text:
\section{\label{Sec_1 copy(1)}Classical relativistic electrodynamics models
revisiting: Lagrangian and Hamiltonian analysis}

\subsection{\label{Subsec_1.1}Introductory setting}

Classical electrodynamics is nowadays considered \cite{LL,Pa} as the most
fundamental physical theory, largely owing to the depth of its theoretical
foundations and wealth of experimental verifications. In the work we
describe a new approach to the classical Maxwell theory, based on a vacuum
field medium model, and reanalyze some of the modern classical
electrodynamics problems related with the description of a charged point
particle dynamics under external electromagnetic field. We remark here that
under \textit{"a charged point particle"} we as usually understand an
elementary material charged particle whose internal spatial structure is
assumed to be unimportant and is not taken into account, if the contrary is
not specified.

The important physical principles, characterizing the related
electrodynamical vacuum field structure and based on the least action
principle, we discuss subject to different charged point particle dynamics,
based on the fundamental least action principle. In particular, the main
classical relativistic relationships, characterizing the charge point
particle dynamics, we obtain by means of the \ least action principle within
the Feynman's approach to the Maxwell electromagnetic equations and the
Lorentz type force derivation. \ Moreover, for each least action principle
constructed in the work, we describe the corresponding Hamiltonian pictures
and present the related energy conservation laws. Making use of the
developed modified least action approach a classical hadronic string model
is analyzed in detail.

As the classical Lorentz force expression with respect to an arbitrary
inertial reference frame is related with many theoretical and experimental
controversies, \ such as the relativistic potential energy impact into the
charged point particle mass, the Aharonov-Bohm effect \cite{AB} and the
Abraham-Lorentz-Dirac radiation force \cite{Jack,Bar,LL} expression, the
analysis of its structure subject to \ the assumed vacuum field medium
structure is \ a very interesting and important problem, which was discussed
by many physicists including E. Fermi, G. Schott, R. Feynman, F. Dyson \cit
{Fer,Sch,Fe1,Dy1,Dy2,Gl} and many others. To describe the essence of the
electrodynamic problems related with the description of a charged point
particle dynamics under external electromagnetic field, let us begin with
analyzing the classical Lorentz force expression
\begin{equation}
dp/dt=F_{\xi }:=\xi E+\xi u\times B,  \label{L0.1}
\end{equation
where $\xi \in \mathbb{R}$ is a particle electric charge, $u\in T(\mathbb{R
^{3})$ is its velocity \cite{AM,BPS} vector, expressed here in the light
speed $c$ units
\begin{equation}
E:=-\partial A/\partial t-\nabla \varphi  \label{L0.2}
\end{equation
is the corresponding external electric field and
\begin{equation}
B:=\nabla \times A  \label{L0.3}
\end{equation
is the corresponding external magnetic field, acting on the charged
particle, expressed in terms of suitable vector $A:M^{4}\rightarrow \mathbb{
}^{3}$ and scalar $\varphi :M^{4}\rightarrow \mathbb{R}$ potentials. Here "
\nabla $" is the standard gradient operator with respect to the spatial
variable $r\in \mathbb{E}^{3},$ $"\times "$is the usual vector product in
three-dimensional Euclidean vector space $\mathbb{E}^{3}:=(\mathbb{R
^{3},<\cdot ,\cdot >),$ which is naturally endowed with the classical scalar
product $\ <\cdot ,\cdot >.$ These potentials are defined on the Minkowski
space $M^{4}\simeq \mathbb{R}\times \mathbb{E}^{3}$, which models a chosen
laboratory reference frame $\mathcal{K}.$ \ Now, it is a well known fact
\cite{LL,Pa,Fe1,Th} that the force expression (\ref{L0.1}) does not take
into account the dual influence of the charged particle on the
electromagnetic field and should be considered valid only if the particle
charge $\xi \rightarrow 0.$ This also means that expression (\ref{L0.1})
cannot be used for studying the interaction between two different moving
charged point particles, as was pedagogically demonstrated in classical
manuals \cite{LL,Fe1}. As the classical Lorentz force \ expression (\re
{L0.1}) is a natural consequence of the interaction of a charged point
particle with an ambient electromagnetic field, its corresponding derivation
based on the general principles of dynamics, was deeply analyzed by R.
Feynman and F. Dyson \cite{Fe1,Dy1,Dy2}.

Taking this into account, it is natural to reanalyze this problem from the
classical, taking only into account the Maxwell-Faraday wave theory aspect,
specifying the corresponding vacuum field medium. \ Other questionable
inferences from the classical electrodynamics theory, which strongly
motivated the analysis in this work, are related both with an alternative
interpretation of the well-known \textit{Lorenz condition}, imposed on the
four-vector of electromagnetic observable potentials $(\varphi
,A):M^{4}\rightarrow T^{\ast }(M^{4})$ and the classical Lagrangian
formulation \cite{LL} of charged particle dynamics under external
electromagnetic field. The Lagrangian approach latter is strongly dependent
on an important Einsteinian notion of the rest reference frame $\mathcal{K
_{\tau }$ and the related least action principle, so before explaining it in
more detail, we first to analyze the classical Maxwell electromagnetic
theory from a strictly dynamical point of view.

Let us consider with respect to a laboratory reference frame $\mathcal{K}$\
the additional \textit{Lorenz condition}
\begin{equation}
\partial \varphi /\partial t+<\nabla ,A>=0,  \label{L0.4}
\end{equation
\ \textit{a priori} assumed the Lorentz invariant wave scalar field equation
\begin{equation}
\partial ^{2}\varphi /\partial t^{2}-\nabla ^{2}\varphi =\rho \text{ \ \ }
\label{L0.5}
\end{equation
and the charge continuity equation
\begin{equation}
\partial \rho /\partial t+<\nabla ,J>=0,  \label{L0.6}
\end{equation
where $\rho :M^{4}\rightarrow \mathbb{R}$ and $J:M^{4}\rightarrow \mathbb{E
^{3}$ are, respectively, the charge and current densities of the ambient
matter. Then one can derive \cite{BPT,BPT0} that the Lorentz invariant wave
equation
\begin{equation}
\partial ^{2}A/\partial t^{2}-\nabla ^{2}A=J\   \label{L0.5a}
\end{equation
and the classical electromagnetic Maxwell field equations \cit
{Jack,LL,Fe1,Pa,Th
\begin{align}
\nabla \times E+\partial B/\partial t& =0,\text{ \ \ \ }<\nabla ,E>=\rho ,
\label{L0.7} \\
\nabla \times B-\partial E/\partial t& =J,\text{ \ \ \ \ \ \ }<\nabla ,B>=0,
\notag
\end{align
hold for all $(t,r)\in M^{4}$ with respect to the chosen laboratory
reference frame $\mathcal{K}.$

Notice here that, inversely, Maxwell's equations (\ref{L0.7}) do not
directly reduce, via definitions (\ref{L0.2}) and (\ref{L0.3}), to the wave
field equations (\ref{L0.5}) and \ (\ref{L0.5a}) without the Lorenz
condition (\ref{L0.4}). This fact \ is very important and suggests that when
it comes to a choice of governing equations, it may be reasonable to replace
Maxwell's equations (\ref{L0.7}) with the Lorenz condition (\ref{L0.4}) and
the charge continuity equation (\ref{L0.6}). To make the equivalence
statement, claimed above, more transparent we formulate it as the following
proposition.

\begin{proposition}
The Lorentz invariant wave equation (\ref{L0.5}) together with the Lorenz
condition (\ref{L0.4}) for the observable potentials $(\varphi
,A):M^{4}\rightarrow T^{\ast }(M^{4})$ and the charge continuity
relationship (\ref{L0.6}) are completely equivalent to the Maxwell field
equations (\ref{L0.7}).
\end{proposition}

\begin{proof}
Substituting (\ref{L0.4}), into (\ref{L0.5}), one easily obtains
\begin{equation}
\partial ^{2}\varphi /\partial t^{2}=-<\nabla ,\partial A/\partial
t>=<\nabla ,\nabla \varphi >+\rho ,  \label{L0.8}
\end{equation
which implies the gradient expression
\begin{equation}
<\nabla ,-\partial A/\partial t-\nabla \varphi >=\rho .  \label{L0.9}
\end{equation
Taking into account the electric field definition (\ref{L0.2}), expression
\ref{L0.9}) reduces to
\begin{equation}
<\nabla ,E>=\rho ,  \label{L0.10}
\end{equation
which is the second of the first pair of Maxwell's equations (\ref{L0.7}).

Now upon applying $\nabla \times $ to definition (\ref{L0.2}), we find,
owing to definition (\ref{L0.3}), that
\begin{equation}
\nabla \times E+\partial B/\partial t=0,  \label{L0.11}
\end{equation
which is the first pair of the Maxwell equations (\ref{L0.7}). Having
differentiated with respect to the temporal variable $t\in \mathbb{R}$ the
equation \ (\ref{L0.5}) and taken into account the charge continuity
equation \ (\ref{L0.6}), one finds that
\begin{equation}
<\nabla ,\partial ^{2}A/\partial t^{2}-\nabla ^{2}A-J>=0.  \label{L0.11a}
\end{equation
The latter is equivalent to the wave equation \ (\ref{L0.5a}) if to observe
that the current vector $J:M^{4}\rightarrow \mathbb{E}^{3}$ is defined by
means of the charge continuity equation \ (\ref{L0.6}) up to a vector
function $\nabla \times S:M^{4}\rightarrow \mathbb{E}^{3}.\ $Now applying
operation $\nabla \times $ to the definition (\ref{L0.3}), owing to the wave
equation \ (\ref{L0.5a}) one obtain
\begin{align}
\nabla \times B& =\nabla \times (\nabla \times A)=\nabla <\nabla ,A>-\nabla
^{2}A=  \notag \\
& =-\nabla (\partial \varphi /\partial t)-\partial ^{2}A/\partial
t^{2}+(\partial ^{2}A/\partial t^{2}-\nabla ^{2}A)=  \notag \\
& =\frac{\partial }{\partial t}(-\nabla \varphi -\partial A/\partial
t)+J=\partial E/\partial t+J,  \label{L0.12}
\end{align
which leads directly to
\begin{equation*}
\nabla \times B=\partial E/\partial t+J,
\end{equation*
which is the first of the second pair of the Maxwell equations (\ref{L0.7}).
The final \textit{"no magnetic charge}" equation
\begin{equation*}
<\nabla ,B>=<\nabla ,\nabla \times A>=0,
\end{equation*
in (\ref{L0.7}) \ follows directly from the elementary identity $<\nabla
,\nabla \times >=0,$ thereby completing the proof.
\end{proof}

This proposition allows us to consider the observable potential functions
(\varphi ,A):M^{4}\rightarrow T^{\ast }(M^{4})$ as fundamental ingredients
of the ambient \textit{vacuum field medium}, by means of which we can try to
describe the related physical behavior of charged point particles imbedded
in space-time $M^{4}.$ The following observation provides strong support for
this approach:

\textbf{Observation.} \textit{The Lorenz condition (\ref{L0.4}) actually
means that the scalar potential field }$\varphi :$\textit{\ }
M^{4}\rightarrow \mathbb{R}$\textit{\ continuity relationship, whose origin
lies in some new field conservation law, characterizes the deep intrinsic
structure of the vacuum field medium.}

To make this observation more transparent and precise, let us recall the
definition \cite{LL,Pa,Fe1,Th} of the electric current $J:$\textit{\ }
M^{4}\rightarrow \mathbb{E}^{3}$ in the dynamical form
\begin{equation}
J:=\rho u,  \label{L0.13}
\end{equation
where the vector $u\in T(\mathbb{R}^{3})$ is the corresponding charge
velocity. Thus, the following continuity relationshi
\begin{equation}
\partial \rho /\partial t+<\nabla ,\rho u>=0  \label{L0.14}
\end{equation
holds, which can easily be rewritten \cite{MC} \ as the integral
conservation law
\begin{equation}
\frac{d}{dt}\int_{\Omega _{t}}\rho (t,r)d^{3}r=0  \label{L0.15}
\end{equation
for the charge inside of any bounded domain $\Omega _{t}\subset \mathbb{E
^{3},$ moving in the space-time $M^{4}$ with respect to the natural
evolution equation
\begin{equation}
dr\ /dt:=u.  \label{L0.16}
\end{equation
Following the above reasoning, we obtain the following result.

\begin{proposition}
The Lorenz condition (\ref{L0.4}) is equivalent to the integral conservation
la
\begin{equation}
\frac{d}{dt}\int_{\Omega _{t}}\varphi (t,r)d^{3}r=0,  \label{L0.17}
\end{equation
where $\Omega _{t}\subset \mathbb{E}^{3}$ \ is any bounded domain, moving
with respect to the charged point particle $\xi $ evolution equation
\begin{equation}
dr/dt=u(t,r),  \label{L0.18}
\end{equation
which represents the velocity vector of related local potential field
changes propagating in the Minkowski space-time $M^{4}.$ Moreover, for a
particle$\ $with the distributed $\ $charge density $\rho :M^{4}\rightarrow
\mathbb{R},$ the following Umov type local energy conservation relationship
\begin{equation}
\frac{d}{dt}\int_{\Omega _{t}}\frac{\rho (t,r)\varphi (t,r)}
(1-|u(t,r)|^{2})^{1/2}}d^{3}r=0\   \label{L0.18a}
\end{equation
holds for any $t\in \mathbb{R}.$
\end{proposition}

\begin{proof}
Consider first the corresponding solutions to potential field equations (\re
{L0.5}), taking into account condition (\ref{L0.13}). Owing to the standard
results from \cite{Fe1,LL}, one finds that
\begin{equation}
A=\varphi u,  \label{L0.19}
\end{equation
which gives rise to the following form of the Lorenz condition (\ref{L0.4})
\begin{equation}
\partial \varphi /\partial t+<\nabla ,\varphi u>=0,  \label{L0.20}
\end{equation
This obviously can be rewritten \cite{MC} as the integral conservation law
\ref{L0.17}), so the expression \ (\ref{L0.17}) is stated.

