frac{i}{\hbar}pu}\langle q+u/2|\psi\rangle\langle\psi|\lambda\rangle\langle\l|q - u/2\rangle \nonumber  \\
                      &=& |\langle \l|\psi\rangle|^2,
\end{eqnarray}
which coincides with that of ordinary quantum mechanics.


This completes the formal proof of the equivalence of the Moyal scheme and the ordinary quantum mechanics. 
We comment on the associativity of the star product.
The associative law of the star product seems to  be guaranteed by the 
corresponding operator relation of the quantum mechanics through the 
relation (\ref{AstarB}).
But it is not always the case.
We know that the annihilation variable, $a$, corresponds to the 
annihilation operator, $\hat{a}$, which eliminates the ground state, $|0\rangle$. 
When the inverse of $\hat{a}$ exists, the product of the three 
operators,  $\hat{a}^{-1}, \hat{a}$ and $|0\rangle\langle 0|$, does not 
satisfy the associativity, because $\hat{a}$ eliminates $|0\rangle\langle 0|$.
Contrastingly,   we can define the function, $a^{-1}$, which is a well-behaved one 
except at the origin, in the 
framework on the function space.
The existence of $a^{-1}$ brings about unwanted state in the physical spectrum, 
but it cannot be excluded by the requirement of the normalization of the distribution 
functions.
With respect to the exclusion of unphysical states, the associativity of the 
star product plays a crucial role.
The definition of the star product and the defining function space satisfying the axioms in the 
previous section, completes the construction of the star quantization.
Therefore it is necessary to  exclude carefully such functions that violate the associativity. 
This is discussed in detail in the following section.

 


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\section{Application to Harmonic Oscillator}


In this section we demonstrate how the Moyal quantization 
procedure in section 2, which does not refer to the state vectors and operators of 
ordinary quantum mechanics, is 
applicable to the  harmonic oscillator in $D$-dimensions \cite{Koi}\cite{CUZ}. 
We show the correct 
physical spectrum consistent with the ordinary 
quantum mechanics. 
When we solve the stargen-value equation, we obtain solutions with the undesirable spectrum 
together with physical spectrum.
In order to exclude those undesirable solutions, the associativity of the star product plays 
an important role.


The Hamiltonian is given by
\begin{eqnarray}
     H = \frac12\sum_{i=1}^{D}(p_i^2 + q_i^2).
\end{eqnarray}
For the sake of later convenience we introduce the creation and annihilation coordinates:
\begin{eqnarray}
     a_i = \frac{1}{\sqrt{2}}(q_i + ip_i), \quad
     a_i^* = \frac{1}{\sqrt{2}}(q_i - ip_i).
\end{eqnarray}
The star product is written in terms of these coordinates as
\begin{eqnarray}
   A\star B(a, a^*) &=& e^{\frac{\hbar}{2}\sum_i\left(\frac{\partial}{\partial a_i}\frac{\partial}{\partial a'^*_i} - \frac{\partial}{\partial {a^*_i}}\frac{\partial}{\partial a'_i}\right)}A(a,a^*)B(a',a'^*)\biggm|_{a=a', a^*=a'^*}.
\end{eqnarray}
In order to solve the left stargenvalue equation, $H\star f = Ef$, let us introduce the variables
\begin{eqnarray}
     z_i = \frac{4}{\hbar}a_ia_i^*,
\end{eqnarray}
and transform the independent variables from $(a_i, a_i^*)$ to $(a_i, z_i)$.\\
Then the left stargen-value equation becomes
\begin{eqnarray}
  \left[H - E - \hbar\sum_{i=1}^{D}\left(\frac{a_i}{2}\frac{\partial}{\partial a_i} + \left(a_i\frac{\partial}{\partial a_i} + 1 +  z_i\frac{\partial}{\partial z_i}\right)\frac{\partial}{\partial z_i}\right)\right]f = 0.\label{sv2}
\end{eqnarray}
A special solution with no dependence on $a_i$ is given by
\begin{eqnarray}
    f_0 = e^{-\sum_iz_i/2}, \qquad E = \frac{\hbar D}{2},\label{f_0}
\end{eqnarray}
which is interpreted as the ground state function.
The general solution is obtained by setting  the $a_i$ dependence of $f$ as $f\sim (a_i)^{k_i}$ with 
integer constants, $k_i$, which is the base functions of the Laurent expansion of the solution. 
Put
\begin{eqnarray}
  f(a_i, z_i) = f_0(z_i)\left(\prod_i(a_i)^{k_i}\right)F(z_i),
\end{eqnarray}
then (\ref{sv2}) becomes
\begin{eqnarray}
    \left[ \sum_{i=1}^{D}\left(z_i\frac{\partial^2}{\partial z_i^2} + (k_i + 1 - z_i)\frac{\partial}{\partial z_i} \right)  + \frac{E}{\hbar} -   \frac{D}{2} \right]F = 0.
\end{eqnarray}
This equation is separable with respect to all $z_i$, so if we put $F = \prod_{i=1}^{D}F_i(z_i)$,
then we get
\begin{eqnarray}
          z_iF''_i + (k_i + 1 - z_i)F'_i + n_iF_i = 0, \label{sv3}\\
        \sum_{i=1}^{D}n_i = \frac{E}{\hbar} -   \frac{D}{2}. \qquad \qquad \label{sum}
\end{eqnarray}
where $n_i$ are the separation constants.

