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%%%%%%%%%%%%%%%%%%%%Fig.1'1%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%Fig.1'3%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%Fig.1a%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%Fig.1b%%%%%%%%%%%%%%%%%%%%%
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\documentclass[12pt]{article}
\usepackage{amsfonts,amssymb,icom}
\usepackage[eqsecnum]{pre2e}
\oddsidemargin=30mm
\evensidemargin=30mm
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\def\Bbb{\mathbb}
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\title[Isotropic forms of dynamics\hfill A.Duviryak, V.Shpytko, V.Tretyak]
{Isotropic forms of dynamics in the relativistic direct interaction
theory}
\author[A.Duviryak, V.Shpytko, V.Tretyak]{A.Duviryak, V.Shpytko, V.Tretyak}
\address{Institute for Condensed Matter Physics of
\newline the National Academy of Sciences of Ukraine
\newline 1 Svientsitskii St., 290011 Lviv--11, Ukraine}
\date{ }

\begin{document}

\maketitle
\setcounter{page}{1}

\begin{abstract}
The Lagrangian relativistic direct interaction theory in the
various forms of dynamics is formulated and its connections
with the Fokker-type action theory and with the constrained
Hamiltonian mechanics are established. The motion of classical
two-particle system with relativistic direct interaction is
analysed within the framework of isotropic forms of dynamics
in the two- and four-dimensional space-time.
Some relativistic
exactly solvable quantum-mechanical models are also discussed.
\keywords relativistic mechanics, Lagrangian formalism,
Hamiltonian formalism, constrained system, Fokker-type action.
%Up to six keywords.
\pacs 03.20.+i, 03.30.+p, 03.65.Ge, 11.30.Cp
\end{abstract}

\section{Introduction}

The relativistic direct interaction theory arises from the
expectation that the dynamics of an interacting particle system can
be constructed in a consistent Poincar\' e-invariant way without
introducing the notion of the field as an independent object with
its own degrees of freedom \cite{[1],[2],[3],[4],[5]}.
At present the principal possibility of
such  a theory is evident in the classical and quantum domains. Its
application to the description of particle systems is most effective
when processes of radiation and particle creation may be neglected.

Among various more or less equivalent approaches to the construction of the
relativistic direct interaction theory, the single-time Lagrangian formalism
\cite{[1],[13],[10]} proposed by Professor Gaida
more than twenty years ago, seems to be the most convenient for the
consideration of the general problem of relativistic dynamics, as well as
for the investigation of various approximations. This formalism has been
extended to an arbitrary form of relativistic dynamics \cite{[14]} defined
geometrically by means of space-like foliations of the Minkowski space
\cite{[15],[10],nlmp}. The conditions of the Poincar\' e-invariance were
reformulated in an arbitrary form of dynamics and a wide class of
exact solutions to the equations expressing these conditions were established
for the interactions originally described by a Fokker-type action.
The transition from the classical Lagrangian to the Hamiltonian description
allows one to consider the relativistic effects in the statistical and
quantum mechanical properties of the particle systems.

The purpose of the present paper is to review some relatively recent
generalizations and specifications of this development.
The transition from a non-relativistic interacting particle system to its
relativistic counterpart, which on a more formal level can be
considered as the replacement of the Galilei group by the Poincar\'e
group as a symmetry group of the system, leads to profound changes in
the structure of the theory. Within the Lagrangian formalism such a change
manifests itself in the necessity of using the interaction
Lagrangians depending on derivatives of an infinitely high order: in the
general case the exact relativistic Lagrangian must be defined on the
infinite order jet space \cite{[13]}. This fact is the Lagrangian counterpart of the
famous no-interaction theorem in the Hamiltonian relativistic mechanics
\cite{Car63} and has with the latter a common cause lying in the very
structure of the Poincar\'e group. It also reflects the time non-locality
inherent to relativistic interactions. All the aforementioned exact solutions
of Poincar\'e-invariance conditions corresponding to time-symmetric
Fokker-type actions have such kind of non-locality in any form of
relativistic mechanics \cite{[8],[9],nlmp}. Although there are elaborated
several methods of dealing with such systems (expansions in various
parameters \cite{[1],[11],[12]}, transition to the center-of-mass variables
\cite{gyt}), it is evident that such a drastic change in the structure of
mechanical description leads to serious difficulties in the physical
interpretation of the formalism, as well as in proving its
mathematical consistency.

But there are important exceptions from the general rule.
If the form of dynamics defines a simultaneity relation in the
Poincar\'e-invariant way (i.e. the Poincar\' e group transforms
simultaneous events into simultaneous ones), then the
corresponding invariance conditions of the Lagrangian description
allow a large class of exact solutions containing derivatives of any
finite order (not less than unity). Particularly, in such forms of
dynamics we can construct in the closed form a variety of nontrivial
interaction Lagrangians depending on the first order derivatives.

This fact was first established for an $N$-particle system in the
two-di\-men\-si\-o\-nal space-time $\M_2$ within the framework of the front
form of dynamics \cite{[20]}. Then it was extended to the case of a
two-particle system in the four-dimensional Minkowski space $\M_4$ by means
of isotropic forms of  dynamics with simultaneity between the events of
particle world lines defined by a light cone \cite{dt}. The existence of
such ``standard'' relativistic  Lagrangians brings the problem of describing
such kind of systems within the scope of the usual analytical (and, probably,
quantum) mechanics. It allows the formulation of various exact models of
relativistic direct interactions which admit more or less explicit
investigations. Such models are the main subject of this paper.

It is organized as follows.
In section~1 we begin with introducing the notion of the form of relativistic
dynamics within the framework of the Lagrangian formalism. The general features
of the relativistic Lagrangian description in a two-dimensional version
of the front form of dynamics
and in isotropic forms of dynamics are discussed
in sections~3 and~4, respectively.
The Fokker-type action integrals which correspond to time-asymmetric
interactions are considered in section~5. Section~6 is devoted to
the construction of the Hamiltonian formalism with constraints in the
isotropic form of dynamics. On this basis in section~7 we investigate
in the most explicit form the motions of two particles under the influence
of a time-asymmetric scalar, vector, and other interactions of physical interest.
The limiting case of
straight line motions of such systems is considered in section~8
within the framework of the front form of dynamics.
Finally, in section~9 we present certain exactly solvable
relativistic quantum models of interacting particle systems in
the two-dimensional space-time.

\input{iso1}
\input{iso2}
\input{iso3}
\input{iso4}

\section{Conclusion}

	We have considered the class of isotropic forms of dynamics which
admit the construction of a variety of exactly solvable relativistic
models of interacting particle systems. Most of the models originate from
the Fokker-type actions with the time-asymmetric (retarded or advanced)
Green function of the d'Alembert equation. These models reflect not only
the relativistic kinematics but also certain field-theory aspects of the
particle interaction. They demonstrate a complexity of the relativistic
particle dynamics in comparison with its non-relativistic counterpart.
The study of such a dynamics in detail is possible because of
the fact that the considered
forms of dynamics allow reformulation of the theory  in
terms of various formalisms and approaches, both three-dimensional
and  manifestly  covariant  four-dimensional.

The physical meaning of time-asymmetric  interactions is  not  so  clear.
Nevertheless, the corresponding models may be regarded as the first step
to some approximation scheme for finding solutions of more  physically
acceptable models, for  example, the Wheeler-Feynman  electrodynamics
and the related theories. Particularly, in the linear approximation in the coupling
constant the time-asymmetric, time-symmetric and purely retarded (field)
approaches yield the same result.
On the other hand, exact solutions  of  such
models  provide  a better  understanding  of the special  features   of
relativistic  interactions  and  interrelations  between   various
descriptions of relativistic interacting particles.\\


We wish to thank Yuriy Kluchkovsky, S.~N.~Sokolov,
V.~I.~Lengel, and J.~Llosa for the stimulating discussions.
The ideas and influence of the late Professor Roman Gaida are evident
throughout all the reported investigations.

\input{iso5}

\end{document}
%2
\section{Geometrical concept of the forms of dynamics}

Let us consider a dynamical system consisting of
$N$ interacting point particles. It is convenient to describe the evolution
of this system in the 4-dimensional Minkowski space ${\Bbb M}_4$
with coordinates $x^{\mu}$, $\mu =0,1,2,3$.
We use the metric $\| \eta _{\mu \nu}\| ={\rm diag}(1,-1,-1,-1)$.
The motion of the particles is described by the world lines
$\gamma _a:{\Bbb R}\rightarrow{\Bbb M}_4$, $a=1,...,N$,
which can be parametrized by arbitrary parameters $\tau _{a}$.
In the coordinates we have
%(2.1)
\begin{equation}
\gamma _a:\tau _a\mapsto x_a^{\mu}(\tau _{a}).
\end{equation}
The velocity of light is taken to be unity.

Since in the Poincar\' e-invariant theory no particle can move with
the velocity greater than the velocity of light, the world lines
$\gamma_{a}$ must be time-like lines, and the tangent vectors
%(2.2)
\begin{equation}
u_a^{\mu}=\frac{\d x_a^{\mu}}{\d\tau_a}
\end{equation}
obey the inequality
%(2.3)
\begin{equation}
u_a^2\equiv \eta _{\mu \nu}u_a^{\mu}u_a^{\nu}\equiv u_a\cdot u_a>0.
\end{equation}

It is well known that the whole physical information about the motion of
the system is contained in the world lines $\gamma _{a}$ considered as
unparametrized paths in the Minkowski space. Therefore, freedom in the
choice of parameters ${\tau _a}$ may be used for the simplification of the
description. Particularly, we can choose common parameter $t$ for all the
world lines of the $N$-particle system. This parameter is defined by a set
of $N$ relations of the following general form:
%(2.4)
\begin{equation}
\Phi _a(x_1(t),\ldots ,x_N(t),u_1(t),\ldots ,u_N(t),t)=0.
\end{equation}

The geometrical concept of the forms of relativistic dynamics
originated by Dirac \cite{[14],[17]} can be introduced within the
framework of the single-time Lagrangian or Hamiltonian
descriptions in the following way \cite{[15],[10],nlmp}.
Let us consider the foliation $\Sigma = \{ \Sigma _t|t\in {\Bbb R}\}$
of the Minkowski space ${\Bbb M}_4$ by the hypersurfaces $\Sigma _t$ with
the equation
%(2.5)
\begin{equation}\label{1.2}
t=\sigma (x),\qquad t\in {\Bbb R}.
\end{equation}
We require that every hypersurface
$\Sigma _{t} = \{ x\in {\Bbb M}_{n+1}|\sigma (x)=t\}$
must intersect  the  world lines $\gamma _{a}$ of all the particles
at one and only one point
%(2.6)
\begin{equation}
x_{a}(t)=\gamma _{a}\bigcap \Sigma _{t} .
\end{equation}
This allows us to consider $t$ as an evolution parameter of the
system \cite{[17],[18]}. In the Poincar\' e-invariant theory, when we
consider only time-like world lines, the hypersurfaces (2.5) must be
space-like or isotropic,
%(2.7)
\begin{equation}
\eta _{\mu \nu}(\partial ^{\mu}\sigma )(\partial ^{\nu}\sigma )\geq 0,
\end{equation}
where $\partial ^{\mu}=\partial /\partial x_{\mu}$.
Then we have $\partial _0\sigma >0$, and the hypersurface equation (2.5)
has the unique solution $x^0=\varphi (t,{\bf x})$,
where ${\bf x}=(x^i)$, $i=1,2,3$.
Therefore, the constraint $x_{a}(t)\in \Sigma _{t}$ enables us to determine
the zeroth component of $x_a(t)$ in terms of $t$ and $x_a^i(t)$.
The parametric equations (2.1) of the world lines of
the particles in the given form of dynamics are as follows:
%(2.8)
\begin{equation}
x^0=\varphi (t,{\bf x}_a(t))\equiv \varphi _a ,\qquad
x^i=x^i_a(t).
\end{equation}

The evolution of the system is determined by  $3N$ functions
$t\mapsto x_a^i(t)$. They may be considered as representatives
(in some local chart) for the sections
$s:{\Bbb R}\rightarrow{\Bbb F},t\mapsto (t,x_a^i(t))$ of
the trivial fibre bundle $\pi :{\Bbb F}\rightarrow {\Bbb R}$ with
$3N$-dimensional fibre space $\cM={\Bbb R}^{3N}$ \cite{28}.
The latter constitutes the configuration space of our system.

Three Dirac forms of relativistic dynamics correspond to the following
hypersurfaces (2.5): $x^0=t$ (instant form), $x^0-x^3=t$ or $x^0+x^3=t$
(front form),
and $\eta _{\mu \nu}x^{\mu}x^{\nu} =t^2$ (point form). Other
examples may be found in \cite{[15]}.

Now we assume that the evolution of the system under consideration is
completely determined by specifying the action functional
%(2.9)
\begin{equation}
S=\int \d tL.
\end{equation}
The Lagrangian function $L:J^{\infty}\pi \rightarrow {\Bbb R}$ is defined
on the infinite order jet space of the fibre bundle
$\pi :{\Bbb F}\rightarrow {\Bbb R}$ with the standard coordinates
$x_a^{i(s)}$ \cite{[22],[24]}. The values of these coordinates for the
section
$s:t\mapsto (t,x_a^i(t))$ belonging to the corresponding equivalence
class from $J^{\infty}\pi$ are
$x_a^{i(s)}(t)=\d^sx_a^i(t)/\d t^s$ , $s=0,1,2,\ldots$.
The variational principle $\delta S=0$ with the action
(2.9) gives Euler-Lagrange equations of motion
%(2.10)
\begin{equation}
\sum _{s=0}^{\infty}(-D)^{s}\frac {\partial L}
{\partial x_{a}^{i(s)}}=0,
\end{equation}
where $D$ is an operator of the total time derivative
%(2.11)
\begin{equation}
D=\sum _{a}\sum _{s=0}^{\infty}x_{a}^{i(s+1)}\frac {\partial }{\partial x_{a}^{i(s)}}+
\frac{ \partial }{\partial t} .
\end{equation}

Let us consider an arbitrary $r$-parametric Lie group ${\cal G}$
acting on ${\Bbb M}_4$ by the point transformations
$g:{\Bbb M}_4\rightarrow {\Bbb M}_4$:
%(2.12)
\begin{equation}
x^{\mu}\mapsto (gx)^{\mu}=x^{\mu}+ \lambda ^{\alpha}\zeta _{\alpha}^{\mu}(x)
+o(\lambda) ,
\end{equation}
where $\lambda ^{\alpha}$, $\alpha =1,\ldots ,r$ are the parameters of
the group. The vector fields
%(2.13)
\begin{equation}
{\cal X}_{\alpha}=\zeta _{\alpha}^{\mu}\partial _{\mu}
\end{equation}
satisfy the commutation relations of the Lie algebra of
group ${\cal G}$,
%(2.14)
\begin{equation}
[{\cal X}_{\alpha},{\cal X}_{\beta}]=c_{\alpha \beta}^{\gamma}{\cal X}_{\gamma},
\qquad \alpha ,\beta ,\gamma =1,\ldots ,r ,
\end{equation}
with the structure constants $c_{\alpha \beta}^{\gamma}$.