To state the local energy conservation relationship \ (\ref{L0.18a}) it is
necessary to combine the conditions \ \ (\ref{L0.14}), (\ref{L0.20}) and
find that
\begin{equation}
\partial (\rho \varphi )/\partial t+<u,\nabla (\rho \varphi u)>+2\rho
\varphi <\nabla ,u>=0.  \label{L0.20a}
\end{equation
Taking into account that the infinitesimal volume transformation
d^{3}r=J(t,r)d^{3}r_{0},$ where the Jacobian $J(t,r):=|\partial
r(t;r_{0})/\partial r_{0}|$ of the corresponding transformation $r:\mathbb{\
}\Omega _{t_{0}}\rightarrow $ $\Omega _{t},$\ induced by the Cauchy problem
for the differential relationship \ (\ref{L0.18}) for any $t\in \mathbb{R},$
satisfies the evolution equation
\begin{equation}
dJ/dt=<\nabla ,u>J,  \label{L0.20b}
\end{equation
easily following from \ (\ref{L0.18}), and applying to the equality \ (\re
{L0.20a}) the operator $\int_{\Omega _{t_{0}}}(...)J^{2}d^{3}r_{0},$ one
obtains that
\begin{equation}
\begin{array}{c}
\int_{\Omega _{t_{0}}}\frac{d}{dt}(\rho \varphi J^{2})d^{3}r_{0}=\frac{d}{dt
\int_{\Omega _{t_{0}}}(\rho \varphi J)Jd^{3}r_{0}= \\
=\frac{d}{dt}\int_{\Omega _{t}}(\rho \varphi J^{\ })d^{3}r\ :=\ \frac{d}{dt
\mathcal{E}(\xi ;\Omega _{t})
\end{array}
\label{L0.20c}
\end{equation
Here we denoted \ the conserved charge $\xi :=\int_{\Omega _{t}}\rho
(t,r)d^{3}r$ and the local energy conservation quantity $\ \mathcal{E}(\xi
;\Omega _{t}):$ $=\int_{\Omega _{t}}(\rho \varphi J^{\ })d^{3}r.$ The latter
quantity can be simplified, owing to the infinitesimal Lorentz invariance
four-volume measure relationship\ $d^{3}r(t,r_{0})\wedge dt=d^{3}r_{0}\wedge
dt_{0},$ where variables $(t,r)\in \mathbb{R}_{t}\times \Omega _{t}\subset
M^{4}$ are, within the present context, taken with respect to the moving
reference frame $\mathcal{K}_{t},$ related to the infinitesimal charge
quantity $d\xi (t,r):=\rho (t,r)d^{3}r,$\ and variables $(t_{0},r_{0})\in
\mathbb{R}_{t_{0}}\times \Omega _{t_{0}}\subset M^{4}$ are taken with
respect to the laboratory reference frame $\ \mathcal{K}_{t_{0}},$ related
to the infinitesimal charge quantity $d\xi (t_{0},r_{0})=\rho
(t_{0},r_{0})d^{3}r_{0},$ satisfying the charge conservation invariance
d\xi (t,r)=d\xi (t_{0},r_{0}).$ The mentioned above infinitesimal Lorentz
invariance relationships make it possible to calculate the local energy
conservation quantity $\mathcal{E}(\xi ;\Omega _{0})$ as $\ $
\begin{eqnarray}
\mathcal{E}(\xi ;\Omega _{0}) &=&\int_{\Omega _{t}}(\rho \varphi J^{\
})d^{3}r=\int_{\Omega _{t}}(\rho \varphi \frac{d^{3}r}{d^{3}r_{0}}^{\
})d^{3}r=  \label{L0.20d} \\
&=&\int_{\Omega _{t}}(\rho \varphi \frac{d^{3}rdt}{d^{3}r_{0}dt}^{\
})d^{3}r=\int_{\Omega _{t}}(\rho \varphi \frac{d^{3}r_{0}dt_{0}}{d^{3}r_{0}d
}^{\ })d^{3}r=  \notag \\
&=&\int_{\Omega _{t}}(\rho \varphi \frac{\ dt_{0}}{\ dt})^{\
}d^{3}r=\int_{\Omega _{t}}\frac{\rho \varphi \ d^{3}r\ }{\ (1-|u|^{2})^{1/2}
,  \notag
\end{eqnarray
where we took into account that $dt=dt_{0}(1-|u|^{2})^{1/2}.$\ Thus, owing
to \ (\ref{L0.20c}) and \ (\ref{L0.20d}) the local energy conservation
relationship \ (\ref{L0.18a}) is satisfied, proving the proposition.
\end{proof}

The constructed above local energy conservation quantity \ (\ref{L0.20d})
can be rewritten as
\begin{equation}
\mathcal{E}(\xi ;\Omega _{t})=\int_{\Omega _{t}}\frac{d\xi (t,r)\varphi
(t,r)\ \ \ }{\ (1-|u|^{2})^{1/2}}:=\int_{\Omega _{t}}d\mathcal{E}(t,r)
\label{L0.20e}
\end{equation
where $d\mathcal{E}(t,r)=d\xi (t,r)\varphi (t,r)(1-|u|^{2})^{-1/2}$ is the
distributed in vacuum electromagnetic field energy density, related with the
electric charge $d\xi (t,r),$ located at point $(t,r)\in M^{4}.$

The above proposition suggests a physically motivated interpretation of
electrodynamic phenomena in terms of what should naturally be called \textit
the vacuum potential field}, which determines the observable interactions
between charged point particles. More precisely, we can \textit{a priori }\
endow the ambient vacuum medium with a scalar potential energy field density
function $W:=\xi \varphi :M^{4}\rightarrow \mathbb{R},$ where $\xi \in
\mathbb{R}_{+}$ is the value of an elementary charge quantity, and
satisfying the governing \textit{vacuum field equations}
\begin{equation}
\begin{array}{c}
\partial ^{2}W/\partial t^{2}-\nabla ^{2}W=\rho \xi ,\text{ \ \ }\partial
W/\partial t+<\nabla ,\hat{A}>=0, \\
\partial ^{2}\hat{A}/\partial t^{2}-\nabla ^{2}\hat{A}=\xi \rho v,\text{ \ \
}\hat{A}=Wv
\end{array}
\label{L0.21}
\end{equation
taking into account the external charged sources, which possess a virtual
capability for disturbing the vacuum field medium. Moreover, \ this vacuum
potential field function $W:M^{4}\rightarrow \mathbb{R}$ allows the natural
potential energy interpretation, whose origin should be assigned not only to
the charged interacting medium, but also to any other medium possessing
interaction capabilities, including for instance, material particles,
interacting through the gravity.

The latter leads naturally to the next important step, consisting in
deriving the equation governing the corresponding potential field $\bar{W
:M^{4}\rightarrow \mathbb{R},$ assigned to a charged point particle moving
in the vacuum field medium with velocity $u\in \ \ T(\mathbb{R}^{3})$ and
located at point $r(t)=R(t)\in \mathbb{E}^{3}$ at time $t\in \mathbb{R}.$ As
can be readily shown \cite{BPT,BPT1,Rep}, the corresponding evolution
equation governing the related potential field function $\bar{W
:M^{4}\rightarrow \mathbb{R},$ assigned to a \ moving in the space $\mathbb{
}^{3}$ charged particle $\xi $ \ has the form
\begin{equation}
\frac{d}{dt}(-\bar{W}u)=-\nabla \bar{W},  \label{L0.22}
\end{equation
where $\bar{W}:=W(r,t)|_{r\rightarrow R(t)},$ $u(t):=dR(t)/dt$ at point
particle location $(t,R(t))\in M^{4}.$

Similarly, if there are two interacting charged point particles, located at
points $r(t)=R(t)$ and $r_{f}(t)=R_{f}(t)\in \mathbb{E}^{3}$ at time $t\in
\mathbb{R}$ and moving, respectively, with velocities $u:=dR(t)/dt$ \ and \
u_{f}:=dR_{f}(t)/dt,$ the corresponding potential field function $\bar{W
^{\prime }:M^{4}\rightarrow \mathbb{R},$ considered with respect to the
reference frame $\mathcal{K}^{\prime }$ specified by Euclidean coordinates
(t^{\prime },r-r_{f})\in \mathbb{E}^{4}$ and moving with the velocity
u_{f}\in \ T(\mathbb{R}^{3})$ \ subject to the laboratory reference frame
\mathcal{K},$ should satisfy \cite{BPT,BPT1} the dynamical equality
\begin{equation}
\frac{d}{dt^{\prime }}[-\bar{W}^{\prime }(u^{\prime }-u_{f}^{\prime
})]=-\nabla \bar{W}^{\prime },  \label{L0.23}
\end{equation
where, by definition, we have denoted the velocity vectors $u^{\prime
}:=dr/dt^{\prime },u_{f}^{\prime }:=dr_{f}/dt^{\prime }\in T(\mathbb{R
^{3}). $ \ The dynamical potential field equations (\ref{L0.22}) and (\re
{L0.23}) appear to have important properties and can be used as means for
representing classical electrodynamic phenomena. Consequently, we shall
proceed to investigate their physical properties in more detail and compare
them with classical results for Lorentz type forces arising in the
electrodynamics of moving charged point particles in an external
electromagnetic field.

In this investigation, we were in part inspired by works \cite{DJ,DuJ,Wi}
and interesting studies \cite{Bu,BPZZ,Ba,B-W} devoted to solving the
classical problem of reconciling gravitational and electrodynamic charges
within the Mach-Einstein ether paradigm. First, we will revisit the
classical Mach-Einstein relativistic electrodynamics of a moving charged
point particle, and second, we study the resulting electrodynamic theories
associated with our vacuum potential field dynamical equations (\ref{L0.22})
and (\ref{L0.23}), making use of the fundamental Lagrangian and Hamiltonian
formalisms which were specially devised in \cite{BP0,BPT0}.

\subsubsection{Classical relativistic electrodynamics revisited}

The classical relativistic electrodynamics of a freely moving charged point
particle in the Minkowski space-time $M^{4}\simeq \mathbb{R}\times \mathbb{E
^{3}$ is based \ on the Lagrangian approach \cite{LL,Fe1,Pa,Th,Bar} with
Lagrangian function
\begin{equation}
\mathcal{L}_{0}:=-m_{0}(1-|u|^{2})^{1/2},  \label{L1.1}
\end{equation
where $m_{0}\in \mathbb{R}_{+}$ is the so-called particle rest mass
parameter and $u\in T(\mathbb{R}^{3})$ is its spatial velocity in the
Euclidean space $\mathbb{E}^{3},$ expressed here and in the sequel in light
speed units (with light speed $c=1$). The least action principle in the form
\begin{equation}
\delta S=0,\text{ \ \ }S:=-\int_{t_{1}}^{t_{2}}m_{0}(1-|u|^{2})^{1/2}dt
\label{L1.2}
\end{equation
for any fixed \ temporal interval $[t_{1},t_{2}]\subset \mathbb{R}$ gives
rise to the well-known relativistic relationships for the mass of the
particle
\begin{equation}
m=m_{0}(1-|u|^{2})^{-1/2},  \label{L1.3}
\end{equation
the momentum of the particle