In order to find the solutions to Eq.(\ref{sv3}), we discuss the solutions to the confluent hypergeometric equation in general. The parameters $n_i$ appearing there can be non-integers at this stage. 
The quantum mechanics is characterized by quantum numbers or discrete integers. How the integers characterizing the quantum mechanics emerge in the Moyal quantization is the present question to be discussed here. 

The solutions to the confluent hypergeometric equation,
\begin{eqnarray}
z\frac{d^2u}{dz^2}+(\gamma-z)\frac{du}{dz}-\alpha u=0,
\end{eqnarray}
are written symbolically by the confluent $\tilde P$ function defined by
\begin{eqnarray}
u=\tilde P \left\{
		\matrix{\infty & 0 & {}          \cr
                  \overbrace{0 \quad \quad \alpha} & 0 & z \cr
                  ~~1  \quad \gamma-\alpha  & 1-\gamma & {}
                }
            \right\}.\label{cnflP}
\end{eqnarray}
This shows that the explicit four solutions are given by either analytic functions (except poles) around $z=0$ or those around $z=\infty$, which are the singularities of the equation. Two of them are independent at most.


The solutions around $z=0$ are given by $u_1=F(\alpha,\gamma;z)$ and $u_2=z^{1-\gamma}F(\alpha-\gamma+1,2-\gamma;z)$ for $\gamma\notin Z_+$, where $Z_+$($Z_-$) is the set of positive(negative) integers. 
(The case $\gamma\in Z_+$ will be discussed shortly.)
Here the hypergeometric function of fluent type $F(\alpha,\gamma;z)$ is defined  by
\begin{eqnarray}
F(\alpha,\gamma;z)=\sum_{l=0}^{\infty}\frac{(\alpha)_n}{(\gamma)_n}\frac{z^n}{n!},
\end{eqnarray}
where $(\alpha)_n =  \alpha(\alpha+1)\cdots(\alpha+n-1)$.
We restrict ourselves to the case with an integer $\gamma$ for the application of the present paper, 
which constitute the basis of the Laurent expansions as mentioned before.

Note that these solutions play the role of the probability distribution functions when multiplied by $f_0=e^{-z/2}$. 
Therefore we should impose a condition that the solutions should diverge more weakly 
than $e^{z/2}$ at $z=\infty$ so that their integrations over phasespace are finite, 
which turns out to restrict the parameter region. 
A sufficient condition is the truncation of the power series of $z$, and it is in fact necessary, since otherwise they diverge as $e^{z}$ at  $z=\infty$.
This restricts the parameters such that either $\alpha\in Z_-\cup\{0\}$ or $\alpha - \gamma + 1\in Z_-\cup\{0\}$.
Introducing $n=-\alpha$ and $k=\gamma-1$ (in accordance with (\ref{sv3})), the solutions are written in terms of the associated Laguerre 
polynomials, $L^{(k)}_n(z)$, as
\begin{eqnarray}
    u_1 &=& F(-n,k+1;z) = L_n^{(k)}(z), \qquad \qquad \qquad {\rm for}\quad n \ge 0,  \\
    u_2 &=& z^{-k}F(-(n+k),1-k;z)=z^{-k}L_{n+k}^{(-k)}(z), \quad {\rm for}\quad  n \ge -k, \label{sol_01}
\end{eqnarray}
where $k\le -1$.
These two solutions look different, but they are in fact identical in the range, $n \ge -k \ge 1$, 
which is seen by the identity
\begin{eqnarray}
z^{-k}L_{n+k}^{(-k)}(z)=(-1)^k\frac{n!}{(n+k)!}L_n^{(k)}(z),\qquad  {\rm for}\quad n, \quad n+k \ge 0.
\end{eqnarray}

