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% Jacobson generators, Fock representations and statistics of sl(n+1)
% T.D. Palev and J. Van der Jeugt
% submitted to J. Math. Phys.
%   JMP # 1-0768; 2nd revised version
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\begin{center}
{\Large \bf
Jacobson generators, Fock representations\\
and statistics of $sl(n+1)$}\\[5mm]
{\bf T.D.\ Palev}\footnote{Permanent address~:
Institute for Nuclear Research and Nuclear Energy,
Boul.\ Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria;
E-mail~: tpalev@inrne.bas.bg}\\
International Centre for Theoretical Physics,
34100 Trieste, Italy\\[2mm]
{\bf J.\ Van der Jeugt}\footnote{E-mail~:
Joris.VanderJeugt@rug.ac.be.}\\
Department of Applied Mathematics and Computer Science,
University of Ghent,
Krijgslaan 281-S9, B-9000 Gent, Belgium.
\end{center}

\vskip 10mm
\noindent Corresponding author~: J.\ Van der Jeugt.
Tel~: ++32 9 2644812. Fax~: ++32 9 2644995.
E-mail~: Joris.VanderJeugt@rug.ac.be
\vskip 3mm
\noindent Running title~: Jacobson generators of $sl(n+1)$
\vskip 10mm

\begin{abstract}
The properties of $A$-statistics, related to the class $A$ of
simple Lie algebras (Palev, T.D.: Preprint JINR E17-10550 (1977);
hep-th/9705032), are further investigated. The description of
each $sl(n+1)$ is carried out via generators
$a_1^\pm,\ldots,a_n^\pm$, which we call Jacobson generators.
With respect to these generators, the definition of
a Fock space of $sl(n+1)$ is given. It is proved
that the Fock spaces $W_p$, $p\in \N$ are the simple symmetric
(finite-dimensional) modules of $sl(n+1)$. The Pauli principle of
the underlying statistics is formulated. Within each $W_p$
operators $B(p)_i^\pm=a_i^\pm/ {\sqrt p}$ $(i=1,\ldots,n)$, called
quasi-Bose creation and annihilation operators (CAOs), are
defined. Then $\displaystyle\lim_{p\to \infty} B(p)_i^\pm
=B_i^\pm$, where $B_i^\pm$ are ordinary Bose CAOs. Therefore
$A$-statistics appears as an approximation of Bose statistics
with CAOs acting in finite-dimensional state spaces. We indicate
that the $p=1$ quasi-Bose operators $B(1)_1^\pm,\ldots,
B(1)_n^\pm$ are natural operators for the description of
hard-core Bose models and of the related Heisenberg spin models.
We argue that (up to a certain natural assumption) $A$-statistics
can be interpreted as an exclusion statistics.
\end{abstract}
\vskip 2mm

\renewcommand{\thesection}{\Roman{section}}

\section{Introduction}

During the last two decades quantum statistics became a field
of increasing interest among field theorists and condensed matter
theorists.  Various new statistics were suggested, leading to
generalizations or deviations from some of the first principles
in quantum physics, such as the Heisenberg commutation relations,
the Pauli exclusion principle and the commutativity of
space-time.

The literature on the subject is vast, especially in the part
related to quantum groups~\cite{dr87,ji85,fa89,ma88,wo87}.
In a paper entitled ``Twisted
Second Quantization"~\cite{pu89} Pusz and Woronowicz introduced multimode
deformed Bose creation and annihilation operators (CAOs), covariant under
the action of the quantum group $U_q[sl(n)]$ (for $n$ pairs of them).
Another deformation with commuting modes of CAOs was proposed
in~\cite{ma89}; the link between them was established in~\cite{ku90}.  A
third deformation, which for one mode of CAOs was known for many
years~\cite{co72}, the so called quon algebra~\cite{gr91}, was defined as
an associative algebra, subject to relations $ a_i^-a_j^+ -q
a_j^+a_i^-=\delta_{ij}. $ This generalization (note that no relations
among only creation operators or among only annihilation operators are
required) was in the origin of a model proposed for a verification of
small violations of Bose-Fermi statistics in quantum field theory
(QFT)~\cite{gr90}.  The quon statistics, which in the classification of
Doplicher, Haag and Roberts~\cite{do71} belongs to the class of ``infinite
statistics", was studied by several authors~\cite{go89} from different
points of view (see~\cite{gr99} for further discussions and references).

Recently string theory was also involved in discussions on
quantum statistics, the latter related to its prediction that
Heisenberg's uncertainty principle has to be corrected at
distances of order of the Plank length $k_P=10^{-32}$~cm.
Consequently there emerges an absolute minimum uncertainty in the
measuring of any length~\cite{ve86}. These predictions motivated several
authors to search for model independent arguments, leading to the same
conclusions as string theory does (we refer to~\cite{ga95} for a survey on
the subject). In particular it has been shown that the above results can
be reproduced on a purely kinematical level with appropriate deformations
of the Heisenberg commutation relations~\cite{ma93,ke94,hi96,ad98}, i.e.,
of canonical quantum statistics. In all such cases the coordinates do not
commute (see also ~\cite{luk97,sti00,nair01,chai01}), a result which is
consistent with the spirit of non-commutative geometry~\cite{co86}.

Turning to condensed matter physics we refer to
anyons, ``particles" with fractional statistics (FS) in two-dimensional
(2D) systems~\cite{le77}. The theoretical studies of this and other
noncanonical statistics were strongly  pushed  forward after the discovery
of the fractional quantum Hall effect (FQHE) in two-dimensional electron
gases~\cite{ts82}. Its theoretical explanation led Laughlin~\cite{la83} to
the conclusion that there exist quasiparticles carrying fractional
electric charges. The statistics of these particles (we write ``particles"
for the elementary excitations, the ``quasiparticles", when no confusion
can arise) also turned out to be fractional statistics~\cite{ha84}.

A further breakthrough in the area of quantum statistics was
marked with the 1991 paper of Haldane~\cite{ha91}, who proposed a
generalized version of the Pauli exclusion principle. For only
one kind of identical particles this new statistics, now called
(fractional) exclusion statistics (ES), asserts that the change
$\Delta d$ in the dimension $d$ of the single-particle Hilbert
space is defined via the relation 
\beq {\Delta d}=-g \cdot{\Delta N}.  
\label{e1-1} 
\eeq 
Here $\Delta N$ is  an allowed
increase of the number of particles. The constant $g$ is called
an exclusion statistics parameter.

Our approach to quantum statistics is strongly influenced by the
ideas of Wigner, outlined in his 1950's work ``Do the equations of
motion determine the quantum mechanical commutation
relations?"~\cite{wi50}.
This was the first paper where it was clearly indicated
that the canonical quantum statistics may, in principle, be
generalized in a logically consistent way. Wigner demonstrated
this on the example of a one-dimensional oscillator with a
Hamiltonian ($m=\omega=\hbar=1$) $H={1\over 2}(p^2 + q^2)$.
Abandoning the requirement $[p,q]=-i$, Wigner was searching for
all operators $q$ and $p$, such that the ``classical" equations of
motion ${\dot p}=-q$, ${\dot q}=p$ are identical with the
Heisenberg equations ${\dot p}=-i[p,H]$, ${\dot q}=-i[q,H]$.
Apart from the canonical solution he found infinitely many other
solutions.  Let ${\sqrt 2}B_1^\pm=q \mp i p$. It turns out~\cite{pa81}
that all these different operators satisfy one and the same triple
relation, namely~(\ref{e1-3}) below with $i=j=k=1,$ (see the end of this
Introduction for the notation)~: 
\beq
[\{B_i^\xi,B_j^\eta\},B_k^\varepsilon]=
\delta_{ik}(\varepsilon-\xi)B_j^\eta +
\delta_{jk}(\varepsilon-\eta)B_i^\xi, \quad i,j,k \in \N, \quad
\xi,\eta,\varepsilon= \pm,~\pm 1. 
\label{e1-3} 
\eeq 
The operators
$B_i^\pm$, $i=1,2,\ldots$ are para-Bose (pB) operators, discovered by
Green~\cite{gr53} three years later as a possible generalization of
statistics of tensor fields in QFT. Thus the infinitely many different
solutions found by Wigner were in fact the Fock representations of one
pair of para-Bose operators.

It is known that the linear span of all operators
$B_i^\xi$, $\{B_j^\eta, B_k^\varepsilon\}$ is a Lie
superalgebra~\cite{om76}
isomorphic to the orthosymplectic Lie superalgebra
$osp(1/2n)$ for $i,j,k=1,\ldots,n$ and $ \xi,\eta,\varepsilon=
\pm$~\cite{ga80}. The para-Bose operators constitute a basis in the odd
subspace of this superalgebra and generate it. Consequently the
representation theory of $n$ pairs of pB operators is completely
equivalent to the representation theory of $osp(1/2n)$. Hence Wigner found
all Fock representations of $osp(1/2)$ long before Lie superalgebras (and
supersymmetry) became of interest in physics.


Similarly, any $n$ pairs of para-Fermi CAOs
$F_1^\pm,F_2^\pm,\ldots,F_n^\pm$~\cite{gr53}, defined by relations
\beq
[[F_i^\xi,F_j^\eta],F_k^\varepsilon]= {1\over 2}
\delta_{jk}(\varepsilon-\eta)^2 F_i^\xi -{1\over
2}\delta_{ik}(\varepsilon-\xi)^2 F_j^\eta, \quad i,j,k \in \N,\quad
\xi,\eta,\varepsilon= \pm,~\pm 1,
\label{e1-4}
\eeq
generate the Lie algebra $so(2n+1)$~\cite{ka62, ry63}. The key observation
here is that both $so(2n+1)$ and $osp(1/2n)$ belong to class $B$ of the
basic Lie superalgebras in the classification of Kac~\cite{ka79}. Hence
parastatistics (and in particular Bose and Fermi statistics) appear as
particular Fock representations of Lie superalgebras from one and the same
class, the Lie superalgebras of class $B$. In this sense Green's
parastatistics could be called $B$-(para)statistics.

The clarification of the mathematical structure, hidden in
parastatistics, provides a natural background for further
searches of new quantum statistics. One such possibility is to
consider deformations of parastatistics, namely deformations of
$so(2n+1)$ and $osp(1/2n)$ in the sense of quantum groups. We
refer to~\cite{pa98} for discussions and results along this line.
Note that parastatistics associated with $so(2n+1)$ (parafermions~:
finite dimensional representations) and with $osp(1/2n)$ (parabosons~:
infinite dimensional representations) are not related to the known
correspondence between $so(2n+1)$ and 
$osp(1/2n)$~\cite{RittenbergScheunert,Sergeev} where only finite 
dimensional representations play a role.

In another approach, initiated in~\cite{pa76}, it was shown that to each
infinite class $A$, $B$, $C$ and $D$ of simple Lie algebras there
corresponds quantum statistics. Examples from classes $A$ and $B$ of
proper Lie superalgebras are also available. We have in mind Wigner
quantum systems (WQSs)~\cite{pa81}. Some such systems possess quite
unconventional physical features. As an example we mention the
$(n+1)$-particle WQS, based on the Lie superalgebra $sl(1/3n)$ from class
$A$~\cite{pa97}. This WQS exhibits a quark-like structure~: the composite
system occupies a small volume $V$ around the centre of mass and no
particles can be extracted out of $V$. Moreover the geometry within $V$ is
noncommutative. Another example is the $osp(3/2)$ WQS from class
$B$~\cite{pa94}. It leads to a picture where two spinless point particles,
``curling" around each other, produce an orbital (internal angular)
momentum $1/2$, a result which cannot be obtained in canonical quantum
mechanics.

The present paper is also in the frame of quantum statistics. We
study further the (microscopic) properties of $A$-statistics,
introduced in~\cite{pa76} (see also~\cite{pa77}), namely the
statistics of Lie algebras $A_n\equiv sl(n+1)$, $n=1,2,\ldots$.
Since Refs.~\cite{pa76} and~\cite{pa77} are not available as
journal publications, we review the main issues of $A$-statistics
in Sections~2,~3 and partially in Section~4 omitting most of the
proofs.

We begin (Section~2) by recalling how the Lie algebra $sl(n+1)$
can be described via generators $a_1^\pm,\ldots,a_n^\pm$ and
relations, see~(\ref{e2-5}). These generators, which we call
Jacobson generators (JG), provide an alternative to the Chevalley
description of $sl(n+1)$.

