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\title {Representations of Wick CCR algebra}

\author { D.P. Proskurin}
\date { }
\def\address{Institute of Mathematics, Ukrainian National Academy of
Sciences,
Tereshchinkivs'ka, 3, Kiev, 252601, Ukraine}

\def\email{prosk@imath.kiev.ua}
\sloppy

\begin {document}

\maketitle

\section{Introduction}
    
In this note we consider the Wick algebra connected with
classical CCR algebra and it representations.

Recall that classical CCR algebra with $d$
generators is the *-algebra generated by set $\{a_i\ ,a_i^*\
,i=1,\ldots ,d\}$ and the basic relations hold:
     \begin{align*}
     a_i^*a_i & = 1+a_i a_i^*\\
     a_i^*a_j & = a_j a_i^*\ \ \ i\neq j\\
     a_i a_j  & = a_j a_i
     \end{align*}
It is known that Fock representation is the unique
``good'' *-representation of these relations.

Let us consider CCR algebra as a Wick algebra (see [1]).
     \begin{definition}
     Let $\{T_{ij}^{kl}\}\subset\mathbb{C},\
     T_{ij}^{kl}=\bar{T}_{ji}^{lk} $. Wick algebra with coefficients
$\{T_{ij}^{kl}\}$ is the *-algebra with generators $\{a_i^*,\ a_i\
,i=1,\ldots ,d\}$ which satisfy following relations:
     \[
a_i^* a_j = \delta_{ij}1 +\sum_{k,l=1}^d T_{ij}^{kl} a_l a_k^*
     \]
     \end{definition}

Since, to transform CCR algebra to Wick algebra one must consider
only relations between $a_i,\ a_j^*$.

     \begin{remark}
  Let us note that the set of elements $\{a_i a_j - a_j a_i =
A_{ij}\}$ generates two sided ideal $I_2$ in the subalgebra generated by $
a_i $ with the distinguish property:
     \[
   a_i^* I_2 \subset I_2 + I_2 <a_k^*>
     \]
  Ideals with such property is called Wick ideals (see [1]).
Moreover, ideal $I_2$ is generated by noncommutative homogeneus polinomials
of degree $2$. So, in this case we deal with the homogeneous Wick
ideal of degree $2$, i.e. quadratic ideal.
     \end{remark}

Thus, the classical CCR may be obtained as a quotient of the Wick
CCR algebra by the largest quadratic ideal.

\section {Homogeneous ideals}
Let $\mathcal{H}=<e_1,\ldots ,e_d>$ -- finite dimensional Hilbert
space. Then algebra generated by $a_i$ may be identefied with full
tenzor algebra $\mathcal{T}(\mathcal{H})$, and Wick algebra identifies
with quotient $\mathcal{T}(\mathcal{H},\mathcal{H}^*)$ by two-sided
ideal generated by basic relations.

Following [1] introduce operators:
\begin{gather*}
T\colon  \mathcal{H} \otimes \mathcal{H}\mapsto \mathcal{H}
 \otimes\mathcal{H}, \quad  T e_{k}\otimes e_{l} =
\sum_{i,j} T_{ik}^{lj}e_{i}\otimes e_{j},\\
T_{i}\colon  \mathcal{H}^{\otimes n}\mapsto\mathcal{H}^{\otimes n},
\quad 
T_{i}=\underbrace{1\otimes \cdots\otimes 1}_{i-1}\otimes T
\otimes\underbrace{1\otimes\cdots\otimes 1}_{n-i-1},
\\
R_{n}\colon \mathcal{H}^{\otimes n}\mapsto \mathcal{H}^{\otimes n},
\quad  
R_{n}=1+T_1+T_1 T_2+\cdots + T_1 T_2\cdots T_{n-1},
\end{gather*}

Let us present now several propositions (see [2]).
\begin{proposition}
Let $P\colon\mathcal{H}^{\otimes {n}}
\mapsto\mathcal{H}^{\otimes {n}}$ - projection. Then
$I_{n}= \langle P \mathcal{H}^{\otimes n} \rangle$ detemine Wick ideal
if and only if the following conditions hold:

$1.$ $R_{n}P=0,$

$2.$ $[1\otimes (1-P)]T_{1}T_{2}\cdots T_{n}[P\otimes 1]=0.$

Moreover, if $T$ satisfies the braid relation $T_{1}T_{2}T_{1}=
T_{2}T_{1}T_{2}$ and  $P$ projection on $\ker R_{n}$,
then condition $2$ holds automatically.
\end{proposition}
     \begin{remark}
In the case of CCR operator $T$ have the following form:
     \begin{gather*}
     T e_i\otimes e_i = e_i\otimes e_i \\
     T e_i\otimes e_j = e_j\otimes e_i
     \end{gather*}
And $T$ satisfies the braid relation. So, in this case $Ker R_n$
generates the largest homogeneous ideal of degree $n$
     \end{remark}

In the general case we have not effective method to describe
generators of homogeneous ideals of higher degrees. But in the braided
case with restriction on norm of $T$ we have the following information 
(see [2]):
     \begin{theorem}
     Let $T$ satisfies the braid condition and $-1\le T \le 1$ than
     \[
     Ker R_3 = (1-T_1 T_2)(Ker R_2\otimes\mathcal{H})
     \]
     \end{theorem}

In  other words, we can describe generators of cubic ideal then
the quadratic ideal is known.