The action (2.12) of group ${\cal G}$ on ${\Bbb M}_4$
can be easily extended on the world lines $\gamma _a$ by the rule:
%(2.15)
\begin{equation}
\gamma _a\mapsto g\gamma _a=\{ gx|x\in {\rm Im}\gamma _a\}.
\end{equation}
But in the given form of dynamics the world lines $\gamma _a$ are
determined by the functions $t\mapsto x_{a}^{i}(t)$ or, in other words,
by sections $s$ of the bundle $\pi$. Therefore, (2.15) induces
an action of group $G$ on $J^{\infty}\pi $ by the Lie-B\" acklund
transformations \cite{[22],[23],[24]}. As it was shown in \cite{[15]}, the
generators of such transformations have the form:
%(2.16)
\begin{equation}
X_{\alpha}=\sum _{a}\sum _{s=0}^{\infty}(D^{s}\xi _{a\alpha}^{i})
\frac {\partial }{\partial x_{a}^{i(s)}} ,
\end{equation}
where
%(2.17)
\begin{equation}
\xi _{a\alpha}^{i}=\zeta _{a\alpha}^{i}-v_{a}^{i}\eta _{a\alpha} ,
\end{equation}
and
%(2.18)
\begin{equation}
\zeta _{a\alpha}^{i}=\zeta _{\alpha}^{i}(t,{\bf x_{a}}) ,\qquad
\eta _{a\alpha}=({\it X}_{\alpha}\sigma)(t,{\bf x_{a}}) ,\qquad
v_{a}^{i}=x_{a}^{i(1)} .
\end{equation}
The Lie-B\" acklund vector fields (2.16) obey the same commutation
relations as (2.14),
%(2.19)
\begin{equation}
[X_{\alpha},X_{\beta}]=c_{\alpha \beta}^{\gamma}X_{\gamma},
\end{equation}
and commute with the total time derivative (2.11)
%(2.20)
\begin{equation}
[X_{\alpha},D]=0 .
\end{equation}

For the Poincar\' e group we have the following ten vector fields
corresponding to the natural  action of ${\cal P}(1,3)$ on ${\Bbb M}_4$:
%(2.21)
\begin{equation}
{\cal X}_{\mu}^{T}=\partial _{\mu} ,
\end{equation}
%(2.22)
\begin{equation}
{\cal X}_{\mu \nu}^{L}=x_{\mu}\partial _{\nu}-x_{\nu}\partial _{\mu} ,
\end{equation}
with the commutation relations
%(2.23)
\begin{equation}
[{\cal X}_{\mu}^{T},{\cal X}_{\mu}^{T}]=0 ,
\end{equation}
%(2.24)
\begin{equation}
[{\cal X}_{\mu}^{T},{\cal X}_{\rho \sigma}^{L}]=
\eta _{\mu \rho}{\cal X}_{\sigma}^{T}-
\eta _{\mu \sigma}{\cal X}_{\rho}^{T} ,
\end{equation}
%(2.25)
\begin{equation}
[{\cal X}_{\mu \nu}^{L},{\cal X}_{\rho \sigma}^{L}]=
\eta _{\nu \rho}{\cal X}_{\mu \sigma}^{L}+
\eta _{\mu \sigma}{\cal X}_{\nu \rho}^{L}-
\eta _{\mu \rho}{\cal X}_{\nu \sigma}^{L}-
\eta _{\nu \sigma}{\cal X}_{\mu \rho}^{L} .
\end{equation}
Thus, we obtain the following realization of the Poincar\' e algebra in terms
of Lie-B\" acklund vector fields (2.16):
%(2.26)
\begin{equation}
X^{T}_{\mu}=\sum _{a}\sum _{s=0}^{\infty}D^{s}
[\delta _{\mu}^{i}-v^{i}_{a}\sigma _{a\mu}]
\frac {\partial }{\partial x_{a}^{i(s)}} ,
\end{equation}
%(2.27)
\begin{equation}
X^{L}_{\mu \nu}=\sum _{a}\sum _{s=0}^{\infty}D^{s}
[x_{a\mu}\delta _{\nu}^{i}-x_{a\nu}\delta _{\mu}^{i}-
v^{i}_{a}(x_{a\mu}\sigma _{a\nu}-x_{a\nu}\sigma _{a\mu}]
\frac {\partial }{\partial x_{a}^{i(s)}} ,
\end{equation}
where we must use (2.8) for the elimination of $x^0_a$, and we denote
%(2.28)
\begin{equation}
\sigma _{a\mu}\equiv (\partial _{\mu}\sigma )(t, {\bf x}_{a}) .
\end{equation}
Making use of the hypersurface equation (2.5) we find:
%(2.29)
\begin{equation}
\sigma _{a0}=(\partial \varphi _{a}/\partial t)^{-1}
\equiv \varphi _{at}^{-1},
\end{equation}
%(2.30)
\begin{equation}
\sigma _{ai}=-\varphi _{at}^{-1}(\partial \varphi _{a}/\partial x_{ai})
\equiv -\varphi _{at}^{-1}\varphi _{ai} .
\end{equation}

It is convenient to introduce the vector fields
%(2.31)
\begin{equation}
{\cal H}=-X^T_0,\quad {\cal P}_{i}=X^{T}_{i},\quad
{\cal J}_{i}=-\frac {1}{2}\varepsilon _{ijk}X^{L}_{jk},\quad
{\cal K}=X^L_{i0},
\end{equation}
obeying the following commutation relations:
%(2.32)
\begin{equation}
[{\cal H},{\cal P}_{i}]=0 ,\qquad [{\cal P}_{i},{\cal P}_{j}]=0 ,\qquad
[{\cal H},{\cal J}_{i}]=0 ,\qquad
[{\cal P}_{i},{\cal J}_{k}]=-\varepsilon _{ikl}{\cal P}_{l} ,
\end{equation}
%(2.33)
\begin{equation}
[{\cal J}_{i},{\cal J}_{k}]=-\varepsilon _{ikl}{\cal J}_{l} ,\qquad
[{\cal K}_{i},{\cal J}_{k}]=-\varepsilon _{ikl}{\cal K}_{l} ,\qquad
[{\cal K}_{i},{\cal K}_{j}]=\varepsilon _{ijk}{\cal J}_{k} ,
\end{equation}
%(2.34)
\begin{equation}
[{\cal H},{\cal K}_{i}]={\cal P}_{i} ,\qquad
[{\cal P}_{i},{\cal K}_{j}]=\delta _{ij}{\cal H} .
\end{equation}
Inserting (2.29), (2.30) into (2.26), (2.27), we obtain the realization
of the Poincar\'e algebra which is convenient for the consideration of the
symmetries of a single-time three-dimensional Lagrangian description
\cite{[15]}:
%(2.35)
\begin{equation}
{\cal H}=\sum _{a}\sum _{s=0}^{\infty}D^{s}
[v^{i}_{a}\varphi ^{-1}_{at}]
\frac {\partial }{\partial x_{a}^{i(s)}} ,
\end{equation}
%(2.36)
\begin{equation}
{\cal P}_{i}=\sum _{a}\sum _{s=0}^{\infty}D^{s}
[\delta ^{j}_{i}+v^{j}_{a}\varphi _{ai}\varphi ^{-1}_{at}]
\frac {\partial }{\partial x_{a}^{j(s)}} ,
\end{equation}
%(2.37)
\begin{equation}
{\cal J}_{i}=\varepsilon _{ikl}\sum _{a}\sum _{s=0}^{\infty}D^{s}
[x^{k}_{a}(\delta ^{j}_{l}+v^{j}_{a}\varphi _{al}\varphi ^{-1}_{at})]
\frac {\partial }{\partial x_{a}^{j(s)}} ,
\end{equation}
%(2.38)
\begin{equation}
{\cal K}_{i}=\sum _{a}\sum _{s=0}^{\infty}D^{s}
[-\varphi _{a}\delta ^{j}_{i}+v^{j}_{a}(x_{ai}-\varphi _{a}\varphi _{ai})
\varphi ^{-1}_{at}]
\frac {\partial }{\partial x_{a}^{j(s)}} .
\end{equation}

   The symmetry of the Lagrangian description of an interacting particle
system under group ${\cal G}$ means the invariance of the
Euler-Lagrange equation (2.10) under corresponding Lie-B\" acklund
transformations generated by the vector fields (2.16). The sufficient
conditions for the symmetry under the Poincar\' e group have the form
\cite{[13],[10]}:
%(2.39)
\begin{equation}
X_{\alpha}L=D\Omega _{\alpha},\qquad \alpha =1,\ldots ,10 ,
\end{equation}
with auxiliary functions $\Omega _{\alpha}$, satisfying the consistency
relations
%(2.40)
\begin{equation}
X_{\alpha}\Omega _{\beta}-X_{\beta}\Omega _{\alpha}=
c_{\alpha \beta}^{\gamma}\Omega _{\gamma} .
\end{equation}

An important corollary of symmetry conditions (2.39), (2.40)
for an arbitrary $r$-parametric Lie group is the existence of $r$
conservation laws
%(2.41)
\begin{equation}
DG_{\alpha}=0 ,\qquad \alpha =1,\ldots ,r ,
\end{equation}
for quantities $G_{\alpha}$ which can be explicitly determined in
terms of the Lagrangian function $L$ and auxiliary functions $\Omega
_{\alpha}$. This statement, which is well known as the N\" other theorem,
follows immediately from the identity \cite{[22],[23]}
%(2.42)
\begin{equation}
X_{\alpha}L=\sum _{a}\xi ^{i}_{a\alpha}{\cal E}_{ai}L+D\sum _{a}
\sum _{s=o}^{\infty}\pi _{ai,s}D^{s}\xi ^{i}_{a\alpha} ,
\end{equation}
which holds for an arbitrary Lie-B\" acklund vector field (2.16). Here,
%(2.43)
\begin{equation}
\pi _{ai,s}=\sum _{n=s}^{\infty}(-D)^{n-s}
\frac {\partial L}{\partial x_{a}^{i(n+1)}}
\end{equation}
are the Ostrogradskyj momenta. Making use of the identity (2.42) in
symmetry conditions (2.39), one readily checks that for the solutions of
Euler-Lagrange equation (2.10) the conservation laws (2.41) hold with
%(2.44)
\begin{equation}
G_{\alpha}=\sum _{a}\sum _{s=o}^{\infty}\pi _{ai,s}D^{s}\xi ^{i}_{a\alpha}-
\Omega _{\alpha} .
\end{equation}

In the general case the Poincar\' e-invariance conditions forbid the
existence  of interaction Lagrangians which are defined on the jet-space
$J^r\pi $ with some finite $r$ (for example, with $r=1$). This leads to
serious difficulties in the physical interpretation of the formalism, and,
in fact, makes it impossible to obtain a closed form of the corresponding
Hamiltonian functions.

In the following we shall consider some exceptions from this rule.
The first is offered by the front form of dynamics in the
two-dimensional Minkowski space. In this case there exists a wide class
of interaction Lagrangians for an $N$-particle system,
which are defined on the first-order jet-space $J^1\pi $ \cite{[20]}.
The second consists in the consideration of a more general definition of
the form of dynamics, than (2.5).
%3
\section{Front form of dynamics in ${\Bbb M}_2$}

In the two-dimensional space-time ${\Bbb M}_2$ the front form of dynamics
corresponds to the foliation of ${\Bbb M}_2$ by isotropic hyperplanes
(i.e., lines):
%(3.1)
\begin{equation}\label{s-1}% 3.1
x^0+x=t.
\end{equation}
In this form of dynamics for an $N$-particle system Poincar\' e-invariance
conditions allow the existence of interaction Lagrangians which do not
contain derivatives higher than the first order. The general form of such a
Lagrangian function including only pairwise interactions is
given by \cite{[20]}:
%(3.2)
\begin{equation}
\label{s-2}
L =-\sum_am_ak_a +
\sumab r_{ab}V_{ab}(r_{ab}k_a^{-1}, r_{ab}k_b^{-1}),
\end{equation}
where $k_a = \sqrt {1-2v_a}$, $r_{ab}\equiv x_a-x_b$, $a,b=\overline{1,N}$,
and $V_{ab}$ are arbitrary functions of the indicated arguments. As a result
of the Poincar\' e invariance of the Lagrangian function \re{s-2}, there exist
three conserved quantities: energy $E$, total momentum $P$, and the
center-of-inertia integral of motion $K$. They have the form \cite{[20]}:
%(3.3)
\begin{eqnarray}
\label{s-3}
E&=&\sum_{a=1}^{N}v_{a}\frac{\partial L}{\partial v_{a}}- L,\quad
P=\sum_{a=1}^{N}\frac{\partial L}{\partial v_{a}}-E,
\nonumber\\
K&=&-tP+\sum_{a=1}^{N}x_a\frac{\partial L}{\partial v_{a}}.
\end{eqnarray}

The existence of the interaction Lagrangians \re{s-2} permits one to trace
quite easily the relations between various formalisms of relativistic
dynamics and to find out special features of relativistic particle systems.
In spite of the fact that the Lagrangian function (\ref{s-2}) does not
contain higher derivatives and the transition to the Hamiltonian
description is a usual Legendre transformation, the investigation of exactly
solvable models shows some new features which do not occur in the
non-relativistic mechanics.

In the classical mechanics, the Lagrangian function
is  determined on the  tangent bundle $T\cM$, $L:T\cM\to \R$
\cite{28}. If the configuration space $\cM$ is diffeomorphic
to  ${\Bbb R}^N$, then the tangent bundle is a trivial one:
$T\cM={\Bbb R}^N\times{\Bbb R}^N$.
This means that a single chart with coordinates $(x_1,...,x_N, v_1,...,v_N)$
covers the whole $T\cM$.

For the Lagrangian \re{s-2} the configuration
space $\cM$ coincides with the whole ${\Bbb R}^N$ or at least with the
disconnected union of open  sets in ${\Bbb R}^N$.
Hence, one can expect that there should not be any complications connected with
the global structure. But the  Lagrangian function (\ref{s-2}) is
determined on submanifold $\cQ _f$ of $T\cM$. This submanifold is
defined by the inequalities
\begin{equation}\label{s-2.8}% 3.4
%-\infty <
 v_a  < 1/2,
\end{equation}
which reflect the time-like character of particle world lines in ${\Bbb M}_2$.
Submanifold $\cQ _f$ does not have the structure of a tangent bundle.

Moreover, we do restrict the Lagrangian description to the smaller region
than $T\cM$ for another reason. The Hamilton principle $\delta  S=0$
leads to Euler-Lagrange equations if the Hessian is positively defined:
\begin{equation}% 3.5
\label{s-3-1}
{\sf h}={\rm det}||\pl^2L/\pl v_a\pl v_b||>0.
\end{equation}
For the  Lagrangian function \re{s-2} the Hessian is, in general, a
complicated function on coordinate differences and velocities:
${\sf h}={\sf h}(r_{ab},k_c)$. Therefore, inequality \re{s-3-1} defines
an open region $\cQ \subset TM\approx {\Bbb R}^{2N}$. This region also does
not have the structure of a tangent bundle and for a free-particle system
coincides with $\cQ _f$.

It could be unimportant if the system moves inside the region (3.5) and
does not reach the boundary
\begin{equation}% 3.5a
\partial\cQ =\{(x_a,v_a)\in T\cM | {\sf h}=0, {\sf h}^{-1}=0\}.
\end{equation}
In contrast, the difficulty arises when the system
reaches the points of the boundary region (singular points)
at a finite value of the evolution parameter $t$ \cite{Sh}.
The theorem of existence and uniqueness for Euler-Lagrange differential
equations breaks at singular points and the Lagrangian system is not defined.
Therefore, we cannot prolong the evolution of the system beyond the critical
points within the framework of the basic Lagrangian description.

The way of overcoming this difficulty is offered by the Hamiltonian
description. It is well known that the Legendre transformation is a
differentiable mapping ${\pounds}:T\cM\to T^*\cM$. The transition
from the Lagrangian \re{s-2} to the Hamiltonian formalism  may be performed
by the usual Legendre transformation. But this transformation is a diffeomorphism
only in the  region $\cQ$.
It maps the open region $\cQ \subset {\Bbb R}^{2N}$
to the open one $\pounds\cQ\subset T^\ast \cM\approx{\Bbb R}^4$.
%$\pounds : \cQ \to \pounds\cQ$.
The Hamiltonian description is equivalent to the Lagrangian one
only in the region $\pounds\cQ$ \cite{28}.
In a strict sense the motion in the Hamiltonian
case is well defined  on $\pounds\cQ$ only. In other words, we
should consider $\pounds\cQ$ as a whole phase space of the system.

After the Legendre transformation is performed, the conserved quantities
\re{s-3} become canonical generators of the Poincar\' e group ${\cal P}(1,1)$:
\begin{eqnarray}% 3.6 - 3.7
\label{s-4}
P_+&=&\sum ^N_{a=1}p_a ,\qquad K=\sum ^N_ {a=1}x_ap_a,\\
\label{s-5}
P_-&=& \sum ^N_{a=1} \frac {m^2_a}{p_a} + \frac {1}{P_+} V(rp_b, r_{1c}/r)~.
\end{eqnarray}
They satisfy the following Poisson bracket relations:
\begin{equation}% 3.8
\label{s-6}
\{ P_{+},P_{-}\} =0 ,\qquad \{ K,P_{\pm}\}=\pm P_{\pm} .
\end{equation}
Here we have introduced more convenient in the front form quantities
$P_{\pm}=E\pm P$. The classical  total mass squared function
$M^{2}=P_{+}P_{-}$ has vanishing Poisson brackets with all the generators
(\ref{s-4}), (\ref{s-5}).

If we deal with  the Lagrangian region $\pounds\cQ $
within the Hamiltonian description,
we shall obtain the same results as in the Lagrangian case.
For  systems which reach the points of $\partial\cQ$,
the Lagrangian description leads  to disconnected segments
of world lines \cite{MST86}. To obtain the whole evolution of such  systems
we have to determine the motion of the system beyond the Lagrangian region.
In the following we shall demonstrate for certain relativistic models that
the Hamiltonian formalism permits one to prolong the evolution of the system
beyond singular points and obtain  smooth world lines in $\M_2$, as well as
in the  four-dimensional space-time $\M_4$ (see sections 8 and 7.1,
respectively).

%4
\section{Isotropic forms of dynamics}

For a two-particle system in ${\Bbb M}_4$ the class of isotropic forms
of dynamics corresponds to the following definition of simultaneity between
the events of particle world lines \cite{dt}:
\begin{equation}\label{3.1}% 4.1
[x_1(t)-x_2(t)]^2=0
\end{equation}
with the supplementary condition
\begin{equation}\label{3.2}% 4.2
{\rm sgn}[x^0_1(t)-x^0_2(t)]=\epsilon,
\end{equation}
where $\epsilon =\pm 1$.
Such a description has been developed within the framework of the predictive
relativistic mechanics in a series of papers by K\"unzle \cite{k1,k2,k3}.
The idea of this definition of simultaneity was formulated in the classic
Van Dam-Wigner's work \cite{vvw}. In the contents of relativistic Lagrangian
and Hamiltonian mechanics the descriptions based on equation~(\ref{3.1}) were
elaborated in \cite{dt,Duv96,Duv97}.

equations~(\ref{3.1}), (\ref{3.2}) determine the difference of the zeroth components:
\begin{equation}\label{3.3}% 4.3
x^0_1(t)-x^0_2(t)=\epsilon |{\bf x}_1(t)-{\bf x}_2(t)|.
\end{equation}
For the definition of the value of the common evolution parameter $t$
we choose the relation
\begin{equation}\label{3.4}% 4.4
\sigma \left (\frac {x_1(t)+x_2(t)}{2}\right )=t,
\end{equation}
where $\sigma (x)$ is the same function as in the definition
of the geometrical forms of dynamics (\ref{1.2}). Therefore, we have
\begin{equation}\label{3.5}% 4.5
\frac {x^0_1(t)+x^0_2(t)}{2}=
\varphi \left (t, \frac {{\bf x}_1(t)+{\bf x}_2(t)}{2}\right ),
\end{equation}
and
\begin{equation}\label{3.6}% 4.6
x^0_1=\varphi (t, {\bf y})+ \frac 12\epsilon |{\bf r}|, \quad
x^0_2=\varphi (t, {\bf y})- \frac 12\epsilon |{\bf r}|.
\end{equation}
Here and henceforth the variables $y^{\mu }\equiv (x^{\mu }_1+x^{\mu }_2)/2$
and $r^{\mu }\equiv x^{\mu }_1-x^{\mu }_2$ are used.

