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\begin{document}

\vspace*{2.5cm} \noindent
{\bf A NOTE ON RELATION BETWEEN QUANTUM MECHANICS AND ALGEBRAIC INVARIANTS }%
\vspace{1.3cm}\\ \noindent
\hspace*{1in}
\begin{minipage}{13cm}
Alex A. Samoletov \vspace{0.3cm}\\
Department of Theoretical Physics\\
Institute for Physics and Technology, Natl. Acad. Sci. Ukraine \\
72 Luxembourg St., 340114 Donetsk \\
Ukraine
\end{minipage}

\vspace*{0.5cm}

\begin{abstract}
\noindent
We propose a program for construction of nonclassical algebraic structures
and quantization by analogy with Klein geometric program. The group of
affine canonical transformations is considered in this context in detail.
\end{abstract}

% section 1
\section{\hspace{-4mm}.\hspace{2mm}INTRODUCTION}

The old quantum theory was developing in the direction: The realization of
observables of the classical mechanics and their algebra remains without
modification and only realization of states to be changed. As the result the
theory has been presented a wealthy of material \cite{bo}. However, the
quantum theory was setting off by the other way: It was constructed in such
a way that the theory remains the algebraic structure of classical mechanics
but fully renounced idea of phase space and of observables as smooth
functions on it \cite{ma},\cite{fa}. Nevertheless, the idea of phase space
appeared in the quantum mechanics at its early period \cite{wey},\cite{wi}
and then took clear form in the work \cite{moy}, then in the theory of
deformational quantization \cite{bay},\cite{fr} and in a number of works
\cite{aga},\cite{shi},\cite{be}. Besides the Weyl-Wigner-Moyal
representation there are others representations of quantum mechanics in
phase space including rare representations \cite{bl},\cite{bbl},\cite{te},%
\cite{mo}. However, the Weyl-Wigner-Moyal (WWM) representation occupy a
special position among them: In its basis lies a maximal group (more precise
definition will be given in what follows). For this reason the WWM
representation is most appropriate initial point for algebro-geometric
speculations.

\vspace{\baselineskip}

The main Klein idea \cite{kl} lies in the correspondence to any geometry a
group which acts in its space. In general, every group of transformations
determines its own geometry. This geometry studies properties of figures
which are invariant under the action of a given group transformations. So,
by Klein, group is first notion of geometry and it can be interpreted as
group of symmetry for geometry which is arising from. All it is well known
\cite{kl},\cite{ale}. The Euclidean geometry is typical example for the
group which has the structure of semidirect product of the group of
orthogonal matrixes and the additive group of vectors (in an $n$-dimentional
space).

\vspace{\baselineskip}

In this work we propose an application of the basic idea of geometric Klein
program, which is understood in a wide sense, to the problem of construction
of nonclassical algebraic structures on the set of classical observables.
The starting point are the WWM phase space representation of quantum
mechanics and the group of affine canonical transformations of the phase
space.

% section 2
\section{\hspace{-4mm}.\hspace{2mm}CLASSICAL AND QUANTUM MECHANICS IN PHASE
SPACE}

This paper has the primary purpose of first presenting the program in as
simple terms as possible, so we deal with the case of systems with one
degree of freedom. It is not the exhaustive exposition, of course, it is the
occasion to discussion only.

In this section we take as a starting point a brief reminiscence of phase
space classical and quantum mechanics notions \cite{ar},\cite{bo},\cite{shi}%
, \cite{wey},\cite{wi},\cite{moy}.

\subsection{\hspace{-5mm}.\hspace{2mm}Classical Mechanics }

Let $x = (q,p)$ denote coordinates of the phase space ${\cal M} = {\bf R}^2$%
, identified with canonical variables of a classical mechanical system, and
let ${\cal A}$ denote the set of smooth functions on ${\cal M}$. The
observables of a classical system are identified with elements $f, g, ...$
of ${\cal A}$. ${\cal A}$ is equipped with two algebraic structures: the
pointwise multiplication (Jordan product), and the Poisson bracket operation

\begin{equation}
\{ f,g \} = {\frac{{\partial f} }{{\partial q}}}{\frac{{\partial g} }{{%
\partial p}}} -{\frac{{\partial f} }{{\partial p}}}{\frac{{\partial g} }{{%
\partial q}}} =\omega^{ij} {\frac{{\partial f} }{{\partial {x^i}}}} {\frac{{%
\partial g} }{{\partial {x^j}}}},
\end{equation}
which makes ${\cal A}$ into a Lie algebra. The notion $\omega$ is used for
simplectic matrix.