The Fock modules of the Jacobson generators, extended also to
$gl(n+1)$-modules, are defined and classified in Section~3. It is
shown how they can be selected out of all irreducible
$gl(n+1)$-modules on the ground of natural physical requirements,
see Definition~\ref{def1}. All Fock modules $W_p$ are
finite-dimensional and are labelled by one positive integer $p\in
\N$. More precisely, the signature (the highest weight) of the
$gl(n+1)$-module $W_p$ is $(p,0,0,\ldots,0)$, or equivalently,
the representation of $gl(n+1)$ in $W_p$ is a symmetric
representation. The definition of the Fock spaces is given in
such a way that within $W_p$ each generator $a_i^+$ (resp.\
$a_i^-$) can be interpreted as an operator creating (resp.\
annihilating) a ``particle" in a state~$i$ (on the orbital $i$).


The Pauli principle for $A$-statistics in $W_p$ is formulated in
Section~4 (Corollary~\ref{cor3}). It states that any number of
particles up to $p$ and no more than $p$ can be distributed in an
arbitrary way along the orbitals. This restriction leads to
properties typical for exclusion statistics \cite{ha91}. We show
that under a certain natural assumption the $A$-statistics can be
interpreted as an exclusion statistics in the form of Wu
\cite{wu94}.

Next, in Section~5, representation dependent creation and
annihilation operators $B(p)_i^\pm=a_i^\pm/ {\sqrt p}$
$(i=1,\ldots,n)$ in $W_p$ are defined. We prove that in an
appropriate topology $\displaystyle\lim_{p\to
\infty} B(p)_i^\pm$ = $B_i^\pm$, where $B_1^\pm,\ldots,B_n^\pm$
are Bose creation and annihilation operators. The operators
$B(p)_1^\pm,\ldots,B(p)_n^\pm$ possess also other Bose-like
properties. For these reasons $B(p)_1^\pm,\ldots,B(p)_n^\pm$ are
referred to as quasi-Bose operators (of order $p$), the
representations of $sl(n+1)$ and $gl(n+1)$ in $W_p$ as quasiboson
representations and the statistics as quasi-Bose statistics.

The Jacobson CAOs $a_1^\pm,\ldots,a_n^\pm$ are ``bosonized" in
Section~6. These operators are expressed via $n$ pairs of Bose CAOs
$B_1^\pm,\ldots,B_n^\pm$. The related realization of $gl(n+1)$ in
$W_p$ turns to be the known Holstein-Primakoff realization~\cite{ok75}.

In Section~7 we point out that the quasi-Bose operators can also
be of more general interest. On the example of a two-leg $S=1/2$
Heisenberg spin ladder we show that the Bose realization of the
Hamiltonian~\cite{go94, su98} together with the restrictions
selecting the physical subspace simply means that the
Bose operators related to each site  have to be replaced by
quasi-Bose operators of order $p=1$. This conclusion is of a more
general nature. It holds for any hard-core Bose
model~\cite{fi89}, since the $p=1$ particles are hard-core bosons
(Proposition~\ref{prop5}).

The final Section~8 is devoted to some conclusions and
discussions.

Throughout the paper we use the following abbreviations and
notation (some of them standard)~:
\begin{itemize}
\item[]
JGs -- Jacobson generators;
\item[]
CAOs -- creation and annihilation operators;
\item[]
UEA -- universal enveloping algebra;
\item[]
$\N$ -- all positive integers;
\item[]
$\Z_+$ -- all non-negative integers;
\item[]
$[a,b]=ab-ba,\qquad \{a,b\}=ab+ba$;
\item[]
$\oplus$ -- direct sum of linear
spaces or of Lie algebras.
\end{itemize}

\section{Jacobson generators of $sl(n+1)$}
\setcounter{equation}{0}

The $sl(n+1)$-statistics, including $n=\infty$, was introduced
in~\cite{pa76} (see also~\cite{pa77})
as an alternative way for quantization of
spinor fields in quantum field theory. Refs.~\cite{pa76} and~\cite{pa77}
are not available as journal publications. Therefore here and in Section~3
we outline the main features of this statistics in somewhat more details.

In order to define the Jacobson generators,
it is convenient to consider
$sl(n+1)$  as a subalgebra of the Lie algebra $gl(n+1)$.  The
universal enveloping algebra $U[gl(n+1)]$ of the latter can be
defined as an associative algebra with unity of the generators
$\{e_{ij}|i,j=0,1,\ldots,n\}$ subject to the relations
\beq
[e_{ij},e_{kl}]=\delta_{jk}e_{il}-\delta_{il}e_{kj}.
\label{e2-1}
\eeq
Then $gl(n+1)$ is a subalgebra of $U[gl(n+1)]$, considered as a
Lie algebra, with generators $e_{ij}$, $i,j=0,1,\ldots,n$ and
commutation relations~(\ref{e2-1}).

The Cartan subalgebra $H'$ of $gl(n+1)$ has a basis
$h_{i}\equiv e_{ii}$, $i=0,1,\ldots,n$. Let $h^0,h^1, \ldots,h^n$
be the dual basis, $h^i(h_j)=\delta_{ij}$.
The root vectors of both $gl(n+1)$ and $sl(n+1)$ are
$e_{ij}$, $i\ne j=0,1,\ldots,n$.  The root of each $e_{ij}$ is
$h^{i} -h^{j}$. Then
\beq
sl(n+1)=\hbox{span}\{e_{ij}, e_{ii}-e_{jj}|i\ne j=0,1,\ldots,n\}.
\label{e2-2}
\eeq
The Jacobson generators (JGs) of $sl(n+1)$ are part of the
generators $e_{ij}$, namely 
\beq
a_i^+=e_{i0},\qquad a_i^-=e_{0i},\qquad i=1,\ldots,n.
\label{e2-3}
\eeq
The correspondence with their roots reads
\beq
a_i^\pm \; \leftrightarrow \; \mp(h^0-h^i), \q i=1,\ldots,n,
\label{e2-4}
\eeq
and therefore the JGs $a_i^+$ ($a_i^-$) are negative
(positive) root vectors with respect to the natural ordering
$h^0,h^1,\ldots,h^n$. Since any other root is a sum of the roots of
$a_j^-$ and $a_i^+$, namely
\[
h^i-h^j=(h^0-h^j)-(h^0-h^i),\q  i\ne j=1,\ldots,n,
\]
the JGs~(\ref{e2-3}) generate
$sl(n+1)$ in the sense of a Lie algebra.

{}From~(\ref{e2-1}) and~(\ref{e2-3}) one derives the triple relations 
\bea
(a) && [[a_i^+,a_j^-],a_k^+]=\delta_{kj}a_i^+ + \delta_{ij}a_k^+ , \nn\\
(b) && [[a_i^+,a_j^-],a_k^-]=-\delta_{ki}a_j^- - \delta_{ij}a_k^-,
\label{e2-5}\\ 
(c) && [a_i^+,a_j^+]=[a_i^-,a_j^-]=0. \nn 
\eea 
On the
contrary, setting $e_{ij}-\delta_{ij}e_{00}=[a_i^+,a_j^-]$, one derives
from~(\ref{e2-5}) the commutation relation between all $sl(n+1)$
generators $e_{ij}$, $e_{ii}-e_{jj}$, $i\ne j=0,1,\ldots,n$.

The above description of $sl(n+1)$ via generators and relations
is a particular case of describing Lie algebras via Lie triple
systems, initiated by Jacobson~\cite{ja49}. For this reason the
elements $a_i^\pm$ are referred to as Jacobson generators of
$sl(n+1)$.

The presentation of simple Lie algebras in terms of generators and
relations (as illustrated here by the JGs for $sl(n+1)$) is a
topic of interest to physicists~\cite{BGLS,LS}. In fact, any
simple finite dimensional Lie algebra can be generated by two
elements only; this was first claimed by N.\ Jacobson 
(see also~\cite{Kuranishi}) and proved
in~\cite{BO}. For this reason, these two generators are sometimes
referred to as ``Jacobson's generators''. We shall not use this
terminology here, in order not to confuse with the Jacobson
generators defined in this section. Observe that the nature of
the relations becomes extremely complicated when using only these
two generators. A simpler description was given in~\cite{GL}, in
terms of three generators (by adding a third generator to the
earlier two, which are related to a principal $sl(2)$ embedding).
The description of $sl(n)$, and other simple Lie algebras or
superalgebras, is explicitly given in~\cite{GL} in terms of such
three generators and the corresponding relations (which are not
triple relations, but of higher degree). Such a description (with
three generators and a number of higher order relations) is
appropriate to present the so-called Lie algebra of matrices of
complex size $gl(\lambda)$, see~\cite{GL,LS} , since for
$gl(\lambda)$ or $sl(\lambda)$ there is no analogue of a Cartan
matrix or of Chevalley generators.

In the present paper, however, we shall only use the presentation
of $sl(n+1)$ by means of the JGs~(\ref{e2-3}) and the triple
relations~(\ref{e2-5}), which is essentially a description by
means of Jacobson's Lie triple systems (LTSs)~\cite{ja49}. The
approach by means of LTSs was further developed to the
$\Z_2$-graded case by Okubo~\cite{ok94}. Let us be more concrete.
By definition~\cite{ja49} a Lie triple system ${\cal L}$ is a
subspace of an associative algebra $U$, so that ${\cal L}$ is
closed under the ternary operation $\omega : {\cal L}\otimes
{\cal L}\otimes {\cal L}\rightarrow {\cal L}$  defined as $\omega
(a\otimes b \otimes c)= [[a,b],c]$, $a,b,c\in {\cal L}$. The
definiton of a Lie supertriple system (equivalent to the
definition in~\cite{ok94}) is similar. The difference is that
${\cal L}$ is a $\Z_2$-graded subspace of an associative
superalgebra $U$ and the commutators in the definition of
$\omega$ are replaced by supercommutators.

The JGs of $sl(n+1)$ are closely related to the above definition.
More precisely, let  ${\cal L}_{sl}$ be the linear span of
the generators~(\ref{e2-3}) and $U_{sl}$ be the associative unital
algebra of the JGs subject to the relations~(\ref{e2-5}). Then
${\cal L}_{sl}$ is a subspace of $U_{sl}$. Moreover
$\omega: {\cal L}_{sl}\otimes {\cal L}_{sl}\otimes
{\cal L}_{sl}\rightarrow {\cal L}_{sl}$ as a consequence
of~(\ref{e2-5}). Hence ${\cal L}_{sl}$ is a Lie triple system
with a basis consisting of the JGs~(\ref{e2-3}) and $U_{sl}$ is the
UEA of $sl(n+1)$. Similarly, the linear span ${\cal L}_{pf}$
of para-Fermi CAOs $F_1^\pm,F_2^\pm,\ldots,F_n^\pm$ together
with the associative algebra $U_{pf}$ of these operators
(subject to the relations~(\ref{e1-4})) is another example of
a LTS.  Hence the para-Fermi operators $F_1^\pm,\ldots,F_n^\pm$
could be called JGs of $so(2n+1)$. In the same spirit
the para-Bose operators $B_1^\pm,\ldots,B_n^\pm$ are JGs
of $osp(1/2n)$.

{}From a purely algebraic point of view the
Jacobson generators provide an alternative
to the Chevalley description of $sl(n+1)$, $so(2n+1)$ and
$osp(1/2n)$. The JGs of  $so(2n+1)$ and  $osp(1/2n)$
however (contrary to the Chevalley generators)
have a direct physical significance. These operators
extend the canonical Fermi and Bose statistics
to the more general parastatistics.
Below we proceed to show that
the JGs of $sl(n+1)$ also introduce a new quantum
statistics, different from Bose and Fermi statistics and their
generalization -- parastatistics.
This statistics is intrinsically related to class $A$
of simple Lie algebras in the same way as the para-Fermi statistics is
related to class $B$ of simple Lie algebras.

Typically the ``commutation relations" between the creation and the
annihilation operators (or the related position and momentum
operators in case of finite degrees of freedom) are derived from
(more precisely, are required to be consistent with) {\it the
main quantization equation}
\beq
[H, a_i^\pm]=\pm \e_ia_i^\pm, \label{e2-6}
\eeq
where $H$ is the Hamiltonian and $i$ replaces all indices that
may appear (momentum, spin, charge, etc.). In quantum field
theory~(\ref{e2-6}) expresses the translation invariance of the field (in
infinitesimal form). In quantum mechanics the same equation appears as a
compatibility condition (in the sense of Wigner~\cite{wi50}) between the
Heisenberg equations of motion and the classical equations, if the system
has a classical analogue (for more details see~\cite{pa81,oh82}). There
are certainly several other conditions to be satisfied (Galilean or
relativistic invariance, causality, etc.; we refer to~\cite{pa97} for
discussions in case of noncanonical quantum mechanics). The possibility
for choosing different statistics essentially depends upon the way one
represents the Hamiltonian $H$. We are going to illustrate this on the
example of para-Fermi statistics.