Using this theorem it is easy to verify that in the case of CCR
     \[
     I_3 = <a_i A_{ij} - A_{ij}a_i>
     \]
where $A_{ij}$ are generators of quadratic ideal.

\section{Representations.}
In this section we describe the classes of unitary equevalent
*-representations of Wick CCR algebra with additional assumption that
these representations annihilate ideal $I_3$. I.e. we consider the
representations of the quotient of CCR by $I_3$.

So, consider the *-algebra of the form:
     \begin{align*}
     a_i^* a_i & =1+ a_i a_i^* ,\ i=1,\ldots,d\\
     a_i^*a_j & = a_j a_i^*\\
     a_i(a_l a_k - a_k a_l) & =(a_la_k - a_k a_l)a_i\quad
i,k,l=1,\ldots,d
     \end{align*}
In the following we denote $a_l a_k - a_k a_l= A_{lk}$. It is easy
follows from the basic relations that for any $i,k,l$
     \[
     a_i^* A_{lk} = A_{lk} a_i^*\ ,\quad A_{lk}^*A_{lk}=A_{lk} A_{lk}^*
     \]
and $A_{lk}$ are in the centre. Consequently in the irreducible
representation $\pi(\cdot)$ $\pi(A_{lk})=\lambda_{lk}\in\mathbb{C}$.
Let us note that the set
$\{\lambda_{lk} - A_{lk}\}$ determines a Wick ideal and we have the
interesting fact:
annihilation of the cubic ideal implies annihilation of the seria of
Wick ideals generated by some polinomials of degree $2$.

In the following theorem we show that for any family of complex number
$\{\lambda_{lk}\}$ there exists the unique representation of Wick CCR with
     \[
     a_l a_k-a_ka_l=\lambda_{lk}\mathbf{1} \ \ (1)
     \]
\begin{theorem}
     Let $\{\lambda_{lk}\}\subset\mathbb{C}$, then the following operators
determine irreducible representation of Wick CCR with additional condition
(1) and every irreducible representation is unitary equivalent to the
presented one:
\begin{align*}
A_1   = & \widehat{a}\otimes_{i=2}^d\mathbf{1} \\
A_k
= &\prod_{i=1}^{k-1}(1+\mid\lambda_{ik}^{(i)}\mid^2)^{\frac{1}{2}}\cdot
\otimes_{j=1}^{k-1}\mathbf{1}\otimes \widehat{a} \otimes_{j=k+1}^d
\mathbf{1}+\\
+ & \sum_{i=1}^{k-1}\lambda_{ik}^{(i)}\prod_{j=1}^{i-1}
(1+\mid\lambda_{jk}^{(j)}\mid^2)^{\frac{1}{2}}\otimes_{j=1}^{i-1}
\mathbf{1}\otimes\widehat{a}^* \otimes_{j=i+1}^d\mathbf{1}
\end{align*}
where the numbers $\{\lambda_{ij}^{(k)},\ k\le i<j\}$ are determined
inductively:
\begin{align*}
\lambda_{ij}^{(1)} & = \lambda_{ij}\\
\lambda_{ij}^{(k+1)} & = \frac{\lambda_{ij}^{(k)}}
{(1+\mid\lambda_{ki}^{(k)}\mid^2)^{\frac{1}{2}}
(1+\mid\lambda_{kj}^{(k)}\mid^2)^{\frac{1}{2}}}
\end{align*}
and operator $\widehat{a}$ is the unique irreducible solution of the
relation $\widehat{a}^*\widehat{a}-\widehat{a}\widehat{a}^*=1$.
\end{theorem}

\begin{thebibliography}{99}

\bibitem{JSW}
{\sc P. E. T. J{\o}rgensen,\sc L. M. Schmitt and R. F. Werner.}
{Positive representations of general commutation relations
allowing Wick ordering}. {\em J. Funct. Anal.} \textbf{134},
no.~1 (1995), 33--99.

\bibitem {Pr}
{\sc D.P. Proskurin.} {Homogeneous ideals in Wick *-algebras}.
{\em Proc. AMS} {\em Proc. of AMS} \textbf{126}, no~1 (1998),
3371--3376.

\bibitem{} {\sc Yu.\ S. Samo\u\i{}lenko and E.Ye. Va\u\i{}sleb}, 
Representations of operator relations by unbounded operators and 
multi-dimensional dynamical systems, 
{\em Ukrain.\ Math.\ J}.\ {\bf 42} (1990), 1011-1019.

\end{thebibliography}
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\address
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\bigskip\noindent{\em e-mail: }\email



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