The states of a classical mechanical system are probability distributions on
${\cal M}$.

\subsection{\hspace{-5mm}.\hspace{2mm}Phase Space Representations of Quantum
Mechanics}

The main example here is the WWM representation of quantum mechanics in
phase space \cite{we},\cite{wi},\cite{moy}.

Let ${\cal M}$ and ${\cal A}$ are the same as in section 2.1. For each $k
\geq 0$ define bilinear partial Moyal bracket of degree $k$
\begin{equation}
{\{ f,g \}}^{(k)} = {\omega}^{i_1 j_1} \cdots {\omega}^{i_k j_k} {\frac{{{%
\partial}^k f} }{{\partial x^{i_1} \cdots \partial x^{i_k}}}} {\frac{{{%
\partial}^k g} }{{\partial x^{j_1} \cdots \partial x^{j_k}}}}.
\end{equation}
For $k=0$ it is usual pointwise multiplication and for $k=1$ it is the
Poisson bracket (2.1).

It is useful and important to remark that the $\{f,g\}^{(k)}$ has a form of
the $k$th {\em transvection} operator of the invariant theory \cite{gu},\cite
{we}. % \cite{die}.

The main algebraic structures of the WWM phase space representation of
quantum mechanics, the Jordan-Moyal product and the Poisson-Moyal bracket,
are of the form
\begin{equation}
f \circ g = {\sum_{n=0}^\infty}{\ {\frac{{(-1)^n} }{{(2n)!}}} {\frac{%
\left(\hbar}{2\right)}}^{2n}{\{f,g\}}^{(2n)}}
\end{equation}
and
\begin{equation}
{\{f,g\}}_M = {\sum_{n=0}^\infty}{\ {\frac{{(-1)^n} }{{(2n+1)!}}} {\frac{%
\left(\hbar}{2\right)}}^{2n} {\{f,g\}}^{(2n+1)}}.
\end{equation}
Here $(\cdot\circ\cdot)$ is a genuine Jordan algebra structure and ${%
\{\cdot,\cdot\}}_M$ is a genuine structure of Lie algebra on ${\cal A}$. $%
\hbar$ is the Planck constant.

This representation differs from the usual operator formulation of quantum
mechanics \cite{ma},\cite{fa} by form (and only by form) but is of the most
close to classical mechanics. Algebraic structures of the classical
mechanics is a limited case of the algebraic structures (2.3) , (2.4): the
pointwise product and the Poisson bracket may be derived from the
Jordan-Moyal product and the Poisson-Moyal bracket by passing to the limit $%
\hbar \to 0$. The case of quantum states is not so simple with respect to
limit $\hbar \to 0$ but it is the case of particular importance which lies
outside the paper.

The algebraic structures of standard operator quantum mechanics and the WWM
representation are connected by the Weyl-Wigner correspondence rule
\begin{equation}
-{\frac{i}{\hbar}} [\hat f, \hat g ] \to {\{f,g\}}_M ; \qquad {\frac{1}{2}} {%
[\hat f, \hat g ]}_+ \to f \circ g.
\end{equation}
Here $[\cdot,\cdot]$ is the commutator and $[\hat f, \hat g ]_+ = (\hat f +
\hat g)^2 - {\hat f}^2 - {\hat g}^2$.

There is a lot of correspondence rules. And for every correspondence rule
there is the associate phase space representation. Rare phase space
representations are existing \cite{bl},\cite{bbl},\cite{mo},\cite{te}. For
example, the representation which is connected with the Blokhintsev bracket
\begin{equation}
\{ f, g \}_B = {\sum_{n=1}^\infty} {\frac{(-i\hbar)^{n-1} }{{n!}}} {\omega}%
^{ij} {\frac{{\partial^n f} }{{\partial {(x^i)^n}}}} {\frac{{\partial^n g} }{%
{\partial {(x^j)^n}}}}.
\end{equation}
\vspace{2mm}

It should be noted that in the reminiscences of this section no mention is
made of the states of the mechanicses considered, because the algebraic
structures are the main object of consideration here, and this is the
question of other program \cite{emp}. However, in the final section we will
consider the connection between the basic group and coherent states notions.