If we put $\varphi (t,{\bf y})=t$ as in the instant form of dynamics,
we obtain
\begin{equation}\label{3.7}% 4.7
x^0_a=t+\frac 12(-1)^{\bar a}\epsilon |{\bf r}|,
\qquad a = 1,2;\ \ \bar a \equiv 3-a.
\end{equation}
These relations have been used in \cite{k1,k2}.
When we choose $\sigma (x)$ as in the front form
$[\varphi (t, {\bf y})=t-y^3]$, we obtain
\begin{equation}\label{3.8}% 4.8
x^0_a=t+ y^3 +\frac 12 (-1)^{\bar a}\epsilon |{\bf r}|.
\end{equation}
In the two-dimensional space-time (\ref{3.8}) reduces to the geometrical
definition of the front form provided $\epsilon ={\rm sgn}(x_2-x_1)$.

The general structure of the Lagrange function is again determined by
the Poincar\'e-invariance conditions. Their formulation requires the
realization of algebra ${\frak p}(1,3)$ by the Lie-B\"acklund vector
fields (2.16). In paper \cite{dt} it was shown, that the components of
the corresponding fields have the form (2.17), where
\begin{equation}\label{3.13}% 4.9
\zeta ^i_{a\alpha }=\zeta ^i_{\alpha }[x_a(t)]
\end{equation}
and
\begin{eqnarray}\label{3.11}% 4.10
\eta _{a\alpha }&=&\frac 12
[\zeta ^{\nu }_{\alpha }(x_1)+ \zeta ^{\nu }_{\alpha }(x_2)]
\partial_{\nu }\sigma \left (\frac {x_1+x_2}{2}\right ) \nonumber \\
&=&(\zeta ^{\nu }_{\alpha }\partial_{\nu }\sigma )
\left (\frac {x_1+x_2}{2}\right )=\eta _{\alpha }(t,{\bf y}).
\end{eqnarray}
All the zeroth components here must be excluded by means of relations
\re{3.6}. Let us note the independence of $\eta _{\alpha }$ on the particle
labels.

It is a matter of simple calculation to verify that such vector fields
satisfy the commutation relations (2.19).

The Poincar\'e-invariance conditions have the form (2.39) where
we can put
\begin{equation}\label{3.14}% 4.11
\Omega _{\alpha }=- \eta _{\alpha }L.
\end{equation}
Such a choice  of auxiliary functions $\Omega _{\alpha }$ enables
(\ref{3.14}) to be expressed in the form:
\begin{equation}\label{3.15}% 4.12
{\hat X}_{\alpha }L + LD\eta _{\alpha }=0,
\end{equation}
where the vector fields
\begin{equation}\label{3.15a}% 4.13
{\hat X}_{\alpha }=X_{\alpha } + \eta _{\alpha }D.
\end{equation}
generate the point transformation of the extended configuration space
${\Bbb F}={\Bbb R}\times \cM$.

As in the case of the front form of dynamics in ${\Bbb M}_2$,
equations~(\ref{3.15}) allow a large class of exact solutions depending on the
derivatives of any finite order.
If we suppose that the Lagrangian contains only the first derivatives,
i.e. it is defined on the space $J^1\pi$, we obtain
\begin{equation}\label{3.17}% 4.14
\eta _{\alpha }\frac{\partial L}{\partial t}+
\sum _{a=1}^2\left ( \zeta ^i_{a\alpha }\frac{\partial L}{\partial x_a^i}+
(D\zeta ^i_{a\alpha } -
v^i_aD\eta _{\alpha })\frac{\partial L}{\partial v_a^i}\right )
+LD\eta _{\alpha}=0,
\end{equation}
where $\zeta ^i_{a\alpha }\equiv \zeta ^i_{\alpha }[x_a(t)]$.

The general solution to these equations can be presented in the form
\cite{dt}:
\begin{equation}\label{3.18}% 4.15
L=\vartheta F(\sigma _1,\sigma _2,\omega ),
\end{equation}
where
$$
\vartheta = (x_1^{\mu }-x_2^{\mu })u_{1\mu}=
(x_1^{\mu }-x_2^{\mu })u_{2\mu}=
\epsilon |{\bf r}|D\varphi (t, {\bf y})-{\bf r}\cdot \dot {\bf y};
$$
$$
\Gamma _a^{-2}=u_a^{\mu}u_{a\mu}=
\left (D\varphi (t, {\bf y})-\frac 12 (-1)^a\epsilon {\bf n}\cdot {\bf v}
\right )^2-v_a^2;
$$
$$
\qquad {\bf n}\equiv {\bf r}/r,\qquad r\equiv |{\bf r}|,\qquad
{\bf v}\equiv {\bf v}_1- {\bf v}_2;
$$
$$
\sigma _a=\Gamma _a\vartheta = r^{\nu }\hat u_{a\nu}, \qquad
\hat u_{a\nu}\equiv u_{a\nu} /\sqrt{u_a^2};
$$
$$
\omega =\Gamma _1\Gamma _2\left [(D\varphi (t, {\bf y}))^2
-{\bf v}_1\cdot {\bf v}_2-\frac 14({\bf n}\cdot {\bf v})^2\right ]=
\hat u_{1\nu}\hat u_2^{\nu},
$$
and $F$ being an arbitrary (smooth) function on three variables.

In the front form of dynamics in ${\Bbb M}_2$ we have $\vartheta  = r$,
$\Gamma _a= (1 - 2v_a)^{-1/2}=k_a^{-1}$, and $\omega $ is a function
on the invariants $\sigma _1$, $\sigma _2$:
\begin{equation}\label{3.20}% 4.16
\omega =\frac 12\left (\frac {\Gamma _1}{\Gamma _2}+
\frac {\Gamma _2}{\Gamma _1}\right )=
\frac 12\left (\frac {\sigma _1}{\sigma _2}+
\frac {\sigma _2}{\sigma _1}\right ).
\end{equation}

Invariance conditions (\ref{3.15}) lead to the conservation laws for the
quantities (2.44). In our case they have the form:
\begin{equation}\label{3.21}% 4.17
G_{\alpha }=\sum_{a=1}^2(\zeta ^i_{a\alpha } - v^i_a\eta _{\alpha })
\frac{\partial L}{\partial v_a^i} - \Omega _{\alpha }.
\end{equation}
Taking into account (\ref{3.14}) they can be expressed as
\begin{equation}\label{3.22}% 4.18
G_{\alpha }=\sum_{a=1}^2\zeta ^i_{a\alpha }
\frac{\partial L}{\partial v_a^i} - \eta _{\alpha }H,
\end{equation}
where
\begin{equation}\label{3.23}% 4.19
H=\sum_{a=1}^2v^i_a\frac{\partial L}{\partial v_a^i} - L.
\end{equation}

Let us introduce the Poincar\'e-invariant functions:
\begin{equation}\label{3.24}% 4.20
A_a=\sigma ^2_a\frac{\partial F}{\partial \sigma _a}+
(\omega \sigma _a-\sigma_{\bar a})\frac{\partial F}{\partial \omega },
\end{equation}
\begin{equation}\label{3.25}% 4.21
B_a=\sigma ^2_a\frac{\partial F}{\partial \sigma _a}+
(\omega \sigma _a+\sigma_{\bar a})\frac{\partial F}{\partial \omega }.
\end{equation}
They are not independent,
\begin{equation}\label{3.26}% 4.22
\sigma _1(A_1 - B_1)=\sigma _2(A_2- B_2).
\end{equation}
In terms of these functions we have:
$$
\frac{\partial L}{\partial v_a^i}=
\frac 12(r\varphi _i(t,{\bf y}) -\epsilon r_i)\tilde F +
v_{ai}\Gamma _a\sigma _a \left(\sigma _a\frac{\partial F}{\partial \sigma _a}
+\omega \frac{\partial F}{\partial \omega }\right )-
v_{bi}\Gamma_{\bar a}\sigma _a\frac{\partial F}{\partial \omega }
$$
\begin{equation}\label{3.27}% 4.23
-\frac 12\varphi _i(t,{\bf y})(\Gamma _1u^0_1A_1+\Gamma _2u^0_2A_2)+
\frac 12(-1)^a\epsilon n_i(\Gamma _1u^0_1B_1-\Gamma _2u^0_2B_2).
\end{equation}
where
\begin{equation}\label{3.28}% 4.24
\tilde F=F+\sum _{a=1}^2\sigma _a\frac{\partial F}{\partial \sigma _a }.
\end{equation}
The function (\ref{3.23}) is easily found to be
\begin{equation}\label{3.29}% 4.25
H=\varphi _t(t,{\bf y})(-r\tilde F+\Gamma _1u^0_1A_1+\Gamma _2u^0_2A_2).
\end{equation}
Explicitly, the integrals of motion (\ref{3.22}) are given by
$$
G_{\alpha }=-\zeta ^0_{\alpha }(t,{\bf y})\varphi _t(t,{\bf y})^{-1}H+
\zeta ^i_{\alpha }(t,{\bf y})(-\epsilon r_i\tilde F+\Gamma _1v_{1i}A_1+
\Gamma _2v_{2i}A_2) +
$$
\begin{equation}\label{3.30}% 4.26
+\frac 12(\zeta ^i_{1\alpha }-\zeta ^i_{2\alpha })[\Gamma _2(\epsilon n_iu^0_2-
v_{2i})B_2 -\Gamma _1(\epsilon n_iu^0_1- v_{1i})B_1].
\end{equation}
Inserting the expressions for functions $\zeta ^{\nu }_{\alpha }$
which correspond to the generators (2.21), (2.22) of the Poincar\'e group,
we obtain the following formulae for conserved
energy $E$, momentum ${\bf P}$, angular momentum  ${\bf J}$
and the center-of-inertia integral of motion ${\bf K}$:
\begin{equation}\label{3.31}% 4.27
E=\varphi ^{-1}_tH= -r\tilde F+\Gamma _1u^0_1A_1+\Gamma _2u^0_2A_2,
\end{equation}
\begin{equation}\label{3.32}% 4.28
{\bf P}=-\epsilon {\bf r}\tilde F+\Gamma _1{\bf v}_1A_1+\Gamma _2{\bf v}_2A_2,
\end{equation}
\begin{equation}\label{3.33}% 4.29
{\bf J}={\bf y\times P} +\frac 12 {\bf r\times }(\Gamma _1{\bf v}_1B_1-
\Gamma _2{\bf v}_2B_2),
\end{equation}
\begin{equation}\label{3.34}% 4.30
{\bf K}={\bf y}E - \varphi (t,{\bf y}){\bf P}
-\frac 12[\Gamma _2({\bf r}u^0_2-\epsilon r{\bf v}_2)B_2-
\Gamma _1({\bf r}u^0_1-\epsilon r{\bf v}_1B_1].
\end{equation}
We note that the expressions (\ref{3.31}), (\ref{3.32}) can be united into
a 4-vector of momentum $P_{\mu }$, as well as equations~(\ref{3.33}), (\ref{3.34})
represent a 4-tensor of angular momentum $J_{\mu \nu }$
\begin{equation}\label{3.35}% 4.31
P_{\mu }=\epsilon r_{\mu }\tilde F-\hat{u}{_1\mu }A_1-\hat{u}_{2\mu }A_2,
\end{equation}
\begin{equation}\label{3.36}% 4.32
J_{\mu \nu }=\frac 12 \left (y_{\nu }P_{\mu }-y_{\mu }P_{\nu }-
r_{\nu }(\hat{u}_{1\mu }B_1-\hat{u}_{2\mu }B_2)
-r_{\mu }(\hat{u}_{1\nu }B_1-\hat{u}_{2\nu }B_2)\right ).
\end{equation}
Here
\begin{equation}\label{3.37}% 4.33
E=-P_0, \quad  J_i=\epsilon _{ilk}J^{lk},\quad K_i=J_{0i},
\end{equation}
and the zeroth components of the 4-vectors $x_a$ and $\hat{u}_a$  must be
excluded with the help of relations (\ref{3.6}).

The structure of the motion integrals (\ref{3.35}), (\ref{3.36}) agrees with
the results of Refs.~\cite{k1,k2} which were derived within the framework of
the predictive relativistic dynamics.

Ten integrals of motion can be used to reduce the integration of
equations of motion to quadratures. But it is more convenient to preform
such a reduction by means of the techniques of the constrained Hamiltonian
mechanics.

%5
\section{Fokker-type action and single-time Lagrangians}

One of the possible ways to specify the form of the arbitrary functions
entering the general solution of the Poincar\'e-invariance conditions
is the comparison with the Fokker-type relativistic mechanics
\cite{[4],[5],[6]}, the oldest attempt to construct the relativistic direct
interaction theory which has a relation to the field description.
It is based on the manifestly Poincar\' e-invariant
variational principle formulated in terms of four-dimensional coordinates and
velocities of the particles. Such a variational principle was first introduced
for the electromagnetic interaction by Schwarzschild, Tetrode, and Fokker at
the beginning of this century and developed by various authors (see Refs.
\cite{[1],[5],[6]} and references therein). Later this description was
extended to other relativistic interactions. The equations of motion following
from such a variational principle explicitly satisfy the demand of relativistic
invariance and can be compared with the corresponding field theory
expressions. However, this approach is not free of difficulties
both on  physical and  mathematical levels. The cost for a
manifestly Poincar\' e-invariant four-dimensional description is the necessity
to use a many-time formalism which complicates the physical interpretation
of its results. Mathematically, it is hard to motivate the obtaining of the
equations of motion from the action integrals which are obviously
divergent because the integration is carried out on the whole length of the
world lines of the particles \cite{[4]}.

Within the framework of Fokker-type mechanics the dynamics of a relativistic
particle  system is specified in a manifestly Poincar\' e- and
reparametrization-invariant way on the basis of the  variational
principle $\delta S=0$ with the action being given by
%(2)
\begin{equation}\label{2}
S=S_f-S_{int} ,
\end{equation}
where
%(3)
\begin{equation}
S_f=-\sum_{a}m_a\int \d\tau _a\sqrt {u_a^2} ,
\end{equation}
corresponds to a free-particle system and
%(4)
\begin{equation}
S_{int}=\sumab \int \d\tau _a\int \d\tau _b \Lambda _{ab}(x_a,x_b,u_a,u_b).
\end{equation}
determines two-particle interactions. Here  $\Lambda_{ab}$ are some
functions depending on the four-dimensional particle coordinates
$x_{a}^{\mu}$ and on the first derivatives $u_{a}^{\mu}$.
They have the form \cite{[6],Ram73}:
%(7)
\begin{equation}
\Lambda_{ab}=\sqrt {u_{a}^{2}u_{b}^{2}}U_{ab}(x_{a},x_{b},
\hat u_{a},\hat u_{b}),
\end{equation}
where $\hat u_{a}^{\mu}=u_{a}^{\mu}/\sqrt {u_{a}^{2}}$ and
function $U_{ab}$ (which we shall call the {\it Fokker potential\/})
depends on the following set of the two-body
Lorentz scalars:
%(8)
\begin{equation}
\varrho _{ab}=(x_{a}-x_{b})^{2},\qquad
\sigma _{ab}=(x_{a}-x_{b})\cdot \hat u_{a},\qquad
\omega _{ab}=\hat u_{a}\cdot \hat u_{b};
\end{equation}
that is
%(9)
\begin{equation}
U_{ab}=U_{ab}(\varrho _{ab},\sigma _{ab},\sigma _{ba},\omega _{ab}).
\end{equation}

In papers \cite{[8],[9]} it was shown that many-time Fokker-type action
integrals can be transformed into single-time actions with non-local
Lagrangians depending on the three-dimensional coordinates of the particles
and on all the derivatives of the coordinates with respect to parameter $t$.
Such Lagrangians provide us with a useful tool for the consideration of
various approximations \cite{[8],[9],[10]}, as well as for the transition to the
predictive relativistic mechanics and Hamiltonian formalism \cite{[11],[12]}.
It was demonstrated \cite{[9]} that non-local Lagrangians corresponding to the manifestly
Poincar\' e-invariant action integrals satisfy the Poincar\' e-invariance
conditions within the framework of the three-dimensional Lagrangian
description of interacting particle systems \cite{[13]}. The conservation laws
which follow from such an invariance were investigated via the N\" other theorem.
Moreover, the non-local single-time Lagrangians
which are found on the basis of the Fokker-type action integrals represent a
closed form for a wide class of solutions of equations~(2.29)
expressing the requirements of the invariance of the Lagrangian description
of particle systems under the Poincar\' e group \cite{[9],nlmp}.