In the case $\gamma\in Z_+$, the solutions are $\hat{u}_1=F(\alpha,\gamma;z)$ and $\hat{u}_2$, where 
$\hat{u}_2 = u_1\log{z} + z^{1-\gamma}F^*$ for  $\alpha\in Z_-\cup\{0\}\cup\{1\}$ or 
$\alpha - \gamma + 1\in Z_+\cup\{0\}$, and $\hat{u}_2 =z^{1-\gamma}F(\alpha-\gamma+1,2-\gamma;z)$ 
otherwise. 
Here $F^*$ is an analytic function around $z=0$.
The solution containing $F^*$ is excluded, since it diverges as $e^{z}$ at $z=\infty$. Then, the admissible solutions are
\begin{eqnarray}
        \hat{u}_1 &=& L_n^{(k)}(z), \qquad \qquad {\rm for}\quad n \ge 0, \\
        \hat{u}_2 &=& z^{-k}L_{n+k}^{(-k)}(z), \quad {\rm for}\quad 1 > n \ge -k,
\end{eqnarray}
where $k\ge 0$.


Other solutions shown in Eq.(\ref{cnflP}) are those around $z=\infty$, which might include solutions 
having essential singularities at $z=0$. One of which diverges as $e^{z}$ at $z=\infty$, and 
we exclude it. 
Another solution is obtained by setting $u=(1/z)^{\alpha}f(1/z)$. By solving the  equation for $f(1/z)$, 
we obtain polynomial solutions obtained above.



We now summarize the solutions to Eq.(\ref{sv3}). The solutions for $n_i \ge 0$ are the associated Laguerre polynomials, $L^{(k_i)}_{n_i}(z_i)$, 
and ones for $0 > n_i \ge -k_i$ are $z^{-k_i}L^{(-k_i)}_{n+k}(z_i)$. 
Defining
\begin{eqnarray}
         \tilde{L}^{(k)}_{n}(z) = \left\{ \begin{array}{c}  L^{(k)}_{n}(z) \qquad \qquad \qquad {\rm for}\quad n \ge 0, ~~~~~~~{}\\
              z^{-k}L^{(-k)}_{n+k}(z) \qquad \qquad {\rm for}\quad 0 > n \ge -k, \end{array} \right. \label{Ltild}
\end{eqnarray}
the general solution of the left stargen-value equation, with total energy
\begin{eqnarray}
      E_{left} = \left(n + \frac{D}{2}\right)\hbar,  \label{leftE}
\end{eqnarray}
is written as a linear combination of 
\begin{eqnarray}
    W(n_1,.., n_D;k_1,.., k_D) = f_0(z)(a_1)^{k_1}\cdot\cdot\cdot(a_D)^{k_D}\tilde{L}^{(k_1)}_{n_1}(z_1)\cdot\cdot\cdot\tilde{L}^{(k_D)}_{n_D}(z_D), \label{solution}
\end{eqnarray}
where $k_i$ are arbitrary integers  and  $n_i$ are integers greater than or equal to $-k_i$, satisfying 
$\sum_{i=1}^{D}n_i = n$. (As for the case $D=1$ see \cite{fair}.)




The parameters $n_i$ designate the quantum number for the sub-Hamiltonian, $H_i=(q_i^2+p_i^2)/2$, of $i$-th dimension, and $n_i+k_i$ are those of the right stargen-value equation, $f\star H = Ef$. 
The solution to the right stargen-value equation is also given by (\ref{solution}),  with 
the energy eigenvalue
\begin{eqnarray}
      E_{right} = \left(n + k + \frac{D}{2}\right)\hbar,\label{rightE}
\end{eqnarray}
where $k = \sum_i k_i$. 