\begin{equation}
p:=mu=m_{0}u(1-|u|^{2})^{-1/2}  \label{L1.4}
\end{equation
and the energy of the particl
\begin{equation}
\mathcal{E}_{0}=m=m_{0}(1-|u|^{2})^{-1/2}.  \label{L1.5}
\end{equation
It follows from \cite{LL,Pa}, that the origin of the Lagrangian (\ref{L1.1})
can be extracted from the actio
\begin{equation}
S:=-\underset{t_{1}}{\overset{t_{2}}{\int }}m_{0}(1-|u|^{2})^{1/2}dt=
\underset{\tau _{1}}{\overset{\tau _{2}}{\int }}m_{0}d\tau ,  \label{L1.6}
\end{equation
on the suitable temporal interval $[\tau _{1,}\tau _{2}]\subset \mathbb{R
\mathbf{.}$ Here $\ m_{0}\in \mathbb{R}_{+}$ is considered \textit{\ }as a
constant positive parameter \textit{a priori }attributed to the point
particle,
\begin{equation}
d\tau :=dt(1-|u|^{2})^{1/2}  \label{L1.6a}
\end{equation
and $\tau \in \mathbb{R}$ is the so-called, proper temporal parameter
assigned to a freely moving particle with respect to the \ rest \ reference
frame $\mathcal{K}_{\tau }.$ \ The action (\ref{L1.6}) is rather
questionable from the dynamical point of view, since it is physically
defined with respect to the \ rest \ reference frame $\mathcal{K}_{\tau },$
giving rise to the constant action $S=-m_{0}(\tau _{2}-\tau _{1}),$ as the
limits of integrations $\tau _{1}<\tau _{2}\in \mathbb{R}$ were taken to be
fixed from the very beginning. Moreover, considering this particle to have a
charge $\xi \in \mathbb{R}$ and be moving in the Minkowski space-time $M^{4}$
under action of an external electromagnetic field $(\varphi ,A)\in M^{4},$
the corresponding classical (relativistic) action functional is chosen\ (see
\cite{LL,Fe1,BN,BC,Pa,Th,BP0,BPT0}) with respect to the \textit{rest
reference system }$\mathcal{K}_{\tau }$\ \ as follows:
\begin{equation}
S:=\underset{\tau _{1}}{\overset{\tau _{2}}{\int }}[-m_{0}d\tau +\xi <A,\dot
r}>d\tau -\xi \varphi (1+|\dot{r}|^{2})^{-1/2}d\tau ],  \label{L1.7}
\end{equation
being parameterized by the \textit{Euclidean }space-time variables $(\tau
,r)\in \mathbb{E}^{4}\ $ satisfying the infinitesimal relationship $d\tau
^{2}+|dr|^{2}=dt^{2},$ where we have denoted $\dot{r}:=dr/d\tau $ in
contrast to the definition $u:=dr/dt.$ The action (\ref{L1.7}) can be
rewritten with respect to the laboratory reference frame $\mathcal{K}$ as
\begin{equation}
S=\underset{t_{1}}{\overset{t_{2}}{\int }}\mathcal{L}dt,\ \mathcal{L
:=-m_{0}(1-|u|^{2})^{1/2}+\xi <A,u>-\xi \varphi ,  \label{L1.8}
\end{equation
defined on the suitable temporal interval $[t_{1},t_{2}]\subset \mathbb{R}.$
The action function \ (\ref{L1.8}) contains two physically incompatible
sub-integral parts - the first one $-m_{0}(1-|u|^{2})^{1/2}dt=-m_{0}d\tau ,$
having sense with respect to the rest reference frame $\mathcal{K}_{\tau }$
and the second one, $\ $equivalent to $<A,dr>-\xi \varphi dt,$ having sense
with respect to the laboratory reference frame $\mathcal{K}.$ Nonetheless,
the least action principle applied to the functional \ (\ref{L1.8}) gives
rise to the following \cite{LL,Fe1,Pa,Th} \ dynamical equation
\begin{equation}
dP/dt=-\nabla (\xi \varphi -<\xi A,u>),  \label{L1.9}
\end{equation
where, by definition, the generalized particle-field momentum
\begin{equation}
P=p+\xi A,\text{ \ \ \ \ }  \label{L1.9a}
\end{equation
the own particle momentum
\begin{equation}
p=mu\text{ }=m_{0}u(1-|u|^{2})^{-1/2}\text{\ \ }  \label{L1.9b}
\end{equation
and its so called "\textit{inertial}" mass
\begin{equation}
m=m_{0}(1-|u|^{2})^{-1/2}.  \label{L1.c}
\end{equation
The corresponding particle conserved energy equals
\begin{equation}
\mathcal{E}_{\ }\ =(m_{0}^{2}+|p|^{2})^{1/2}+\xi \varphi ,  \label{L1.10}
\end{equation
that is
\begin{equation}
d\mathcal{E}_{\ }/dt=0=d\mathcal{E}\ /d\tau  \label{L1.10a}
\end{equation
with respect to both the laboratory reference frame $\mathcal{K}$ and the
rest reference frame $\mathcal{K}_{\tau }.$

The obtained \ above expression (\ref{L1.10}) for the particle energy
\mathcal{E}\in \mathbb{R}$ appears to be open to question, since the
electrical potential energy $\xi \varphi ,$ entering additively, has no
affect on the relativistic particle mass $m=m_{0}(1-|u|^{2})^{-1/2},$
contradicting the experimental facts \cite{Fe1,Jack} that some part of the
observable charged particle mass is of the electromagnetic origin. This fact
was also underlined by L. Brillouin \cite{Br}, who remarked that the fact
that the potential energy has no affect on the particle mass tells us that
"...\textit{\ any possibility of existence of a particle mass related with
an external potential energy, is completely excluded}". Moreover, it is
necessary to stress here that the least action principle, based on the
action functional (\ref{L1.8}) and formulated with respect to the laboratory
reference frame $\mathcal{K}$ time parameter $t\in \mathbb{R},$ appears to \
be logically inadequate, for there is a strong physical inconsistency with
other time parameters of the Lorentz equivalent laboratory reference frames
depending simultaneously both on the spatial and temporal coordinates. This
was first mentioned by R. Feynman in \cite{Fe1}, in his efforts to rewrite
the Lorentz force expression with respect to the rest reference frame
\mathcal{K}_{\tau }.$ This and other special relativity theory and
electrodynamics problems stimulated many prominent physicists\ of the past
\cite{Br,Fe1,We,Pa,BD} and present \cit
{BM,Me,Me1,Wi,DJ,DuJ,Lo2,Lo3,Rep,Ne,Gr1,B-W} to try to develop alternative
relativity theories based on completely different space-time and matter
structure principles. In particular, in \cite{BM} authors have analyzed a
super-symmetric version of the classical relativistic charged particle
electrodynamics and shown that its physically reasonable least action
formulation is achieved only by means of the reformulation of the
corresponding Lagrangian function with respect to the proper rest reference
frame $\mathcal{K}_{\tau }.$

There also is another controversial inference from the action expression
\ref{L1.8}) and resulting dynamical equation \ (\ref{L1.9}): the force
F_{\xi }=dP/dt,$ exerted by the external electromagnetic field on the
\textit{particle-field cluster} \cite{Fe1} carrying the momentum $P=p+\xi A,$
appears to be the\ standard gradient expression
\begin{equation}
F_{\xi }=-\nabla W_{\xi },  \label{L1.10b}
\end{equation
where the generalized "\textit{potential energy}"\
\begin{equation}
W_{\xi }:=\xi \varphi -<\xi A,u>.  \label{L1.10c}
\end{equation
Its first part $\ \xi \varphi \in \mathbb{R}$ equals the classical \cit
{Fe1,Jack} electrical potential energy, but its second part $-<\xi A,u>$ \
is strictly related with magnetic vector potential $A\in \mathbb{E}^{3}$ and
has nowadays no reasonable physical explanation. As one can easily show \cit
{LL,Pa,Fe1,Th,BPS} from \ (\ref{L1.9}), the corresponding expression \ for
\textit{the classical Lorentz force} is given as
\begin{equation}
dp/dt=F:=\xi E+\xi u\times B,  \label{L1.11}
\end{equation
where we have defined, as before,
\begin{equation}
E:=-\partial A/\partial t-\nabla \varphi  \label{L1.12}
\end{equation
for the corresponding electric field and
\begin{equation}
B:=\nabla \times A  \label{L1.13}
\end{equation
for the related magnetic field, acting on the point particle with the
electric charge $\xi \in \mathbb{R}.$ The expression (\ref{L1.11}) means, in
particular, that the Lorentz force \ (\ref{L1.11}) \ depends linearly on the
particle velocity vector $u\in T(\mathbb{R}^{3}),$ and so there is a strong
dependence on the reference frame with respect to which the charged point
particle $\xi $ moves. Attempts to reconcile this and some related
controversies \cite{Br,Fe1,Rep,Kl} forced Einstein to devise his special
relativity theory and proceed further to creating his general relativity
theory trying to explain the gravity by means of geometrization of
space-time and matter in the Universe.

Here we once more mention that the classical Lagrangian function $\mathcal{L
:$ $T(M^{4})\rightarrow \mathbb{R}$\ in (\ref{L1.8}) is simultaneously
written as a combination of terms incompatibly expressed from the physical
point of view by means of both the Euclidean rest reference frame variables
(\tau ,r)\in \mathbb{E}^{4},$\ naturally attributed to the charged point
particle, and arbitrarily chosen Minkowski reference frame variables
(t,r)\in M^{4}.$

These problems were recently analyzed within a completely different "\textit
no-geometry}" \ approach \cite{BPT,BPT1}, where new dynamical equations were
derived, which were free of the controversial elements mentioned above.
Moreover, this approach avoided the introduction of the well known Lorentz
transformations of the space-time reference frames with respect to which the
action functional (\ref{L1.8}) is invariant. From this point of view, there
are interesting for discussion of conclusions in \cite{Sant,Haj,Aqui,Ba,BPZZ
, \ and where some electrodynamic models, possessing intrinsic Galilean and
Poincar\'{e}-Lorentz symmetries, are reanalyzed from diverse geometrical
points of view. Subject to a possible geometric space-type structure and the
related vacuum field background, exerting the decisive influence on the
particle dynamics, we need to mention here recent works \cite{Smol,Smol-1}
and the closely related with their ideas the classical articles \cit
{Kib,Penr}. Next, we shall revisit the results obtained recently in \cit
{BPT,BPT1,BPT,BPS} from the classical Lagrangian and Hamiltonian formalisms
\cite{BP0} in order to shed new light on the physical underpinnings of the
vacuum field theory approach to the study of combined electromagnetic and,
eventually, also gravitational effects.

\subsection{The vacuum field theory electrodynamics equations: Lagrangian
analysis}

\subsubsection{\label{Subsec_1.2.1}A point particle moving in vacuum - an
alternative electrodynamic model}

Within the vacuum field theory approach to electromagnetism, devised in \cit
{BPT,BPT1}, the main vacuum potential field function $\bar{W}:M^{4}\mathbb
\rightarrow R},$ related to a charged point particle $\xi ,$ satisfies the
differential evolution equation \ (\ref{L0.22}), namely
\begin{equation}
\frac{d}{dt}(-\bar{W}u)=-\nabla \bar{W},\text{ \ \ }  \label{L2.1}
\end{equation
in the case when all of the external charged particles are at rest, \ that
is $\partial \bar{W}/\partial t=0,$ and as above, $u:=dr/dt$ is the particle
velocity with respect to some laboratory reference system $\mathcal{K},$
specified by the Minkowski coordinates $(t,r)\in $ $M^{4}.$

To analyze the dynamical equation (\ref{L2.1}) from the Lagrangian point of
view, we write the corresponding action functional as
\begin{equation}
S:=-\underset{t_{1}}{\overset{t_{2}}{\int }}\bar{W}dt=-\underset{\tau _{1}}
\overset{\tau _{2}}{\int }}\bar{W}(1+|\dot{r}|^{2})^{1/2}\text{ }d\tau ,
\label{L2.2}
\end{equation
expressed with respect to the rest reference frame $\mathcal{K}_{\tau },$
specified by the Euclidean coordinates $(\tau ,r)\in $ $\mathbb{E}^{4}.$
Fixing the proper temporal parameters $\tau _{1}<\tau _{2}\in \mathbb{R},$
one finds from the least action principle $\delta S=0$ that
\begin{align}
p& :=\partial \mathcal{L}/\partial \dot{r}=-\bar{W}\dot{r}(1+|\dot{r
|^{2})^{-1/2}=-\bar{W}u,  \label{L2.3} \\
\dot{p}& :=dp/d\tau =\partial \mathcal{L}/\partial r=-\nabla \bar{W}(1+|\dot
r}|^{2})^{1/2},  \notag
\end{align
where, owing to (\ref{L2.2}), the corresponding Lagrangian function is
\begin{equation}
\mathcal{L}:=-\bar{W}(1+|\dot{r}|^{2})^{1/2}.  \label{L2.4}
\end{equation
Recalling now the definition of the particle "\textit{inertial}" mass
\begin{equation}
m:=-\bar{W}  \label{L2.5}
\end{equation
and \ the relationship
\begin{equation}
d\tau =dt(1-|u|^{2})^{1/2}=dt(1+|\dot{r}|^{2})^{-1/2},\text{ }\dot{r}d\tau
=udt,  \label{L2.6}
\end{equation
from (\ref{L2.3}) we easily obtain the classical dynamical equation exactly
coinciding with (\ref{L2.1})
\begin{equation}
dp/dt=-\nabla \bar{W}.  \label{L2.6a}
\end{equation
Moreover, one now readily finds that the corresponding dynamical mass,
defined by means of expression (\ref{L2.5}), is given as
\begin{equation}
m=m_{0}(1-|u|^{2})^{-1/2},\text{ \ }m_{0}:=-\bar{W}(R(t_{0})),  \label{L2.6b}
\end{equation
where $\left. u(t)\right\vert _{t=t_{0}}=0$ at the spatial point
r=R(t_{0})\in \mathbb{E}^{3},$ and which completely coincides with
expression (\ref{L1.3}) of the preceding section. Now one can formulate the
following proposition using the results obtained above.

\begin{proposition}
The alternative freely moving point particle electrodynamic model (\ref{L2.1
) allows the physically reasonable least action formulation based on the
action functional (\ref{L2.2}) with respect to the "rest\textquotedblright\
\ reference frame variables, where the Lagrangian function is given by
expression (\ref{L2.4}). The related electrodynamics is completely
equivalent to that of a classical relativistic freely moving point particle,
described in Subsection \ \ref{Subsec_1.1}.
\end{proposition}

\subsubsection{\label{Subsec_1.2.2}A moving in vacuum interacting two charge
system - an alternative electrodynamic model}