If $U_{ab}$ happens to have the special form
%(10)
\begin{equation}
U_{ab}=e_ae_b\omega _{ab}\delta (\varrho _{ab}) ,
\end{equation}
then action (\ref{2}) describes the electromagnetic interaction of charges
$e_a$ within the framework of the Tetrode-Fokker-Wheeler-Feynman
electrodynamics. Such an approach has been extended to the interactions
which are mediated by massive scalar and vector fields
\cite{[4],[6],Ram73}:
%(11)
\begin{equation}
{\rm the~scalar~case }\qquad
U_{ab}=g_ag_bG^{sym}(\varrho _{ab}) ,
\end{equation}
%(12)
\begin{equation}
{\rm the~vector~case }\qquad
U_{ab}=g_ag_b\omega _{ab}G^{sym}(\varrho _{ab}) .
\end{equation}
In the above, $g_a$ is a coupling constant of particle $a$ and
$G^{sym}(x)=G^{sym}(x^2)$ is a time-symmetric Green function of the
Klein-Gordon equation
%(13)
\begin{equation}
(\Box+\kappa ^2)G^{sym}(x)=4\pi \delta (x) ,
\end{equation}
where $\Box\equiv \eta _{\mu \nu}\partial ^\mu \partial ^\nu$ and
$\kappa $ is a mass of the field quanta. Explicitly, we have
%(14)
\begin{equation}
G^{sym}(x)=\delta (x^2)-\Theta (x^2)
\frac {\kappa}{2\sqrt{x^2}}J_1(\kappa \sqrt{x^2}) ,
\end{equation}
where $\Theta (x)$ is the Heaviside step function and $J_1(x)$ is the Bessel
function of order~1.

There exists a wider class of physically important Fokker-type
integrals which permit a field-theoretical interpretation of
interaction between particles. It corresponds to
Fokker potentials of the following form:
%	                5.2
\begin{equation}\label{5.2}
U_{ab} =g_ag_bf(\omega _{ab})G(\varrho _{ab}),
\end{equation}
where $f(\omega)$ depends upon the tensor structure of the fields mediating the
interaction, and $G(x)$ is a  symmetrical  Green
function of the relevant wave equation. In the case of massless fields
$G(x) = \delta (x^2)$. Especially, for interactions mediated by the
massless field with the given helicity $\lambda =\pm n$ we have \cite{Tre98}
\begin{equation}
f(\omega )= T_n(\omega).
\end{equation}
One more example is a model of confinement interaction \cite{ri}, for which
\begin{equation}
U_{ab} =g_ag_b\sigma _{ab}\sigma_{ba}\delta (\varrho_{ab}).
\end{equation}
Equivalently, this model can be presented in the form (\ref{5.2}) with
$f(\omega )=\omega$ and the Green function $G(x)$ replaced by the
``phenomenological propagator'' $\Theta (x^2)$.

Generally, the Fokker-type action with a time-symmetric Green function
leads to non-local in time Lagrangians and integral- or
difference-differential  equations  of  motion. It makes  the
analysis of particle motions a complicated task (except for the  case  of
circular motion when the solution may  be  constructed  explicitly
\cite{Shi63,And70}). An interesting  possibility to obtain ordinary
differential equations of motion is to replace $G$ in the right-hand side
of equation~(\ref{5.2}) by the retarded (advanced) Green function of
d'Alambert equation \cite{[7]}:
%	                5.3
\begin{equation}
G_{\epsilon}(x)=2\Theta(\epsilon x^0)\delta(x^2), \qquad \epsilon= \pm 1.
\end{equation}
This choice in the case of a two-particle system corresponds to the model with the
following particle interaction: the advanced field of the first particle acts on the
second particle and the retarded field of the second particle acts on the
first particle. Such interactions correspond to the exact solutions of the
Poincar\'e-invariance conditions considered above in the front and isotropic
forms of dynamics \cite{[20],dt,Duv97}.

In such models a one-to-one correspondence of points of two
particle world lines appears naturally, namely, of those points which
satisfy the {\em light cone condition}:
%		        5.4
\begin{equation}\label{5.4}
r^2 = 0 , \quad \epsilon r^0>0,\qquad {\rm i.e.,} \qquad
\epsilon r^0 = |\bf r|,
\end{equation}
%
This correspondence allows one to reduce
the Fokker-type integral to a manifestly covariant single-time action,
%		        5.5
\begin{equation}\label{5.5}
S_I = \int\!\!d\tau\,(L + \lambda r^2),
\end{equation}
%
where the Lagrangian multiplier $\lambda$ is introduced to take into
account condition (\ref{5.4}) as a holonomic constraint (the
boundary constraint $\epsilon r^0 > 0$ is also meant).

An action of this kind occurs when the  Fokker potential has a
more general structure:
%		        5.6
\begin{equation}
U = \tilde f(\omega, \sigma_1, \sigma_2)G_{\epsilon}(r), \qquad
\sigma_1 \equiv \sigma_{12}, \quad \sigma_2 \equiv \sigma_{21}.
\end{equation}
%
The relevant Lagrangian function reads:
%		        5.7
\begin{equation}\label{5.7}
L = - \sum^2_{a=1}m_a \lr{a} - \frac{\lr{a}\lr{b}}{\midl\dot y\cdot r\midr}
\tilde f,
\end{equation}
where the dot denotes a derivative on parameter $\tau$.
It creates a sufficiently  wide  class  of  two-particle
{\em time-asymmetric} models. Their study would not be successful
without an appropriate Hamiltonian description.
%6
\section{Hamiltonian description in the isotropic form of dynamics}

The Lagrangian description in the configuration space ${\Bbb M}_4^2$
allows a natural transition to the manifestly covariant Hamiltonian
description with constraints \cite{Dir50,[7]}. The corresponding phase space
${\rm T}^*\IM_4^2$ is a 16-dimensional one with the Poisson brackets
$[...,...]$. They have  a standard form in terms of covariant coordinates
$x^{\mu}_a$ and conjugated momenta defined in a usual manner:
%			6.1
\begin{equation}
p_{a\mu} = \partial L/\partial\dot x^{\mu}_a.
\end{equation}
%

Since the Lagrangian \re{5.7} and the constraint \re{5.4} are
Poincar\'e-invariant, there exist ten N\"other integrals of motion,
%			6.2
\begin{equation}
P_{\mu} = \sum^2_{a=1} p_{a\mu},\qquad
J_{\mu\nu} = \sum^2_{a=1} \left(x_{a\mu}p_{a\nu} - x_{a\nu}p_{a\mu}\right).
\end{equation}
%
In the Hamiltonian description these $P_{\mu}$ and $J_{\mu\nu}$ are
generators of the canonical realization of the Poincar\'e group.

By  virtue  of parametric invariance of the action (\ref{5.5}),
the  Lagrangian
(\ref{5.7}) is singular. Hence the canonical Hamiltonian vanishes, while the
dynamics of the system is determined by the {\em dynamical} constraint of
the following general form:
%              		6.3
\begin{equation}
\phi(P^2,~p_\bot^2, ~P \cdot r, ~p_\bot \cdot r) = 0,
\end{equation}
%
which appears together with the holonomic constraint (\ref{5.4}); here
$p_{\bot\mu} \equiv p_{\mu} - r_{\mu}P \cdot p/P \cdot r; ~P_{\mu}$
and $p_{\mu} = \ha(p_{1\mu} - p_{2\mu})$ are canonical momenta conjugated
to $y^{\mu}$ i $r^{\mu}$, respectively. Both the constraints are of the
first class, and they unambiguously determine the particle dynamics in
${\Bbb M}_4$ (i.e. the particle world lines).

Since no secondary  constraints  occur, the system possesses  12
physically essential  degrees  of  freedom.  In  order to single them out
explicitly, two subsidiary {\em gauge fixing} constraints are needed. They
can be given in the general form:
%			6.4
\begin{equation}
\chi (y,r,P,p_\bot,t) = 0, \qquad [\chi ,\phi ] \not= 0, \quad
\partial \chi /\partial t \not= 0.
\end{equation}
%
%			6.5
\begin{equation}
\psi (y,r,P,p) = 0, \qquad  [\psi , r^2] \not= 0.
\end{equation}
These constraints permit one to eliminate redundant time-like variables
$x^0_a$ and the corresponding momenta $p_{a0}$,
and then to pass to the three-dimensional Hamiltonian description.

	The gauge fixing constraints do not influence the dynamics of the
model, but their choice determines specific features of the final description,
namely, the reduced phase space $\IP$ (as a submanifold of ${\rm
T}^*\IM_4^2$), the induced Poisson brackets, and a possible choice of
variables, in terms of which these brackets take the canonical form. An
explicit form of observables (i.e. the covariant particle positions,
the generators of the Poincar\'e group etc.), being functions of the
canonical variables of space $\IP$, depends on a choice of the gauge
fixing constraints, too. Thus, using the arbitrariness of this choice one
can make an effective influence on the structure of the final description.

A special choice of the constraint (6.4) in the form
%			6.6
\begin{equation}
\chi = \chi (y,r,P_0,\tau),
\end{equation}
%
allows one to avoid a well-known {\em no--interaction theorem}
\cite{Car63}, that is,
to pass to such a three-dimensional Hamiltonian description of
time-asymmetric models in which the spatial covariant particle positions
$x_a^i\ (a=1,2;\ i=1,2,3)$ become  canonical variables.

	The three-dimensional Hamiltonian description in terms of covariant
variables is desirable in various aspects. For example, it simplifies the
introduction of the interaction with external fields and allows a
position representation on the quantum-mechanical level. But this
description is not convenient for solving a two-body problem,
because it does not provide a relevant separation of external and internal
degrees of freedom.

	Another choice of the gauge fixing constraint (6.4),
%			6.7
\begin{equation}
\chi = y^0 + {\rm tr}(\Lambda\,\Omega\,\partial\Lambda^{\rm T}\!/
\partial P_0) - \tau = 0,
\end{equation}
%
where
%			6.8
\begin{equation}
\Omega_{\mu\nu} \equiv r_\mu p_\nu - r_\nu p_\mu,
\end{equation}
%
$|P| \equiv \sqrt{P^2}$, and matrix $\|\Lambda(P/|P|)^\nu_{\ \mu}\| \in
{\cS\cO(1,3)}$, $\Lambda^\mu_{\ \nu}P^\nu = \delta_0^\mu|P|$
describes the Lorentz transformation into the centre-of-mass (CM) reference
frame, leads to a three-dimensional Hamiltonian description within the framework of
the Ba\-kam\-ji\-an-Tho\-mas model \cite{B-T53,D-K92,D-K93}. Within this
description ten generators of the Poincar\'e group $P_{\mu}$,
$J_{\mu\nu}$, as well as the covariant particle positions $x_a^{\mu}$ are the
functions of canonical variables ${\bf Q,~P}, ~\brho, ~\bpi$. The only
arbitrary function entering into expressions for canonical generators is
the total mass $\midl P \midr=  M(\brho, \bpi)$ of the system which
determines its internal dynamics. For time-asymmetric models this
function is defined by the mass-shell equation \cite{dt,Duv97} which can
be derived from the dynamical constraint via the following substitution of
arguments on the l.-h.s.\ of (6.3):
%
%       		6.9
\begin{equation}
\hspace{-2.5em} P^2\!\to M^2,~~p_\bot^2\!\to -\bpi^2,
~~P\!\cdot\!r\!\to\!\epsilon M\rho,~~
p_\bot\!\cdot\!r\!\to -
\bpi\!\cdot\!\brho;~~~{\rm here}~~ \rho \equiv \midl \brho
\midr\!.
\end{equation}%

	Due to the Poincar\'e-invariance of the description, it is sufficient
to choose the CM reference frame in which
${\bf P=0,~Q=0}$. Accordingly,  $P_0 = M$,
$J_{0i}=0~ (i=1,2,3)$, and the components $S_i \equiv \frac{1}{2}
\varepsilon_i^{\ jk}J_{jk}$ form a 3-vector of the total spin of the system
(internal angular momentum) which is an integral of motion. At this
point the problem is reduced to a rotation-invariant problem of some
effective single particle which is integrable in terms of polar coordinates,
%
%           		6.10
\begin{equation}
\brho = \rho{\bf e}_\rho,~~~~\bpi = \pi_\rho{\bf e}_\rho + S{\bf
e}_{\varphi}/\rho.
\end{equation}
%
Here $S \equiv {}\midl {\bf S} \midr$; the unit vectors ${\bf e}_\rho$,
~${\bf e_{\varphi}}$ are orthogonal to ${\bf S}$, they form together
with ${\bf S}$ a right-oriented triplet and can be decomposed in terms of the
Cartesian unit vectors ${\bf i,~j}$:
%
%         		6.11
\begin{equation}
{\bf e}_\rho = {\bf i} \cos \varphi + {\bf j} \sin \varphi,~~~~
{\bf e}_{\varphi} = - {\bf i} \sin \varphi + {\bf j} \cos \varphi,
\end{equation}
%
where $\varphi$ is a polar angle.

	The corresponding quadratures read:
%
%                   	6.12-13
\begin{eqnarray}
t - t_0 = \inta \d\rho\ \partial\pi_\rho(\rho,M,S)/\partial M,&& \\
\varphi - \varphi_0 = - \inta \d\rho\ \partial\pi_\rho(\rho,M,S)/\partial S,&&
\end{eqnarray}
%
where $t$ is an evolution parameter fixed by constraint (6.7) in
the CM reference frame, and the radial momentum $\pi_\rho$ as a function
of $\rho,~M,~S$ is defined by the mass-shell equation written down in
terms of these variables,
%
%             		6.14
\begin{equation}
\phi\left(M^2,\ - \bpi^2,\ \epsilon M\rho,\ -\bpi\cdot\brho\right) \equiv
\phi\left(M^2,\ - \pi_\rho^2 - \frac{S^2}{\rho^2},\
\epsilon M\rho,\ -\pi_\rho\rho\right) = 0.
\end{equation}
%

The solution of the problem given in terms of canonical variables
enables one to obtain particle world lines in the Minkowski space using the
following formulae \cite{dt,Duv97}:
%
%             		6.15-16
\begin{equation}
x_a^0 = t + \frac{1}{2}(-)^{\bar a} \epsilon\rho,
\end{equation}
\begin{equation}
{\bf x}_a = \frac{1}{2}(-)^{\bar a}\brho +
\epsilon\rho\frac{\bpi}{M} \equiv
\left(\frac{1}{2}(-)^{\bar a} +
\epsilon \frac{\pi_\rho}{M} \right)\rho
{\bf e}_\rho + \epsilon \frac{S}{M} {\bf e}_{\varphi}.
\end{equation}
%
Particularly, vector ${\bf r} = {\bf x}_1 - {\bf x}_2 = \brho$
characterizes the relative motion of particles.

%7
\section{Time-asymmetric models of particle interactions with
long-range and confining potentials}

	The explicit form of $\phi$ (6.3) depends in a complicated manner
on the choice of the original Fokker potential.
Its construction is the main difficulty which occurs in the analysis of
time-asymmetric models.
	Let us split function $\phi$ into two parts:
%              		7.1
\begin{equation}
\phi_f + \phi_{int}=0,
\end{equation}
%
where
%             		7.2
\begin{equation}
\phi_f= \frac{1}{4}P^2 - \frac{1}{2}(m^2_1 + m_2^2) + (m_1^2 - m_2^2)
\frac{p_\bot \cdot r}{P \cdot r} + p_\bot^2
\end{equation}
%
corresponds to a free-particle system, and $\phi_{int}$ is to be found.
Hereafter we refer to $\phi_{int}$ as the {\em Hamiltonian potential}.

	Only few cases are known when function $\phi_{int}$
can be constructed explicitly. They correspond to the three-parametric
Fokker potential
%             		7.3
\begin{equation}
U = U_s + U_v + U_c =
(\alpha_s + \alpha_v\omega + \alpha_c\sigma_1\sigma_2)G_{\epsilon}(r)~,
\end{equation}
%
where $\alpha_s,\alpha_v,\alpha_c$ are arbitrary constants. The first and the
second terms on the r.-h.s. of equation (7.3) correspond to the scalar
and vector field-type interactions with the coupling constants $\alpha_s$ and
$\alpha_v$, respectively, and the third term describes the confinement
interaction (when $\alpha_c>0$). In the non-relativistic limit this model
leads to the potential $U^{(0)} = (\alpha_s+\alpha_v)/r + \alpha_c r$, where
$r$ is the distance between the particles.

	The corresponding Hamiltonian potential has the form:
%			7.4
\begin{eqnarray}
\phi _{int} & = & -\,\frac{2\alpha_s m_{1}m_{2} + \alpha_v
(P^{2}-m_{1}^{2}- m_{2}^{2})}{\epsilon P\cdot r} -
2\alpha_c\left(\frac{b_1 b_2}{\epsilon P\cdot r} - \alpha_v\right)  \nonumber \\
&& -\,(\alpha_s ^2 -\alpha_v ^2)
\frac {2\alpha_s m_1m_2 + (b_{1}-\alpha_v)m_{2}^{2}+(b_{2}-\alpha_v)m_{1}^{2}}
{\epsilon P\cdot r\bigl((b_{1}-\alpha_v)(b_{2}-\alpha_v) -\alpha_s^2\bigr)}
\end{eqnarray}
where
%			7.5
\begin{equation}
b_a \equiv \epsilon(\ha P\cdot r + \na p_\bot\!\cdot r).
\end{equation}
It is worth noting that the interactions are combined in terms of
the Hamiltonian potential in a non-linear manner.