Consider a nonrelativistic free field locked in a
finite volume. In the case of a Fermi field the Hamiltonian $\H$ is
written in a normal-product form
\beq
\H=\sum_i \e_i f_i^+ f_i^-, \label{e2-7}
\eeq
so that the energy of the vacuum is zero. Here $f_i^+$ ($f_i^-$)
are Fermi creation (annihilation) operators~: $ \{f_i^\xi,
f_j^\eta\}={1\over 4}(\xi-\eta)^2\delta_{ij}$, $\xi, \eta=\pm$
or $\pm 1$. Then~(\ref{e2-6}) holds,
\beq
[\H, f_i^\pm]=\pm \e_i f_i^\pm , \label{e2-8}
\eeq
and each $f_i^\xi$ can be interpreted as an operator creating
($\xi=+$) or annihilating ($\xi=-$) a particle,
i.e.\ a fermion with energy $\e_i$. Eq.~(\ref{e2-8})
is not fulfilled however, if the Fermi operators in~(\ref{e2-7}) are
replaced by para-Fermi operators~(\ref{e1-4})~:
for $H=\sum_i \e_i F_i^+ F_i^-$ the equation
\beq
[H, F_i^\pm] = \pm \e_i F_i^\pm  \label{e2-9}
\eeq
does not hold. Why? In order to answer this question using
proper Lie algebraic language assume that the sum in~(\ref{e2-7}) is
finite (finite number of Fermi oscillators),
\beq
\H=\sum_{i=1}^n \e_i f_i^+ f_i^-. \label{e2-10}
\eeq
This is only an intermediate step. The considerations below
remain valid for $n=\infty$. Recall now that any $n$ pairs of
Fermi CAOs generate a particular Fermi representation of the Lie
algebra $so(2n+1)\equiv B_n$, whereas the para-Fermi operators
$F_1^{\pm},\ldots,F_n^{\pm}$ are (representation independent)
generators of $so(2n+1)$~\cite{ka62, ry63}.
Eq.~(\ref{e2-8}) is not preserved,
when passing to other representations of $B_n$, because $H$ is
not an element from $B_n$ and hence $[H, F_i^\pm]$ in the LHS
of~(\ref{e2-9}) is not a representation independent commutator.  This
observation suggests also the answer~: one has to rewrite~(\ref{e2-10}) in
a representation independent form.  In order to achieve this,
represent~(\ref{e2-10}) in the following identical form~: 
\beq 
\H={1\over
2}\sum_{i=1}^n \e_i ([f_i^+,f_i^-] + \{f_i^+,f_i^-\}).   
\label{e2-11}
\eeq
Consider the Lie algebra generated from $f_1^\pm,\ldots,f_n^\pm$ and
$\{f_i^+,f_i^-\}$. Since $\{f_i^+,f_i^-\}=1$, we obtain a representation
of the Lie algebra $B_n \oplus I$,
where $I$ is the one-dimensional
center. Now $\H  \in B_n \oplus I $
and therefore the commutation
relations~(\ref{e2-8}) hold for any other representation of
$B_n\oplus I$.
In other words, if we substitute $f_i^\pm \rightarrow
F_i^\pm$ and $\{f_i^+,f_i^-\} \rightarrow {\hat p}$ in~(\ref{e2-11}),
i.e.~set 
\beq 
H={1\over 2}\sum_{i=1}^n \e_i ([F_i^+,F_i^-] + {\hat p}),
\label{e2-12} 
\eeq 
where ${\hat p}$ is a generator of the center $I$, then
the quantization condition~(\ref{e2-8}) will be fulfilled for any
representation of $B_n\oplus I$
and in particular for the para-Fermi
operators~(\ref{e1-4})~: $[H, F_i^\pm]=\pm \e_i F_i^\pm$. The requirement
${\hat p}|0\rangle=p|0\rangle$, $p\in \N $ (and $F_i^-
F_j^+|0\rangle=\delta_{ij}p|0\rangle$, $F_i^- |0\rangle=0$), leads to a
representation with an order of the (para)statistics $p$~\cite{gr65}. Then
the energy of the vacuum is also zero.

We shall now apply a similar approach for the algebra $sl(n+1)$.
Let $E_{ij}$, $i,j=0,1,\ldots,n,$ be the $(n+1)\times (n+1)$ matrix
units.
The map $\pi : e_{ij} \rightarrow E_{ij}$, $i,j=0,1,\ldots,n,$
gives a representation of $gl(n+1)$ (usually referred to as
defining or identity representation). Its restriction to
$sl(n+1)$ gives a representation of
$sl(n+1)$. The operators $ A_i^+=E_{i0}$,
$A_i^-=E_{0i}$, $i=1,2,\ldots,n$ satisfy the triple
relations~(\ref{e2-5}). Set
\beq
{\hat H}=\sum_{i=1}^n \e_i A_i^+A_i^-.  
\label{e2-14}
\eeq
Then
\beq
[{\hat H},A_i^\pm]=\pm \e_i A_i^\pm. 
\label{e2-15}
\eeq
Hence $A_i^\xi$ can be interpreted as an operator creating $(\xi=+)$ or
annihilating $(\xi=-)$ a particle (quasiparticle, excitation) with energy
$\e_i$ for any $i=1,\ldots,n$. The representation $\pi$ is an analog of
the Fermi representation of para-Fermi statistics.

The commutation relations~(\ref{e2-15}) do not hold for other
representations of
$sl(n+1)$. In order to extend the class of
admissible representations we rewrite the Hamiltonian~(\ref{e2-14}),
like in the Fermi case, in the following identical form
\beq
{\hat H}=\sum_{i=1}^n \e_i ([A_i^+,A_i^-]+E_{00}).  
\label{e2-16}
\eeq
The Lie algebra generated from the operators $A_1^\pm,\ldots,A_n^\pm$ and
$E_{00}$ is $gl(n+1)$ (in the representation $\pi$).
Since ${\hat H} \in gl(n+1)$ (in this representation), (\ref{e2-15}) also
holds for any other representation of $gl(n+1)$. In other words the
Hamiltonian 
\beq 
H=\sum_{i=1}^n \e_i ([a_i^+,a_i^-]+e_{00}) = \sum_{i=1}^n
\e_i ([a_i^+,a_i^-]+h_{0})  
\label{e2-17} 
\eeq 
satisfies~(\ref{e2-6}) for
any other representation of $gl(n+1)$.

One may argue that expression~(\ref{e2-17}) is not satisfactory,
because the Hamiltonian $H$ is not a function of the
Jacobson generators only.
Below, in Corollary~\ref{cor1}, we show that within every
irreducible representation $H$ can be written as a function of
the JGs. Here we note that $[a_i^+,a_i^-]+e_{00} = h_i$ and
therefore the Hamiltonian~(\ref{e2-17}) can be represented manifestly as
an element from the Cartan subalgebra of $gl(n+1)$~: 
\beq 
H= \sum_{i=1}^n \e_i h_{i}.    
\label{e2-18} 
\eeq

\section{Fock representations of $sl(n+1)$}
\setcounter{equation}{0}

We proceed to outline those representations of the Jacobson
generators, which possess the main features of Fock space
representations in ordinary quantum theory.  In order to
distinguish between the abstract generators and their
representations, the JGs $a_1^\pm,\ldots,a_n^\pm$, considered as
operators in a certain
$sl(n+1)$-module $W$,
are called (Jacobson) creation and annihilation
operators of
$sl(n+1)$ (abbreviated also as Jacobson CAOs of $sl(n+1)$,
$sl(n+1)$-CAOs, $A$-CAOs or simply CAOs).

\begin{defi}
Let $a_1^\xi,\ldots,a_n^\xi$ be Jacobson creation
$(\xi=+)$ and annihilation $(\xi=-)$ operators. The
$sl(n+1)$-module
$W$ is said to be a Fock space of the algebra
$sl(n+1)$ if it is a
Hilbert space, so that the following conditions hold~:
\begin{enumerate}
\item
Hermiticity condition ($A^*$ denotes the operator conjugate to $A$)
\beq
(a_i^+)^*=a_i^-, \q i=1,\ldots,n. 
\label{e3-1}
\eeq
\item
Existence of vacuum. There exists a vacuum vector
$|0\rangle \in W $ such that
\beq
a_i^-|0\rangle=0, \q i=1,\ldots,n. 
\label{e3-2}
\eeq
\item
The representation space $W$ is spanned
on vectors
\beq
a_{i_1}^+a_{i_2}^+\cdots a_{i_m}^+|0\rangle, \q m\in \Z_+.
\label{e3-3}
\eeq
\end{enumerate}
\label{def1}
\end{defi}

The Fock space of $sl(n+1)$ is also said to be an $A_n$-module of
Fock or simply a Fock module.

Assume that $W$ is a Fock space. Condition~(\ref{e3-1}) asserts
that any Fock representation is unitarizable with respect to this
star operation, considered as an antilinear antiinvolution on $sl(n+1)$.
It is known that all such representations are realized in
direct sums of finite-dimensional irreducible
$sl(n+1)$-modules.
Then~(\ref{e3-3}) yields that any Fock module is a finite-dimensional
irreducible $sl(n+1)$-module.

We list a few propositions, proofs of which can be
found in~\cite{pa76, pa77}.

\begin{prop}
The
$sl(n+1)$-module $W$ is a Fock space if and only
if it is an irreducible finite-dimensional module with a highest
weight $\Lambda$ such that
\beq
a_i^-a_j^+x_\Lambda=0 \q i\ne j=1,\ldots,n. 
\label{e3-4}
\eeq
The vacuum $|0\rangle$ is unique (up to a multiplicative constant)
and can be identified with the highest weight vector $x_\Lambda$
in $W$~ $|0\rangle=x_\Lambda$.
\label{prop1}
\end{prop}

For a proof see Theorem 1 in~\cite{pa77}.

Recall that the Hamiltonian $H$, see~(\ref{e2-18}), does not belong to
$sl(n+1)$. It is an element from $gl(n+1$). In order to define $H$ as an
operator in $W$, we extend each Fock module to an irreducible
$gl(n+1)$-module. To this end we define the action of the $gl(n+1)$
central element (also $gl(n+1)$ Casimir operator) $h_0+h_1+\ldots +h_n$ in
$W$, setting 
\beq 
(h_0+h_1+\ldots +h_n)x=px \q \forall x\in W,
\label{e3-5} 
\eeq 
where $p$ can be any number.

The next proposition classifies the Fock spaces. Unless otherwise
stated, the roots and the weights are represented by their
coordinates in the basis $h^0,h^1,\ldots,h^n$, i.e.,
$
\lambda=\sum_{i=0}^n l_i h^i\equiv (l_0,l_1,\ldots,l_n).
$

\begin{prop}
The irreducible $gl(n+1)$-module $W_p$
is a Fock space, so that the energy of the vacuum is zero
($H|0\rangle=0$), if and only if its highest weight (namely the
weight of $|0\rangle$) is $\Lambda=ph^0\equiv(p,0,\ldots,0)$, i.e., if
\beq h_0|0\rangle =p|0\rangle,\q h_i|0\rangle =0,\q i=1,\ldots,n,
\label{e3-6} 
\eeq 
where $p$ is an arbitrary positive integer.
\label{prop2} 
\end{prop}

Let us add that a representation with a highest weight
$(p,0,\ldots,0)$ is the $p$-th symmetric power of the identity
representation $(1,0,\ldots,0)$. It corresponds to a Young
diagram with one row and $p$ boxes. 

{}From~(\ref{e2-3}) and~(\ref{e3-5}) $h_0+h_1+\cdots+h_n=p$,
$h_0-h_i=[a_i^-,a_i^+]$, $i=1,\ldots,n$, which yields
\beq
h_0={1\over{n+1}}\big(p+\sum_{i=1}^n [a_i^-,a_i^+]\big), \q
h_i={1\over{n+1}}\Big(p+n[a_i^+,a_i^-]- \sum_{k\ne i=1}^n
[a_k^+,a_k^-]\Big) 
\label{e3-7}
\eeq
The last result shows that within any Fock module the
generators $e_{ij}$ can be expressed as functions
of $a_1^\pm,\ldots,a_n^\pm$. In view of this we say that
$a_1^\pm,\ldots,a_n^\pm$ are Jacobson CAOs of both  $sl(n+1)$
and of $gl(n+1)$.