We proceed now to the case when our charged point particle $\xi $ moves in
the space-time with velocity vector $u\in T(\mathbb{R}^{3})$ and interacts
with another external charged point particle $\xi _{f},$ moving with
velocity vector $u_{f}\in T(\mathbb{R}^{3})$ with respect to a common
reference frame $\mathcal{K}.$ As was shown in \cite{BPT,BPT1}, the
respectively modified dynamical equation for the vacuum potential field
function $\bar{W}^{\prime }:M^{4}\mathbb{\rightarrow R}$ subject to the
moving reference frame $\mathcal{K}^{\prime }$ is given by \ equality (\re
{L0.23}), or
\begin{equation}
\frac{d}{dt^{\prime }}[-\bar{W}^{\prime }(u^{\prime }-u_{f}^{\prime
})]=-\nabla \bar{W}^{\prime },  \label{L2.7}
\end{equation
where, as before, the velocity vectors $u^{\prime }:=dr/dt^{\prime
},u_{f}^{\prime }:=dr_{f}/dt^{\prime }\in T(\mathbb{R}^{3}).$ \ Since the
external charged particle $\xi _{f}$ \ moves in the space-time $M^{4},$ it
generates the related magnetic field $B:=\nabla \times A,$ whose magnetic
vector potential $A:M^{4}\mathbb{\rightarrow E}^{3}$ is defined, owing to
the results of \cite{BPT,BPT1,Rep}, as
\begin{equation}
\xi A:=\bar{W}u_{f}.  \label{L2.8}
\end{equation
Whence, taking into account that the field potential
\begin{equation}
\bar{W}=\bar{W}^{\prime }(1-|u_{f}|^{2})^{-1/2}  \label{L2.8a}
\end{equation
and the particle momentum $p^{\prime }=-\bar{W}^{\prime }u^{\prime }=-\bar{W
u,$ equality (\ref{L2.7}) becomes equivalent to
\begin{equation}
\frac{d}{dt^{\prime }}(p^{\prime }+\xi A^{\prime })=-\nabla \bar{W}^{\prime
},  \label{L2.9}
\end{equation
if considered with respect to the moving reference frame $\mathcal{K
^{\prime },$ or to the Lorentz type force equality
\begin{equation}
\frac{d}{dt}(p+\xi A)=-\nabla \bar{W}(1-|u_{f}|^{2}),  \label{L2.9a}
\end{equation
if considered with respect to the laboratory reference frame $\mathcal{K},$
owing to the classical Lorentz invariance relationship (\ref{L2.8a}), as the
corresponding magnetic vector potential, generated by the external charged
point test particle $\xi _{f}$ with respect to the reference frame $\mathcal
K}^{\prime },$ is identically equal to zero. To imbed the dynamical equation
(\ref{L2.9a}) into the classical Lagrangian formalism, we start from the
following action functional, which naturally generalizes the functional (\re
{L2.2})
\begin{equation}
S:=-\underset{\tau _{1}}{\overset{\tau _{2}}{\int }}\bar{W}^{\prime }(1+
\dot{r}-\dot{r}_{f}|^{2})^{1/2}\text{ }d\tau .  \label{L2.10}
\end{equation
Here, as before, $\bar{W}^{\prime }$ is the respectively calculated vacuum
field potential $\bar{W}\ $ subject to the moving reference frame $\mathcal{
}^{\prime },$ $\dot{r}=u^{\prime }dt^{\prime }/d\tau ,\dot{r
_{f}=u_{f}^{\prime }dt^{\prime }/d\tau ,$ $d\tau =dt^{\prime }(1-|u^{\prime
}-u_{f}^{\prime }|^{2})^{1/2},$ which take into account the \ relative
velocity of the charged point particle $\xi $ subject to the reference frame
$\mathcal{K}^{\prime },$ \ specified by the Euclidean coordinates
(t^{\prime },r-r_{f})\in \mathbb{R}^{4},$ and moving simultaneously with
velocity vector $u_{f}\in T(\mathbb{R}^{3})$ \ with respect to the
laboratory reference frame $\mathcal{K},$ specified by the Minkowski
coordinates $(t,r)\in M^{4}$ and related to those of the reference frame
\mathcal{K}^{\prime }$ and $\mathcal{K}_{\tau }$ by means of the following
infinitesimal relationships
\begin{equation}
dt^{2}=(dt^{\prime })^{2}+|dr_{f}|^{2},\text{ \ }(dt^{\prime })^{2}=d\tau
^{2}+\newline
|dr-dr_{f}|^{2}.  \label{L2.10a}
\end{equation
So, it is clear in this case that our charged point particle $\xi $ moves
with the velocity vector $u^{\prime }-u_{f}^{\prime }\in T(\mathbb{R}^{3})$
\ with respect to the reference frame $\mathcal{K}^{\prime }$ in which the
external charged particle $\xi _{f}$ is at rest. Thereby, we have reduced
the problem of deriving the charged point particle $\xi $ \ dynamical
equation to that before solved in Subsection \ \ref{Subsec_1.2.1}.

Now we can compute the least action variational condition $\delta S=0,$
taking into account that, owing to (\ref{L2.10}), the corresponding
Lagrangian function with respect to the rest reference frame $\mathcal{K
_{\tau }$ is given as
\begin{equation}
\mathcal{L}:=-\bar{W}^{\prime }(1+|\dot{r}-\dot{r}_{f}|^{2})^{1/2}.
\label{L2.11}
\end{equation
As a result of simple calculations, the generalized momentum of the charged
particle $\xi $ equals
\begin{align}
P& :=\partial \mathcal{L}/\partial \dot{r}=-\bar{W}^{\prime }(\dot{r}-\dot{r
_{f})(1+|\dot{r}-\dot{r}_{f}|^{2})^{-1/2}=  \label{L2.12} \\
& =-\bar{W}^{\prime }\dot{r}(1+|\dot{r}-\dot{r}_{f}|^{2})^{-1/2}+\bar{W
^{\prime }\dot{r}_{f}(1+|\dot{r}-\dot{r}_{f}|^{2})^{-1/2}=  \notag \\
& =mu^{\prime }+\xi A^{\prime }:=p^{\prime }+\xi A^{\prime }=p+\xi A,  \notag
\end{align
where, owing to (\ref{L2.8a}) the vectors $p^{\prime }:=-\bar{W}^{\prime
}u^{\prime }=-\bar{W}u=p\in \mathbb{E}^{3},$ $\ A^{\prime }=\bar{W}^{\prime
}u_{f}^{\prime }=\bar{W}u_{f}=A\in \mathbb{E}^{3},\ $\ and giving rise to
the dynamical equality
\begin{equation}
\frac{d}{d\tau }(p^{\prime }+\xi A^{\prime })=-\nabla \bar{W}^{\prime }(1+
\dot{r}-\dot{r}_{f}|^{2})^{1/2}\   \label{L2.13}
\end{equation
with respect to the\ rest reference frame $\mathcal{K}_{\tau }.$ As
dt^{\prime }=d\tau (1+|\dot{r}-\dot{r}_{f}|^{2})^{1/2}$ and $(1+|\dot{r}
\dot{r}_{f}|^{2})^{1/2}=(1-|u^{\prime }-u_{f}^{\prime }|^{2})^{-1/2},$ we
obtain from (\ref{L2.13}) the equality
\begin{equation}
\frac{d}{dt^{\prime }}(p^{\prime }+\xi A^{\prime })=-\nabla \bar{W}^{\prime
},  \label{L2.13a}
\end{equation
exactly coinciding with \ equality (\ref{L2.9}) subject to the moving
reference frame $\mathcal{K}^{\prime }.$ Now, making use of \ expressions \
\ref{L2.10a}) and \ (\ref{L2.8a}), one can rewrite \ (\ref{L2.13a}) as that
with respect to the laboratory reference frame $\mathcal{K}:
\begin{equation}
\begin{array}{c}
\frac{d}{dt^{\prime }}(p^{\prime }+\xi A^{\prime })=-\nabla \bar{W}^{\prime
}\Rightarrow \\
\\
\Rightarrow \frac{d}{dt^{\prime }}(\frac{-\bar{W}u^{\prime }}
(1+|u_{f}^{\prime }|^{2})^{1/2}}+\frac{\xi \bar{W}u_{f}^{\prime }}
(1+|u_{f}^{\prime }|^{2})^{1/2}})=-\frac{\nabla \bar{W}}{(1+|u_{f}^{\prime
}|^{2})^{1/2}}\Rightarrow \\
\\
\Rightarrow \frac{d}{dt^{\prime }}(\frac{-\bar{W}dr}{(1+|u_{f}^{\prime
}|^{2})^{1/2}dt^{\prime }}+\frac{\xi \bar{W}dr_{f}/}{(1+|u_{f}^{\prime
}|^{2})^{1/2}})=-\frac{\nabla \bar{W}}{(1+|u_{f}^{\prime }|^{2})^{1/2}
\Rightarrow \\
\\
\Rightarrow \frac{d}{dt}(-\bar{W}\frac{dr}{dt}+\xi \bar{W}\frac{dr_{f}}{dt
)=-\nabla \bar{W}(1-|u_{f}|^{2})
\end{array}
\label{L2.13b}
\end{equation
exactly coinciding with (\ref{L2.9a})
\begin{equation}
\frac{d}{dt}(p+\xi A)=-\nabla \bar{W}(1-|u_{f}|^{2}).  \label{L2.13c}
\end{equation}

\begin{remark}
The equation \ (\ref{L2.13c}) allows to infer the following \ important and
physically reasonable phenomenon: if the test charged point particle
velocity \ $u_{f}\in T(\mathbb{R}^{3})$ tends to the light velocity $c=1,$
the corresponding acceleration force $F_{ac}:=-\nabla \bar{W}(1-|u_{f}|^{2})$
is vanishing. Thereby, the electromagnetic fields, generated by such rapidly
moving charged point particles, have no influence on the dynamics of charged
objects if observed with respect to an arbitrarily chosen laboratory
reference frame $\mathcal{K}.$
\end{remark}

The latter equation \ (\ref{L2.13c}) can be easily rewritten a
\begin{eqnarray}
dp/dt &=&-\nabla \bar{W}-\xi dA/dt+\nabla \bar{W}|u_{f}|^{2}=  \label{L2.13d}
\\
&=&\xi (-\xi ^{-1}\nabla \bar{W}\ -\ \partial A/\partial t)-\xi <u,\nabla
>A+\xi \nabla <A,u_{f}>,  \notag
\end{eqnarray
or, \ using the well-known \cite{LL} identity
\begin{equation}
\nabla <a,b>=<a,\nabla >b+<b,\nabla >a+b\times (\nabla \times a)+a\times
(\nabla \times b),  \label{L2.13e}
\end{equation
where $a,b\in \mathbb{E}^{3}$ are arbitrary vector functions, in the
standard Lorentz type form
\begin{equation}
dp/dt=\xi E+\xi u\times B-\ \nabla <\xi A,u-u_{f}>.  \label{L2.13f}
\end{equation}

The result \ (\ref{L2.13f}), being before found with respect to the moving
reference frame $\mathcal{K}^{\prime }\mathcal{\ }$in \cite{BPT,BPT1,Rep}
and \cite{MarPin}, makes it possible to formulate the next important
proposition.

\begin{proposition}
The alternative classical relativistic electrodynamic model (\ref{L2.9})
allows the least action formulation based on the action functional (\re
{L2.10}) with respect to the rest \ reference frame $\mathcal{K}_{\tau },$
where the Lagrangian function is given by expression (\ref{L2.11}). The
resulting Lorentz type force expression equals \ (\ref{L2.13f}), being
modified by the additional force component $F_{c}:=-\nabla <\xi A,u-u_{f}>$,
\ important for explanation \cite{AB,Boy,Tr} of the well known Aharonov-Bohm
effect.
\end{proposition}

\subsubsection{A moving charged point particle formulation dual to the
classical alternative electrodynamic model}

It is easy to see that the action functional (\ref{L2.10}) is written
utilizing the standard classical Lorentz transformations of reference
frames. If we now consider the action functional (\ref{L2.2}) for a charged
point particle moving with respect to the rest reference frame $\mathcal{K
_{\tau },$ and take into\textit{\ }account its interaction with an external
magnetic field generated by the vector potential $A:$ $M^{4}\rightarrow
\mathbb{E}^{3},$ it can be naturally generalized as
\begin{equation}
S:=\underset{t_{1}}{\overset{t_{2}}{\int }}(-\bar{W}dt+\xi <A,dr>)=\underset
\tau _{1}}{\overset{\tau _{2}}{\int }}(-\bar{W}(1+|\dot{r}|^{2})^{1/2}\text{
}+\xi <A,\dot{r}>)d\tau ,  \label{L2.14}
\end{equation
where $d\tau =dt(1-|u|^{2})^{1/2}.$ The chosen form of functional \ (\re
{L2.14}) can be explained by means of the following physically motivated
reasonings. Consider an action functional \ like (\ref{L2.14}) and calculate
its value along any smooth arbitrarily chosen and dynamically admissible
closed path $l\subset M^{4},$ which should be naturally put to be zero:
\begin{equation}
0=\int\limits_{l}(-\bar{W}dt+\xi <A,dr>).  \label{L2.14a}
\end{equation
Having applied to the right-hand side of \ (\ref{L2.14a}) the standard
Stokes theorem \cite{AM}, one easily obtains that
\begin{eqnarray}
\oint\limits_{l}(-\bar{W}dt+\xi &<&A,dr>)=\ \int\limits_{S(l)}(-<\nabla \bar
W},dr\wedge dt>-  \label{L2.14b} \\
- &<&\xi \partial A/\partial t,dr\wedge dt)=\int\limits_{S(l)}<-\nabla \bar{
}-\xi \partial A/\partial t,dr\wedge dt>=  \notag \\
&=&\int\limits_{S(l)}<\xi E,dr\wedge dt>=\int\limits_{S(l)}<F,dr\wedge dt>=
\notag \\
&=&\int\limits_{S(l)}<F_{\xi }\ ,dr\wedge dt>=-\int\limits_{S(l)}<d\mathcal{
}\wedge dt>=  \notag \\
&=&-\int\limits_{S(l)}d\mathcal{\ }(\mathcal{E}dt)=-\oint\limits_{l}\mathcal
E}dt=0,\   \notag
\end{eqnarray
if and only if the charged point particle energy $\mathcal{E}$ \ is
conserved along this arbitrarily chosen and admissible path $l\subset M^{4}.$
As a simple consequence of \ (\ref{L2.14a}) the work performed by the
electromagnetic force $F_{\xi }$ depends only on the electric field $E\in
\mathbb{E}^{3},$ not depending on the related magnetic field $B=\nabla
\times A\in \mathbb{E}^{3}.$\ Thus, having assumed that the corresponding
charged point particle dynamical equations conform the energy conservation
condition mentioned above, the action functional \ (\ref{L2.14}) can be
accepted as reasonable from physical point of view.