If one observes only the total energy, the left and right total eigen-values should coincide, hence $k=0$. This state is an incoherent 
mixture of states, each of which can take a {\it negative} eigen-value of $H_i$   in the case of $D\ge 2$ as well as  positive one. Although these negative eigen-value states are not observable as far as only the total energy is concerned,
the states with negative energy should be excluded in order to be consistent 
with the ordinary quantum mechanics.




These solutions with negative eigenvalues might be studied from a different viewpoint. We are able to construct solutions in an algebraic way.
Because of the relations, 
\begin{eqnarray}
         a_i\star a^*_j - a^*_j\star a_i =  \hbar\delta_{ij}, \quad
         a_i\star f_0 = f_0\star a^*_i = 0, \quad 
         H = \sum_ia_ia^*_i, 
\end{eqnarray}
we see that the function
\begin{eqnarray}
     \tilde{W}(n_1,.., n_D;k_1,.., k_D) = \prod_{i=1}^D(a_i^*\star)^{n_i}f_0\prod_{i=1}^D(\star a_i)^{n_i + k_i} \label{solution2}
\end{eqnarray}
is the  eigenfunction of  the Hamiltonian, 
with the left and right eigenvalues (\ref{leftE}) and (\ref{rightE}), respectively.
In fact one can verify that $\tilde{W}$ coincides with $W$ defined by (\ref{solution}) except the normalization constant (see Appendix).



In this algebraic construction of solutions, one might think that the creation and annihilation functions by use of the star product are completely isomorphic to those by operators through the relation (\ref{AstarB}). However, here appear peculiar functions whose correspondence can not be found in the ordinary operator formalism. The parameters $n_i$ and $n_i+k_i$ in Eq.(\ref{solution2}) can take negative integers, which correspond to the solutions with negative energy mentioned above. 



We study functions that correspond to ``the inverse'' of creation and annihilation operators. We denote the reciprocals of the creation and annihilation functions, $a$ and $a^*$, as $a^{-1}$ and $a^{*-1}$, respectively, implying that they are the inverses in the sense of the star product. Those ``inverse" functions are well defined except at the phasespace origin, $z=2(q^2+p^2)/\hbar=0$, and  satisfy
\begin{eqnarray}
a^{-1} \star a= a \star a^{-1}=1,\qquad 
a^{*-1}\star a^*=a^* \star a^{*-1}=1.
\end{eqnarray}
Then, it is easy to show that $a^*$ raises the superscript of the Laguerre function, 
and $a^{*-1}$ lowers the superscript as 
\begin{eqnarray}
(a^*\star )^kL^{(k')}_n(z)=a^{*k}L_n^{(k'+k)}(z), \qquad 
(a^{*-1}\star )^kL^{(k')}_n(z)=a^{*-k}L_n^{(k'-k)}(z),
\end{eqnarray}
for $k\ge 0$. This admits us to write as  $(a^{*-1}\star)^k = (a^*\star)^{-k}$ for $k\ge 0$ 
(and $(a^{-1}\star)^k = (a\star)^{-k}$) at least if they act to the Laguerre polynomials. 
This shows that the introduction of functions $a^{-1}$ and $a^{*-1}$ enables one 
to adjust the parameters $n_i$ and $k_i$ freely in (\ref{solution2}) beyond its innate positive values.



Although the ``inverse" functions are useful for practical use, as shown above (also see Appendix), 
they have peculiar features coming from the singularity at the origin in the phasespace. 
Since $a^{*-1}=4a/(\hbar z)$ is the inverse of $a^*$, it behaves like a function $a$ in lowering the eigenvalues, and $a^{-1}$ is similar to $a^*$. However, the peculiar features of $a^{-1}$ and $(a^*)^{-1}$ appear in computation by using them. The commutation relation is given by
\begin{eqnarray}
[a^{-1},(a^*)^{-1}]_{\star} := a^{-1}\star (a^*)^{-1}-(a^*)^{-1}\star a^{-1}=\frac{4}{\hbar}\sum_{n=1}^{\infty}(2n!) \Big(\frac{2}{z}\Big)^{2n}.
\end{eqnarray}
Since r.h.s.  has no range of convergence, the function in l.h.s. belongs to the outside of the admissible range of 
the function space.
Correspondingly to the relation $a \star f_0(z)=0$, we find that
\begin{eqnarray}
(a^*)^{-1} \star f_0=(a^*)^{-1} f_0\Big(1-1+1-1+\cdots \Big).
\end{eqnarray}
Since r.h.s. does not converge, $a^{*-1}$ has not meaningful star operation at least to the specific 
function, $f_0$.