An immediate consequence of~(\ref{e2-17}) and~(\ref{e3-7}) is
the following

\begin{coro}
Within every Fock module $W_p$ the
Hamiltonian~(\ref{e2-17}) can be expressed entirely via the Jacobson
creation and annihilation operators~:
\beq
H={1\over{n+1}}\sum_{i=1}^n \e_i\Big(p+n[a_i^+,a_i^-]- \sum_{k\ne
i=1}^n [a_k^+,a_k^-]\Big).  
\label{e3-8}
\eeq
\label{cor1}
\end{coro}

{}From~(\ref{e3-4}), (\ref{e3-6}) and (\ref{e3-7}) one concludes~:

\begin{coro}
The Fock module $W_p$ with a highest
weight $\Lambda=(p,0,\ldots,0)$ is completely defined by the relations
\beq a_i^-a_j^+|0\rangle=\delta_{ij}p|0\rangle, \q a_k^-|0\rangle=0,\q
p\in \N, \q i,j,k=1,\ldots,n. 
\label{e3-9} 
\eeq 
\label{cor2} 
\end{coro}

The above two conditions are the same as in the case of Green's
parastatistics of order $p$~\cite{gr53}. Therefore $p$ is referred to as
an order of
$sl(n+1)$-statistics (or $A$-statistics). The conclusion is that
like in parastatistics the Fock spaces are labelled by a positive integer
$p\in\N$. The representations corresponding to different orders of
statistics have different highest weights and are therefore inequivalent.

Taking into account the second relation $a_k^-|0\rangle=0$
in~(\ref{e3-9}),
one can also define the Fock module $W_p$ by means of the relations
\beq
[a_i^-a_j^+]|0\rangle=\delta_{ij}p|0\rangle, \q
a_k^-|0\rangle=0, \q p\in \N, \q i,j,k=1,\ldots,n. 
\label{e3-10}
\eeq
In view of this $A$-statistics and its Fock representations
can be formulated in a somewhat more mathematical terminology.
The latter is based on the observation that the linear span of
all generators $[a_i^-a_j^+]$, $a_i^-$, $i,j=1,\ldots,n,$ is a subalgebra
$\cal A$ of $gl(n+1)$ (which contains as subalgebra also
$gl(n)=\hbox{span} \{[a_i^-a_j^+]|i,j=1,\ldots,n \}$).
Equations~(\ref{e3-10}) define one-dimensional representations of $\cal
A$, spanned on the vacuum $|0\rangle$. Therefore the Fock modules $W_p$
can be defined as those irreducible finite-dimensional $gl(n+1)$-modules,
which are induced from trivial one-dimensional modules of $\cal A$ via
eqs.~(\ref{e3-10}). Then $p$ labels the different, inequivalent
one-dimensional modules of $\cal A$.

On the other hand one can define $A$-statistics by means of the
triple relations~(\ref{e2-5}). Then eqs.~(\ref{e3-9})
define completely the Fock
modules $W_p$. All calculations can be carried out without even
mentioning the underlying Lie algebraic structure of
$A$-statistics (which is usually the case for parastatistics).

Let $W_p$ be a Fock space with order of statistics $p$.
{}From~(\ref{e3-3}) and the fact that the creation operators
commute with each other one concludes that $W_p$ is a linear span
of vectors
$(a_1^+)^{l_1}(a_2^+)^{l_2}\cdots(a_n^+)^{l_n}|0\rangle$,
$l_1,\ldots,l_n\in
\Z_+$. The correspondence weight $\leftrightarrow$ weight vector
is one to one~:
\beq
(a_1^+)^{l_1}(a_2^+)^{l_2}\cdots(a_n^+)^{l_n}|0\rangle \q
\leftrightarrow \q
(p-\sum_{k=1}^n l_k,l_1,l_2,\ldots,l_n), 
\label{e3-11}
\eeq
i.e.\ all weight subspaces are one-dimensional.

\begin{prop}
Let $W_p$ be an
$sl(n+1)$-module of Fock
with order of statistics $p$. The vector
\beq
(a_1^+)^{l_1}(a_2^+)^{l_2}\cdots(a_n^+)^{l_n}|0\rangle  
\label{e3-12} 
\eeq
is not zero if and only if 
\beq 
l_1+l_2+\cdots+l_n\le p. 
\label{e3-13}
\eeq 
\label{prop3} 
\end{prop}

The proof is a consequence of the properties of the roots in any
finite-dimensional irreducible $sl(n+1)$-module $W$. If $\Lambda
=(L_0,L_1,\ldots,L_n)$ is the highest weight in $W$, then for any other
weight $\lambda =(l_0,l_1,\ldots,l_n)$ the following inequality holds~:
\beq 
l_{i_0}+l_{i_1}+\cdots+l_{i_m} \le L_0+L_1+\cdots+L_m, 
\label{e3-14}
\eeq 
where $i_0\ne i_1\ne \ldots,\ne i_m=0,1,\ldots,n$ and
$m=0,1,\ldots,n$. Equation~(\ref{e3-14}) is an equality for $m=n$. If
$W_p$ is a Fock space, $L_0+L_1+\ldots+L_m=p$.

Proposition~\ref{prop3} can be proved also by a direct, but rather long
computation.  One verifies that the infinite-dimensional module ${\hat
W_p}$ spanned on all vectors~(\ref{e3-12}) with $l_1,\ldots,l_n$ being
arbitrary non-negative integers contains an invariant subspace $V_p$
spanned on all vectors~(\ref{e3-12}) with $l_1+l_2+\ldots+l_n > p$. Then
$W_p$ is the factor module ${\hat W_p}/V_p$ and all vectors~(\ref{e3-12}),
subject to~(\ref{e3-13}) are (representatives of) the basis vectors in
$W_p={\hat W_p}/V_p$.

We proceed to recall how one defines a metric in $W_p$, so that it
is a Hilbert space and the hermiticity condition~(\ref{e3-1}) holds.
Consider the vectors
\beq
(a_1^+)^{l_1}(a_2^+)^{l_2}\cdots(a_n^+)^{l_n}|0\rangle, \q
l_1+l_2+\cdots+l_n\le p  
\label{e3-15}
\eeq
from $W_p$.  All such vectors have different weights.
Consequently they are linearly independent and can be considered
as a basis in $W_p$. Define a Hermitian form $(~,~)$ on $W_p$ in
the usual way (for quantum theory), postulating (in addition to
$a_i^-|0\rangle=0$, see~(\ref{e3-2}))~:
\bea
(a) &\ & (|0\rangle,|0\rangle)\equiv \langle 0|0\rangle=1, \nn\\
(b) && \langle 0|a_i^+ =0, \q i=1,\ldots,n, \label{e3-16}\\
(c) && \left((a_1^+)^{m_1}(a_2^+)^{m_2}\cdots(a_n^+)^{m_n}|0\rangle,
     (a_1^+)^{l_1}(a_2^+)^{l_2}\cdots(a_n^+)^{l_n}|0\rangle\right)=\nn\\
&&   \langle 0| (a_n^-)^{m_n}\cdots(a_2^-)^{m_2}(a_1^-)^{m_1}
     (a_1^+)^{l_1}(a_2^+)^{l_2}\cdots(a_n^+)^{l_n}|0\rangle.\nn
\eea
With respect to this form the vectors~(\ref{e3-15}) are orthogonal.
Moreover,
\beq
\Big((a_1^+)^{l_1}(a_2^+)^{l_2}\cdots(a_n^+)^{l_n}|0\rangle,~
(a_1^+)^{l_1}(a_2^+)^{l_2}\cdots(a_n^+)^{l_n}|0\rangle \Big)=
{p!\over
(p-\sum_{j=1}^n l_j )!}\prod_{i=1}^n l_i!>0.  
\label{e3-17}
\eeq
Therefore all vectors
\beq
|p;l_1,\ldots,l_n\rangle=\sqrt{(p-\sum_{j=1}^n l_j )!\over p!}
{(a_1^+)^{l_1}\ldots(a_n^+)^{l_n}\over{\sqrt{l_1!l_2!\ldots
l_n!}}}|0\rangle, \q l_1+l_2+\cdots+l_n\le p
\label{e3-18}
\eeq
constitute an orthonormal basis in $W_p$, i.e.\ $(~,~)$ is a
scalar product. Then by construction the hermiticity
condition~(\ref{e3-1}) holds too.

The transformation of the basis~(\ref{e3-18}) under the action of the
Jacobson CAOs reads~: 
\bea 
a_i^+|p;l_1,\ldots ,l_i,\ldots,l_n\rangle&=&
  \sqrt{(l_i+1)(p-\sum_{j=1}^n l_j  )}~
  |p;l_1,\ldots,l_{i-1},l_i+1,l_{i+1}\ldots,l_n\rangle,
\label{e3-19a}\\
a_i^-|p;l_1,\ldots,l_i,\ldots,l_n\rangle&=&
  \sqrt{l_i(p-\sum_{j=1}^n l_j +1  )}~
  |p;l_1,\ldots,l_{i-1},l_i-1,l_{i+1}\ldots,l_n\rangle.
\label{e3-19b}
\eea
Moreover,
\bea
&& h_0 |p;l_1,l_2,\ldots,l_n\rangle=(p-\sum_{i=1}^n
l_i)|p;l_1,l_2,\ldots,l_n\rangle, \label{e3-20a} \\
&& h_i |p;l_1,l_2,\ldots,l_n\rangle=l_i|p;l_1,l_2,\ldots,l_n\rangle,
\q i=1,\ldots,n. 
\label{e3-20b}
\eea

Let us consider in some more detail the $p=1$ representation.
Denote by $b_i^\pm$ the Jacobson CAOs $a_i^\pm$ in this representation. In
this particular case the representation space $W_1$ is $(n+1)$-dimensional
with a basis 
\beq 
|1;l_1,\ldots,l_n\rangle, \q l_1+\cdots+l_n\le 1,
\label{e3-21} 
\eeq 
i.e.\ at most one of the labels $l_1,\ldots,l_n$ in
$|1;l_1,\ldots,l_n\rangle$ is equal to 1 and all other are zeros.
Then~(\ref{e3-19a})-(\ref{e3-19b}) reduces to 
\beq 
\begin{array}{l}
b_i^+|1;l_1,\ldots,l_{i-1},l_i,l_{i+1},\ldots,l_n\rangle
  =(1-l_i)|1;l_1,\ldots,l_{i-1},l_i+1,l_{i+1},\ldots,l_n\rangle, \\[2mm]
b_i^-|1;l_1,\ldots,l_{i-1},l_i,l_{i+1},\ldots,l_n\rangle=
  l_i|1;l_1,\ldots,l_{i-1},l_i-1,l_{i+1},\ldots,l_n\rangle.
\end{array}
\label{e3-22}
\eeq
The matrix elements of $b_i^+$ and $b_i^-$, in the basis ordered
as $|1;0,0,0,\ldots,0\rangle$, $|1;1,0,0,\ldots,0\rangle$,
$|1;0,1,0,\ldots,0\rangle$, $|1;0,0,1,\ldots,0\rangle$, $\ldots$,
$|1;0,0,0,\ldots,1\rangle$ are the same as those of the matrix
units $E_{i 0}$ and $E_{0 i }$ in the defining
$(n+1)$-dimensional matrix representation of $gl(n+1)$. Hence the $p=1$
representation is the same as the defining representation and one
can think of the operators $b_{i}^\pm$ as of matrices, 
\beq 
E_{i 0}=b_{i}^+, \q E_{0 i}=b_{i}^-, \q i=1,\ldots,n.  
\label{e3-23}
\eeq 
{}From here and~(\ref{e3-7}) (with $p=1$) one can express
also the rest of the generators via $p=1$ Jacobson creation and
annihilation operators~:
\beq 
E_{00}={1\over{n+1}}(1-\sum_{i=1}^n
[b_{i}^+,b_{i}^-]), \q E_{ij}=[b_{i}^+,b_{j}^-]+ {\delta_{ij}\over{n+1}}
(1-\sum_{k=1}^n [b_{k}^+,b_{k}^-]), \q i,j=1,\ldots,n. 
\label{e3-24} 
\eeq

\section{The Pauli principle for $A$-statistics}
\setcounter{equation}{0}

The results obtained so far justify the terminology used.
Equations~(\ref{e2-18}) and~(\ref{e3-6}) yield 
\beq
H|p;l_1,\ldots,l_i,\ldots,l_n\rangle =\sum_{i=1}^n
l_i\e_i|p;l_1,\ldots,l_i,\ldots,l_n\rangle.  
\label{e4-1} 
\eeq
Therefore the state $|p;l_1,\ldots,l_i,\ldots,l_n\rangle$ can be
interpreted as a many-particle state with $l_1$ particles on the
first orbital, $l_2$ particles on the second orbital, etc. For
reasons that will become clear soon, we refer to these particles
as  $A$-particles or simply particles.
The operator $h_i$, $i=1,\ldots,n,$ see~(\ref{e3-20b}), is the
number operator for the $A$-particles on the $i^{th}$ orbital,
whereas ${\hat N}=h_1+\cdots+h_n$ counts all particles,
accommodated in the state $|p;l_1,\ldots,l_i,\ldots,l_n\rangle$.