\begin{remark}
It is also interesting to remark that a condition $\int\limits_{l}\mathcal{L
dt=0,$ similar to \ (\ref{L2.14a}), calculated for the Lagrangian function
\ \mathcal{L}=$ $\frac{m|\dot{r}|^{2}}{2}-\bar{W}$ in the classical
mechanics of a point particle with mass $m\in \mathbb{R}_{+}$ moving under
an external potential $\bar{W}:\mathbb{R}^{3}\rightarrow \mathbb{R},$ gives
rise to the true classical Newton's mechanics
\begin{equation*}
\oint\limits_{l}(\frac{m|\dot{r}|^{2}}{2}-\bar{W})dt=\oint\limits_{l}(\frac{
}{2}<\dot{r},\dot{r}>\ -\bar{W})dt=
\end{equation*
\
\begin{eqnarray}
&=&\int\limits_{S(l)}(<m\dot{r},d\dot{r}\wedge dt>-<\nabla \bar{W},dr\wedge
dt>)=  \notag \\
&=&\int\limits_{S(l)}(-<mdr,\wedge d\dot{r}\ \ >-<\nabla \bar{W},dr\wedge
dt>)=  \notag \\
&=&\int\limits_{S(l)}(-<dr,\wedge m\ddot{r}dt\ \ >-<\nabla \bar{W},dr\wedge
dt>)=  \label{L2.14c} \\
&=&-\int\limits_{S(l)}(<m\ddot{r},dr\wedge dt\ \ >+<\nabla \bar{W},dr\wedge
dt>)=  \notag \\
&=&-\int\limits_{S(l)}<m\ddot{r}+\nabla \bar{W},dr\wedge dt\ \ >=0  \notag
\end{eqnarray
if and only if the Newton's equation
\begin{equation}
m\ddot{r}=-\nabla \bar{W}  \label{L2.14d}
\end{equation
holds.
\end{remark}

The least action condition $\delta S=0,$ as calculated with respect to the
rest reference frame $\mathcal{K}_{\tau },$ states in the Feynman's spirit
\cite{Fe1} of reasonings that the charged point particle $\xi $ chooses in
the Minkowski space-time $M^{4}$ such a trajectory of its motion, which
realizes the least action value of the functional \ (\ref{L2.14}),
calculated namely with respect to its own rest reference time parameter
\tau \in \mathbb{R},$ being \textit{a unique physically sensible quantity
attributed to the charged point particle dynamics}. Really, as it was
stressed by R. Feynman \cite{Fe1}, the least action principle, as applied to
the functional \ (\ref{L2.14}) with respect to the laboratory reference
frame time parameter, gives rise to a senseless expression, whose value is
both ambiguous and physically not well defined. \ Thus, the corresponding
common generalized particle-field momentum takes the form
\begin{align}
P& :=\partial \mathcal{L}/\partial \dot{r}=-\bar{W}\dot{r}(1+|\dot{r
|^{2})^{-1/2}+\xi A=  \label{L2.15} \\
& =mu+\xi A:=p+\xi A,  \notag
\end{align
and satisfies the equatio
\begin{align}
\dot{P}& :=dP/d\tau =\partial \mathcal{L}/\partial r=-\nabla \bar{W}(1+|\dot
r}|^{2})^{1/2}\text{ }+\xi \nabla <A,\dot{r}>=  \label{L2.16} \\
& =-\nabla \bar{W}(1-|u|^{2})^{-1/2}+\xi \nabla <A,u>(1-|u|^{2})^{-1/2},
\notag
\end{align
where
\begin{equation}
\mathcal{L}:=-\bar{W}(1+|\dot{r}|^{2})^{1/2}\text{ }+\xi <A,\dot{r}>
\label{L2.16a}
\end{equation
is the corresponding Lagrangian function. Since $d\tau =dt(1-|u|^{2})^{1/2},$
one easily finds from (\ref{L2.16}) that
\begin{equation}
dP/dt=-\nabla (\bar{W}-<\xi A,u>).  \label{L2.17}
\end{equation
Upon substituting (\ref{L2.15}) into (\ref{L2.17}) and making use of the
identity \ (\ref{L2.13e}), we obtain the classical expression for the
Lorentz force $F_{\xi },$ \ acting on the moving charged point particle $\xi
:
\begin{equation}
dp/dt:=F_{\xi }=\xi E+\xi u\times B,  \label{L2.19}
\end{equation
where, by definition,
\begin{equation}
E:=-\xi ^{-1}\nabla \bar{W}-\partial A/\partial t  \label{L2.20}
\end{equation
is its associated electric field and
\begin{equation}
B:=\nabla \times A  \label{L2.21}
\end{equation
is the corresponding magnetic field. This wondering result can be summarized
as follows.

\begin{proposition}
The classical relativistic Lorentz force (\ref{L2.19}) allows the least
action formulation based on the action functional (\ref{L2.14}) with respect
to the rest \ \ reference frame $\mathcal{K}_{\mathcal{\tau }},$ where the
Lagrangian function is given by formula (\ref{L2.16a}).
\end{proposition}

Concerning the related electrodynamics of a charged point particle $\xi ,$ \
described by the dual classical Lorentz force (\ref{L2.19}), we need to
state that it is \textit{not equivalent} to that described by means of the
classical Lorentz force (\ref{L1.11}). Moreover, one can easily observe that
the classical Lorentz force $F_{\xi }=\xi E+\xi u\times B,$ exerted on the
charged point particle $\xi $ \ by an external charged point test particle
\xi _{f}$ is not \textit{a priori} vanishing as it should follow from the
relativistic physics point of view. The details of these aspects will be
analyzed in more details in the next Section to follow.

Comparing the obtained above Lorentz type forces expressions (\ref{L2.19})
and (\ref{L2.13f}), differing by the gradient term $F_{c}:=-\xi \nabla
<A,u-u_{f}>,$ which reconciles the dual Lorentz force acting on a moving
charged point particle $\xi $ \ with respect to an arbitrarily chosen
laboratory reference frames $\mathcal{K}$ and, as it is was mentioned below,
is responsible \cite{AB,Boy,Tr} for the Aharonov-Bohm effect. This fact is
important for our vacuum field theory approach since it uses no special
geometry and makes it possible to analyze electromagnetic and, under some
conditions, also gravitational fields simultaneously by employing the new
definition of the dynamical mass by means of expression (\ref{L2.5}).

\subsection{The vacuum field theory electrodynamics equations: Hamiltonian
analysis}

Any Lagrangian theory has an equivalent canonical Hamiltonian representation
via the classical Legendre transformation \ \cite{Ar,Th,AM,PM,BPS}. As we
have already formulated our vacuum field theory of a moving particle with a
charge $\xi \in \mathbb{R}$ \ in Lagrangian form, we proceed now to its
Hamiltonian analysis making use of the action functionals (\ref{L2.2}), (\re
{L2.11}) and (\ref{L2.14}).

Take, first, the Lagrangian function (\ref{L2.4}) and the momentum
expression (\ref{L2.3}) for defining the corresponding Hamiltonian functio
\begin{align}
H& :=<p,\dot{r}>-\mathcal{L}=  \notag \\
& =-|p|^{2}\bar{W}^{-1}(1-|p|^{2}/\bar{W}^{2})^{-1/2}+\bar{W}(1-|p|^{2}/\bar
W}^{2})^{-1/2}=  \notag \\
& =-|p|^{2}\bar{W}^{-1}(1-|p|^{2}/\bar{W}^{2})^{-1/2}+\bar{W}^{2}\bar{W
^{-1}(1-|p|^{2}/\bar{W}^{2})^{-1/2}=  \label{L3.1} \\
& =-(\bar{W}^{2}-|p|^{2})(\bar{W}^{2}-|p|^{2})^{-1/2}=-(\bar{W
^{2}-|p|^{2})^{1/2}.  \notag
\end{align
Consequently, it is easy to show \cite{AM,Ar,Th,PM} \ that the Hamiltonian
function (\ref{L3.1}) is a conservation law of the dynamical field equation
\ref{L2.1}); that is, for all $\tau ,t\in \mathbb{R}
\begin{equation}
dH/dt=0=dH/d\tau ,  \label{L3.2}
\end{equation
which naturally leads to an energy interpretation of $H.$ Thus, we can
represent the particle energy as
\begin{equation}
\mathcal{E}=(\bar{W}^{2}-|p|^{2})^{1/2}.  \label{L3.3}
\end{equation
The corresponding Hamiltonian system equivalent to the vacuum field equation
(\ref{L2.1}) can be written a
\begin{align}
\dot{r}& :=dr/d\tau =\partial H/\partial p=p(\bar{W}^{2}-|p|^{2})^{-1/2}
\label{L3.4} \\
\dot{p}& :=dp/d\tau =-\partial H/\partial r=\bar{W}\nabla \bar{W}(\bar{W
^{2}-|p|^{2})^{-1/2},  \notag
\end{align
and we have the following result.

\begin{proposition}
The alternative freely moving point particle electrodynamic model, based on
the action functional (\ref{L2.2}), allows the canonical Hamiltonian
formulation (\ref{L3.4}) with respect to the rest\ reference frame $\mathcal
K},$ where the Hamiltonian function is given by expression (\ref{L3.1}).
\end{proposition}

Concerning the charged point particle electrodynamics, based on the
dynamical equations \ (\ref{L3.4}), we state that it is completely
equivalent to the classical relativistic freely moving point particle
electrodynamics described above in Subsection \ref{Subsec_1.1}.

In an analogous manner, one can now use the Lagrangian (\ref{L2.11}) and
equation \ (\ref{L2.16}) to construct the Hamiltonian function for the
dynamical field equation (\ref{L2.9a}), describing the motion of a charged
point particle $\xi $ in an external electromagnetic field\ as
\begin{equation}
\dot{r}:=dr/d\tau =\partial H/\partial P,\text{ \ \ \ \ \ }\dot{P}:=dP/d\tau
=-\partial H/\partial r,  \label{L3.5a}
\end{equation
where, by definition,
\begin{align}
H& :=<P,\dot{r}>-\mathcal{L}=  \notag \\
& =<P,\dot{r}_{f}-P\bar{W}^{\prime -1}(1-|P|^{2}/\bar{W}^{\prime 2})^{-1/2}>
\bar{W}^{\prime }[\bar{W}^{\prime }{}^{2}(\bar{W}^{\prime
}{}^{2}-|P|^{2})^{-1}]^{1/2}=  \notag \\
& =<P,\dot{r}_{f}>+|P|^{2}(\bar{W}^{\prime }{}^{2}-|P|^{2})^{-1/2}-\bar{W
^{\prime }{}^{2}(\bar{W}^{\prime }{}^{2}-|P|^{2})^{-1/2}=  \notag \\
& =-(\bar{W}^{\prime }{}^{2}-|P|^{2})(\bar{W}^{\prime
}{}^{2}-|P|^{2})^{-1/2}+<P,\dot{r}_{f}>=  \label{L3.5} \\
& =-(\bar{W}^{\prime }{}^{2}-|P|^{2})^{1/2}-<\xi A,P>(\bar{W}^{\prime
}{}^{2}-|P|^{2})^{-1/2}.  \notag
\end{align
Here we took into account that, owing to definitions (\ref{L2.8}) and (\re
{L2.12})
\begin{align}
\xi _{f}A^{\prime }& :=\bar{W}^{\prime }u_{f}^{\prime }=\bar{W}^{\prime
}dr_{f}/dt^{\prime }=\bar{W}u_{f}=\xi _{f}A=  \label{L3.6} \\
& =\bar{W}^{\prime }\frac{dr_{f}}{d\tau }\cdot \frac{d\tau }{dt^{\prime }}
\bar{W}\dot{r}_{f}(1-|u^{\prime }-u_{f}^{\prime }|)^{1/2}=  \notag \\
& =\bar{W}^{\prime }\dot{r}_{f}(1+|\dot{r}-\dot{r}_{f}|^{2})^{-1/2}=  \notag
\\
& =-\bar{W}^{\prime }\dot{r}_{f}(\bar{W}^{\prime }{}^{2}-|P|^{2})^{1/2}\bar{
}^{\prime -1}=-\dot{r}_{f}(\bar{W}^{\prime }{}^{2}-|P|^{2})^{1/2},  \notag
\end{align
o
\begin{equation}
\dot{r}_{f}=-\xi _{f}A(\bar{W}^{\prime }{}^{2}-|P|^{2})^{-1/2},  \label{L3.7}
\end{equation
where $A:M^{4}\mathbb{\rightarrow R}^{3}$ is the related magnetic vector
potential generated by the moving external charged particle $\xi _{f}.$
Equations (\ref{L3.5a}) can be easily rewritten with respect to the
laboratory reference frame $\mathcal{K}$ in the form
\begin{equation}
dr/dt=u,\text{ \ \ \ }dp/dt=\xi E+\xi u\times B-\nabla <\xi A,u-u_{f}>,
\label{L3.7a}
\end{equation
which coincide with the result (\ref{L2.13f}).