	For other Fokker potentials approximation methods (such as coupling
constant expansion) should be applied for the Hamiltonization procedure.
Especially for the time-asymmetric analogue of the
Fokker potential (5.12)  the Hamiltonian potential in the second order
approximation in coupling constant $\alpha=g_1g_2$ reads:
%         		7.6
\begin{equation}
\phi_{int} = - \frac{2m_1m_2}{\epsilon P \cdot r}\alpha f(\nu) -
\frac{\alpha^2h(\nu)}{\epsilon P \cdot r} \left(\frac{m_1^2}{b_1} +
\frac{m_2^2}{b_2} \right) +O(\alpha^3),
\end{equation}
%
where
%
%          		7.7
\begin{equation}
h(\nu) \equiv \left( (f(\nu) - \lambda f'(\nu)\right)^2 -
\left(f' (\nu)\right)^2
\end{equation}
%
and
%         		7.8
\begin{equation}
\nu \equiv \frac{P^2 - m_1^2 - m_2^2}{2m_1m_2}.
\end{equation}
%

	We note that particular cases of (5.12) are the Fokker potentials
which correspond to the particle interaction via
massless linear tensor fields of an arbitrary rank (see equation (5.13))
and their superpositions.

Below we consider some most interesting features of time-asymmetric models
described in this section.


\subsection{Vector and scalar models}

	We begin with vector and scalar
time-asymmetric  interactions.  These  models  are  based  on  the
Fokker-type integrals (5.1) with the Fokker potentials $U = U_v$ and
$U = U_s$, respectively (see (7.3)).  Both
scalar and vector models were  partly  considered  earlier  (the
former in the two-dimensional  space-time  only) \cite{Sta70,R-H70,Ste85,%
MST86,[18],k2,Fah81}. Our
results  obtained by means of both the Lagrangian and especially  the
Hamiltonian formulations of these models complete the analysis of
their classical dynamics.

	The vector and scalar time-asymmetric models present two-body
problems lying near the border line of those problems the solution of which
can be presented in a closed form. A lot of analyses can be made
analytically. Especially, turning and other important for the integration
points are solutions of the third and fourth order algebraic
equations, while the quadratures cannot be expressed even in terms of
elliptic and other special functions and, thus, they need computer work.
For simplicity here we limit ourselves to the case of equal particle rest
masses $m_0$.

	In the non-relativistic limit the vector and scalar interactions
reduce to the Coulomb interaction with the coupling constant $\alpha$
(namely, $\alpha_v$ and $\alpha_s$, respectively). Thus, it is convenient
to present the specific features of vector and scalar models
in comparison with the non-relativistic Coulomb system.

	The variety of solutions to the equations of motion of a two-particle
system consists of a 12--parametric family. The Poincar\'e transformations
(which form a 10--parametric group) change only the motion of system
considered as the whole. This motion does not reflect specific features
of the models. Here we do not distinguish solutions which differ from one
another by the Poincar\'e transformations. So, non-equivalent solutions form
a two--parametric family. It can be parametrized  by values of total
mass $M$ (or energy $E$ in the non-relativistic case) and spin (internal
angular momentum) $S$, the pair of integrals of motion. Thus, a variety of
all the possible solutions is reduced to some subset of ($M,S$)--plane. We note
that parameters $m_0$ and $|\alpha|$ become unessential when using $m_0$ and
$r_0 \equiv \midl\alpha\midr/m_0$ as units of measurement for momentum-- and
position--like variables, respectively. We also introduce dimensionless
integrals of motion $\mu = \ha M/m_0$ and $\sigma = S/\midl\alpha\midr$.
For the convenience we will speak about various solutions (namely,
phase trajectories, particle trajectories and world lines) as if each of them
is placed at the corresponding point of ($\mu,\sigma$)--plane.

	First of all we shall consider a vector model.
	Qualitatively different types of the phase trajectory (three top
graphs of figure 1) correspond to three different domains $\cD(+)$,
$\cD(-)$ and $\cD(0)$ of ($\mu,\sigma$)--plane
(the bottom graph of figure 1).
\begin{figure}[p]
%\vspace{1in}
\input fig1a.pic
%\vspace{1in}
\input fig1b.pic
%\vspace{1in}
\input fig1c.pic
%\vspace{1in}
\input fig1.pic
\vspace{-2.5in}

\hangindent=2.6in\hangafter=0
\noindent
{\small{\sf Figure 1.} Vector model. Various doma\-ins
%{\small{\hvsf\bfseries Figure 1.} Vector model. Various doma\-ins
of ($\mu,\sigma$)-plane (bottom graph) and the corresponding types of
the phase trajectory (three top graphs).\\
Curve $\cal F$ on the ($\mu,\sigma$)-plane is
determined by the parametric equations:\\
$$\sigma^2 = \frac{p(p+2)^2}{2p^2+5p+4},$$
$$\mu^2 = \frac{p(2p^2+5p+4)}{2(p+4)^2},$$
$$p\in[0,\infty[.$$}\vspace{-0.5in}
\end{figure}
\begin{figure}[p]
%\vspace{2.0in}
\input fig2.pic
\vspace{-2.2in}

\hangindent=3.2in\hangafter=0
\noindent
{\small{\sf Figure 2.} Vector model, $\alpha<0$ (attraction).
Regular
particle trajectories for various values of $\mu$, $\sigma$.\\
a) $\cal D(+)$: $\mu = 1.01$, $\sigma = 1.0$;\\
b) $\cal D(-)$: $\mu = 0.95$, $\sigma = 0.68$;\\
c) $\cal D(-)$: $\mu = 0.05$, $\sigma = 0.01$.\\
\vspace*{0.8in}
}

\hangindent=-4.0in\hangafter=0
\noindent
{\small{\sf Figure 3.} Vector model. Pathological
world lines $\gamma_+^a$.\\
Critical points:\\
$\Box$ is a turning point;\\
$\bullet$ is a collision point;\\
$\circ$ is start/end of the evolution.
}
\vspace{-1.3in}

%\vspace{2in}
\input fig3.pic
\vspace{-0.2in}
\end{figure}

The number and the position of
$(\mu,\sigma)$--domains on ($\mu,\sigma$)--plane are roughly in accordance
with the non-relativistic case, while the phase trajectories are more
complicated: they consist of few disconnected branches. It does means that
there exist few solutions of the Hamiltonian equations of motion at the
same values of the integrals of motion $\mu,\sigma$.

	Only one branch, namely,
$\gamma_-^a$ for the attraction case ($\alpha<0$) and $\gamma_-^r$ for the
repulsion case ($\alpha>0$), is {\em regular}, {\em i.e.} it is a
relativistic analogue of the phase trajectory of
the Coulomb system and coincides with the latter in a weakly
relativistic domain of ($\mu,\sigma$)--plane ({\em i.e.}, $\mu\approx1$,
and $\sigma\gg1$ for the attraction case). If $\mu>1$ ($\cal D(+)$ domain of
($\mu,\sigma$)--plane), both $\gamma_-^a$ and $\gamma_-^r$ exist and
correspond to unbounded particle trajectories which are analogues to the
hyperbolas of the Coulomb problem. We note that particle trajectories
$\gamma_-^a$ have a loop-like shape at the points of ($\mu,\sigma$)--plane
which are close to $\mu=1$, $\sigma=1$ (figure   2a). This effect becomes
more evident for bounded states $\gamma_-^a$ (existing in $\cal D(-)$)
as the appearance of the perihelion advance (figure   2b) (states
$\gamma_-^r$ of the repulsion case disappear in this domain). In the
ultrarelativistic case $\mu\to0$ (the left lower corner of $\cal D(-)$) the
particles "stick" together, so that the distance between them becomes far
less than the distance to the centre-of-mass (figure   2c). On curve
${\cal F}$ the regular states correspond to a circular motion of particles,
so that in domain ${\cal D}(0)$ (from below  curve $\cal F$) the
regular motion is forbidden.

The remaining branches $\gamma_+^0$, $\gamma_+^+$, $\gamma_+^-$, and
$\gamma_+^r$ of the phase trajectory do not have non-relativistic counterparts.
They exist and are qualitatively similar on the whole ($\mu,\sigma$)--plane.
These branches present a rather strange motion of particles, so that
the sign of $\alpha$ does not characterize the interaction as attractive or
repulsive. Moreover, it turns out natural to sew up the three
branches $\gamma_+^0$, $\gamma_+^+$, and $\gamma_+^-$ into a unique one
$\gamma_+^a$ (this is shown in figure 1 for the phase trajectory in $\cD(0)$),
so that the resulting motion is as follows: the particles move from an infinite
distance between them to their collision, go through one another and go away
to the distance $\sim r_0$, draw closer to one another, collide again,
and go away to an infinite distance (figure 3). Branches  $\gamma_+^r$ and
$\gamma_+^a$ and the corresponding world lines are {\em pathological} in the
sense that the velocities of massive particles tend asymptotically up to the
light speed. Besides, these solutions contain critical points (namely,
collision and turning points) in which massive particles reach
but not exceed the light speed at a finite time. Nevertheless, the particle
world lines turn out smooth both at these points and
everywhere. Another specific feature of the pathological states is that the
evolution of particles is spread over a semiinfinite interval of the
coordinate time while the evolution parameter covers the whole real axes
(figure   3).

	The scalar model is more intricate, especially for an attractive
interaction. There are
more qualitatively different types of the phase trajectory which correspond
to a larger number of ($\mu,\sigma$)--domains and which consist of more
branches (figure   4).

	Among them only one branch is regular, {\em i.e.} analogous to the
Coulomb phase trajectory. It exists in the domains ${\cal D}(1\pm;1)$.
Bounded states (in ${\cal D}(1-;1)$) present a
motion of particles with the perihelion retardance (unlike the advance in
the vector model). They disappear from below
curve ${\cal F}$, $\sigma>1/\sqrt{5}$, on which the particle trajectories
become circular.

\begin{figure}[p]
%\vspace{1in}
\input fig4a.pic
\vspace{0.2in}
%\vspace{1in}
\input fig4b.pic
\vspace{0.2in}
%\vspace{1in}
\input fig4c.pic
\vspace{0.2in}
%\vspace{1in}
\input fig4.pic
\vspace{-4.0in}

\hangindent=3.5in\hangafter=0
\noindent
{\small{\sf Figure 4.} Scalar model. Various domains
of the ($\mu,\sigma$)-plane (bottom graph) and the corresponding types of the
phase trajectory (six top graphs).\\ The curves $\cal F$, $\cal J$,
${\cal H}_\pm$, and ${\cal X}_\pm$ on the
($\mu,\sigma$)-plane are defined by the equations:
\begin{eqnarray}
{\cal F}:&\mu^2 = \frac{27\sigma}{2\left((3+\sigma^2)^{3/2} +
\sigma(9-\sigma^2)\right)},&\nonumber\\
{\cal J}:&\mu^2 = \frac{1}{2(1-\sigma^2)},&\nonumber\\
{\cal H}_\pm:&\mu = \frac{1}{2(1\pm\sigma)},&\nonumber\\
{\cal X}_\pm:&\mu = \left(\sqrt{1+\sigma^2}\pm\sigma\right)/2.&\nonumber
\end{eqnarray}
}
\vspace{1in}
\end{figure}
\begin{figure}[t]
\vspace{-0.35in}
%\vspace{1in}
\input fig5.pic
\vspace{2.3in}
{\small{\sf Figure 5.}
Scalar model, $\alpha<0$ (attraction). Various types of bounded particle
trajectories.\\
a) ${\cal D}(1-;2):\ \mu=0.95,\ \sigma=0.68;$
b) ${\cal D}(1-;3):\ \mu=0.9,\ \sigma=0.5;$\\
c) ${\cal D}(1-;3)$ near ${\cal J}:\ \mu=0.75,\ \sigma=0.3;$
d) ${\cal D}(2)$ near ${\cal H_-}:\ \mu=0.6,\ \sigma=0.16666;$\\
e) ${\cal D}(4;2):\ \mu=0.3,\ \sigma=0.15.$}
\end{figure}

In contrast to the case of a vector model, the domain of
regular states is bounded not only from below, but also from the left
where a motion is not forbidden. The border lines ${\cal X}_+$ and
${\cal J},\ \mu>\sqrt{5/8}$ indicate no special changes in the
particle motion except the appearance of critical points (which corresponds
to reaching  the light speed) on the particle world lines.
These {\em provisionally regular} states exist in the
domains ${\cal D}(1\pm;2)$ and ${\cal D}(1\pm;3)$. The effect of the
perihelion retardance grows for them (figure   5a), especially in the domain
${\cal D}(1\pm;3)$; here the particles move as if they attract one another at
a large distance, while at a small distance $\sim r_0$ each particle
repulses another one by a very (but not absolutely) hard core. The particles
bounce back off this core with the light speed, but their world lines
are smooth at this critical point (figure   5b).
Going to  curve ${\cal J}$, $\sigma<1/\sqrt{5}$,
the particle trajectories tend (as in the regular case)
to circular ones, but in a very strange manner: the particles rebound
more frequently (figure 5c), so that in the limiting
circular trajectories (which corresponds to ${\cal J}$ itself)
the set of critical points becomes dense everywhere.

Apart from the regular or provisionally regular states (which present
a reasonable behaviour of the particles on the whole) and the pathological
ones (which are roughly similar to those in the vector model), the
attraction (i.e. $\alpha<0$) scalar model possesses some {\em exotic}
states which correspond to a bounded particle motion at a relative
distance of order $r_0$. These states exist in the domains ${\cal D}(2)$,
${\cal D}(3\pm)$, and ${\cal D}(4;1)-{\cal D}(4;2)$, i.e., far from
the weakly relativistic domain, and thus they have no non-relativistic
analogues. For example, in the domain ${\cal D}(2)$ the particles move as if
each particle repulses another one by the hard exterior of an empty core
inside (figure 5d); in the domains ${\cal D}(4;1)-{\cal D}(4;2)$
the trajectory of one of the particles always lies inside the
trajectory of another particle (figure 5e).

	The variety of solutions described above is obtained within the
Hamiltonian formulation of the vector and scalar models. Within the
framework of the Lagrangian formalism only regular solutions can
be reconstructed completely. Besides, this framework partially recovers
provisionally regular solutions, namely, some segments of world lines
between the critical points. Other solutions disappear within the
Lagrangian formalism.

	In our consideration the Lagrangian formalism is primary with
respect to the Hamiltonian one. Thus, one can conclude at first sight
that non-Lagrangian solutions have no physical meaning. On the other
hand, the Hamiltonian formulation of the models is an important link toward
their quantization, and non-Lagrangian solutions may contribute to the
resulting quantum-mechanical picture.

	These complicated questions are discussed in more detail in
sections 8 and 9.1 where we study the classical and quantum mechanics of the
vector and scalar models in ${\Bbb M}_2$.


\subsection{Scalar--vector model}

	The purely vector and scalar time-asymmetric models are
calculatingly cumbersome and present a rather intricate particle dynamics.
The case of arbitrary superposition of the scalar and  vector
interaction is not expected to be simpler (though it is also solvable).
It follows from the complicated structure of the Hamiltonian potential (see
equation  (7.4) with $\alpha_c=0$).

      In a special case of superposition, $\alpha_v =\kappa\alpha_s
\equiv\kappa \alpha,
\ \ \kappa =\pm 1$, the second term of $\phi _{int}$ (7.4) vanishes.
This structure of the dynamical constraint simplifies to a great extent
the dynamics of the model and makes it similar, in the mathematical respect,
to the dynamics of a non-relativistic system with the Coulomb interaction.
In this case one can expect the existence of an additional integral
of motion, the relativistic analogue of the Runge-Lenz vector.

	Actually, it is easy to guess the structure of this integral
of motion working within the framework of manifestly covariant Hamiltonian
mechanics \cite{Duv96}.
For this purpose it is convenient to simplify the free-particle
term $\phi _f$ (10) of the dynamical constraint whose cumbersome form
obscures  the following
treatment of the model and is caused by a descriptional rather than dynamical
reason. Let us perform the canonical transformation
$(y^\mu ,\ \ P_\mu ,\ \ r^\mu ,\ \ p_\mu )\longmapsto
(z^\mu ,\ \ P_\mu ,\ \ r^\mu ,\ \ q_\mu )$,
%			7.9
\begin{equation}
q_{\mu} =
p_{\mu } - \frac {m_{1}^2 - m_{2}^2}{2P^2}P_{\mu }, \qquad
z^{\mu } =
y^{\mu } + \frac {m_{1}^2 - m_{2}^2}{2P^2}
\Bigl( r^{\mu } - 2\frac {P\cdot r}{P^2}P^{\mu }\Bigr) ,
\end{equation}
(the variables $r^\mu $ and $P_\mu $ remain unchanged). In terms of new
variables the dynamical constraint takes the form:
%			7.10
\begin{equation}
\phi = \frac {1}{4}P^2 -  \frac {1}{2}(m_{1}^2 + m_{2}^2) +
\frac {(m_{1}^2 - m_{2}^2)^2}{4P^2} + q_\bot^2
-\frac{\alpha \bigl(P^2 - (m_1 - \kappa m_2)^2\bigr)}{\epsilon P\cdot r} = 0,
\end{equation}
where
%			7.11
\begin{equation}
q_{\bot\mu} \equiv P^{\nu}\Xi _{\nu \mu}/P\cdot r,\quad
\Xi _{\mu \nu} = r_\mu q_\nu - r_\nu q_\mu ;\quad q_\bot\!\cdot P\equiv 0.
\end{equation}

Then, it is easy to examine that the relativistic analogue of the Runge-Lenz
vector has the following form:
%			7.12
\begin{equation}
R_\mu = \Pi _\mu ^\nu \Bigl( q_\bot^\lambda \Xi_{\lambda \nu} +
\frac{\alpha \bigl(P^2 - (m_1 - \kappa m_2)^2\bigr)}
{2\epsilon P\cdot r}r_\nu \Bigr) ,
\end{equation}
where $\Pi _\mu ^\nu \equiv \delta _\mu ^\nu - P_\mu P^\nu /P^2$.
It is indeed an integral of motion, i.e.
%			7.13
\begin{equation}
\lbrack R_\mu ,\phi \rbrack \approx 0, \qquad
\lbrack R_\mu ,r^2\rbrack = 0
\end{equation}
and satisfies the relations:
%			7.14
\begin{equation}
\lbrack R_\mu ,P_\nu \rbrack = 0 , \qquad \ \
\lbrack R_\mu ,J_{\lambda \sigma } \rbrack =
- \eta _{\mu \lambda} R_\sigma + \eta _{\mu \sigma} R_\lambda ,
\end{equation}
%			7.15
\begin{equation}
\lbrack R_\mu ,R_\nu \rbrack \approx
\Big( \frac {1}{4}P^2 -  \frac {1}{2}(m_{1}^2 + m_{2}^2) +
\frac {(m_{1}^2 - m_{2}^2)^2}{4P^2}\Bigr)
\Pi _\mu ^\lambda \Pi _\nu ^\sigma J _{\lambda \sigma },
\end{equation}
where the Dirac symbol $\approx $ denotes a weak equality.