Since, see~(\ref{e3-19a}),
\beq a_i^+|p;l_1,\ldots
,l_i,\ldots,l_n\rangle \sim |p;l_1,\ldots,
l_{i-1},l_i+1,l_{i+1},\ldots,l_n\rangle, \ \hbox{ if }\
\sum_{i=1}^n l_i<p,   
\label{e4-2}
\eeq
the operator $a_i^+$ creates an $A$-particle on the $i^{th}$ orbital, a
particle with energy $\e_i$, if the state contains less than $p$
particles. On the other hand, $a_i^+|p;l_1,\ldots,
l_{i-1},l_i,l_{i+1},\ldots,l_n\rangle=0$, if $\sum_{i=1}^n l_i=p$, i.e.\
no more than $p$ particles can be accommodated. Similarly, if $l_i>0$,
$a_i^-$ ``kills" a particle with energy $\e_i$. Therefore, reformulating
Proposition~\ref{prop3}, one obtains~:

\begin{coro}[Pauli principle for $A$-statistics]
Let $W_p$ be a Fock space of
$sl(n+1)$, corresponding to an order of
statistics $p$.  Within $W_p$ all states containing no more than
$p$ $A$-particles, namely all states 
\beq
|p;l_1,\ldots,l_i,\ldots,l_n\rangle\ \hbox{ with }\
0\le\sum_{i=1}^n l_i \le p, 
\label{e4-3} 
\eeq 
are allowed. There
are no states accommodating more than $p$ $A$-particles.
\label{cor3}
\end{coro}

Let us consider, as an example, $A$-statistics of order $p=4$
with $n=6$ orbitals (for instance with 6 different energy levels).
{}From~(\ref{e4-3}), it follows that there is no restriction on
the number of particles to be accommodated on a certain orbital as
long as the total number of particles in any configuration does
not exceed $p$. Hence, the following three states or
configurations are allowed (the orbitals, for instance the energy
levels, are represented by lines, and the particles by dots)~:
\[
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\put(45.00,150.00){\makebox(0,0)[cc]{$(b)$}}
\put(75.00,150.00){\makebox(0,0)[cc]{$(c)$}}
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\vskip -122mm
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\]
Note that the last two configurations $(b)$ and $(c)$ are already
``saturated'' in the sense that no more particles can be added,
since the total number of them is already equal to $p=4$. The
following two configurations correspond to forbidden states~:
\[
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\vskip -122mm
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\]
None of the states $(d)$ and $(e)$ is allowed since the total number of
particles in the configuration exceeds $p=4$.

This example illustrates the statistical interaction between the
orbitals~: the filling of an orbital depends on how many particles
are already accommodated on the other orbitals. This property is
typical for Haldane's exclusion statistics \cite{ha91}.
Although Haldane's relation (\ref{e1-1}) does not hold for
$A$-statistics, up to a certain natural assumption
$A$-statistics can be viewed as a special case of exclusion
statistics in the sense of Wu.

In~\cite{wu94} Wu proposed an ``integral form" compatible with
Haldane's relation (\ref{e1-1})~:
\beq
d(N)=n-g(N-1). \label{e4-4}
\eeq
This should be interpreted as follows~: let $n$ be the
total number of orbitals that are available for the first
particle, and suppose $N-1$ particles are already accommodated in
the configuration, then $d(N)$ expresses the dimension of the
single-particle space for the $N^{\rm th}$ particle (or the
number of orbitals where the $N^{\rm th}$ particle can be
``loaded''). Bose statistics has $g=0$, and Fermi statistics has
$g=1$.

The natural assumption mentioned above is that the domain of
definition of the function $d(N)$ consists of all {\it
admissible} values of $N$, i.e.\ one does not
require~(\ref{e4-4}) to be applicable for values of $N$ which the
system cannot accommodate. Under this assumption
$A$-statistics is a particular case of exclusion statistics, also
with $g=0$~: 
\beq 
d(N)=n, \q \forall\, N \in \{1,2,\ldots,p\}.
\label{e4-5} 
\eeq 
If however one drops the condition for $N$ to
be an admissible value, one cannot satisfy equation~(\ref{e4-4}).
Indeed, (\ref{e4-4}) with $g=0$, does not hold for $N=p+1$, since
$d(p+1)=0$~\cite{me98, me99}.

$A$-statistics is similar to Bose statistics in the sense that
there is no restriction on the number of particles on an orbital
apart from the general requirement that the total configuration
should contain no more than $p$ particles. This restriction leads
perhaps to the simplest statistical interaction between the
orbitals. Nevertheless the situation is not as easy as it sounds.
As in any (nontrivial) model of an exclusion statistics (so also in
this case) the orbitals cannot be considered anymore as independent (or
quasiclosed) subsystems. In particular the grand partition function of the
system cannot be represented anymore as a product of the partition
functions of the orbitals. The latter makes the problem of studying the
statistical properties of $A$-statistics more involved as compared to Bose
or Fermi systems. From the point of view of statistical mechanics this is
the main difference between $A$-statistics and Bose
statistics~\cite{Jellaletal}.


\section{Quasi-Bose creation and annihilation operators}
\setcounter{equation}{0}

In the present section we show first approximately and then in a
strict sense that $A$-statistics can be viewed as a
good finite-dimensional approximation to Bose statistics for
large values of order of statistics $p$. The terminology {\it
finite-dimensional approximation} comes to remind that the Fock
spaces $W_p$ of $A$-statistics are finite-dimensional linear
spaces, whereas any Bose Fock space is infinite-dimensional.

Introduce new, representation dependent, creation and
annihilation operators
\beq
B(p)_i^\pm = {a_i^\pm\over \sqrt{p}}, \q i=1,\ldots,n, \q p\in \N,
\label{e5-1}
\eeq
in $W_p$. The transformations following
from~(\ref{e3-19a})-(\ref{e3-19b}) read~:
\bea
&& B(p)_i^+|p;l_1,\ldots,l_i,\ldots,l_n\rangle=
  \sqrt{(l_i+1)(1-{{\sum_{k=1}^n l_k}\over p})}~
|p;l_1,\ldots,l_i+1,\ldots,l_n\rangle, \label{e5-2a} \\
&& B(p)_i^-|p;l_1,\ldots,l_i,\ldots,l_n\rangle=
  \sqrt{l_i(1+{{1-\sum_{k=1}^n l_k}\over p})}~
|p;l_1,\ldots,l_i-1,\ldots,l_n\rangle. 
\label{e5-2b} 
\eea
Consider the above equations for values of the order of
statistics $p$, which are much greater than the number of
accommodated particles, namely $l_1+l_2+\cdots+l_n \ll p.$ In this
approximation one obtains~: 
\beq
\begin{array}{l}
B(p)_i^-|p;l_1,\ldots, l_{i-1},l_i,l_{i+1},\ldots,l_n\rangle \simeq
 \sqrt{l_i}\;
 |p;l_1,\ldots,l_{i-1},l_i-1,l_{i+1},\ldots,l_n\rangle,\\[2mm]
B(p)_i^+|p;l_1,\ldots, l_{i-1},l_i,l_{i+1},\ldots,l_n\rangle \simeq
 \sqrt{l_i+1}\;|p;l_1,\ldots,l_{i-1},l_i+1,l_{i+1}\ldots,l_n\rangle,
\end{array}
\label{e5-3}
\eeq
which yields (an approximation to) the Bose commutation relations~:
\bea
&& [B(p)_i^+,B(p)_j^+]=[B(p)_i^-,B(p)_j^-]=0,\q
\hbox{(exact commutators)}, \label{e54-a} \\
&& [B(p)_i^-,B(p)_j^+]\simeq \delta_{ij},
\q \hbox{ if } l_1+l_2+\cdots+l_n \ll p.   \label{e5-4b}
\eea
Since for $l_1+l_2+\cdots+l_n\equiv\sum_k l_k \ll p$
\[
{{(p-\sum_k l_k)!}\over{p!}}~p^{\sum_k l_k}=
{p\over{p-\sum_k l_k+1}}~{p\over{p-\sum_k l_k+2}}\ldots{p\over
p}\simeq 1,
\]
in a first approximation~(\ref{e3-18}) reduces also to the well
known expressions for the orthonormed basis in a Fock space of $n$
pairs of Bose creation and annihilation operators~: 
\beq
|p;l_1,\ldots,l_n\rangle=
{(B(p)_1^+)^{l_1}\cdots(B(p)_n^+)^{l_n}\over{\sqrt{l_1!l_2!\cdots
l_n!}}}|0\rangle. 
\label{e5-5} 
\eeq 
The conclusion is that the
representations of $B(p)_i^\pm$ in (finite-dimensional) state
spaces $W_p$ with large values of $p$, restricted to states with
a small amount $l_1+l_2+\cdots+l_n \ll p$ of accommodated
particles, provide a good approximation to Bose creation and
annihilation operators~\cite{pa76, pa77}. For this  reason we
refer to the operators $B(p)_i^\pm$ as {\it quasi-Bose creation
and annihilation operators (of order $p$)}
and to the corresponding particles as quasibosons.

In the remaining part of this section we will prove that in the
limit $p \rightarrow \infty$ the quasi-Bose operators reduce to
Bose creation and annihilation operators. To this end we proceed
to introduce first an appropriate topology.

Let $W$ be a Hilbert space with an orthonormed basis
\beq
|l_1,\ldots,l_i,\ldots,l_n\rangle\equiv |L\rangle, \q \forall\,
l_1,\ldots,l_n \in \Z_+.
\label{e5-6}
\eeq
Whenever possible we write $|L\rangle$ as an abbreviation
for $|l_1,\ldots,l_i,\ldots,l_n\rangle$ and denote
by $|L\rangle_{\pm i}$ a vector obtained from $|L\rangle$
by replacing $l_i$ with $l_i\pm 1$, namely
\beq
|L\rangle_{\pm i}=|l_1,\ldots,l_{i-1},l_i\pm 1,l_{i+1},
\ldots,l_n\rangle.
\label{e5-7}
\eeq
The space $W$ consists of all vectors
\beq
\Phi=\sum_{l_1=0}^\infty \cdots \sum_{l_n=0}^\infty
c(l_1,\ldots,l_n)|l_1,\ldots,l_n\rangle \equiv \sum_{L}c(L)|L\rangle,
\label{e5-8} 
\eeq 
where $c(l_1,\ldots,l_n)\equiv c(L)$ are complex numbers
such that 
\beq 
\sum_{l_1=0}^\infty \cdots \sum_{l_n=0}^\infty
|c(l_1,\ldots,l_n)|^2 \equiv \sum_{l_1,\ldots,l_n=0}^\infty
|c(l_1,\ldots,l_n)|^2 \equiv \sum_{L} |c(L)|^2 < \infty , 
\label{e5-9}
\eeq 
and this is in fact the square of the Hilbert space norm
$(|\Phi|_0)^2$ of $\Phi$.

Embed the $sl(n+1)$-module $W_p$ in $W$ via an identification
of the basis vectors
\beq
|p;l_1,\ldots,l_i,\ldots,l_n\rangle\equiv
|l_1,\ldots,l_i,\ldots,l_n\rangle\equiv
|L\rangle \q \forall~ l_1+\ldots+l_n\le p.
\label{e5-10}
\eeq
In order to turn the entire space $W$ into an
$sl(n+1)$-module, so that the restriction on $W_p\subset W$
coincides with~(\ref{e5-2a})-(\ref{e5-2b}), we set~:
\bea
&& B(p)_i^+\Phi=\sum_{l_1+\cdots+l_n \le p}c(L)
\sqrt{(l_i+1)(1-{{\sum_{k=1}^n l_k}\over p})}~|L\rangle_i,
\label{e5-11a} \\
&& B(p)_i^-\Phi=\sum_{l_1+\cdots+l_n \le p}c(L)
\sqrt{l_i(1+{{1-\sum_{k=1}^n l_k}\over
p})}~|L\rangle_{-i},
\label{e5-11b}
\eea
where $\Phi$ is any vector~(\ref{e5-8}) from $W$ and
$\sum_{l_1+\cdots+l_n
\le p}$ is a sum over all possible $l_1,\ldots,l_n \in \Z_+$ such that
$l_1+\cdots+l_n \le p$. Note that the sums
in~(\ref{e5-11a})-(\ref{e5-11b}) are finite.