The most serious trouble is that the associativity rule of the star products, when the ``inverse" functions are involved, does not hold as in the following example;
\begin{eqnarray}
f_0(z)=(a^{-1} \star a)\star f_0(z) \ne a^{-1} \star (a \star f_0(z))=0.
\end{eqnarray}
We thus find that the origin of emergence of all solutions with negative eigenvalues which are forbidden in the 
ordinary quantum mechanics is the existence of functions which violate the associative law.
In other words the axiom of the associative law forbids the states corresponding to 
the solutions with negative eigenvalues.



Note that we cannot exclude the negative eigenvalue solutions solely by the requirement of the absence of 
the ``infrared divergences'' at $z=0$, which might be yielded in the phasespace integrations.
This is seen, {\it e.g.}, by the fact that the phase space integration of the state function, 
containing the factor $z^{-1}L^{(-1)}_{n+1}(z)$ with  $n = -1$,  
vanishes (without singularity at $z=0$) in consistence with the normalization condition (\ref{norm1}).



Summary of this section is that the physically and mathematically consistent solution to the left and right eigenvalue equation is obtained if and only if we restrict the range of the function space such that the 
star product operation is consistently defined. This brings about the physically admissible 
non-negative values of $n_i$ and $n_i+k_i$ in (\ref{solution}).




 \vspace{.5cm}



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\section{Summary and Discussion}

In this paper, we have pursued a quantization scheme, where 
a quantum theory is described entirely in terms of the function space without any 
reference to operators in the Hilbert space.
Demonstrating the procedure obtaining the solutions of the stargen-value equation 
in the $D$-dimensional harmonic oscillator, 
we showed the importance of the role played by  the star product operation, which enables us to 
extract the proper content of the theory.
We have shown that a choice of the star product in the general framework leads to classical or quantum mechanics.
We find that not all the solutions of the stargen-value equation are proper as physical solutions. In an example of D-dimensional harmonic oscillator case, we find that negative energy solutions emerge. However, the inclusion of such solutions brings about the violation of the associativity rule of the star product.
Interestingly, such solutions are shown to be related to the inverse of the creation and annihilation functions.


In the algebraic construction of D-dimensional harmonic oscillator solutions, integer properties of 
$n_i$ and $n_i+k_i,(i=1,2,\cdots,D)$ are assumed at first.
However, the fact that the creation(annihilation) coordinate raises(lowers) the superscript, $k$, in the 
Laguerre polynomials, $L^{(k)}_n$, permit us to write as ${a^*}^k\star = (a^*\star)^k$, where $k$ 
is {\it not} restricted to integer numbers. It is difficult to treat such non-integer numbers in algebraic manipulations. We exclude the negative energy state corresponding to negative $k$ by requiring consistency of the star product operations.
Contrastingly, when we solve the confluent hypergeometric equation in the analytic method, the quantum feature characterized by an integer number is a result derived from the requirement 
of finiteness of the probability distribution functions of the solutions.
However, the exclusion of the negative energy state is difficult solely in the analytic method.



 In one dimension, only the solutions expressed by the Laguerre function $L_n^{(k)}$ with $k=0$ are allowed. These are the functions of $z$ and so real. In the dimensions larger than or equal to two, the solutions with $k_i \ne 0$ are allowed, since the harmonic oscillator Hamiltonian can distinguish only the sum of the quantum number in each dimension, or the eigenvalues are degenerate. Note that these are functions of real variables $z_i,(i=1,2,\cdots,D)$
 together with complex variables $a_i,(i=1,2,\cdots,D)$, and so they are no more real. Although the sum of complex conjugate functions can be real, the coefficients can not be determined by Hamiltonian only. We need other observables which resolve the degeneracy in order to fix them.