	The relations (7.14)--(7.15) are similar to those obtained
for the Runge--Lenz vector of a simple relativistic oscillator and
Coulomb models in  \cite{DVN90}. These relations are essentially
nonlinear and thus their group theoretical treatment is complicated.
In the present paper we limit our study to the case of the CM reference
frame in which the corresponding Poisson bracket relation can be
linearized.

 	For this purpose we reformulate (as in the previous cases) the
present time-asymmetrical model into the framework of the Bakamjian-Thomas
model. Then the Runge-Lenz vector becomes $R_\mu = (0,\bf R)$, where
%			7.16
\begin{equation}
{\bf R} = \bpi \times {\bf S} + g(M)\brho/\rho,
\end{equation}
$\bf S = \brho \times \bpi$ is a spin of the system, and the total mass
satisfies the equation
%			7.17
\begin{equation}
d(M) - \bpi^2 - 2g(M)/\rho = 0.
\end{equation}
Here
%			7.18
\begin{equation}
d(M) \equiv \frac{1}{4M^2}\Bigl( M^2 - (m_1 + m_2)^2\Bigr)
\Bigl( M^2 - (m_1 - m_2)^2\Bigr) ,
\end{equation}
%			7.19
\begin{equation}
g(M) \equiv \frac{\alpha}{2M}\Bigl( M^2 - (m_1 - \kappa m_2)^2\Bigr) .
\end{equation}
Besides, in the CM reference frame the covariant
particle positions are the following functions of the canonical variables:
%			7.20
\begin{equation}
{\bf x}_a =
\frac{(-)^{\bar a}}{2}\Bigl( 1 +
\frac{m^2_{\bar a} - m^2_a}{M^2}\Bigr)\brho +
\epsilon\rho\frac{\bpi}{M},
\qquad a = 1,2;\ \ \bar a \equiv 3-a.
\end{equation}

        The Poisson bracket relations for the internal
angular momentum (spin) of the system $\bf S$ and the Runge-Lenz vector
$\bf R$ are similar to those in the non-relativistic Coulomb problem:
%			7.21
\begin{equation}
\lbrace S_i,S_j\rbrace = \varepsilon_{ij}^{\ \ k}S_k,\ \
\lbrace R_i,S_j\rbrace = \varepsilon_{ij}^{\ \ k}R_k,\ \
\lbrace R_i,R_j\rbrace = -d(M)\varepsilon_{ij}^{\ \ k}S_k.
\end{equation}
Indeed, when $d(M) = 0$, equations (7.21) are the relations for
generators of the Euclidian group $\cE(3)$. In the case $d(M) \not= 0$ the $S_i$
and the normalized
${\hat R}_i \equiv R_i/\sqrt {\vert d\vert }$ generate the group
$\cS\cO(4)$, when $d(M) < 0$, and the group $\cS\cO(1,3)$, when $d(M) > 0$.
Taking into account equation (7.21) we obtain the following cases for the
algebra of internal symmetries:\\

\hspace*{1cm}$\gs\go(4)\ \ \ $ for $\vert m_1 - m_2\vert < M < m_1 + m_2$,

\hspace*{1cm}$\gge(3)\ \ \ \ \ $ for $M = \vert m_1 - m_2\vert $
and $M = m_1 + m_2$,

\hspace*{1cm}$\gs\go(1,3)\ $  for $0 < M < \vert m_1 - m_2\vert $ and $M >
m_1 + m_2$.\\

The existence of the Runge--Lenz vector makes it possible
to obtain both the relative and particle trajectories
traced by vectors $\brho$ and ${\bf x}_a$, respectively, without an integration.
At first we note that these trajectories  are  flat  curves  placed  on
the  plane orthogonal to the spin
of the system, i.e. ${\bf{\brho\cdot S}} = {\bf{x}}_a{\bf{\cdot S}} = 0$.
Vector $\bf R$
lies on the same plane, i.e. ${\bf{R\cdot S}} = 0$. Multiplying equation (7.16)
by $\brho$ one can obtain the relation:
%			7.22
\begin{equation}
{\bf{R\cdot\brho}} = g\rho + S^2,
\end{equation}
where $S \equiv \vert \bf{S}\vert$. Let $\varphi $ be an angle between
$\bf R$ and $\brho$, i.e.
${\bf{R\cdot\brho}} = R\rho\cos \varphi$.
Then equation (7.22) can be reduced to the
canonical equation of a conic section
%			7.23
\begin{equation}
p/\rho = e\cos{\varphi} - {\rm{sgn}}\hspace*{0.05cm}g
\end{equation}
with the following canonical parameter $p$ and eccentricity $e$:
%			7.24
\begin{equation}
p = \frac{S^2}{\vert g\vert} = \frac{2MS^2}
{\vert \alpha \vert \vert M^2 -(m_1 - \kappa m_2)^2\vert },\ \
\ \ e = \frac{R}{\vert g\vert} = \sqrt{1 + \frac{S^2}{\alpha ^2}
\hspace*{0.06cm}\frac{M^2 - (m_1 + \kappa m_2)^2}
{M^2 - (m_1 - \kappa m_2)^2}}.
\end{equation}

Searching for the equations of particle trajectories
is a similar but somewhat complicated task. Let us define the vectors:
%			7.25
\begin{equation}
{\bf r}_a \equiv {\bf x}_a - c_a{\bf R},
\end{equation}
where
%			7.26
\begin{equation}
c_a = \frac{2(-)^{\bar a}}{(M + m_{\bar a})^2 - m_a^2}.
\end{equation}
Then, one can obtain the relations
%			7.27
\begin{equation}
(-)^{\bar a}{\bf{R\cdot r}}_a = gr_a + \frac{m_{\bar a}}{M}S^2,
\end{equation}
which are similar to equation (7.22) and hence can be written down as follows:
%			7.28
\begin{equation}
p_a/r_a = e\cos \varphi_a - {\rm{sgn}}\hspace*{0.05cm}g,
\end{equation}
where $\varphi_a$ are angles between $(-)^{\bar a}\bf R$ and ${\bf r}_a$.
Equations (7.28) describe the particle trajectories as being
conic sections
of the same shape as the relative trajectory, i.e. with the same
eccentricity $e$ (7.25) but with other canonical parameters $p_a =
\frac{m_{\bar a}}
{M}p$. The foci of these conic sections are shifted with respect to the
centre of mass by vectors $c_a\bf R$. On the contrary,
the non-relativistic particle trajectories have a common focus which is
located in the centre of mass.

\subsection{Models with higher rank tensor interactions.}

	As it was pointed out above, among time-asymmetric field-type models
only those corresponding
to the (arbitrary) superposition of scalar and vector interactions permit
the exact hamiltonization. In the case when the
rank of the field $n \ge 2$, the transition to the Hamiltonian
description and the construction of quadratures can be done by means of
the method of expansion in a coupling constant.

	The structure of the second order Fokker potential (7.6) is common
for linear field-type interactions of various tensor dimensions. It
specifies the sort of interaction by the functions $f(\nu)$ and
$h(\nu)$ which depend on the integral of motion $\nu$ only.
Moreover, the nonlinear gravitational interaction can be also  described
(at least in a slow motion approximation) by this potential (7.6) (see
\cite{Tur82,Duv96P}) with
%       		7.29
\begin{eqnarray}
&f_{gr}(\nu) = 2 \nu^2-1,&\\
%
%			7.30
&h_{gr}(\nu) = - 2(2 \nu^2+1),&
\end{eqnarray}
%
and $\alpha_{gr} = -\Upsilon m_1m_2$ where $\Upsilon$ is the gravitational
constant. It is possible to integrate a two-body problem considering
$f$ and $h$ as arbitrary first and second order functions, respectively.

	We note that in the second order approximation the quadratures
for the present case can be expressed in terms of elementary functions.
For bounded states they lead to the relative motion trajectory of
a very simple form,
%			7.31
\begin{equation}
1/\rho\ =\ \midl a\midr\ +\
b\cos\left((1-\delta)\varphi\right)~~~~~~(b < \midl a \midr),
\end{equation}
%
where $a$, $b$, and $\delta$ are functions of the integrals of motion.
It describes an ellipse which precesses with the perihelion advance
%
%			7.32
\begin{equation}
\Delta \varphi\ =\ 2\pi \delta\ =\ -\pi\alpha^2 h(1)/S^2.
\end{equation}
%
In the case of a linear purely tensor interaction of arbitrary rank $n$
the perihelion advance $\Delta \varphi$ can be calculated by means of the
formulae (7.7), (5.12)--(5.13),
%
%			7.33
\begin{equation}
\Delta \varphi\ =\ \pi(2n^2-1)(g_1g_2/S)^2.
\end{equation}
%
For the gravitational interaction, using (7.30), we obtain
%
%			7.34
\begin{equation}
\Delta \varphi\ =\ 6\pi( \Upsilon m_1m_2/S)^2.
\end{equation}
%

	The spatial particle trajectories calculated by means of (7.16)
turn out to be more intricate than the relative trajectory
which is the typical feature of time-asymmetric models. Nevertheless, their
analysis leads to the same value of the perihelion advance.

	We note that these relations for the perihelion advance fit those
obtained within the various quasirelativistic approaches to the
relativistic direct interactions
\cite{Dar20,Fic50,Rya79,[1],Yar90}.

\subsection{Confinement models}

	Our simplest version of a confinement model \cite{Duv98} is based on
the Fokker
potential $U_c$ (see equation  (7.3)), the time-asymmetric counterpart of
which is proposed in \cite{ri}. This model could be regarded as a
classical relativisation of the primitive quarkonium model with the linear
non-relativistic potential. Of course, the relativisation of any
non-relativistic system is not unique. There exists in the literature a
wide variety of relativistic versions of the potential confinement model.
The present model has a number of features which are expected for the models
of this kind but which usually are not realized together.

	1. The model is a self-consistent relativistic two-particle model.
The quantities in terms of which it is built have a clear physical meaning.
Solutions of this model are free of any
critical point and lead to timelike particle world lines.

	2. It is well known that a non-relativistic potential model
with the linear potential leads to the Regge trajectory with the
unsatisfactory asymptote $M \sim S^{2/3}$. Here we do not propose
a quantum version of the present model, but we make the estimates of the
Regge trajectory from what follows.

	Usually the Regge trajectories in the potential models are
calculated in the oscillator approximation \cite{L-S89}. Then, the leading
Regge trajectory originates from the classical mechanics: it coincides
with the curve of circular motions on the ($M,S$)--plane. In our case
this curve is described by the following equation:
%			7.35
\begin{equation}
S = \frac{M^2(1 - 4m_0^2/M^2)^{3/2}}{6\sqrt{3}\alpha_c}
\end{equation}
(we consider the case of equal particle rest masses $m_0$).
In the ultrarelativistic limit $M\to\infty$ it leads to the desirable
linear asymptote:
%			7.36
\begin{equation}
M^2 \approx 6\sqrt{3}\alpha_c S.
\end{equation}
It is remarkable that this asymptote is achieved only by taking account
of relativity.

	3. The present model permits the interpretation of an interaction in
terms of some classical fields. It follows from the fact that the Fokker
potential $U_c$ can be transformed into an equivalent form,
%			7.37
\begin{equation}
\tilde U_c = - 2\alpha_c\omega D_\epsilon(x),
\end{equation}
where function $D_\epsilon(x)$,
%			7.38
\begin{equation}
D_\epsilon(x) = \ha\Theta(\epsilon x^0) \Theta(x^2),
\end{equation}
is the fundamental solution of the equation
%			7.39
\begin{equation}
\Box^2 D_\epsilon(x) = 4\pi\delta(x).
\end{equation}
Thus, the interaction of particles can be considered as mediated by the vector
field obeying some fourth order equation. Gauge invariant nonlinear
equations of this kind arise when considering the behaviour of a gluon
propagator in the infrared region \cite{AAB82}. Static solutions of such
equations are used in a sort of the bag model of confinement \cite{Ale88}.
\begin{figure}[h,t]
%\vspace{1in}
\input fig6.pic
\vspace{0.2in}
{\small{\sf Figure 6.}
Confinement model. Classical Regge trajectories at various rates
of the coupling constants and the rest mass.}
\end{figure}

	The simplest version of the relativistic confinement model
can be appropriate for the description of light mesons for which the
confinement interaction dominates. To include into consideration also heavy
mesons one can modify the present model by adding to $U_c$ the usual
vector potential $U_v$ (with the appropriate coupling constant $\alpha_v<0$)
\cite{Duv98}.
In the non-relativistic limit this mixture leads to the well known
potential $U^{(0)} = -\midl\alpha_v\midr/r + \alpha_c r$.
The resulting model becomes appreciably cumbersome but still remains
solvable. Pathological solutions which occur in this model can be
unambiguously separated from its regular solutions (which are free of critical
points). As an illustration we present the classical Regge trajectories
for various rates of the coupling constants and the rest mass (figure   6).
We note that all the trajectories tend asymptotically to straight lines.
Moreover, the vector correction does not influence their asymptotic
behaviour which is still described by equation  (7.36).


\endinput
\section{Vector and scalar models in $\M_2$}                %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{figure}{6}
\renewcommand{\thefigure}{\arabic{figure}}

	The  analysis of the vector and scalar time-asymmetric models
in the four-di\-men\-si\-o\-nal space-time $\M_4$
was carried out  in the previous section  for the case of
equal particle masses. The
cumbersome form of the expressions
and a large set of possible motions
obscures the physical
understanding of the obtained results.
In the two-dimensional space-time $\M_2$ the analysis of dynamics becomes
considerably simpler even for different particle masses.
The dynamics in  $\M_2$ seems to
correspond to the motions with the inner angular momentum (spin)
$S=0$. But as it turns out, the limit
$S\to 0$ is a singular one.
Therefore, the consideration of the dynamics of
such models in the two-dimensional Minkowski space $\M_2$
appears to be interesting.


The Fokker-type action integral with a
time-asymmetric variant of the
Fokker potential \re{5.2}
in the front form in $\M_2$ leads to the Lagrangian \cite{[20]}
\beql{s-7}
L=-\sum_{a=1}^{N} m_{a}k_{a} -
\frac{\al k_{1}k_{2}f(\om)}{r} ,\qq r>0,
\eeq
where
\beql{s-8}
\om=\fr{1}{2}\left(\frac{k_1}{k_{2}}+\frac{k_{2}}{k_{1}}
\right).
\eeq
The existence of three integrals of motion, which for the Lagrangian
\re{s-7}
have the form
\begin{eqnarray}
\lab{s-9}
\!P_+& =&\fr{m_1}{k_1} + \fr{m_2}{k_2}
- \fr{\al B (\om)}{ r }~, \\
\lab{s-10}
P_-& =& m_1 k_1 + m_2k_2, \qqqq\qqq\qqqq\qq\\
\lab{s-11}
K& =& - t(P_+ + P_-)/2 - \sul{a=1}{2}\fr{x_am_a}{k_a}-\nn\\
&&-\ \fr{\al}{r} \left[
\left(\fr{x_1k_2}{k_1}+\fr{x_2k_1}{k_2}\right)f+
\fr12\left(\fr{k_1}{k_2}-\fr{k_2}{k_1}\right)
\left(\fr{x_1k_2}{k_1}-\fr{x_2k_1}{k_2}\right)f'
\right],
%\!K{=} {-}\fr{t(P_+ {+} P_-)}{2} {-}\fr{x_1 m_1}{k_1} {-}
%\fr{x_2m_2}{k_2}-\qqqq\qqq\nn\\
%\!\!\!\!\!\fr{2\al\om^{\cl-1}}{\vert r
%\vert}\left[(1{+}\cl)(x_1{+}x_2){+}(1{-}\cl)\left(\fr{x_1k_2^2}{k_1^2}
%{+}\fr{x_2k_1^2}{k_2^2}\right)\right],\q
\end{eqnarray}
where
\beql{s-11-1}
B(\om) = 2 \left(-\om f + (\om^2 - 1)f' \right),
\eeq
permits one to reduce the solutions of Euler-Lagrange equations to
quadrature \cite{MST86}. But
solutions of Euler-Lagrange equations exist only in the region
$\cQ \subset
T\cM\approx\R^4$ which is defined by the inequalities \re{s-3-1}:
\begin{equation}
\lab{s-12}
%\infty >
{\sf h}_f
=\frac{m_1m_2}{k_1^3k_2^3} -
\al \fr{(m_2 \k_2 +
m_1 \k_1) A(\om)}{ rk_1^3k_2^3 }
>0~,
\end{equation}
where
\beql{s-12-1}
A(\om) = - f + \om f' + (\om^2 - 1)f''~.
\eeq

The investigation of two-particle models with the time-asymmetric
field-like interactions (see \cite{MST86,Sh}) shows that
for some values of the parameters  the system  reaches the boundary of the
Lagrangian region
$\pl\cQ=\{(x_a,x_b,v_a,v_b)\in\R^{4}|{\sf h}_\ell=0;\\
{\sf h}_\ell^{-1}=0\}$.
An exception is  the repulsion case ($\al>0$)
if the total mass  of the system
$M>m_1+m_2=m$, where  $m_1, m_2$ are  particle rest masses.
 Then the system does not reach the
singular points and %it is possible to construct smooth
the world lines are smooth timelike curves in ${\Bbb M}_2$
\cite{Ste85,R-H70}. The Hamiltonian description allows one
to prolong the evolution of the system beyond the
critical points for other values of the parameters
and, as a result, to obtain continuous world lines in the following
way \cite{Sh}.