The transformation of the basis, following
from~(\ref{e5-11a})-(\ref{e5-11b}), reads~:
\bea
&& B(p)_i^+|L\rangle=
  \sqrt{(l_i+1)(1-{{\sum_{k=1}^n l_k}\over p})}~ |L\rangle_{i},
\q \forall~L\hbox{ such that }{{\sum_{k=1}^n l_k}}\le p,
\label{e5-12a}\\
&& B(p)_i^-|L\rangle=
  \sqrt{l_i(1+{{1-\sum_{k=1}^n l_k}\over p})}~|L\rangle_{-i},
\q \forall~L\hbox{ such that }{{\sum_{k=1}^n l_k}}\le p,
\label{e5-12b}\\
&& B(p)_i^\pm|L\rangle=0,\q \forall~L\hbox{ such that }
{{\sum_{k=1}^n l_k}}> p.
\label{e5-12c}
\eea
The relations~(\ref{e5-12a})-(\ref{e5-12b}) are the same
as~(\ref{e5-2a})-(\ref{e5-2b}) (via the
identification~(\ref{e5-10})).

Since the quasi-Bose operators $B(p)_i^\pm$ take values in a
finite-dimensional subspace of $W$,
see~(\ref{e5-11a})-(\ref{e5-11b}), they are bounded
and hence continuous linear operators in $W$.
In view of this, see~(\ref{e5-8}), $ B(p)_i^\pm \Phi=B(p)_i^\pm
\sum_{L}c(L)|L\rangle=\sum_{L}c(L)B(p)_i^\pm|L\rangle $ and
therefore~(\ref{e5-11a})-(\ref{e5-11b}) are a consequence
of~(\ref{e5-12a})-(\ref{e5-12c}).

Next we proceed to define $n$ pairs of Bose operators
$B_i^\pm$, $i=1,\ldots,n$, in $W$. It is known that such operators
cannot be realized as bounded operators in $W$ (so that the
corresponding position and momentum operators  are selfadjoint
operators in $W$;
see, for instance, \cite{re72} or~\cite{em72}).
Therefore care has to be taken about the common domain of
definition $\O$ of the Bose operators. Following~\cite{bo75} we set $\O$
to be a dense subspace of $W$ (with respect to the Hilbert space
topology), consisting of all vectors~(\ref{e5-8}) for which the series
\beq 
(|\Phi|_r)^2=\sum_{l_1,\ldots,l_n=0}^\infty (1+\sum_{k=1}^n
l_k)^r|c(l_1,\ldots,l_n)|^2 
\label{e5-13} 
\eeq 
is convergent for any
$r=0,1,2,\ldots$. Then the relations 
\beq 
B_i^-|L\rangle =  \sqrt{l_i}\;
|L\rangle_{-i}, \qquad B_i^+|L\rangle =  \sqrt{l_i+1}\;|L\rangle_{i},
\label{e5-14} 
\eeq 
define a representation of $n$ pairs of bosons
$B_1^\pm,\ldots,B_n^\pm$, namely of operators, which satisfy the relations
\beq 
[B_i^-,B_j^+]= \delta_{ij},\q [B_i^+,B_j^+]=[B_i^-,B_j^-]=0, \q
i,j=1,\ldots,n, 
\label{e5-15} 
\eeq 
in $\Omega$ (with $\Omega$ being a
common domain of definition for all them). In terms of these operators
\beq 
(|\Phi|_r)^2=(\Phi,(1+\sum_{k=1}^n B_k^+B_k^-)^r\Phi). 
\label{e5-16}
\eeq 
The norms $|\Phi|_r$, $r=0,1,2,\ldots,$ turn $\Omega$ into a
countably normed topological space (which can be viewed also as a metric
space~\cite{ge68}). All balls 
\beq 
B(\Phi_0;r,\epsilon)=\{\Phi\in
\O~|~|\Phi-\Phi_0|_r < \epsilon \}, \q\forall~\Phi_0\in \O, \q\forall~r\in
\Z_+, \q\forall~\epsilon >0, 
\label{e5-17} 
\eeq 
constitute a basis of open
sets in the countably normed topological space $\O$, whereas the
balls~(\ref{e5-17}) with a fixed $r$ yield a basis in $\O$, viewed as a
$|\,\cdot\,|_r$-normed topological space. Clearly any
$|\,\cdot\,|_r$-normed topology ({\it $r$-normed topology}) is weaker than
the countably normed topology ({\it $cn$-topology}).

{}From now on we restrict the domain of definition of all
quasi-Bose operators~(\ref{e5-1}) to be $\O$.  The fact that each
quasi-Bose operator maps $\O$ into a finite-dimensional subspace
of $\O$, see~(\ref{e5-11a})-(\ref{e5-11b}),
indicates that each such operator is a
bounded and hence a continuous linear operator with respect to the
$r$-normed topology for any $r\in\Z_+$. A similar property however
does not hold for the Bose creation and annihilation
operators~(\ref{e5-14}).
These operators are not continuous with respect to any of
the $r$-normed topologies in $\O$. Therefore, if
$\sum_{i=1}^\infty \Phi_i=\Phi$ converges in the sense of a certain
$r$-normed topology, for instance in the Hilbert space topology
($r=0$), one cannot in general use relations like
\beq
B_i^\pm \sum_{i=1}^\infty \Phi_i= \sum_{i=1}^\infty B_i^\pm\Phi_i.
\label{e5-18}
\eeq
One of the advantages of the $cn$-topology is that it avoids the
above difficulties.  Here are some of the properties of this
topology, which will be relevant for the rest of the
exposition~\cite{bo75}~:
\begin{itemize}
\item
$\Omega$ is stable under the action of any polynomial of Bose
operators,
\beq
P(B_1^\pm,\ldots,B_n^\pm)\Omega~\subset~\Omega;
\label{e5-19a}
\eeq
\item
Any polynomial of Bose CAOs is a continuous linear
operator in $\Omega$ with respect to the
$cn-$topology;
\vskip -11mm 
\beq 
\label{e5-19b} 
\eeq
\item
The scalar product in $\Omega$ is continuous with
respect to the convergence defined by the $cn$-topology.
\vskip -11mm 
\beq 
\label{e5-19c} 
\eeq
\end{itemize}

As a consequence, (\ref{e5-18}) holds for any series
$\sum_{i=1}^\infty \Phi_i$ which converges in the $cn$-topology;
moreover~(\ref{e5-19c}) yields $(\sum_{i=1}^\infty
\Phi_i,\Psi)=\sum_{i=1}^\infty (\Phi_i,\Psi)$.
The relevance of
the $cn$-topology however goes far beyond the above
considerations. This topology, called nuclear topology, is of
prime importance in the theory of generalized functions~\cite{ge68,ge64},
and their applications in quantum theory (see, for instance~\cite{bo75}).

Let $\P$ be the set of all linear operators in $\O$ defined
everywhere in $\Omega$, which are continuous in the
$cn$-topology. With respect to the usual operations between
operators $\P$ is an associative algebra~\cite{ge68}.
According to~(\ref{e5-19b}) the Bose operators belong to $\P$.
The quasi-Bose operators~(\ref{e5-1}) (with domain of definition
restricted to $\Omega$) also belong to $\P$.  Indeed $B(p)_i^\pm$
are bounded and hence continuous operators in $\O$ with respect
to any $r$-normed topology. Let $B(\Phi_0;r,\epsilon)$ be an arbitrary
open ball in the $cn-$topology, see~(\ref{e5-17}). $B(\Phi_0;r,\epsilon)$
is an open ball also in the $r$-normed topology. Therefore the inverse
image $O=[B(p)_i^\pm]^{-1}B(\Phi_0;r,\epsilon)$ of $B(\Phi_0;r,\epsilon)$
is an open set in the $r$-normed topology. Since the latter is weaker than
the $cn$-topology, $O$ is an open set also in the $cn$-topology.  Thus,
the inverse image $O=[B(p)_i^\pm]^{-1}B(\Phi_0;r,\epsilon)$ of any open
ball (i.e.\ of any open set from the basis) in the $cn$-topology is an
open set with respect to the same topology.  Therefore $B(p)_i^\pm$ is a
continuous operator in the $cn$-topology.

Introduce a topology on $\P$ in a way similar to the strong
topology in the algebra ${\cal B(H)}$ of all bounded linear
operators on a Hilbert space ${\cal H}$~\cite{na72}.  Let
$\Phi_1,\ldots,\Phi_s$ be $s$ different elements from $\Omega$
and $\epsilon$ be a positive number. A strong neighborhood
$U(A_0;\Phi_1,\ldots,\Phi_s;\epsilon)$ of the operator $A_0\in \P$ is
(defined as) the set of all operators $A\in \P$, which satisfy the
inequalities 
\beq 
|(A-A_0)\Phi_k|_0<\epsilon,\q \forall\,k=1,\ldots,s.
\label{e5-20} 
\eeq

\begin{defi}
A {\em strong topology} on $\P$ is the
topology with a basis of open sets consisting of all possible
strong neighborhoods $U(A_0;\Phi_1,\ldots,\Phi_s;\epsilon)$ (namely the
collection of strong neighborhoods, corresponding to any $A_0\in \P$, to
any $\epsilon>0$, to any $s\in \N$ and to any sequence
$\Phi_1,\ldots,\Phi_s$ of different elements from $\O$). \label{def2}
\end{defi}

\begin{prop}
In the strong topology
\beq
\displaystyle\lim_{p\to \infty} B(p)_i^\pm =B_i^\pm,\q i=1,\ldots,n.
\label{e5-21}
\eeq
\label{prop4}
\end{prop}

\noindent {\it Proof.}
In order to prove that~(\ref{e5-21}) holds it is sufficient
to show that every strong neighborhood
$U(B_i^\pm;\Phi_1,\ldots,\Phi_s;\epsilon)$ of $B_i^\pm$ contains all
elements of the sequence $B(1)_i^\pm,B(2)_i^\pm,\ldots$ apart from a
finite number of them.  Since
$U(B_i^\pm;\Phi_1,\ldots,\Phi_s;\epsilon)=\cap_{k=1}^s
U(B_i^\pm;\Phi_k;\epsilon)$, it is sufficient to show that for any
neighborhood $U(B_i^\pm;\Phi;\epsilon)$ there exists an integer $N$
such that $B(p)_i^\pm \in U(B_i^\pm;\Phi;\epsilon)$ for any $p> N$ or,
which is the same, see~(\ref{e5-20}), that 
\beq 
|(B(p)_i^\pm  -  B_i^\pm)
\Phi|_0 < \epsilon, \q\forall~p>N. 
\label{e5-22} 
\eeq 
The above equation
has to hold for any $\Phi$ and any $\epsilon$. In general $N$ depends on
$\Phi$ and $\epsilon$, $N=N(\Phi,\epsilon)$.