A theory can be quantized once a consistent star product and a function space are defined. A quantization via star product is promising for the quantization of not only the flat space as in the Moyal quantization but also the curved space.
Lately the deformation quantization theory is studied extensively both by physicists and  mathematicians\cite{deform}. We expect to extract physically interesting results by future study in this direction 

 \vspace{1cm}


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\noindent {\Large {\bf Appendix}}\vspace{.5cm}




\noindent (1) {\it Laguerre polynomials}\\
We verify that the two expressions, (\ref{solution}) and 
(\ref{solution2}), for the solution of the stargen-value equation 
with respect to the total energy, coincide for integer $n$ greater than or equal to $0$, up to the normalization constant.
First we consider the one dimensional case. We compute the quantity given by
\begin{eqnarray}
               W_n=(a^{*}\star)^nf_0(z)(\star a)^n,
\end{eqnarray}
with $n \ge 0$. 
Introducing $V_n$ by $W_n=\hbar^nn!f_0V_n$, we find the iteration equation 
\begin{eqnarray}
     V_n &=& \frac{1}{n\hbar f_0}(a^* \star f_0V_{n-1}\star a) \nonumber  \\
         &=& \frac{1}{n}\left(z\frac{d^2}{dz^2}+ (1-2z)\frac{d}{dz}+(z-1) \right)V_{n-1}.
\end{eqnarray}
The solution with $V_0=1$ is give by the Laguerre polynomials, $L_n(z)$, 
and we obtain
\begin{eqnarray}
              W_n = (-1)^n\hbar^nn!f_0(z)L_n(z).
\end{eqnarray}
Successive operations of $a^*\star$ to $W_n$ from left and $\star a$ from right 
lead to 
\begin{eqnarray}
         (a^{*}\star)^nf_0(z)(\star a)^m =  (-1)^nn!\hbar^n(2a)^{m-n}f_0(z)L^{(m-n)}_n(z), \label{Wnm}
\end{eqnarray}
where $L^{(k)}_n(z) = (1 - d/dz)^kL_n(z)$ are the associated Laguerre polynomials. 
The Hermiticity property, $(f_1\star f_2)^* = f^*_2\star f^*_1$, defined in section 2 ensures that 
complex conjugate of (\ref{Wnm}) also satisfies the relation, since $n!L_n^{(m-n)}(z)=(-z)^{n-m}m!L_m^{(n-m)}(z)$(see Appendix (4) for the proof) holds.
Extending the above relation to higher dimensions, we obtain the relation between the two 
expressions, (\ref{solution}) and (\ref{solution2}), given by
\begin{eqnarray}
\tilde{W}(n_1,.., n_D;k_1,.., k_D) = (-\hbar)^n(n_1!...n_D!)W(n_1,.., n_D;k_1,.., k_D). 
\end{eqnarray}