The Legendre transformation $\pounds$
associated with the Lagrangian \re{s-7} with
$f(\om)=\om^\cl;~ \cl=0,1,2,...$ has the form:
%(2.3)
\begin{equation}
\lab{s-13}
p_{a}=\frac{\partial L}{\partial v_a}=
\frac{m_{a}}{k_{a}}+\frac{\alpha}{2r}\left( 1+\ell+(1-\ell )
\frac{k_{\bar a}^{2}}
{k_{a}^{2}}\right) \om^{\ell -1}.
\end{equation}
Here $a=1,2,~{\bar a}=3-a$.

In the scalar ($\cl=0$) and vector ($\cl=1$) cases
it is possible
to solve equations \re{s-13} with respect to velocities and obtain
from
the expressions for conserved quantities
\re{s-3} the generators of the Lie algebra of the Poincar\' e group
${\cal P}(1,1)$ in the explicit form \cite{26-1,Sh}.
Se\-pa\-ration of the external and
internal motions is carried out by the choice:
\beql{s-14}
P_{+}=p_1+p_2~,~~Q = K/ P_{+}~;~~\{Q,P_+\}=1
\eeq
as new external canonical variables. As internal variables  we choose
%(2.9)
\begin{equation}
\lab{s-15}
\xi =\frac{m_{2}p_{1}-m_{1}p_{2}}{P_{+}} ,\quad q=r\frac{P_{+}}{m} ;
\quad \{ q,\xi \} =1 ,
\end{equation}
where $m=m_{1}+m_{2}$.
Then the Hamiltonian equations of motion become
%(2.19)
\begin{eqnarray}
\lab{s-2.27}
\dot Q = 1/2 - \frac{M^2}{2P_+},\quad
\dot P_+ = 0,\quad\\
\lab{s-2.28}
\dot q = \frac{1}{2P_+} \frac{\partial M^2}{\partial\xi},\quad
\dot\xi =-\frac{1}{2P_+}\frac{\partial M^2}{\partial q}.
\end{eqnarray}


Solving equations \re{s-13} with respect to velocities and substituting
the solutions into the expression for the Hessian we obtain from  \re{s-12}
inequalities which define the image $\pounds\cQ$
of the Lagrangian region $\cQ$ under the Legendre transformation
\re{s-13}.
 The external canonical variable $P_+$
is an integral of motion.
The Hessian does not depend on the
external  variable $Q$.
 Thus,  all the singularities of the Hessian
are expressed in terms of inner variables and we can transform
 \re{s-12} into an inequality  which defines the region $\widetilde{\pounds\cQ}$
in the inner phase space: $\widetilde{\pounds\cQ}\subset\R^2$.
In the scalar case, if $\al>0$, the
region $\widetilde{\pounds\cQ}$  of the
phase plane  corresponds to the
region $q>0$ restricted by the curves $y_1,~y_2$
(see figure 7) which are defined by the equations:
\begin{eqnarray}
\lab{s-2.24}
y_1:-m_1 \xi + m_1m_2 + m_2 \al/q=0~,\nn\\
\\
y_2: m_2 \xi + m_1m_2 + m_1 \al/q=0~.
\nonumber
\end{eqnarray}
 If $\al<0$, then $\widetilde{\pounds\cQ}$
lies between the curves $y_1,~y_2 $ to the right of
their  intersection point.
%%%%%%%%%%%%%%%%%%%%%%% FIG 7  %%%%%%%%%%%%%%%%
\begin{figure*}[p]
\input fig7.pic
%\vspace*{40mm}
\caption{Scalar interaction. Phase trajectories (continuous curves):
 $(m_2-m_1)/m=0.2;~ M/m=1.2$.
{\bf a)}: $\alpha >0$, {\bf b)}: $\alpha <0$.
Dashed curves $ y_1,
 y_2$ correspond to the singularity of the Hessian.
}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%% FIG 8  %%%%%%%%%%%%%%%%
\begin{figure*}[p]
\input fig8.pic
%\vspace*{40mm}
\caption{Vector interaction. Phase trajectories (continuous curves):
$(m_2-m_1)/m=0.2;~ M/m=1.2$.
{\bf a)}: $\alpha >0$, {\bf b)}: $\alpha <0$.
Dashed curves $ y_1,
 y_2$ correspond to the singularity of the Hessian.
}
\end{figure*}


In the vector case, if $\al<0$, the
region $\widetilde{\pounds\cQ}$
corresponds to the
region bounded  by the curves ${\tilde y}_1 ,
~{\tilde y}_2 ,~q=0$ (see figure 8)
which are defined by the equations:
\begin{eqnarray}
\lab{s-2.25}
\tilde y_1 :  m_1 + \xi - {\al}/{q}=0~,\nn\\
\\
\tilde y_2 :  m_2 - \xi - {\al}/{q}=0~. \nn
\end{eqnarray}
 If $\al>0$, then the indicated region lies between the curves ${\tilde y}_1,
~{\tilde y}_2 $ to the right of their intersection point.
The intersection points of the phase trajectories and the curves
$y_1 $, $y_2 $ (${\tilde y}_1,~{\tilde y}_2 $)
correspond to the case when one of the particles
reaches  the speed of light: $\k_1 =0$ or $ \k_2 =0$.

%The time-asymmetric Fokker-type  action integral
% reduces to the single-time
%Lagrangian function if $r>0$.
% Therefore $q>0$ and
% we can neglect the module sign
%in $\widetilde{\pounds\cQ}$. It does not matter whether we preserve it
%in $\R^2\backslash\widetilde{\pounds\cQ}$, since the Lagrangian and the
%Hamiltonian formalisms are not equivalent there.
To construct smooth world lines in $\M_2$ it is necessary to consider the
inner motion in more detail.
It is determined by the mass-shell equation
\begin{equation}
\lab{s-2.19}
(\xi{-} \xi_M)^2{=}
\fr{  (\nu^2{-} 1)m^2m_1^2m_2^2q^2
{-}2 \al M^2 m_1m_2m \nu^\cl q {+} ({-}1)^{\cl{+}1}M^4 \al^2 }
{M^4q^2},
\end{equation}
where
\beql{s-2.20}
%\hspace*{-8mm}
\xi_M = \frac{(M^2{-}m^2)(m_2{-}m_2)}{2M^2},\quad
\nu = \frac{M^2{-}m^2_1{-}m^2_2}{2m_1m_2}~.
\eeq
%%%%%%%%%%%%%%%%%%%%%%% FIG 9  %%%%%%%%%%%%%%%%
\begin{figure*}[h,t]
\input fig9.pic
\vspace*{15mm}
\caption{World lines in $\M_2$
for an unbounded motion: $(m_2-m_1)/m=0.2,~ M/m=1.2~,
 \alpha <0$. {\bf a)}:   scalar interaction (Stephas case
\protect\cite{Ste85});
{\bf b)}:  vector interaction (Rudd and Hill case
\protect\cite{R-H70}).}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%% FIG 10  %%%%%%%%%%%%%%%%
\begin{figure*}[p]
\input fig10.pic
\vspace*{15mm}
\caption{ Scalar interaction. World lines in $\M_2$
for an unbounded motion: $(m_2-m_1)/m=0.2,~ M/m=1.2~,
 \alpha <0$.}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%% FIG 11  %%%%%%%%%%%%%%%%
\begin{figure*}[p]
\input fig11.pic
%\vspace*{10mm}
\caption{Vector interaction. World lines in $\M_2$
for an unbounded motion: $(m_2-m_1)/m=0.2,~ M/m=1.2~,
 \alpha <0$.}
\end{figure*}


 We assume that equation \re{s-2.19}
is true in the whole phase plane $\R^2$. The motion is possible in the
region where
\beql{s-2.21}
{\sf D}_\cl=
(\nu^2 - 1)m^2m_1^2m_2^2q^2 - 2 \al M^2 m_1m_2m \nu^\cl q +
 (-1)^{\cl+1}M^4 \al^2
\eeq
is non-negative. Then, we see that for a bounded motion
$q$ belongs to the interval $[q_1,
q_2]$, where $q_1$, $q_2$ are real solutions of the quadratic equation
${\sf D}_\cl=0$:
\beql{s-2.22}
\hspace*{-8mm}
q_1 = \frac{2\alpha M^2(-1)^{\ell+1}}{(M^2-(m_1-m_2)^2)m}, \quad
q_2=\frac{2\alpha M^2}{(M^2-m^2)m}.
\eeq
In such a manner we get the phase trajectories which lead to smooth world
lines in $\M_2$  for all the values of the total mass of the system $M>0$
and signs of the coupling constant $\al$
\cite{Sh}.
Using phase trajectory equation \re{s-2.19} and solving equations
\re{s-2.27}, \re{s-2.28} we
obtain a parametric equation for world lines in $\M_2$:
\beql{s-2.31}
x_1^0(q)=t(q)-x_1(q)~,~~x_2^0(q)=t(q)-x_2(q)~;
\eeq
\begin{eqnarray}
\lab{s-2.32}
x_1(q) = {K}/{P_+} + \left({m_2 - \xi (M^2, q)}
\right)q/{P_+}~,\nn\\
\\
x_2(q) = {K}/{P_+} -\left({m_1 + \xi (M^2, q)}
\right)q/{P_+}~.\nn
\end{eqnarray}

\vspace*{6mm}


Figures 7,8  show examples of
the phase trajectories for the scalar
(figure 7) and  vector (figure 8) interactions.  Figures 9-11  show the
corresponding smooth world lines.

Unlike
the scalar interaction, there exist particle collisions  in the
vector  case. At the collision points $(q=0)$ the particles mutually change
their positions
 (figure 8, b)  and the
 phase trajectories break up.
The motion along smooth world lines corresponds to the jumps
along the momentum axis
$-\infty \to \infty$ ($\infty \to -\infty $).







%In Ref. [Shp] it have been demonstrated that the jumps in
%the phase trajectories $(+0,- \infty) \longleftrightarrow (-0,\infty)$;
%$(+0, \infty) \longleftrightarrow (-0, - \infty)$ correspond to the
%particles motion along the
%smooth world lines. This means that such jumps are not observable in the
%two-dimensional Minkowski space
%and
%therefore  we can assume that they are not physically observable events.
% Our canonical
%variables describe the system in the proper way only in some bounded
%region in $\R^2$ and we can identify the points$(+0,- \infty)
%\sim (-0,\infty)$; $(+0, \infty)
%\sim (-0, - \infty)$ because each  pair correspond to the one point
%on the  world lines. In other words our canonical
%variables are only local coordinates and cannot describe the whole
%evolution of the system.


%We  constructed smooth world lines in the
%two-dimensional Minkowski space. But we have not proved the uniqueness of
%such a construction or at least  that it could be caused by physical
%reasons. We have shown that the Hamiltonian formalism is more preferable
%than the Lagrangian one for the two-particle system with field-type
%interactions. To prove that obtained world lines could have physical
%sense it is necessary to connect them with some important physical
%observable. One of such an important observable is mass spectrum of
%the system. One can show \cite{Sh} that considered
%world lines for bounded motion permit to obtain semiclassical
%mass spectra which correlate well with exact
%quantum results. The semiclassical mass spectra are related directly with
%the period of bounded motion and therefore also need information
%about the whole evolution of the system.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Quantum models in $\M_2$}                          %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we  consider a number of exactly solvable
quantum-mechanical models which follow from certain quantization
procedures applied to the corresponding classical counterparts. We construct
a quantum description for the investigated above classical time-asymmetric
scalar and vector models, as well as for the classical models for which
the Lagrangian description is not known.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Vector and scalar interactions}                 %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The classical two-particle system with
 time-asymmetric
scalar and vector interactions can be quantized
in  a purely algebraic way \cite{26-1} regarding the Lie algebra $\gs\go(2,1)$
as the basic algebraic structure.
Let us introduce  the following functions of canonical variables:
\begin{eqnarray}
\lab{s-2.24-1}
J_{0}&=&\frac{1}{2}\left( \Delta q\xi ^{2}+\frac{q}{\Delta}+
\fr{\al^2 \Delta (\al_0^2-\al_1^2)}{q}\right) ,\nn\\
J_{1}&=&\frac{1}{2}\left( \Delta q\xi ^{2}-\frac{q}{\Delta}+
\fr{\al^2 \Delta (\al_0^2-\al_1^2)}{q}\right),\\
J_{2}&=&q\xi~,\nn
\end{eqnarray}
where ${\Delta}$ is an arbitrary constant. They
span, under the Poisson bracketing, the Lie algebra
$\gs\go$(2,1)
%(2.6)
\begin{equation}\lab{s-2.28-1}
\{ J_{0},J_{1}\} =J_{2} ,\qquad \{ J_{1},J_{2}\} =-J_{0} ,\qquad
\{ J_{2},J_{0}\} =J_{1} .
\end{equation}
Then, the mass-shell  equation \re{s-2.19} takes the form:
\beql{s-2.25-1}
 J+C_\cl=0.
\eeq
The quantity
\beql{s-2.26-1}
 J=aJ_0+bJ_1+dJ_2
\eeq
is an element of the Lie algebra of group $\cS\cO(2,1)$ and we use the
following notation:
\begin{eqnarray}
\lab{s-2.27-1}
a{=}\fr{M^{2}}{\Delta}{+}\Delta m_{1}m_{2}(m^{2}{-}M^{2}),\quad
b{=}\fr{M^{2}}{\Delta} {-}\Delta m_{1}m_{2}(m^{2}{-}M^{2}),
\nonumber\\
d{=}(m_{2}{-}m_{1})(m^{2}{-}M^{2}) ,\quad
C_{\cl}{=}2\alpha mm_{1}m_{2}\nu^\cl.\qq
\end{eqnarray}
It would appear  natural  that
the structure of the linear relation on the Lie algebra $\gs\go$(2,1)
must be preserved after quantization.
Then, replacing functions \re{s-2.24-1} with the Hermitian
operators obeying the commutation relations of the
$\gs\go$(2,1) Lie algebra
%(2.7)
\begin{equation}\lab{s-3.2}
[\hat J_{0},\hat J_{1}] =i\hat J_{2} ,\qquad
[\hat J_{1},\hat J_{2}] =-i\hat J_{0} ,\qquad
[\hat J_{2},\hat J_{0}] =i\hat J_{1} ,
\end{equation}
we  obtain the quantum-mechanical equation:
%(2.8)
\begin{equation}\lab{s-3.3}
(\hat J+C_\cl)|\psi\rangle =0  .
\eeq
This equation was considered in \cite{26-1} as the basic one for the
quantum--mechanical problem.
One can obtain in a purely algebraic way on the basis of equation \re{s-3.3} the
mass spectrum
\begin{equation}\lab{s-3.15}
(M_{n}^{\pm})_{\cl}^{2}=m_{1}^{2}+m_{2}^{2}\pm 2m_{1}m_{2}
\bigl( 1-(-1)^{\cl}\alpha^{2}/n^{2}\bigr) ^{(-1)^{\cl}/2} ,
\end{equation}
where
\beql{s-3.12}
n=(-1+\sqrt{1+4(-1)^\cl\alpha^{2}})/2+s ,~ s=1,2,...
\eeq


The branch $(M_{n}^ {+})_{k}^{2}$ has a correct
non-relativistic limit.
% in contrast to the branch $(M_{n,\lambda}^{-})_{k}^{2}$.
Expansion  to the order $1/c^2$ gives the
following correction to the energy spectrum:
%(2.16)
\begin{eqnarray}
\lab{s-3.18}
E \approx - \fr{m_1m_2\al^2}{2ms^2 \hbar^2} - \fr{\al^4m_1m_2}
{4m \hbar^4 s^4c^2} \left[ \left( 1-4\cl + \fr{m_1m_2}{m^2} \right)
\fr{1}{2}
 -  4s(-1)^\cl \right],
 \nonumber\\~s=1,2,\ldots~.
\end{eqnarray}
%\begin{equation}\lab{s-3.18}
%E_s=\frac{\nu\alpha ^2}{2s^2}-\frac{\nu\alpha ^4}{4s^4}\Bigl( -\frac{1}{2}
%\bigl( 1-\frac{\nu}{m}\bigr) +(-1)^\ell\bigl( 1-4s\bigr)\Bigr)+\cdots ;
% \nu =\frac{m_1m_2}{m}, s=1,2,\ldots
%\end{equation}
In the single-particle limit $(m_1/m_2\to 0)$ we obtain
%(2.17)
\begin{equation}\lab{s-3.19}
E=m_1\Bigl( 1-(-1)^\ell\alpha ^2/n^2\Bigr) ^{(-1)^\ell /2}-m_1 ,
\end{equation}
which is in agreement with a one-particle problem in the
external scalar or vector field in the case of states with the zero value
of the quantum orbital number.
The  mass spectrum of a vector type   agrees
with the  result obtained by Barut
on the basis of the infinite component
wave equation \cite{B-R}.