The fact that $B_i^+ - B(p)_i^+$ is a continuous linear
operator in $\Omega$ is essential since relations like~(\ref{e5-18}) can
be used. The latter together with~(\ref{e5-11a})-(\ref{e5-11b})
and~(\ref{e5-14}) yields~: 
\bea 
(B_i^+ - B(p)_i^+)\Phi &=&
\sum_{l_1+\cdots+l_n < p}c(L)(\sqrt{l_i+1} \Biggl(1-\sqrt{1-{{\sum_{k}
l_k}\over p }}\Biggr)|L\rangle_i \nn\\ 
&&+\sum_{l_1+\cdots+l_n\ge
p}c(L)(\sqrt{l_i+1}|L\rangle_i. 
\label{e5-23} 
\eea 
The continuity of the
scalar product with respect to the $cn$-topology and the fact that all
terms in the RHS of~(\ref{e5-23}) are orthogonal to each other yield~:
\beas (|(B_i^+ - B(p)_i^+)\Phi|_0)^2 &=& \sum_{l_1+\cdots+l_n< p}|c(L)|^2
(l_i+1) \Biggl(1-\sqrt{1-{{\sum_{k} l_k}\over p }}\Biggr)^2 \\
&&+\sum_{l_1+\cdots+l_n\ge p}|c(L)|^2(l_i+1). 
\eeas 
Let $\epsilon >0$.
Select $p_0\in \N$ to be fixed. For any $p>p_0$ 
\bea 
&&(|(B_i^+ -
B(p)_i^+)\Phi|_0)^2 =
  \sum_{l_1+\cdots+l_n\le p_0}|c(L)|^2 (l_i+1)
  \Biggl(1-\sqrt{1-{{\sum_{k} l_k}\over p }}\Biggr)^2 \nn\\
&& +\sum_{p_0<l_1+\cdots+l_n < p}|c(L)|^2 (l_i+1)
  \Biggl(1-\sqrt{1-{{\sum_{k} l_k}\over p }}\Biggr)^2
+ \sum_{l_1+\cdots+l_n\ge p}|(1+l_i)c(L)|^2  \nn \\
&& < \sum_{l_1+\cdots+l_n\le p_0}|c(L)|^2 (l_i+1)
  \Biggl(1-\sqrt{1-{{\sum_{k} l_k}\over p }}\Biggr)^2
+ \sum_{l_1+\cdots+l_n>p_0}|(1+l_i)c(L)|^2.   \label{e5-24}
\eea
Since the partial sums of
$\sum_{l_1,\ldots,l_n=0}^\infty(1+l_i)|c(L)|^2$
constitute an increasing sequence of positive numbers,
which is restricted from
above, $\sum_{l_1,\ldots,l_n=0}^\infty(1+l_i)|c(L)|^2\le|\Phi|_1$,
the series $\sum_{l_1,\ldots,l_n=0}^\infty(1+l_i)|c(L)|^2$ converges.
Choose $p_0$ such that
$\sum_{l_1+\ldots+l_n>p_0}(1+l_i)|c(L)|^2<{{\epsilon^2}\over 2}$. Then for
any $p>p_0$ 
\bea 
&& (|(B_i^+ - B(p)_i^+)\Phi|_0)^2 <
  \sum_{l_1+\cdots+l_n\le p_0}|c(L)|^2 (l_i+1)
  \Biggl(1-\sqrt{1-{{\sum_{k} l_k}\over p }}\Biggr)^2 +
  {{\epsilon^2}\over 2}  \nn\\
&& < \sum_{l_1+\cdots+l_n\le p_0}|c(L)|^2 (l_i+1)
    \Biggl(1-\sqrt{1-{{p_0}\over p }}\Biggr)^2 +
  {{\epsilon^2}\over 2}
  <  d \Biggl(1-\sqrt{1-{{p_0}\over p }}\Biggr)^2 +
  {{\epsilon^2}\over 2},
\label{e5-25}
\eea
where
$d=\sum_{l_1+\cdots+l_n\le p_0}|c(L)|^2 (l_i+1)$ is a constant.
Clearly there exists $N\in \N$ such that
$d \Bigl(1-\sqrt{1-{{p_0}\over p }}\Bigr)^2 <
 {{\epsilon^2}\over 2}$ for any $p>N$.
Hence for every $\epsilon > 0$ there exists a positive integer
$N$ such that $|(B_i^+ - B(p)_i^+)\Phi|_0 < \epsilon$, $\forall
p> N$, i.e.\ (\ref{e5-22}) holds.

In a similar way one proves that $\displaystyle\lim_{p\to \infty}
B(p)_i^- =B_i^- $. This completes the proof. \mybox


\section{Bosonization of $A$-statistics}
\setcounter{equation}{0}

A simple comparison of~(\ref{e3-19a})-(\ref{e3-19b})
with~(\ref{e5-14}) suggests
that the Jacobson CAOs of any order $p$ can be bosonized, namely
that they can be expressed as functions of Bose CAOs
$B_1^\pm,\ldots,B_n^\pm$, see~(\ref{e5-15}). Indeed, taking into account
that $B_i^+B_i^-\equiv N_i$ is a number operator for bosons in a state
$i$, 
\beq 
N_i|L\rangle\equiv N_i|l_1,\ldots,l_i,\ldots,l_n\rangle=
l_i|l_1,\ldots,l_i,\ldots,l_n\rangle,\q i=1,\ldots,n, 
\label{e6-1} 
\eeq
one rewrites~(\ref{e3-19a}) as~: 
\[ 
a_i^+|L\rangle=
  \sqrt{(l_i+1)(p-{\sum_{k=1}^n N_k+1})}~
|L\rangle_{i}.
\]
In view of~(\ref{e5-14}) the latter can also be represented as
\beq
a_i^+|L\rangle=
  \sqrt{p+1-\sum_{k=1}^n N_k}~B_i^+|L\rangle=
B_i ^+\sqrt{p-\sum_{k=1}^n B_k^+B_k^-}~|L\rangle.
\label{e6-2}
\eeq
Since~(\ref{e6-2}) holds for any $|L\rangle$,
\beq
a_i^+= B_i ^+\sqrt{p-\sum_{k=1}^n B_k^+B_k^-}, \quad
 i=1,\ldots,n.
\label{e6-3}
\eeq
In a similar way one derives from~(\ref{e3-19b})~:
\beq
a_i^- =\sqrt{p-\sum_{k=1}^n B_k^+B_k^-}~B_i^-,\quad
 i=1,\ldots,n.
\label{e6-4}
\eeq
Evidently also, see~(\ref{e3-20a}),
\beq
h_0=p-\sum_{k=1}^n B_k^+B_k^-.
\label{e6-5}
\eeq
Note that the entire Fock space $W$ is reducible with respect to
the Jacobson CAOs. Its finite-dimensional ``physical" subspace
$W_p$, see~(\ref{e5-10}), is a simple (= irreducible) $gl(n+1)$-module and
within this module $(a_i^+)^*=a_i^- $ holds.

After simple calculations and taking into account that
$a_i^+=e_{i0}$, $a_i^-=e_{0i}$, $i=1,\ldots,n$, see~(\ref{e2-3}), one can
express all
generators $\{e_{ij}|i,j=0,1,\ldots,n\}$ of $gl(n+1)$ via
$n$ pairs of Bose operators~: 
\bea 
(a)&& e_{ij}=B_i^+B_j^-, \quad
i,j=1,\ldots,n, \nn\\ 
(b)&& e_{i0}=B_i^+\sqrt{p-\sum_{k=1}^n B_k^+B_k^-},
\quad e_{0i}=\sqrt{p-\sum_{k=1}^n B_k^+B_k^-}~B_i^-, \quad i=1,\ldots,n,
\label{e6-6} 
\\ (c)&& e_{00}=p-\sum_{k=1}^n B_k^+B_k^-, \nn 
\eea 
where, we
recall, $p$ is any positive integer, $p\in\N$.

The above bosonization of $gl(n+1)$ is not unknown.  Up to a
choice of notation it is the same as the so-called
Holstein-Primakoff (H-P) realization of $gl(n+1)$~\cite{ok75},
initially introduced for $sl(2)$~\cite{ho40, dy56}.
Note that~(\ref{e6-6}a) alone gives
the known Jordan-Schwinger realization of $gl(n)$ via $n$ pairs
of Bose operators.


\section{Quasi-Bose operators in spin models}
\setcounter{equation}{0}

In the present section we show that the Jacobson CAOs are implicitly
present in various models. We demonstrate this on the example of a two-leg
$S=1/2$ Heisenberg spin ladder~\cite{go94, su98}. The considerations below
hold however for several other Heisenberg spin models (examples include
lattice models with dimerization~\cite{sa90, ch91b, ch91}, two-layer
Heisenberg models~\cite{ch95, ko98, sh00}) and more generally for any
hard-core Bose model~\cite{fi89} with degenerated orbitals per site (as
for instance in~\cite{zi94, la98}).

The Hamiltonian of the model reads:
\beq
{\hat H}=\sum_{i} (J{\bf {\hat S}}_{i}^+ {\bf {\hat S}}_{i+1}^+ +
J{\bf {\hat S}}_{i}^- {\bf {\hat S}_{i+1}}^- + J_\bot{\bf {\hat
S}}_{i}^+ {\bf {\hat S}}_{i}^-).
\label{e7-1}
\eeq
Here
${\bf {\hat S}}_{i}^\pm\equiv ({\hat S}_{1 i}^\pm,{\hat S}_{2 i}^\pm,
{\hat S}_{3 i}^\pm)$ are two commuting spin-$1/2$ vector operators
``sitting" on site $i$ of the chain $\pm$ and the Hamiltonian is a scalar
with respect to the total spin operator ${\bf {\hat S}} =\sum_{i}( {\bf
{\hat S}}_{i}^+ + {\bf {\hat S}}_{i}^-)$~: 
\beq 
[{\hat S}_{\alpha
i}^\pm,{\hat S}_{\beta i}^\pm]= i\sum_{\gamma}\epsilon_{\alpha \beta
\gamma}{\hat S}_{\gamma i}^\pm,\q [{\hat S}_{\alpha i}^+,{\hat S}_{\beta
j}^-]=0, \q [{\hat H},{\bf {\hat S}}]=0. 
\label{e7-2} 
\eeq

Every local state space $W_i$ related to site $i$ is
4-dimensional with a basis $|\uparrow, \uparrow \rangle$, $|\uparrow,
\downarrow \rangle$, $|\downarrow, \uparrow \rangle$, $|\downarrow,
\downarrow \rangle$ and $W=W_1\otimes W_2 \otimes \ldots \otimes W_N$ is
the global state space of the system (in the case of a ladder with $N$
sites). The notation is standard~: if $A$ is any operator in $W_i$, then
the corresponding to it operator in $W$ is denoted as $A_i$, where
$A_i\equiv id_1\otimes\ldots \otimes id_{i-1}\otimes A \otimes id_{i+1}
\otimes \ldots \otimes id_N$.

If the system is in a disordered phase ( $J_\bot \gg J$)
its state is well described with the bond operator representation
of spin operators~\cite{ch89, sa90}, which is a particular kind of
bosonization~:
\beq
{\hat S}_{\alpha i}^\pm={1\over 2}(\pm B_{\alpha i}^- \pm
B_{\alpha i}^+ -i\epsilon_{\alpha \beta \gamma}B_{\beta
i}^+B_{\gamma i}^-),\q
\alpha, \beta, \gamma=1,2,3.
\label{e7-3}
\eeq
Here $B_{1i}^\pm$, $B_{2i}^\pm$, $B_{3i}^\pm$ are three pairs of
Bose CAOs related to  site $i$ and the vectors
$|0\rangle_i$, $B_{1i}^+|0\rangle_i$, $B_{2i}^+|0\rangle_i$,
$B_{3i}^+|0\rangle_i$
constitute another basis in $W_i$.

The treatment of the model in terms of bosonic  operators is
advantageous because of the simpler commutation rules of Bose
statistics. It rises however certain problems. As mentioned above,
any local state space $W_i$ is 4-dimensional, whereas the local Bose
Fock space $\Phi_i$ is infinite-dimensional. Moreover $W_i$ is not
invariant in $\Phi_i$ with respect to the Bose CAOs (and, as a
result, with respect to the local spin operators~ (\ref{e7-3})). The
physical state space $W$ is not an invariant subspace of the
global Fock space $\Phi=\Phi_1\otimes \Phi_2 \otimes \ldots \otimes
\Phi_N$ with respect to the Hamiltonian~(\ref{e7-1}).

Various approaches have been proposed in order to overcome the
problem. Following~\cite{sa90},
additional scalar bosons $s_i^\pm$ were
introduced in~\cite{go94}.
Then the physical states are those which satisfy an
additional constraint $s_i^+s_i+ \sum_\alpha
B_{i\alpha}^+B_{i\alpha}=1$.  Another way is to keep the
realization~(\ref{e7-3}) but to introduce ``by hands" a fictitious
infinite on-site repulsion between the ``bosons"~\cite{ko98} (first
proposed in~\cite{fi89} for a nondegenerate case). This forbids
configurations with two or more bosons accommodated on one and the
same site.  The latter leads to the ``hard-core" condition
$B_{\alpha i}^\pm B_{\beta i}^\pm=0$, i.e.\ the hard-core bosons
are not quite bosons, since they satisfy fermionic-like
conditions.