%section 3
\section{\hspace{-4mm}.\hspace{2mm}THE PROBLEM}

The main content of this section is the following. In the first place we
briefly recall a formulation of the Dirac problem. Then as a preparetion to
the solution of this problem we derive the invariance group ${\cal G}$
(group of phase space transformations) of the algebraic structures of the
WWM representation. Then we to pose a question on the solution of the Dirac
problem by means of introducing into consideration of new algebraic
structures on ${\cal A}$ instead of the algebraic structures of classical
mechanics. Exactly, these new algebraic structures to be defined by the
required property to be invariant under the action of the main group ${\cal G%
}$ of the phase space transformations. Then these new algebraic structures
on the set of classical mechanics observables to be used for the solution of
the Dirac problem by means of Weyl-Wigner correspondence.

In such a way, and after obvious generalization, the basic notion of
quantization is a group of the affine phase space transformations. The
quantization amounts to the construction of relevant nonclassical invariant
algebraic structures on the set of classical mechanical observables. It is
clear {\em analog} of the Klein geometric program \cite{kl},\cite{ale}.

\subsection{\hspace{-5mm}.\hspace{2mm}The Dirac Problem}

The Dirac problem (quantization) can be formulated as the correspondence
problem in the following manner \cite{emp}, \cite{hu}: To establish {\em ab
initio} a mapping $Q$, which is defined on the algebra of classical
observables $f, g, ...$, takes values in the algebra of quantum observables $%
\hat f, \hat g, ...$ (self-adjoint operators acting in a Hilbert space $%
{\cal H}$ \cite{fa}, \cite{ma}), and has the following properties:\\
\vspace{0mm}\\ $(Q1) \qquad Q(\lambda f + \mu g) = \lambda Q(f) + \mu Q(g),
\quad \lambda, \mu \in {\bf R}; $\\ $(Q2) \qquad Q(\{ f, g \}) = {\frac{1 }{%
i\hbar}} [Q(f), Q(g)]; $\\ $(Q3) \qquad Q(f^2) = (Q(f))^2; $\\ $(Q4) \qquad
Q(1) = 1_{{\cal H}}. $\\ \vspace{0mm}\\ It is known that such the
correspondence problem leads to a lot of difficulties \cite{emp}, \cite{hu}.

\subsection{\hspace{-5mm}.\hspace{2mm}The Basic Group of the WWM
Representation}

We take as our starting point the set of partial Moyal brackets (2.2). The
Jordan-Moyal product (2.3) and the Poisson-Moyal bracket (2.4) are the
linear combinations of the partial Moyal brackets. It has been outlined
above (section 2.2) that the partial Moyal brackets are connected with
transvections of classical invariant theory. Hence, we may ask the question:
If the set of partial Moyal brackets is the set of bilinear invariant
algebraic structures on the set of classical observables, what is the group
of phase space transformations for? The answer on this question is almost
obvious: It is the group of affine canonical transformations (general form
of affine cononical transformation is: $x \to Cx + \xi$, where $C$ is a
linear canonical transformation, and $\xi \in {\cal M}$; as group it has the
structure of semidirect product of the symplectic group and the group of
vectors ${\cal M}$: $\quad {\cal G}_C = Sp({\cal M}) \dot \times {\cal M}
\quad $). This answer we can find in classical invariant theory (see \cite
{we} or \cite{gu}).

\subsection{\hspace{-5mm}.\hspace{2mm}The Program}

The program for the solution of the Dirac problem can be formulated now in
the following manner.

For a given group ${\cal G}$ of the phase space ${\cal M}$ affine
transformations and a system for which smooth functions from ${\cal A}$ are
the observables, firstly, the set of bilinear maps ${\cal A} \times {\cal A}
\to {\cal A}$ , which are invariant under the transformations ${\cal G}$, to
be constructed. Then, from these invariant maps the multiplication
operations of Lie $\{\cdot,\cdot\}_{{\cal G}}$ and Jordan $(\cdot \circ
\cdot)_{{\cal G}}$ to be constructed. If these operations can be defined
uniquely in a sense, they are useing in the Dirac problem instead of
classical operations of the Poisson bracket and pointwise multiplication:\\
\vspace{0mm}\\ $(q1) \qquad q(\lambda f + \mu g) = \lambda q(f) + \mu q(g),
\quad \lambda, \mu \in {\bf R}; $\\ $(q2) \qquad q(\{f,g\}_{{\cal G}}) = {%
\frac{1 }{{i\hbar}}} [q(f),q(g)]; $\\ $(q3) \qquad q((f \circ f)_{{\cal G}})
= (q(f))^2; $\\ $(q4) \qquad q(1) = 1_{{\cal H}}. $\\ \vspace{0mm}\\ It is
known that in the case of algebraic structures of the WWM representation
such the mapping $q$ is well defined \cite{po}.