Whence, we see that the Hamiltonian function (\ref{L3.5}) satisfies the
energy conservation condition
\begin{equation}
dH/dt=0=dH/d\tau ,  \label{L3.8}
\end{equation
for all $\tau ,t\in \mathbb{R},$ and the suitable energy expression, owing
to \ (\ref{L2.8a}), i
\begin{equation}
\mathcal{E}=\left( \bar{W}^{2}-|\xi A|^{2}-|P|^{2}\right) ^{1/2}+\ <\xi
A,P>\left( \bar{W}^{2}-|\xi A|^{2}-|P|^{2}\right) ^{-1/2},  \label{L3.9}
\end{equation
where the generalized momentum $P=p\ +\xi A.$ \ The result (\ref{L3.9})
differs essentially from that obtained in \cite{LL}, which is strongly based
on the Einsteinian Lagrangian for a moving charged point particle $\xi $ in
the external electromagnetic fields, generated by a charged point test
particle $\xi _{f},$ moving with velocity $u_{f}\in T(\mathbb{R}^{3})$ with
respect to a laboratory reference frame $\mathcal{K}.$ \ Thus, we obtained
the following proposition,

\begin{proposition}
The alternative classical relativistic electrodynamic model (\ref{L3.7a}),
which is intrinsically compatible with the classical Maxwell equations \
\ref{L0.6}), allows the Hamiltonian formulation (\ref{L3.5a}) with respect
to the rest reference frame $\mathcal{K}_{\tau },$ where the Hamiltonian
function is given by expression (\ref{L3.5}).
\end{proposition}

The inference above is a natural candidate for experimental validation of
our vacuum field theory. It is strongly motivated by the following remark.

\begin{remark}
It is necessary to mention here that the Lorentz force expression (\re
{L3.7a}) uses the particle momentum $p=mu,$ where the dynamical "mass" $m:=
\bar{W}$ \ satisfies condition (\ref{L3.9}). The latter gives rise to the
following crucial relationship between the particle energy $\mathcal{E}_{0}$
and its rest mass $m_{0}=$ $-\bar{W}_{0}$ (for the velocity $u\ =0$ at the
initial time moment $t=0):
\begin{equation}
\mathcal{E}_{0}=m_{0}\frac{(1-|\xi A_{0}/m_{0}|^{2})}{(1-2|\xi
A_{0}/m_{0}|^{2})^{1/2}}\ ,  \label{L3.9a}
\end{equation
or, equivalently, at the condition $\ \ |\xi A_{0}/m_{0}|^{2}<1/2$
\begin{equation}
m_{0}=\mathcal{E}_{0}\left( \frac{1}{2}+|\xi A_{0}/\mathcal{E}_{0}|^{2}\pm
\frac{1}{2}\sqrt{1-4|\xi A_{0}/\mathcal{E}_{0}|^{2}}\right) ^{1/2},
\label{L3.9b}
\end{equation
where $A_{0}:=A|_{t=0}\in \mathbb{E}^{3},$ which strongly differs from the
classical expression $m_{0}=\mathcal{E}_{0}-\xi \varphi _{0},$ following
from (\ref{L1.10}) and is not depending a priori on the external potential
energy $\xi \varphi _{0}.$\ As \ \ \ \ the quantity \ \ \ \ $|\xi A_{0}
\mathcal{E}_{0}|\rightarrow 0,$ the following asymptotical mass values
follow from \ (\ref{L3.9b})
\begin{equation}
m_{0}^{(+)}\simeq \ \mathcal{E}_{0},\text{ \ \ \ \ \ \ }m_{0}^{(-)}\simeq \
\pm \sqrt{2}|\xi A_{0}|.  \label{L3.9c}
\end{equation
The first mass value $m_{0}^{(+)}\simeq \ \mathcal{E}_{0}$ is physically
correct, giving rise to the bounded charged particle energy $\mathcal{E
_{0}, $ but the second mass value $m_{0}^{(-)}\simeq \ \pm \sqrt{2}|\xi
A_{0}|$ is not physical, as it gives rise to the vanishing denominator \ \
(1-2|\xi A_{0}/m_{0}^{(-)}|^{2})^{1/2}\simeq 0$ in \ (\ref{L3.9a}), being
equivalent to the unboundedness of the charged particle energy $\mathcal{E
_{0}.$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{remark}

To make this difference more clear, we now will analyze the dual classical
Lorentz force (\ref{L2.19}) from the Hamiltonian point of view, based on the
Lagrangian function (\ref{L2.16a}). Thus, we can easily obtain that the
corresponding Hamiltonian functio
\begin{align}
H& :=<P,\dot{r}>-\mathcal{L}=<P,\dot{r}>+\bar{W}(1+|\dot{r}|^{2})^{1/2}-\xi
<A,\dot{r}>=  \label{L3.10} \\
& =<P-\xi A,\dot{r}>+\bar{W}(1+|\dot{r}|^{2})^{1/2}=  \notag \\
& =-|p|^{2}\bar{W}^{-1}(1-|p|^{2}/\bar{W}^{2})^{-1/2}+\bar{W}(1-|p|^{2}/\bar
W}^{2})^{-1/2}=  \notag \\
& =-(\bar{W}^{2}-|p|^{2})(\bar{W}^{2}-|p|^{2})^{-1/2}=-(\bar{W
^{2}-|p|^{2})^{1/2}.  \notag
\end{align
Since $p=P-\xi A,$ expression (\ref{L3.10}) assumes the final "\textit{no
interaction}" \ \cite{LL,Pa,Kup,PS} form
\begin{equation}
H=-(\bar{W}^{2}-|P-\xi A|^{2})^{1/2},  \label{L3.11}
\end{equation
which is conserved with respect to the evolution equations (\ref{L2.15}) and
(\ref{L2.16}), that is
\begin{equation}
dH/dt=0=dH/d\tau  \label{L3.11a}
\end{equation
for all $\tau ,t\in \mathbb{R}.$ These equations are equivalent to the
following Hamiltonian syste
\begin{align}
\dot{r}& =\partial H/\partial P=(P-\xi A)(\bar{W}^{2}-|P-\xi A|^{2})^{-1/2},
\label{L3.12} \\
\dot{P}& =-\partial H/\partial r=(\bar{W}\nabla \bar{W}-\nabla <\xi A,(P-\xi
A)>)(\bar{W}^{2}-|P-\xi A|^{2})^{-1/2},  \notag
\end{align
as one can readily check by direct calculations. Actually, the first equatio
\begin{align}
\dot{r}& =(P-\xi A)(\bar{W}^{2}-|P-\xi A|^{2})^{-1/2}=p(\bar{W
^{2}-|p|^{2})^{-1/2}=  \label{L3.13} \\
& =mu(\bar{W}^{2}-|p|^{2})^{-1/2}=-\bar{W}u(\bar{W
^{2}-|p|^{2})^{-1/2}=u(1-|u|^{2})^{-1/2},  \notag
\end{align
holds, owing to the condition $d\tau =dt(1-|u|^{2})^{1/2}$ and definitions
p:=mu,$ $m=-\bar{W},$ postulated from the very beginning. Similarly we
obtain that
\begin{align}
\dot{P}& =-\nabla \bar{W}(1-|p|^{2}/\bar{W}^{2})^{-1/2}+\nabla <\xi
A,u>(1-|p|^{2}/\bar{W}^{2})^{-1/2}=  \label{L3.14} \\
& =-\nabla \bar{W}(1-|u|^{2})^{-1/2}+\nabla <\xi A,u>(1-|u|^{2})^{-1/2},
\notag
\end{align
or equivalently, the dual Lorentz dynamical expression
\begin{equation}
dp/dt=\ \xi E+\xi u\times B,  \label{L3.14a}
\end{equation
exactly coinciding with that of equation (\ref{L2.19}). This result can be
reformulated as the next proposition.

\begin{proposition}
The dual to the classical relativistic electrodynamic \ model (\ref{L2.19})
allows the canonical Hamiltonian formulation (\ref{L3.12}) with respect to
the rest reference frame $\mathcal{K}_{\tau },$ where the Hamiltonian
function is given by expression (\ref{L3.11}). Moreover, this formulation
circumvents the "mass-potential" energy controversy attached to the
classical electrodynamic model, based the classical action functional (\re
{L1.8}).
\end{proposition}

The classical Lorentz force expression (\ref{L3.14a}) and the related
conserved energy relationship
\begin{equation}
\mathcal{E}=\ (\bar{W}^{2}-|P-\xi A|^{2})^{1/2}  \label{L3.14b}
\end{equation
are characterized by the following remark.

\begin{remark}
If we make use of the modified relativistic Lorentz force expression (\re
{L3.14a}) as an alternative to the classical one of (\ref{L1.11}), the
corresponding charged particle energy relationship (\ref{L3.14b}) gives rise
to a different energy expression (for the velocity $u\ =0$ at the initial
time moment $t=0$). Namely, one naturally obtains the physically reasonable
Einsteinian mass-energy relationship $\mathcal{E}_{0}=m_{0}$ instead of the
senseless classical expression $\mathcal{E}_{0}=m_{0}+\xi \varphi _{0},$
following from (\ref{L1.10}), where $\varphi _{0}:=\varphi |_{t=0}\ $ and
where the mass parameter $m_{0}$ is a constant parameter not depending on
the external electromagnetic field.
\end{remark}

\subsection{ Comments}

All of dynamical field equations discussed above are canonical Hamiltonian
systems with respect to the corresponding physically proper \ rest reference
frames $\mathcal{K}_{\tau },$ parameterized by the Euclidean coordinates
(\tau ,r)\in \mathbb{E}^{4}.$ Upon passing to the basic laboratory reference
frame $\mathcal{K},$ naturally parameterized by the Minkowski coordinates
(t,r)\in M^{4},$ the related Hamiltonian structure is lost, giving rise to a
suitably altered interpretation of the real particle motion. Namely, as it
was demonstrated above, a least action principle for a charged point
particle dynamics makes sense only with respect to the proper rest \
reference frame $\mathcal{K}_{\tau }$ as, otherwise, it becomes completely
senseless with respect to all other laboratory reference frames. As for the
Hamiltonian expressions (\ref{L3.1}), (\ref{L3.5}) and (\ref{L3.11}), one
observes that they all depend strongly on the vacuum potential field
function $\bar{W}:M^{4}\mathbb{\rightarrow R},$ thereby avoiding the mass
problem related with the well known classical energy expression and pointed
out by L. Brillouin \cite{Br}.

Some comments are also on order concerning the classical relativity
principle and a way of its application to real physical phenomena. We have
obtained our results with using the standard Lorentz transformations of
reference frames - relying only on the natural notion of the rest reference
frame $\mathcal{K}_{\tau }$ and its suitable parametrization with respect to
any other laboratory reference frame $\mathcal{K}.$ It seems physically
reasonable that the true state changing of a moving charged particle $\xi $\
is exactly realized only with respect to its proper rest reference system
\mathcal{K}_{\tau }.$

Thus, the only remaining question would be about the physical justification
of the corresponding relationship between time parameters of the
corresponding laboratory and rest reference frames. The relationship between
these reference frames that we have used through is simply expressed as
\begin{equation}
d\tau =dt(1-|u|^{2})^{1/2},  \label{L4.1}
\end{equation
where $u:=dr/dt\in \mathbb{E}^{3}$ is the velocity vector with which the
rest reference frame $\mathcal{K}_{\tau }$ moves with respect to other
arbitrarily chosen reference frame $\mathcal{K}.$ Expression (\ref{L4.1})
implies, in particular, that
\begin{equation}
dt^{2}-|dr|^{2}=d\tau ^{2},  \label{L4.2}
\end{equation
which is evidently identical to the classical infinitesimal Lorentz
invariant. This is not a coincidence, since all our dynamical vacuum field
equations were derived in turn \cite{BPT,BPT1} \ from the governing
equations of the vacuum potential field function $W:M^{4}\mathbb{\rightarrow
R}$ in the form
\begin{equation}
\partial ^{2}W/\partial t^{2}-\nabla ^{2}W=\rho ,\text{ }\partial W/\partial
t+<\nabla ,vW>=0,\text{ }\partial \rho /\partial t+<\nabla ,v\rho >=0,
\label{L4.3}
\end{equation
which is \textit{a priori }Lorentz invariant. Here $\rho \in $ $\mathbb{R}$
is the charge density and $v:=dr/dt$ the associated local velocity of the
vacuum field potential evolution. Consequently, the dynamical infinitesimal
Lorentz invariant (\ref{L4.2}) reflects this intrinsic structure of
equations (\ref{L4.3}). If it is rewritten in the following slightly
nonstandard Euclidean form
\begin{equation}
dt^{2}=d\tau ^{2}+|dr|^{2}  \label{L4.4}
\end{equation
it gives rise to a completely different relationship between the reference
frames $\mathcal{K}$ and $\mathcal{K}_{\tau },$ namel
\begin{equation}
dt=d\tau (1+|\dot{r}|^{2})^{1/2},  \label{L4.5}
\end{equation
where $\dot{r}:=dr/d\tau $ is the related particle velocity with respect to
the rest reference system $\mathcal{K}_{\tau }.$ Thus, we observe that all
our Lagrangian analysis is strongly related\ to the functional expressions
written in these "Euclidean" space-time coordinates and with respect to
which the least action principle was applied. So we see that there are two
alternatives - the first one is to apply the least action principle to the
corresponding Lagrangian functions, expressed in the Minkowski space-time
variables with respect to an arbitrarily chosen reference frame $\mathcal{K
, $ and the second one is to apply the least action principle to the
corresponding Lagrangian functions expressed in the Euclidean space-time
variables with respect to the rest reference frame $\mathcal{K}_{\tau }.$
But, as it was demonstrated above, the second alternative appeared to be
physically reasonable in contrast to the \ first one, which gives rise to
different physically senseless controversies.