A third approach was worked out in~\cite{ch89}
(see also~\cite{ch91b, ch91, ch95}).
It proposes the Bose operators $B_{\alpha i}^\pm$ in~(\ref{e7-3}) to be
replaced throughout by new operators $b_{\alpha i}^\pm$ as follows~: 
\beq
B_{\alpha i}^+ ~ \rightarrow ~ b_{\alpha i}^+= B_{\alpha i}^+
\sqrt{1-\sum_{\beta=1}^3 B_{\beta i}^+B_{\beta i}^-},\q B_{\alpha i}^- ~
\rightarrow ~ b_{\alpha i}^-= \sqrt{1-\sum_{\beta=1}^3 B_{\beta
i}^+B_{\beta i}^-}~B_{\alpha i}^-. 
\label{e7-4} 
\eeq 
A simple comparison
with~(\ref{e6-3}), (\ref{e6-4}) indicates that 
\begin{itemize} 
\item The Bose operators related to site $i$, i.e.\ $B_{1i}^\pm$, $B_{2i}^\pm$,
$B_{3i}^\pm$, are replaced by $p=1$ Jacobson CAOs (or, which is the same,
by $p=1$ quasi-Bose operators), 
\beq 
B(1)_{\alpha i}^\pm \equiv b_{\alpha
i}^\pm,\q\alpha=1,2,3, 
\label{e7-5} 
\eeq 
in their Holstein-Primakov
realization. Consequently (Proposition~\ref{prop3}) the hard-core
condition $b_{\alpha i}^+b_{\beta i}^+=0$ holds; 
\item The Jacobson CAOs
from different sites commute~: 
\beq 
[b_{\alpha i}^\xi, b_{\beta
j}^\eta]=0,\q \hbox{if } i\ne j\hbox{ for any }\xi,\eta=\pm\hbox{ and }
\alpha, \beta=1,2,3. 
\label{e7-6} 
\eeq 
\end{itemize} 
It is essential that
the substitution~(\ref{e7-4}) does not change the commutation
relations~(\ref{e7-2}) between the new spin operators 
\beq 
S_{\alpha
i}^\pm={1\over 2}(\pm b_{\alpha i}^- \pm b_{\alpha i}^+ -i\epsilon_{\alpha
\beta \gamma}b_{\beta i}^+b_{\gamma i}^-),\q \alpha, \beta, \gamma=1,2,3,
\label{e7-7} 
\eeq 
and the corresponding new Hamiltonian 
\beq 
H=\sum_{i}
(J{\bf S}_{i}^+ {\bf {S}}_{i+1}^+ + J{\bf { S}}_{i}^- {\bf { S}_{i+1}}^- +
J_\bot{\bf { S}}_{i}^+ {\bf { S}}_{i}^-). 
\label{e7-8} 
\eeq 
Moreover each
local state space $W_i$ is an invariant subspace of $\Phi_i$ with respect
to the Jacobson CAOs and hence with respect to any function of them (in
particular with respect to the spin operators~(\ref{e7-7})). The
Hamiltonian~(\ref{e7-8}) is also a well defined operator in $W$.

The conclusion is that replacing throughout the model the Bose
operators with $p=1$ Jacobson CAOs $b_{\alpha i}^\pm$, which
commute at different sites (see~(\ref{e7-6})), one obtains directly the
physical state space and the correct expressions for the spin operators
and the Hamiltonian.

Let us point out that the above results can be also derived from
the following proposition, which is of independent interest.

\begin{prop}
Let $B_\alpha^\pm$, $\alpha=1,\ldots,n$,
be $n$ pairs of Bose CAOs with a Fock space $\cal F$ and a
basis~(\ref{e5-6}).
Denote by ${\cal F}_1$ the subspace of $\cal F$ linearly
spanned on the vacuum and all ``single-particle" states,
\beq
{\cal F}_1=\hbox{span}\{|l_1,\ldots,l_n\rangle |~
l_1+\cdots+l_n\le 1\}.
\label{e7-9}
\eeq
Let $\cal P$ be a projection operator of $\cal F$ onto ${\cal F}_1$~: 
\beq
{\cal P}|l_1,\ldots,l_n\rangle = \left\{ \begin{array}{ll}
|l_1,\ldots,l_n\rangle, &\hbox{ if } l_1+\cdots+l_n\le 1;\\ 0, & \hbox{ if
} l_1+\cdots+l_n> 1. \end{array} \right. 
\label{e7-10} 
\eeq 
Then the
operators ${\cal P}B_\alpha^\pm{\cal P}$, $\alpha=1,\ldots,n$, considered
as operators in ${\cal F}_1$, are $p=1$ Jacobson CAOs, 
\beq 
{\cal P}B_\alpha^\pm{\cal P}=B(1)_\alpha^\pm\equiv b_\alpha^\pm, \q
\alpha=1,\ldots,n. 
\label{e7-11} 
\eeq 
\label{prop5} 
\end{prop}

\noindent {\it Proof.}
One verifies directly that (\ref{e2-5}) and (\ref{e3-22})
hold. \mybox

Coming back to the two-leg spin ladder model, introduce a
projection operator ${\cal P}_w={\cal P}_1\otimes{\cal
P}_2\otimes\ldots\otimes {\cal P}_N$ of $\Phi$ onto $W$, where each
${\cal P}_i$ projects $\Phi_i$ onto $W_i$ according to~(\ref{e7-10})
with $n=3$. The projector ${\cal P}_w$ provides an alternative way
for writing down the expressions for the spin operators~(\ref{e7-7}) and
the Hamiltonian~(\ref{e7-8}). Instead of using the
substitution~(\ref{e7-4}), one can set: 
\beq 
H={\cal P}_w {\hat H} {\cal
P}_w,~~ S_{\alpha i}^\pm={\cal P}_i{\hat S}_{\alpha i}^\pm {\cal P}_i,\q
i=1,\ldots,N. 
\label{e7-12} 
\eeq 
The operator ${\cal P}_w$ is a Bose
analogue of the Gutzwiller projection operators~\cite{gu63}, extensively
used in the $t$-$J$ models in order to exclude the double occupation of
fermions at each site (see, for instance~\cite{ts94} where a similar
problem, a $t$-$J$ two-leg ladder is investigated).



\section{Concluding remarks}
\setcounter{equation}{0}

{}From a mathematical point of view the JGs
$a_1^\pm,\ldots,a_n^\pm$ provide a new description
of the Lie algebra $sl(n+1)$
in terms of generators and relations~(\ref{e2-5}),
based on the concept of Lie triple systems.
For the same reason any $n$ pairs of parafermions (resp.\
parabosons) can be called Jacobson generators of the orthogonal Lie
algebra $so(2n+1)$ (resp.\ of the orthosymplectic Lie
superalgebra $osp(1/2n)$). The JGs provide an alternative to
the Chevalley descriptions of these Lie (super)algebras.

{}From a physical point of view the interest in the JGs of
$sl(n+1)$ stems from the observation that they indicate the
possible existence of a new quantum statistics. Indeed, we have
seen that within each Fock space $W_p$ the operator $a_i^+$
(resp.\ $a_i^-$) can be interpreted as an operator creating
(resp.\ annihilating) a particle in a state $i$ (in particular
with an energy  $\e_i$).

In many respects the quasibosons behave as bosons.
Similar as for bosons, the quasibosons
can be distributed along the orbitals in an
arbitrary way as far as the number of
accommodated particles $M$ does not exceed $p$.
The number of different states of $M\le p$ quasibosons
is the same as for bosons (the $M$-particle subspaces
of quasibosons and bosons have one and the same dimension).
There is however one essential difference~:
quasiboson systems of order $p$ can accommodate at most
$p$ particles.

In order to use a proper Lie algebraic language we have
restricted our considerations to finite-dimensional Lie algebras.
In other words, we were studying systems with a finite number $n$
of orbitals. Such systems certainly do exist. Examples are the
local state spaces of spin systems (in particular the example
considered in Section~7), $su(n)$ lattice models etc.
Nevertheless it is natural to ask whether $A$-statistics can be
extended to incorporate infinitely many orbitals as this is usual
in quantum theory. The answer to this question is positive and it
is in fact evident from the results we have obtained so far.
First of all the description of $sl(n+1)$ via generators
(\ref{e2-3}) and relations~(\ref{e2-5}) is well defined for
$i,j,k \in \Z_+$, namely for $sl(\infty)=\hbox{span}
\{e_{ij}-\delta_{ij}e_{00}|i,j \in \Z_+\}$ (an equivalent
definition is to say that $sl(\infty)$ is the algebra of all
traceless infinite (in one direction) matrices containing no more
than a finite number of nonzero entries).
Secondly, any Fock
module $W_p$ as given in Corollary~\ref{cor2} and in particular
equations~(\ref{e3-9}) are also well defined for
$i,j,k \in \N$. In this case any $W_p$ is an irreducible
$sl(\infty)$ module, generated out of the vacuum by means of the
Jacobson creation operators. Therefore each state
$|p;l_1,\ldots,l_i,\ldots\rangle$ contains no more than a finite
number of nonzero entries $l_i$. Moreover due to
Proposition~\ref{prop3} the physical state space is a linear span
of all vectors $|p;l_1,\ldots,l_i,\ldots\rangle$ with
\beq
\sum_{i=1}^{\infty}l_i \le p.
 \label{e8-1}
\eeq
All such states constitute an (orthonormal) basis in $W_p$. They transform
according to the same relations~(\ref{e3-19a})-(\ref{e3-19b}) with
$n=\infty$. It is straightforward to verify that any $sl(\infty)$ module
$W_p$  is a Fock space in the sense of Definition~\ref{def1}. Finally, the
Pauli principle (Corollary~\ref{cor3}) remains valid also for $n=\infty$~:
despite of the infinitely many available orbitals, the infinitely many
places to be occupied by the quasibosons, the system cannot accommodate
more than $p$ particles.
[For an overview of commonly recognized definitions of $gl(\infty)$,
see Ref.~\cite{Egorov}, where in particular several types of
infinite dimensional Lie superalgebras of type $gl$ are studied.]

We should point out that within $A$-statistics the main
quantization equation~(\ref{e2-6}) does not determine uniquely the
creation and annihilation operators. The Jacobson generators~(\ref{e2-3})
yield one possible solution of~(\ref{e2-6}). For another possible choice
(a causal $A$-statistics), we refer to~\cite{pa79}.

The quasi-Bose operators $B(p)_1^\pm,\ldots, B(p)_n^\pm$,
introduced in Section~5 can be used as an approximation to Bose
statistics for values of the order of statistics $p$, which is
much bigger than the number of accommodated particles. An
additional advantage of the quasi-Bose CAOs of any order $p$ is
that they are bounded linear operators, defined everywhere in the
Fock space $W_p$. This property avoids the rather delicate
questions of whether the operators under consideration can be
defined on a common domain of definition $\Omega$, so that any
polynomial of them  is also well defined in $\Omega$.

The ``opposite" to $p \rightarrow \infty$ case, namely the $p=1$
Jacobson CAOs (or, which is the same, the $p=1$ quasi-Bose
operators) turns out to appear implicitly in various models
simply because the hard-core bosons are $p=1$ quasibosons. We have
illustrated this on a particular example from condensed matter
physics. This observation does not lead to immediate new results
as far as hard-core Bose models are concerned. Even so, we believe
that the clarification of the hidden quasiboson structure of
these models is of some interest. If, for instance, the
Hamiltonian is written explicitly via quasibosons, then the model
is representation independent. Considering instead of the
hard-core bosons quasibosons with order of statistics $p=M$, one is
led to a model where each site can accommodate up to $M$
particles.


For applications of quasiboson representations in nuclear theory
we refer to~\cite{me98}.
As indicated there, the $p=1$ quasi-Bose operators
reduce to Klein-Marshalek algebras~\cite{kl88},
which are extensively used in
nuclear physics.

One way to enlarge the class of statistics studied here is to
deform the relations~(\ref{e2-5}) or, which is the same, to
deform $sl(n+1)$ so that the main quantization
equation~(\ref{e2-6}) remains unaltered. The possibility for such
deformations stems from the observation that the commutation
relations between the Cartan elements (the Hamiltonian is a
Cartan element, see~(\ref{e2-18})) and the root vectors (the
Jacobson generators are root vectors, see~(\ref{e2-4})) remain
unaltered upon quantum deformations ($q$-deformations). Therefore
the problem actually is to express the known $q$-deformations of
$sl(n+1)$ via deformed Jacobson generators. This is the first
step. The second step will be to define the Fock representations
and to write down the deformed analogue
of~(\ref{e3-19a})-(\ref{e3-19b}). Partial results in this respect
were already announced~\cite{par98,pa00}.


\section*{Acknowledgments}
One of us (T.D. Palev) is grateful to Prof.\ Randjbar-Daemi for the
kind hospitality at
the High Energy Section of ICTP. Constructive discussions with
Dr.\ N.I.\ Stoilova are greatly acknowledged.  This work was
supported by the Grant $\Phi$-910 of the Bulgarian Foundation for
Scientific Research, and by Grant PST.CLG.976865 of NATO.

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} % end of small
\end{thebibliography}

\end{document}