In such a way, this program will be indeed the quantization program if, at
least, from the set of bilinear maps, which are invariant under the action
of the group ${\cal G}_C$ of affine canonical transformations, the
multiplication operations of Lie $\{\cdot,\cdot\}_{{\cal M}}$ (2.4) and
Jordan $(\cdot\circ\cdot)$ (2.3) can be constructed uniquely. When this test
case is verivied then we may take into consideration others basic affine
transformation groups.

\section{\hspace{-4mm}.\hspace{2mm}THE BASIC GROUP ${\cal G}_C$: NONCLASSICAL
ALGEBRAIC STRUCTURES}

The purpose of this section is to construct nonclassical Lie and Jordan
operations on the basis of the set $\{\{\cdot,\cdot\}^{(k)} ,\quad k\geq 0\}$
of ${\cal G}_C$-invariant bilinear operations. $\{\cdot,\cdot\}^{(0)}$ and $%
\{\cdot,\cdot\}^{(1)}$ are classical pointwise multiplication (Jordan
product) and Poisson bracket (Lie product) correspondently. It is easy to
see that for $k\geq2$ there is no any $k=m$ such that $\{\cdot,\cdot\}^{(m)}$
is Lie or Jordan product. Hence, for a construction of nonclassical
algebraic structures it is necessary to use an infinite linear combination
of all $\{\cdot,\cdot\}^{(k)},\quad k\geq0$.

\subsection{\hspace{-5mm}.\hspace{2mm}Lie Structure}

Let us consider an infinite series with coefficients $c_k,\quad k=0,1,2,...;$
\begin{equation}
\{f,g\}_{{\cal G}_C} = \sum_{k=0}^\infty c_k \{f,g\}^{(k)}.
\end{equation}
The operation $\{f,g\}_{{\cal G}_C}$ to be considered as Lie product if
\begin{equation}
\{f,g\}_{{\cal G}_C} = - \{f,g\}_{{\cal G}_C},
\end{equation}
and the Jacobi identity
\begin{equation}
\{f,\{g,h\}_{{\cal G}_C}\}_{{\cal G}_C} + \{h,\{f,g\}_{{\cal G}_C}\}_{{\cal G%
}_C} + \{g,\{h,f\}_{{\cal G}_C}\}_{{\cal G}_C} = 0
\end{equation}
are satisfied.

We shall test the hypothesis that the conditions (4.2), (4.3) determine the
coefficients $c_k$. First of all, it is evident that $\{f,g\}^{(k)} = (-1)^k
\{g,f\}^{(k)} $ for all $k $ and it follows
\begin{equation}
\{f,g\}_{{\cal G}_C} = \sum_{n=0}^\infty c_{2n+1} \{f,g\}^{(2n+1)}.
\end{equation}

Let us now consider the observables from ${\cal A}$ with the Fourier
representation
\begin{equation}
f(x) \sim \int \tilde f (\alpha) \exp(i\alpha(x)) d\alpha,
\end{equation}
where $\alpha(x)$ is 1-form on ${\cal M}$. It is easy to see that
\begin{equation}
\{\exp(i\alpha(x)), \exp(i\beta(x))\}_{{\cal G}_C} = \sum_{n=0}^\infty
c_{2n+1} \exp(i\alpha+i\beta) (-\{\alpha, \beta \}^{(1)}) ^{2n+1}.
\end{equation}

Define function $F(z)$ as
\begin{equation}
F(z)= \sum_{n=0}^\infty c_{2n+1} z^{2n+1}, \qquad F(-z)=-F(z).
\end{equation}
Substituting equations (4.6) and (4.7) in the Jacobi identity (4.3), we
derive the functional equation
\begin{equation}
F(z_1 + z_2)F(z_3) + F(-z_2 -z_3)F(z_1) + F(z_3-z_1)F(-z_2) = 0.
\end{equation}
After formal manipulations we find that
\begin{equation}
F^{\prime \prime}(z) F^{\prime}(z) - F^{\prime \prime \prime}(z) F(z) = 0
\end{equation}
since $F(0) = F^{\prime \prime}(0) = 0 $. The equation (4.9) for power
series $F $ satisfying condition (4.7) leads immediately to
%\begin{equation}
$F^{\prime \prime}(z) = - d_1^2 F(z) %\end{equation}
$, and then to
\begin{equation}
F(z) = d_2^{\prime} \sin(d_1 z) = d_2 \sum_{n=0}^\infty {\frac{{(-1)^n} }{{%
(2n+1)!}}} (d_1)^{2n} z^{2n+1},
\end{equation}
where $d_2 = d_2^{\prime} d_1 $. The solution (4.10) define all coefficients
$c_{2k+1} $. Hence, ${{\cal G}_C} $-invariant Lie product has the form
\begin{equation}
\{ f,g \}_{{\cal G}_C} = {d_2} \sum_{n=0}^\infty {\frac{{(-1)^n} }{{(2n+1)!}}%
} (d_1)^{2n} \{ f,g \}^{(2n+1)}.
\end{equation}
The coefficient $d_2 $ is unessential for $\{\cdot,\cdot\}_{{\cal G}_C} $ as
Lie product. The operation (4.11) is the same as Moyal bracket (2.4) when $%
d_2 = 1 $ and $d_1 = (\hbar /2) $.