The existence of an additional algebraic structure of the mass-shell equation
permits one to quantize the classical problem without ambiguities typical
of relativistic mechanics \cite{NTSH}. Furthermore, such a quantization method
allows one to avoid difficulties connected with the choice of certain
representation (coordinate, momentum, etc.) which is very
important for the field-type interactions because of the difficulties of
the global structure of the Hamiltonian description (see above).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{
%{\em A priori}
Relativistic Hamiltonian models in $\M_2$}                         %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



Considering the field-type models we started from the Lagrangian
description. But it is also possible to construct a number of exactly
solvable models immediately within the
framework of the Hamiltonian description \cite{39,40,89}.
Contrary to the models based on the Fokker-type action integral,
relativistic Hamiltonian models
are not connected with the field theory. Nevertheless, they
are also of interest for a variety of reasons.
They  can describe
phenomenological aspects of the inner structure of mesons and baryons
\cite{79,61}.   Besides, these models
can be useful for the verification of
different approximation methods, and may be considered
as an approximation of more realistic models.  It appears to be
significant for the explanation of relativistic effects in the
well-established non-relativistic oscillator-like quark models of
hadrons.


The standard quantization procedure consists in the transition from a set of
canonical generators to a set of Hermitian operators which determine the
unitary representation of the Poincar\' e group. So, in the case of
two-dimensional space-time we must put in correspondence with the canonical
generators of $\cP(1,1)$ the Hermitian operators $\hK,\hP_+,\hP_-$
in some Hilbert space which satisfy the following bracket relations:
\begin{equation}
\lab{s-16}
[ \hP_{+},\hP_{-}] =0 ,\qquad [ \hK,\hP_{\pm}]=\pm i \hP_{\pm} .
\end{equation}
This determines the squared total mass operator $\hM^2=\hP_+\hP_-$ and the
quantum problem
is reduced to the eigenvalue problem \cite{39,40,89}:
\beql{s-17}
\hM^2\psi=M_{n,\la}^2\psi.
\eeq


From a variety of the known  paths  for
such a transition we choose the Weyl quantization  rule  \cite{8}.
It is necessary that typical of the front form  inequalities
\beql{s-18}
p_a>0
\eeq
be satisfied  for this quantization method. It will be noted that
these conditions are destroyed by field-like interactions.
The wave functions $\psi (p)=\langle p|\psi\rangle$ describing the physical
(normalized) states in the front form of dynamics
constitute  the  Hilbert  space $ {\cal H}_N^F ={\cal L}^2
(\R_+^N ,d\mu _N^F )$  with the inner product \cite{39,40,89}:
%(2.8)
\begin{equation}
\lab{s-19}
(\psi _1,\psi ) = \int \d\mu ^F_N (p)\psi ^\ast _1(p) \psi (p),
\end{equation}
where
%(2.9)
\begin{equation}
\lab{s-20}
\d\mu ^F_N (p) = \prod ^N_ {a=1} \frac {\d p_a}{2p_a}\Theta (p_a)
 \end{equation}
is a Poincar\' e--invariant measure and $\Theta(p_a)$ is the Heaviside
function.
 According to the Weyl rule we get
the following operators \cite{39,40,89}:
%(2.10)
\begin{equation}
\lab{s-21}
\hat P_+=\sum ^N_{a=1}p_a ,\quad  \hat K = i \sum_{a=1} ^N p_a \partial /
\partial p_a ,\quad  \hat P_- = \hat M^2 / \hat P_+ \q ,
\end{equation}
which  are Hermitian with respect to the inner product \re{s-19}.
They determine  the
unitary realization of group ${\cal P}(1.1)$ on the Hilbert  space
${\cal H}_N^F$. Here $\hat M $ is determined by
\begin{equation}
\lab{s-22}
\hat M^2=\hat M_f^2 + \hat V,
\end{equation}
where $\hat M_f^2$ is a free-particle part of the square mass operator:
\begin{equation}\lab{s-22-1}
\hat M_f^2=\hat P_+\sum_{a=1}^N\frac{m_a^2}{p_a}.
\end{equation}
Operator $\hat V $ is
an integral operator
\begin{equation}
\lab{s-23}
( \hat V \psi)(p) = \int \d \mu_N^F (p)V (p,p') \psi (p')
\end{equation}
with the kernel
%(2.13)
\begin{eqnarray}
\lab{s-24}
V(p,p') = \left[ \prod _{d=1}^N \sqrt {4p_dp_d'} \right] \delta (P_+-P_+')
\int _{- \infty} ^{\infty} V \left( r \frac {p_b +p_b'}{2}; \frac {r_{1c}}{r}
\right) \times \nonumber \\
\times \exp\left[i \sum_ {a=2}^N r_{1a}(p_a-p_a') \right] \prod_ {a=2} ^N
\frac {\d r_{1a}}{2 \pi}.
\end{eqnarray}
The  general  properties  of  the Weyl  transformation \cite{8}
ensure that in the classical limit these  operators  correspond
to the functions \re{s-4}, \re{s-5}.

The evolution of the quantum system
is described in the front form of dynamics by the Schr\" odinger-type
equation
%(2.14)
\begin{equation}
\lab{s-25}
{\rm i}\frac{\partial\Psi}{\partial t}=\hat H\Psi,
\end{equation}
where $\Psi\in{\cal H}_N^F$ and
%(2.15)
\begin{equation}
\lab{s-26}
\hat H=\frac{1}{2}(\hat P_++\hat P_-)=\frac{1}{2}(\hat P_++\hat M^2/\hat P_+).
\end{equation}
Putting $\Psi =\chi (t,P_+)\psi$, where $\psi$ is a function of some
Poincar\' e-invariant inner variables, we obtain a stationary eigenvalue
problem for the operator $\hat M^2$. In such a way a number of exactly
solvable two-particle systems were considered in Refs.~\cite{39,40}.
It is convenient to introduce for a two-particle system
the following Poincar\' e-invariant
inner momentum variable \cite{79}
\beql{s-1.40}
\e = (p_1 - p_2)/2P_+~,
\eeq
which is linearly related to the variable $\xi=
(m_2 - m_1)/2 + m \e$.
Then the interaction part of the squared total mass of the system
$V$ takes the form:
\beql{s-1.41}
V(rp_1, rp_2) = F(\rho, \e)~,~~~~~~\rho=rP_+~.
\eeq
The conditions \re{s-18} lead to inequality
$
| \e | < 1/2~.
$
The Hilbert space $\cH_2^F$ decomposes into the tensor product
$
\cH_2^F = h_{int} \otimes \cH_{ext}^F~,
$
where "inner" and "external"
spaces are realized, correspondingly, by functions $\psi (\eta )$
and $\chi (P_+)$ with the inner products
\begin{eqnarray}
\lab{s-1.46}
(\psi_1, \psi) = \fr{1}{2} \int \limits_{-1/2}^{1/2} \fr{\d \e}{1/4 - \e^2}
\psi_1^* (\e ) \psi (\e )~, \\
\lab{s-1.47}
(\chi_1, \chi) = \int \limits_0^{\infty} \fr{\d P_+}{2P_+}\chi_1^*(P_+)
\chi(P_+)~.
\end{eqnarray}
Operator $\hat M^2$ acts nontrivially only on $h_{int}$.
It is an integral operator which is determined  by the rule:
\begin{eqnarray}
\lab{s-1.48}
(\hat M^2 \psi)(\e ) = \left( \fr{2m_1^2}{1+2 \e} + \fr{2m_2^2}{1-2 \e}
\right) \psi (\e ) + \nonumber \\
\\
+ \int \limits_{-1/2}^{1/2} \d \e' \sqrt{\fr{1-4 \e^2}
{1-4 \e'^2}}W(\e, \e' ) \psi (\e')~, \nonumber
\end{eqnarray}
where kernel $W(\e , \e')$ has the form:
\beql{s-1.49}
W(\e , \e') = \fr{1}{2 \pi} \int \limits_{- \infty}^{\infty}
\d\rho F \left(\rho,
\fr{\e + \e'}{2} \right) {\rm e}^{-i\rho(\e - \e')}~.
\eeq
The structure of operator
$\hat M^2$ coincides with the one-dimensional variant of the corresponding
expression in Ref. \cite{79}, but in the present treatment kernel
$W(\e , \e')$ is directly related to the classical interaction potential
$V$.


It is convenient to pass from the functions $\psi(\e)$ with the inner
product \re{s-1.46} to the functions
\beql{s-1.50}
\vp (\e ) = \fr{\psi (\e )}{\sqrt{1/2 - 2 \e^2}}
\eeq
with the inner product
\beql{s-1.51}
(\vp_1, \vp) = \int \limits_{-1/2}^{1/2} \d \e \vp_1^* (\e ) \vp (\e )~.
\eeq
The latter differs from the non-relativistic product only by limits of
integration.
The action of $\hat M^2$ on the function $\vp(\e)$ is defined by
the equation
\beql{s-1.52}
(\hat M^2 \vp )(\e ) = \left( \fr{2m_1^2}{1 + 2 \e} + \fr{2m_2^2}{1 - 2 \e}
\right) \vp (\e ) + \int \limits_{-1/2}^{1/2} \d \e' W(\e, \e' ) \vp (\e' )~.
\eeq

Let us consider two simple examples.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

	1. \underline{$\de$--\it like potential} . Let us put $F(\rho,\e)=
\al\de(\rho),~\al=const$. Then the equation for  $\vp(\e)$ has the form
\cite{39}:
\beql{s-1.108}
\left( M^2-\fr{m_1^2}{1/2+\e}-\fr{m_2^2}{1/2-\e}\right)\vp(\e)=
\fr{\al}{2\pi}\intl{-1/2}{1/2}\d\e'\vp(\e')~.
\eeq
Putting
\beql{s-1.109}
\intl{-1/2}{1/2}\d\e\vp(\e)=C~(\ne 0)
\eeq
we get from \re{s-1.108}
\beql{s-1.110}
\vp(\e)=\fr{\al C}{2\pi}
\left( M^2-\fr{m_1^2}{1/2+\e}-\fr{m_2^2}{1/2-\e}\right)^{-1}~.
\eeq
Substituting  \re{s-1.110} into \re{s-1.109}, we obtain the equation
\beql{s-1.111}
\fr{2\pi}{\al}=\intl{-1/2}{1/2}\d\e\left(
M^2-\fr{m_1^2}{1/2+\e}-\fr{m_2^2}{1/2-\e}\right)^{-1}~,
\eeq
which describes the eigenvalues of $M^2$ for bound states.
%%%%%%%%%%%%%%%%%%%%%%-Fig.12-%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\input fig12.pic
%\vspace*{40mm}
\caption{$\de$-like potential.
$f_1(\la)$ is a graph of the r.-h. side of equation
\protect\re{s-1.113},
$f_2(\la)$ is a graph of the r.-h. side of equation
\protect\re{s-1.114}}
\end{figure}


	Let us consider the case of equal particle masses
$(m_1=m_2=m/2)$. Then we get from \re{s-1.111}
\beql{s-1.112}
\fr{2\pi M^2}{\al}=1-\fr{m^2}{2M}\intl{-M}{M}\fr{\d x}{x^2+m^2-M^2}~.
\eeq
If $M<m$, putting $M=m\sin \la,~0\le\la\le\pi/2$, we come to the
following 	transcendental	equation for $\la$:
\beql{s-1.113}
\fr{2\pi m^2}{\al}=
\left( 1-\fr{2\la}{\sin 2\la}\right) \sin^{-2}\la\equiv f_1(\la) ~.
\eeq
The graph of the right-hand side of this equation (figure 12) shows
that there exists
its only solution for $-3\pi m^2<\al<0$.
This corresponds to attraction. The energy of a bound state has a proper
non-relativistic limit.


It is interesting to point out that  there also exists  a
bound state in the case of a strong repulsion. If $M>m$, one can put
$M=m\ch\la,~\la>0$. Then, from  \re{s-1.112} we obtain the following
equation:
\beql{s-1.114}
\fr{2\pi m^2}{\al}=
\left( 1+\fr{2\la}{\sh 2\la}\right) \ch^{-2}\la\equiv f_2(\la) ~,
\eeq
which has the only solution if $\al>\pi m^2$ (figure 12).
This solution does not have a non-relativistic limit.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




	2. \underline{\it Oscillator potential}. Let us consider an interaction
with a quadratical dependence on coordinates
of the following type:
\beql{s-1.88}
V=\om_0^2r^2p_1p_2= \om_0^2(1/4-\e^2)\rho^2,~\om_0 \in \R.
\eeq
Then, equation \re{s-17} transforms into an ordinary differential equation
of the hypergeometric type \cite{39,40}:
\begin{eqnarray}
\lab{s-1.89}
\left(\fr{1}{4}-\e^2\right)\vp''(\e)-2\e\vp'(\e)+\qqqq\qqqq\qqqq\qqq\nn\\
+\left[-\fr{1}{2}+\fr{1}{\om_0^2}
\left( M_n^2-\fr{m_1^2}{1/2+\e}-\fr{m_2^2}{1/2-\e}\right)\right]\vp(\e)=0
\end{eqnarray}
with the boundary conditions
\beql{s-1.86}
\lim\limits _{\e\to\pm 1/2}u(\e)\vp(\e)=0,~
\lim\limits _{\e\to\pm 1/2}u(\e)\vp'(\e)=0~.
\eeq
Equation \re{s-1.89} leads to the mass spectrum
\beql{s-1.96}
M_n^2=\left[ m+\om_0(n+1/2)\right]^2+\fr{\om_0^2}{4}.
\eeq
Its  nontrivial solutions, which are bounded and square-integrable on
the interval $(-1/2,~1/2)$ have the form:
\beql{s-1.97}
\vp_n(\e)=C_n\left(\fr12+\e\right)
^{m_1/\om_0}\left(\fr12-\e\right)^{m_2/\om_0}
P_n^{(2m_2/\om_0,2m_1/\om_0)}(2\e)~.
\eeq
In equation \re{s-1.97} $P_n^{(2m_2/\om_0,2m_1/\om_0)}(2\e)$ are  Jacobi
polynomials \cite{45-2} and $C_{n}$ are normalization constants.

In the non-relativistic limit
$\hbar\om_0 /mc^2\to 0,~M_n\to m+E_n/c^2$ we obtain well--known
wave functions in the momentum
representation and
a non-relativistic energy spectrum of the
harmonic oscillator: $E=\hbar\om_0(n+1/2)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

It is also possible to construct within the framework of the two-dimensional
variant of the front form an exactly solvable quantum-mechanical $N$-particle
model with the oscillator-like interaction
\begin{equation}
\lab{s-27}
V=\om_0^2\sum\sum_{\hspace*{-5mm}a<b}r_{ab}^2p_ap_b.
\end{equation}
Function \re{s-27} gives an $N$-particle generalization of the two-particle
interaction \re{s-1.88}, as well as one of the possible
relativistic generalizations of the N-particle oscillator potential.
For this system by means of the Weyl quantization rule one can
also reduce the
eigenvalue problem \re{s-17} to a differential equation.
The system with interaction \re{s-27} has $N-2$ additional integrals of
motion which mutually commute and provide the exact integrability of the
system in the classical case.
They depend nontrivially on the products of coordinate and momenta
variables \cite{89}.
Therefore, in general, the quantization procedure can
destroy commutation relations between these quantities and,
as a result, the integrability of the quantum problem.
The Weyl quantization rule transforms classical additional integrals of
motion into a set of quantum integrals of motion in involution.
That permits one to solve exactly the eigenvalue problem and to
obtain the eigenfunctions and eigenvalues of $\hM^2$ (see \cite{89}):
\begin{equation}
\lab{s-28}
M_n^2=\left[\sum_{a=1}^Nm_a+\om_0\sum_{b=1}^{N-1}(n_b+1/2) \right]^2+
\frac{N-1}{4}\om_0^2.
\end{equation}

Interaction function \re{s-27} may be generalized by adding terms
which are linear in the coordinates
%(4.18)
\begin{equation}
\lab{s-29}
V\to {\tilde V}=V+\al
\sum\sum_{\hspace*{-5mm}a<b}r_{ab}(p_a-p_b).
\end{equation}
Such a system also has additional integrals of motion and permits exact
solutions in the quantum case \cite{89}.


Thus, the Weyl quantization rule preserves the commutation relation of
Po\-in\-ca\-r\'e group $\cP(1,1)$, as well as additional
symmetries which are responsible for the integrability of this model
\cite{89}. As it was shown in \cite{NTSH} on the example of a
two-particle oscillator-like
model in the two-dimensional variant of the front form,
the Weyl quantization is not the only quantization rule with this
property. The application of different quantization rules preserving the
commutation relation of  $\cP(1,1)$ may result in different observables as,
for instance, a mass spectrum of the system \cite{NTSH}.
%To restrict
%the ambiguity of quantization procedure it is necessary to find additional
%symmetries of the problem or additional algebraic structure
%similar to that which there is in the case of field-like interactions.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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