All that above entails also a next slightly amusing but thought-provoking
observation: "It follows that all of the results of classical special
relativity related with the electrodynamics of charged point particles can
be obtained (but not in a one-to-one correspondence) using our new
physically reasonable definitions of the dynamical particle mass and the
physically motivated least action principle, calculated with respect to the
related Euclidean space-time variables specifying the rest reference frame
\mathcal{K}_{\tau }.$

\subsection{The quantization of electrodynamics models within the vacuum
field theory no-geometry approach}

\subsubsection{The problem setting}

Recently \cite{BPT,BPT1}, we devised a new regular no-geometry approach to
deriving the electrodynamics of a moving charged point particle $q$ in an
external electromagnetic field from first principles. This approach has, in
part, reconciled the mass-energy controversy \cite{Br} in classical
relativistic electrodynamics. Using the vacuum field theory approach
proposed in \cite{BPT,BPT1,Rep}, we reanalyzed this problem above both from
the Lagrangian and Hamiltonian perspective and derived key expressions for
the corresponding energy functions and Lorentz type forces acting on a
moving charge point particle $\xi .$

Since all of our electrodynamics models were represented here in canonical
Hamiltonian form, they are well suited to the application of Dirac
quantization \cite{Di,BS,BS1} and the corresponding derivation of related
Schr\"{o}dinger type evolution equations. We describe these procedures in
this section.

\subsubsection{Free point particle electrodynamics model and its quantizatio
}

The charged point particle electrodynamics models, discussed in detail in
Sections 2 and 3, were also considered in \cite{BPT1} from the dynamical
point of view, where a Dirac quantization of the corresponding conserved
energy expressions was attempted. However, from the canonical point of view,
\ the true quantization procedure should be based on the relevant canonical
Hamiltonian formulation of the models given in (\ref{L3.4}), (\ref{L3.5a})
and (\ref{L3.12}).

In particular, consider a free charged \ point particle electrodynamics
model characterized by (\ref{L3.4}) and having the Hamiltonian equation
\begin{align}
dr/d\tau & :=\partial H/\partial p=-p(\bar{W}^{2}-|p|^{2})^{-1/2},
\label{L5.1} \\
dp/d\tau & :=-\partial H/\partial r=-\bar{W}\nabla \bar{W}(\bar{W
^{2}-|p|^{2})^{-1/2},  \notag
\end{align
where $\bar{W}:M^{4}\rightarrow \mathbb{R}$ defined in the preceding
sections is the corresponding vacuum field potential characterizing medium
field structure, $(r,p)\in T^{\ast }(\mathbb{E}^{3})\simeq \mathbb{E
^{3}\times \mathbb{E}^{3}$ are the standard canonical coordinate-momentum
variables on the cotangent space $T^{\ast }(\mathbb{E}^{3})$, $\tau \in
\mathbb{R}$ is the proper rest reference frame $\mathcal{K}_{r}$ time
parameter of the moving particle, and $H:T^{\ast }(\mathbb{E
^{3})\rightarrow \mathbb{R}$ is the Hamiltonian functio
\begin{equation}
H:=-(\bar{W}^{2}-p^{2})^{1/2},  \label{L5.2}
\end{equation
expressed here and hereafter in light speed units. The rest reference frame
\mathcal{K}_{r},$ parameterized by variables $(\tau ,r)\in \mathbb{E}^{4},$
is related to any other reference frame $\mathcal{K}$ in which our charged
point particle $\xi $ moves with velocity vector $u\in \mathbb{E}^{3}.$ The
frame $\mathcal{K}$ is parameterized by variables $(t,r)\in M^{4}$ via the
Euclidean infinitesimal relationshi
\begin{equation}
dt^{2}=d\tau ^{2}+|dr|^{2},  \label{L5.3}
\end{equation
which is equivalent to the Minkowski infinitesimal relationshi
\begin{equation}
d\tau ^{2}=dt^{2}-|dr|^{2}.  \label{L5.4}
\end{equation
The Hamiltonian function (\ref{L5.2}) clearly satisfies the energy
conservation conditions
\begin{equation}
dH/dt=0=dH/d\tau  \label{L5.5}
\end{equation
for all $t,\tau \in \mathbb{R}.$ This means that the suitable energy
\begin{equation}
\mathcal{E}=(\bar{W}^{2}-|p|^{2})^{1/2}  \label{L5.6}
\end{equation
can be treated by means of the Dirac quantization scheme \cite{Di,Di1,KS} to
obtain, as $\hbar /c\rightarrow 0,$ (or the light speed $c\rightarrow \infty
$) the governing Schr\"{o}dinger type dynamical equation. To do this
following the approach in \cite{BPT,BPT1}, we need to make canonical
operator replacements $\mathcal{E}\rightarrow \hat{\mathcal{E}}:=-\frac
\hbar }{i}\frac{\partial }{\partial \tau },$ \ $p\rightarrow \hat{p}:=\frac
\hbar }{i}\nabla ,$ as $\hbar \rightarrow 0,$ in the following energy
expression
\begin{equation}
\mathcal{E}^{2}:=(\hat{\mathcal{E}}\psi ,\hat{\mathcal{E}}\psi )=(\psi ,\hat
\mathcal{E}}^{2}\psi )=(\psi ,\hat{H}^{+}\hat{H}\psi ),  \label{L5.7}
\end{equation
where $(\cdot ,\cdot )$ is the standard $L_{2}$ - inner product. It follows
from (\ref{L5.6}) that
\begin{equation}
\hat{\mathcal{E}}^{2}=\bar{W}^{2}-|\hat{p}|^{2}=\hat{H}^{+}\hat{H}
\label{L5.8}
\end{equation
is a suitable operator factorization in the Hilbert space $\mathcal{H
:=L_{2}(\mathbb{R}^{3};\mathbb{C})$ and $\psi \in \mathcal{H}$ \ is the
corresponding normalized quantum vector state. Since the following
elementary identity
\begin{equation}
\bar{W}^{2}-|\hat{p}|^{2}=\bar{W}(1-\bar{W}^{-1}|\hat{p}^{2}|\bar{W
^{-1})^{1/2}(1-\bar{W}^{-1}|\hat{p}|^{2}\bar{W}^{-1})^{1/2}\bar{W}
\label{L5.9}
\end{equation
holds, we can use (\ref{L5.8}) and (\ref{L5.9}) to define the operator
\begin{equation}
\hat{H}:=(1-\bar{W}^{-1}|\hat{p}|^{2}\bar{W}^{-1})^{1/2}\bar{W}.
\label{L5.10}
\end{equation
Having calculated the operator expression (\ref{L5.10}) as $\hbar
\rightarrow 0$ up to operator accuracy $O($ $\hbar ^{4}),$ \ it is easy see
that
\begin{equation}
\hat{H}=\frac{|\hat{p}^{2|}}{2m(u)}+\bar{W}:=-\frac{\hbar ^{2}}{2m(u)}\Delta
+\bar{W},  \label{L5.11}
\end{equation
where we have taken into account the dynamical mass definition $m(u):=-\bar{
}$ (in the light speed units). Consequently, using (\ref{L5.7}) and (\re
{L5.11}), we obtain up to operator accuracy $O($ $\hbar ^{4})$ the following
Schr\"{o}dinger type evolution equatio
\begin{equation}
i\hbar \frac{\partial \psi }{\partial \tau }:=\hat{\mathcal{E}}\psi =\hat{H
\psi =-\frac{\hbar ^{2}}{2m(u)}\Delta \psi +\bar{W}\psi  \label{L5.12}
\end{equation
with respect to the rest \ reference frame $\mathcal{K}_{r}$ evolution
parameter $\tau \in \mathbb{R}.$ For a related evolution parameter $t\in
\mathbb{R}$ parameterizing a reference frame $\mathcal{K},$ the equation
\ref{L5.12}) takes the for
\begin{equation}
i\hbar \frac{\partial \psi }{\partial t}=-\frac{\hbar ^{2}m_{0}}{2m(u)^{2}
\Delta \psi -m_{0}\psi .  \label{L5.13}
\end{equation
Here we used the fact that it follows from (\ref{L5.6}) that the classical
inertial mass relationship
\begin{equation}
m(u)=m_{0}(1-|u|^{2})^{-1/2}  \label{L5.14}
\end{equation
holds, where $m_{0}\in \mathbb{R}_{+}$ is the corresponding rest mass of our
point particle $\xi .$

The linear Schr\"{o}dinger equation (\ref{L5.13}) for the case $\hbar
/c\rightarrow0$ actually coincides with the well-known expression \cit
{LL,Di,Fe1} from classical quantum mechanics.

\subsubsection{Classical charged point particle electrodynamics model and
its quantization}

We start here from the first vacuum field theory reformulation of the
classical charged point particle electrodynamics (introduced in Section 3)
and based on the conserved Hamiltonian function (\ref{L3.11})
\begin{equation}
H:=-(\bar{W}^{2}-|P-\xi A|^{2})^{1/2},  \label{L6.1}
\end{equation
where $q\in $ $\mathbb{R}$ is the particle charge, $(\bar{W},A)\in \mathbb
R\times E}^{3}$ is the corresponding representation of the electromagnetic
field potentials and $P\in \mathbb{E}^{3}$ is the common generalized
particle-field momentum
\begin{equation}
P:=p+\xi A,\text{ \ \ \ \ }p:=mu,  \label{L6.2}
\end{equation
which satisfies the classical Lorentz force equation. Here $m:=-\bar{W}$ is
the observable dynamical mass of our charged particle, and $u\in \mathbb{E
^{3}$ is its velocity vector with respect to a chosen reference frame
\mathcal{K},$ all expressed in light speed units.

Our electrodynamics based on (\ref{L6.1}) is canonically Hamiltonian, so the
Dirac quantization scheme
\begin{equation}
P\rightarrow \hat{P}:=\frac{\hbar }{i}\nabla ,\text{ \ \ \ \ \ \ }\mathcal{E
\rightarrow \hat{\mathcal{E}}:=-\frac{\hbar }{i}\frac{\partial }{\partial
\tau }  \label{L6.3}
\end{equation
should be applied to the energy expression
\begin{equation}
\mathcal{E}:=(\bar{W}^{2}-|P-\xi A|^{2})^{1/2},  \label{L6.4}
\end{equation
following from the conservation conditions
\begin{equation}
dH/dt=0=dH/d\tau ,  \label{L6.5}
\end{equation
satisfied for all $\tau ,t\in \mathbb{R}.$

Proceeding as above, we can factorize the operator $\hat{\mathcal{E}}^{2}$
as
\begin{equation*}
\begin{array}{c}
\bar{W}^{2}-|\hat{P}-\xi A|^{2}=\bar{W}(1-\bar{W}^{-1}|\hat{P}-\xi A|^{2
\bar{W})^{1/2}\times \\
\times (1-\bar{W}^{-1}|\hat{P}-\xi A|^{2}\bar{W}^{-1})^{1/2}\bar{W}:=\hat{H
^{+}\hat{H}
\end{array
\end{equation*
where (as $\hbar /c\rightarrow 0,$ $\hbar c=const$
\begin{equation}
\hat{H}:=\frac{1}{2m(u)}|\frac{\hbar }{i}\nabla -\xi A|^{2}+\bar{W}
\label{L6.7}
\end{equation
up to operator accuracy $O(\hbar ^{4}).$ Hence, the related Schr\"{o}dinger
type evolution equation in the Hilbert space $\mathcal{H}=L_{2}(\mathbb{R
^{3};\mathbb{C})$ is
\begin{equation}
i\hbar \frac{\partial \psi }{\partial \tau }:=\hat{\mathcal{E}}\psi =\hat{H
\psi =\frac{1}{2m(u)}|\frac{\hbar }{i}\nabla -\xi A|^{2}\psi +\bar{W}\psi
\label{L6.8}
\end{equation
with respect to the rest reference frame $\mathcal{K}_{r}$ evolution
parameter $\tau \in \mathbb{R}$, and corresponding Schr\"{o}dinger type
evolution equation with respect to the evolution parameter $t\in \mathbb{R}$
takes the form
\begin{equation}
i\hbar \frac{\partial \psi }{\partial t}=\frac{m_{0}}{2m(u)^{2}}|\frac{\hbar
}{i}\nabla -\xi A|^{2}\psi -m_{0}\psi .  \label{L6.9}
\end{equation
The Schr\"{o}dinger equation (\ref{L6.8}) ( as $\hbar /c\rightarrow 0$)
coincides \cite{LL1,Di} with the classical quantum mechanics version.