\subsection{\hspace{-5mm}.\hspace{2mm}Jordan Structure}

Now we consider, once again, an infinite series of the type (4.1) and use
the arguments analogous to those presented in section 4.1 for construction
of nonclassical Jordan product.

Let
\begin{equation}
(f \circ g)_{{\cal G}_C} = \sum_{k=0}^\infty c_k \{ f,g\}^{(k)}.
\end{equation}
The operation $(f\circ g)_{{\cal G}_C} $ to be considered as Jordan product
\cite{emb} if
\begin{equation}
(f\circ g)_{{\cal G}_C} = (g\circ f)_{{\cal G}_C},
\end{equation}
and the Jordan identity
\begin{equation}
((f \circ g)_{{\cal G}_C} \circ (f\circ f)_{{\cal G}_C} )_{{\cal G}_C} - (f
\circ (g \circ (f\circ f)_{{\cal G}_C} )_{{\cal G}_C} )_{{\cal G}_C} = 0
\end{equation}
are satisfied.

We shall test now the hypothesis that the conditions (4.12) and (4.13)
determine the coefficients $c_k $ (4.11). Firstly, we see that
\begin{equation}
(f \circ g)_{{\cal G}_C} = \sum_{n=0}^\infty c_{2n} \{f,g\}^{(2n)}.
\end{equation}
Define function $G(z) $:
\begin{equation}
G(z) = \sum_{n=0}^\infty c_{2n} z^{2n},\qquad G(-z)=G(z).
\end{equation}
Then the repeated application of the section 4.1 procedure gives the
folloing equation
\begin{equation}
G(z_1)G(z_2 +z_3) - G(z_1 +z_2)G(z_3) = 0.
\end{equation}
After formal manipulations we find that $G(z) $ satisfy the equation (4.9) :
$G^{\prime \prime}G^{\prime} - G^{\prime \prime \prime}G = 0 $ with the
condition (4.15) which is another then (4.7). As the result we have
\begin{equation}
G(z)= d_2 \sum_{n=0}^\infty {\frac{{(-1)^n} }{{(2n)!}}} (d_2)^{2n} z^{(2n)}.
\end{equation}
Hence, ${\cal G}_C $-invariant Jordan product has the form
\begin{equation}
(f\circ g)_{{\cal G}_C} = d_2 \sum_{n=0}^\infty {\frac{{(-1)^n} }{{(2n)!}}}
(d_1)^{2n} \{f,g\}^{(2n)}.
\end{equation}
We see that the structure of the Jordan product (4.18) has the same
peculiarity as the structure of the Lie product (4.11) but the coefficients $%
d_1, d_2 $ in equatins (4.11) and (4.18) are not connected. To account for
compatibility between Lie and Jordan product, we can take into consideration
the condition \cite{emb}
\begin{equation}
((f\circ g)_{{\cal G}_C} \circ h )_{{\cal G}_C} - ((f\circ (g\circ h)_{{\cal %
G}_C} )_{{\cal G}_C} \sim \{g,\{f,h\}_{{\cal G}_C} \}_{{\cal G}_C}.
\end{equation}
\vspace{0mm}\\ As the {\em resume} of the sections 4.1 and 4.2, we see that
our program give us the result which coinsides with th WWM algebraic
structures.

For a further consideration it should be kept in mind that the algebras may
be complex.

\section{\hspace{-4mm}.\hspace{2mm}AS A CONCLUSION}

Further, the following case may be considered with use: the group of affine
homothetic transformations. As use we mean Blokhintsev brackets and Klein
principle. On the other hand, it is clear that the program is in its
beginning stages. This work is the part of a step toward the program's
direction. We believe that developments in this area will be forthcoming.

\vspace{\baselineskip}

\vspace{\baselineskip}

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\end{document}