We can express $\tilde{W}$ in terms of only Laguerre polynomials. Introduce the variables
\begin{eqnarray}
z_{ij}&=&\frac{4a_ia^*_j}{\hbar},
\end{eqnarray}
it is modified as
\begin{eqnarray}
\tilde{W} &=&(a_1)^{k_1}\cdot\cdot\cdot(a_D)^{k_D}L^{(k_1)}_{n_1}(z_1)\cdot\cdot\cdot L^{(k_D)}_{n_D}(z_D)\nonumber \\
&=&(a_1/a_D)^{k_1}\cdot\cdot\cdot(a_{D-1}/a_D)^{k_{D-1}}L^{(k_1)}_{n_1}(z_1)\cdot\cdot\cdot L^{(k_D)}_{n_D}(z_D)\nonumber \\
&=&(\frac{4a_1a^*_D}{\hbar z_D})^{k_1}\cdot\cdot\cdot(\frac{4a_{D-1}a^*_D}{\hbar z_D})^{k_{D-1}}L^{(k_1)}_{n_1}(z_1)\cdot\cdot\cdot L^{(k_D)}_{n_D}(z_D)\nonumber \\
&=&(z_{1D})^{k_1}\cdot\cdot\cdot(z_{D-1D})^{k_{D-1}}L^{(k_1)}_{n_1}(z_1)\cdot\cdot\cdot  L^{(k_{D-1})}_{n_{D-1}}(z_{D-1})
\times (z_D)^{-\sum k_i}L^{(-\sum k_i)}_{n-\sum n_i}(z_D),\nonumber \\
\end{eqnarray}
where $\sum k_i=\sum_{i=1}^{D-1} k_i$ and $\sum n_i=\sum_{i=1}^{D-1} n_i$. 
Using
\begin{eqnarray}
z^{-\alpha}L_n^{(-\alpha)}(z)=(-1)^{\alpha}\frac{(n-\alpha)!}{n!}L_{n-\alpha}^{(\alpha)}(z)
\end{eqnarray}
we have
\begin{eqnarray}
(z_D)^{-\sum k_i}L^{(-\sum k_i)}_{n-\sum n_i}(z_D)=(-1)^{\sum k_i}\frac{(n-\sum n_i-\sum k_i)!}{(n-\sum n_i)!}L_{n-\sum n_i-\sum k_i}^{(\sum k_i)}(z_D).
\end{eqnarray}
Since
\begin{eqnarray}
L_n^{(-n)}(x)=(-1)^n\frac{x^n}{n!},
\end{eqnarray}
we also have
\begin{eqnarray}
(z_{iD})^{k_i}=\frac{n!}{(-1)^{k_i}}L_{k_i}^{(-k_i)}(z_{iD}).
\end{eqnarray}
Finally  we obtain
\begin{eqnarray}
    f = f_0\sum_{\{n_i\},\{k_i\}}c^{k_1k_2\cdot\cdot\cdot k_{D-1}}_{n_1n_2\cdot\cdot\cdot n_{D-1}}\Big(\prod_{i=1}^{D-1}L_{k_i}^{(-k_i)}(z_{iD})L_{n_i}^{(k_i)}(z_i)\Big)L_{n-\sum n_i-\sum k_i}^{(\sum k_i)}(z_D),
\end{eqnarray}
where the summations are taken over $i\ne D$: 
\begin{eqnarray}
\{n_i\}=\{n_1,\cdots,n_{D-1}\}, \quad \{k_i\}=\{k_1,\cdots,k_{D-1}\}. 
\end{eqnarray}


 \vspace{.5cm}

\noindent (2) {\it Normalization}\\
The energy eigenfunction, $W$, can be normalized to satisfy the
pure state condition (\ref{pure}).
The normalization constant  is most easily obtained 
by the algebraic expression of $\tilde{W}$.
The normalized one, $W_N$ is
\begin{eqnarray}
       W_N(n_1,.., n_D;k_1,.., k_D) = \prod_{i=1}^D\sqrt{\frac{n_i!}{(n_i+k_i)!}}\tilde{W}(n_1,.., n_D;k_1,.., k_D).
\end{eqnarray}
We have not fixed the normalization constant of the ground state function, $f_0$.
This is done by imposing $f_0\star f_0 = f_0$, which requires analytic calculations.
Explicitly, for the function $f=exp(-(q^2+p^2)/\kappa)$, we have
\begin{eqnarray}
      f\star f(Q, P) &=& \int \frac{dxdx'dpdp'}{(\pi\hbar)^2}~e^{\frac{2i}{\hbar}(xp' - x'p) - \frac{1}{\kappa}\left((Q + x)^2 + (Q + x')^2 + (P + p)^2 + (P + p')^2\right)} \nonumber  \\
    &=& e^{-\frac{2\kappa}{\kappa^2+\hbar^2}(Q^2+P^2)}\int \frac{d\tilde{x}d\tilde{x}'d\tilde{p}d\tilde{p}'}{(\pi\hbar)^2}~e^{-\frac{1}{\kappa}(\tilde{x}^2 + \tilde{x}'^2) - \frac{\kappa^2+\hbar^2}{\kappa\hbar^2}(\tilde{p}^2 + \tilde{p}'^2)},
\end{eqnarray}
where 
$\tilde{x} = x - i\kappa p'/\hbar + Q$,   
$\tilde{x}' = x' + i\kappa p/\hbar + Q$, 
$\tilde{p} = p -  \kappa\hbar^2(iQ/\hbar - P/\kappa)/(\kappa^2+\hbar^2)$ and 
$\tilde{p}' = p' +  \kappa\hbar^2(iQ/\hbar + P/\kappa)/(\kappa^2+\hbar^2)$.
After the Gauss integrations we obtain
\begin{eqnarray}
   f\star f(Q, P) = \frac{\kappa^2}{\kappa^2+\hbar^2}e^{-\frac{2\kappa}{\kappa^2+\hbar^2}(Q^2+P^2)}.
\end{eqnarray}
Thus the normalized ground state function is given by
\begin{eqnarray}
       f_0 = 2^De^{-\sum_iz_i/2}.
\end{eqnarray}

 \vspace{.5cm}

 
\noindent (3) {\it {Schr\"{o}dinger} equation}\\
It has been known that the Wigner function, which is the solution to the stargen function, can be constructed from the solution of the 
ordinary {Schr\"{o}dinger} equation. We apply the construction to the harmonic oscillator case.
The solution of the one dimensional {Schr\"{o}dinger} equation, $\psi'' - \frac{1}{\hbar^2}(q^2 - 2E)\psi = 0$, 
with the energy, $E = (n+1/2)\hbar$, is
\begin{eqnarray}
      \psi_n(q) = \psi_0(q)H_n\left(\sqrt{2/\hbar}~q\right),
\end{eqnarray}
where $H_n$ is the Hermite polynomials and $\psi_0(q)$ is the ground state 
wave function
\begin{eqnarray}
      \psi_0(q) = (\hbar\pi)^{-1/4}e^{-\frac{q^2}{2\hbar}}. \label{groundSr}
\end{eqnarray}
Hence the one dimensional phasespace eigenfunction is
\begin{eqnarray}
   \Big(|n\rangle\langle n+k|\Big)_W &=& \int du~e^{-\frac{i}{\hbar}pu}\psi_n^*(q + u/2)\psi_{n+k}(q - u/2) \nonumber  \\
     &=& (-1)^nn!\left(\frac{2}{\sqrt{\hbar}}\right)^k\!\!e^{-z/2}~a^{k}L_n^{(k)}(z).  \label{wfbsr}
\end{eqnarray}
In deriving the above formula we have used the identity,
\begin{eqnarray}
      \frac{d^{n+m}}{dy^ndy'^m}e^{-yy' + i\omega(y - y')}|_{y=y'}= (-1)^mn!e^{-y^2}(y + i\omega)^{m-n}L_n^{(m-n)}(y^2 + \omega^2).
\end{eqnarray}

The $D$-dimensional phasespace eigenfunction is the product of (\ref{wfbsr}) over all dimensions, and 
is proportional to $W$ in eq.(\ref{solution}).
Of course the state with negative energy, {\it i.e.}, $n<0$ or $n+k<0$, is forbidden by the 
normalization condition of the wave functions, which gives polynomial solutions in (\ref{groundSr}).

 \vspace{.5cm}

\noindent (4) {\it Identity of $L^{(k)}_n$}\\
We prove the following identity, used in Section 4, by the algebraic method,
\begin{eqnarray}
z^{k}L_n^{(k)}(z)=(-1)^{\alpha}\frac{(n+k)!}{n!}L_{n+k}^{(-k)}(z).
\end{eqnarray}
Multiplying both sides by $f_0(z)=e^{-z/2}$, we can prove this in the following way,
\begin{eqnarray}
f_0(z)z^k L_n^{(k)}(z) &=&\frac{(-1)^n}{n! \hbar^n(2a)^k}\Big( \frac{4aa^*}{\hbar}\Big)^{k}
(a^* \star)^n f_0 (\star a)^{n+k}\nonumber \\
% &=&\frac{(-1)^n (2a^*)^k}{n! \hbar^{n+k}}
% (a^* \star)^n f_0 (\star a)^{n+k}\nonumber \\
&=&\frac{(-1)^n (2a^*)^k}{n! \hbar^{n+k}}
(a^* \star)^{-k} (a^* \star)^{n+k}f_0 (\star a)^{n+k}\nonumber \\
&=&\frac{(-1)^n (n+k)!}{n!}f_0 L_{n+k}^{(-k)}(z).
\end{eqnarray}


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\bibitem{fair}D.~Fairlie, Proc. Camb. Phil. Soc. {\bf 60}~(1964),~581.
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\end{thebibliography}

\end{document}