\subsubsection{Modified charged point particle electrodynamics model and its
quantization}

From the canonical viewpoint, \ we now turn to the true quantization
procedure for the electrodynamics model, characterized by (\ref{L2.13}) and
having the Hamiltonian function (\ref{L3.5})
\begin{equation}
H:=-(\bar{W}^{2}-|P|^{2})^{1/2}-\xi <A,P>(\bar{W}^{2}-|P|^{2})^{-1/2}.
\label{L7.1}
\end{equation
Accordingly the suitable energy function is
\begin{equation}
\mathcal{E}:=(\bar{W}^{2}-|P|^{2})^{1/2}+\xi <A,P>(\bar{W
^{2}-|P|^{2})^{-1/2},  \label{L7.2}
\end{equation
where, as before,
\begin{equation}
P:=p+\xi A,\text{ \ \ \ \ }p:=mu,\text{ \ }m:=-\bar{W},  \label{L7.3}
\end{equation
is a conserved quantity for (\ref{L2.13}), which we will canonically
quantize via the Dirac procedure (\ref{L6.3}). Toward this end, let us
consider the quantum conditio
\begin{equation}
\mathcal{E}^{2}:=(\hat{\mathcal{E}}\psi ,\hat{\mathcal{E}}\psi )=(\psi ,\hat
\mathcal{E}}^{2}\psi ),\text{ \ \ \ \ \ }(\psi ,\psi ):=1,  \label{L7.4}
\end{equation
where, by definition, $\hat{\mathcal{E}}:=-\frac{\hbar }{i}\frac{\partial }
\partial t}$ and $\psi \in \mathcal{H}=L_{2}(\mathbb{R}^{3};\mathbb{C})$ is
a respectively normalized quantum state vector. Making now use of the energy
function (\ref{L7.2}), one readily computes that
\begin{equation}
\mathcal{E}^{2}=\bar{W}^{2}-|P-qA|^{2}+\xi ^{2}<A,A>+\xi ^{2}<A,P>(\bar{W
^{2}-|P|^{2})^{-1}<P,A>),  \label{L7.5}
\end{equation
which transforms by the canonical Dirac type quantization $P\rightarrow \hat
P}:=\frac{\hbar }{i}\nabla $ into the symmetrized operator expression
\begin{equation}
\hat{\mathcal{E}}^{2}=\bar{W}^{2}-|\hat{P}-\xi A|^{2}+\xi ^{2}<A,A>+\xi
^{2}<A,\hat{P}>(\bar{W}^{2}-|\hat{P}|^{2})^{-1}<\hat{P},A>.  \label{L7.6}
\end{equation
Factorizing the operator (\ref{L7.6})\ in the form $\hat{\mathcal{E}}^{2}
\hat{H}^{+}\hat{H},$ and retaining only terms up to $O(\hbar ^{4})$ (as
\hbar /c\rightarrow 0$), we compute that
\begin{align}
\hat{H}& :=\frac{1}{2m(u)}|\frac{\hbar }{i}\nabla -\xi A|^{2}-\frac{\xi ^{2
}{2m(u)}<A,A>-  \label{L7.7} \\
-\frac{\xi ^{2}}{2m^{3}(u)}& <A,\frac{\hbar }{i}\nabla ><\frac{\hbar }{i
\nabla ,A>,  \notag
\end{align
where, as before, $m(u)=-\bar{W}$ in light speed units. Thus, owing to (\re
{L7.4}) and (\ref{L7.7}), the resulting Schr\"{o}dinger evolution equation
is
\begin{align}
i\hbar \frac{\partial \psi }{\partial \tau }& :=\hat{H}\psi =\frac{1}{2m(u)}
\frac{\hbar }{i}\nabla -qA|^{2}\psi -  \label{L7.8} \\
-\frac{q^{2}}{2m(u)}& <A,A>\psi -\frac{\xi ^{2}}{2m^{3}(u)}<A,\frac{\hbar }{
}\nabla ><\frac{\hbar }{i}\nabla ,A>\psi  \notag
\end{align
with respect to the rest reference frame proper evolution parameter $\tau
\in \mathbb{R}.$ The latter can be rewritten in the equivalent form as

\begin{eqnarray}
i\hbar \frac{\partial \psi }{\partial \tau } &=&-\frac{\hbar ^{2}}{2m(u)
\Delta \psi -\frac{1}{2m(u)}<[\frac{\hbar }{i}\nabla ,qA]_{+}>\psi -
\label{L7.8a} \\
-\frac{\xi ^{2}}{2m^{3}(u)} &<&A,\frac{\hbar }{i}\nabla ><\frac{\hbar }{i
\nabla ,A>\psi ,  \notag
\end{eqnarray
where $[\cdot ,\cdot ]_{+}$ means the formal anti-commutator of operators.
Similarly one also obtains the related Schr\"{o}dinger equation with respect
to the time parameter $t\in \mathbb{R},$ which we shall not dwell upon here.
The result (\ref{L7.8}) only slightly differs from the classical Schr\"{o
dinger evolution equation (\ref{L6.8}). Simultaneously, its form (\ref{L7.8a
) almost completely coincides with the classical ones from \cite{LL1,Pa,Di}
modulo the evolution considered with respect to the rest reference time
parameter $\tau \in \mathbb{R}.$ This suggests that we must more thoroughly
reexamine the physical motivation of the principles underlying the classical
electrodynamic models, described by the Hamiltonian functions (\ref{L6.1})
and (\ref{L7.1}), giving rise to different Lorentz type force expressions. A
more deeply considered and extended analysis of this matter is forthcoming
in a paper now in preparation.

\subsection{ Comments}

All of dynamical field equations discussed above are canonical Hamiltonian
systems with respect to the corresponding proper \ rest reference frames
\mathcal{K}_{r},$ parameterized by suitable time parameters $\tau \in
\mathbb{R}.$ Upon passing to the basic laboratory reference frame $\mathcal{
}$ with the time parameter $t\in \mathbb{R},$naturally the related
Hamiltonian structure is lost, giving rise to a new interpretation of the
real particle motion. Namely, one that has an absolute sense only with
respect to the proper rest \ reference system, and otherwise being
completely relative with respect to all other reference frames. As for the
Hamiltonian expressions (\ref{L3.1}), (\ref{L3.5}) and (\ref{L3.11}), one
observes that they all depend strongly on the vacuum potential field
function $\bar{W}:M^{4}\mathbb{\rightarrow R},$ thereby avoiding the mass
problem of the classical energy expression pointed out by L. Brillouin \cit
{Br}. It should be noted that the canonical Dirac quantization procedure can
be applied only to the corresponding dynamical field systems considered with
respect to their proper rest reference frames.

An additional remark concerning the quantization procedure of the proposed
electrodynamics models is in order: If the dynamical vacuum field equations
are expressed in canonical Hamiltonian form, as we have done in this paper,
only straightforward technical details are required to quantize the
equations and obtain the corresponding Schr\"{o}dinger evolution equations
in suitable Hilbert spaces of quantum states. There is another striking
implication from our approach: the Einsteinian equivalence principle \cit
{LL,Pa,Fe1,Fe,Kl} is rendered superfluous for our vacuum field theory of
electromagnetism and gravity.

Using the canonical Hamiltonian formalism devised here for the alternative
charged point particle electrodynamics models, we found it rather easy to
treat the Dirac quantization. The results obtained compared favorably with
classical quantization, but it must be admitted that we still have not given
a compelling physical motivation for our new models. This is something that
we plan to revisit in future investigations. Another important aspect of our
vacuum field theory no-geometry (geometry-free) approach to combining the
electrodynamics with the gravity, is the manner in which it singles out the
decisive role of the rest \ reference frame $\mathcal{K}_{r}.$ More
precisely, all of our electrodynamics models allow both the Lagrangian and
Hamiltonian formulations with respect to the rest reference \ system
evolution parameter $\tau \in \mathbb{R}$, which are well suited the to
canonical quantization. The physical nature of this fact remains is as yet
not quite clear. In fact, as far as we know \cite{Pa,LL,Kl,Lo2,Lo3}, there
is no physically reasonable explanation of this decisive role of the rest \
reference system, except for that given by R. Feynman who argued in \cit
{Fe1} that the relativistic expression for the classical Lorentz force (\re
{L1.11}) has physical sense only with respect to \ the rest reference frame
variables $(\tau ,r)\in \mathbb{E}^{4}.$ In future research we plan to
analyze the quantization scheme in more detail and begin work on formulating
a vacuum quantum field theory of infinitely many particle systems.

\section{\label{Sec_3}The modified Lorentz force and charge radiation
problem within the vacuum field theory approach}

\subsection{Introductory setting}

The Maxwell equations, being a one of physical fundamental theories nowadays
allow, as is well known two main forms of representations: either by means
of the electric and magnetic fields or by means of the electric and magnetic
potentials. The latter were mainly considered as a mathematically motivated
representation useful for different applications but having no physical
significance.

That the situation is not so simple and the evidence that the magnetic
potential demonstrates the physical properties was doubtless, the physics
community understood when Y. Aharonov and \ D. Bohm \cite{AB} formulated
their "paradox" concerning the measurement of magnetic field outside a
separated region where it is completely vanishing. Later such similar
effects were also revealed in the superconductivity theory of Josephson
media. As the existence of any electromagnetic field in the ambient space
can be tested only owing to its interaction with electric charges, their
dynamical behavior, being of great importance, was deeply studied by M.
Faraday, A. Ampere and H. Lorentz subject to its classical second Newton law
form. Namely, the classical Lorentz force
\begin{equation}
dp/dt=\xi E+\xi \frac{u}{c}\times B  \label{R1}
\end{equation
was derived, where $E$ and $B\in \ \mathbb{E}^{3}$ are, respectively,
electric and magnetic fields, acting on a point charged particle $\xi \in $
\mathbb{R},$ possessing the momentum $p=mu,$ $m\in $ $\mathbb{R}_{+}$ is the
observed particle mass and $u\in T(\mathbb{R}^{3})$ is its velocity,
measured with respect to a suitably chosen laboratory reference frame $\ \
\mathcal{K}.$

That the Lorentz force (\ref{R1}) is not a completely satisfactory
expression was well known still for Lorentz himself, as the nonuniform
Maxwell equations describe also the electromagnetic fields, radiated by any
accelerated charged particle, easily seen from well-known expressions for
the Lienard-Wiechert electromagnetic four-potential $(\varphi
,A):M^{4}\rightarrow T^{\ast }\mathbb{(}M^{4}),$ related to the
electromagnetic fields by means of the well known \cite{LL,Jack,Bar}
relationships
\begin{equation}
E:=-\nabla \varphi -\frac{1}{c}\frac{\partial A\ }{\partial t},B:=\nabla
\times A.\   \label{R1a}
\end{equation}

This fact had inspired many physicists to "improve" the classical Lorentz
force expression (\ref{R1}) and its modification was then suggested by G.A.
Shott \cite{Sch} and later by M. Abraham and P.A.M. Dirac (see \cit
{Jack,Bar}), who found that the so called \ classical "\textit{radiation
reaction}" force, owing to the self-interaction of a charged particle with
charge $\xi \in $ $\mathbb{R},$ equal
\begin{equation}
\frac{dp}{dt}=\xi E+\xi \frac{u}{c}\times B+\frac{2\xi ^{2}}{3c^{3}}\frac
d^{2}u}{dt^{2}}.  \label{R2}
\end{equation
The additional self-reaction force expressio
\begin{equation}
F_{r}:=\frac{2\xi ^{2}}{3c^{3}}\frac{d^{2}u}{dt^{2}},  \label{R3}
\end{equation
depending on the particle acceleration had entailed right away many
questions concerning its physical meaning, since for instance, a uniformly
accelerated charged particle, owing to the expression (\ref{R2}), does feel
no radiation reaction, contradicting the fact that any accelerated charged
particle always radiates electromagnetic waves. This "paradox" was a
challenging problem during the XX century \cite{Sch,Dir,Jack,BSt,Pegg,Jack}
and still remains to be not explained completely \cite{Roh,Med,MarPin} up to
present days. As there exist different approaches to explanation this
reaction radiation phenomenon, we mention here only such most popular ones
as the Wheeler-Feynman's \cite{WF} \ "absorber radiation" theory, based on a
very sophisticated elaboration of the retarded and advanced solutions to the
nonuniform Maxwell equations, the vacuum Casimir effect approach devised in
\cite{Mor,Sidh}, and the construction of Teitelbom \cite{Tei} which
extensively exploits the intrinsic structure \ of the electromagnetic energy
tensor subject to the advanced and retarded solutions to the nonuniform
Maxwell equations.

It is also worth of mentioning here very nontrivial development of the
Teitelbom's \ theory devised recently in \cite{Kos,Sim} and applied to the
non-abelian Yang-Mills equations, naturally generalizing the classical
Maxwell equations. Nonetheless, all of these explanations do not prove to be
satisfactory from the modern physics of view. Taking this state of art into
account we will reanalyze once more the structure of the "radiative" Lorentz
type force \ (\ref{R2}) from the vacuum field theory approach of the Section
\ref{Sec_1} and obtain that this force allows some natural slight
modification.

\subsection{The radiation reaction force: the vacuum-field theory approach}

In the Section to proceed below, we will develop further our vacuum field
theory approach devised before in \cite{BP0,BPT0,BPT,BPT1} to the
electromagnetic Maxwell and Lorentz electron theories and will show that it
is in complete agreement with the classical results and even more, it allows
some nontrivial generalizations, which may have some nontrivial physical
applications. It will be also shown that the closely related electron mass
problem can be satisfactorily explained within the devised vacuum field
theory approach and the spatial electron structure assumption.

The modified Lorentz force, acting on a particle of charge $\xi \in \mathbb{
}$ and exerted by a moving with velocity $u_{f}\in T(\mathbb{R}^{3})$ \
charged particle $\xi _{f}\in \mathbb{R},$ \ was derived in Section \re
{Sec_1}\ and equal
\begin{equation}
dp/dt:=F_{s}=\xi E+\xi \frac{u}{c}\times B-\nabla <\xi A/c,u-u_{f}>,
\label{R4}
\end{equation
where $(\varphi ,A)\in T^{\ast }\mathbb{(}M^{4}\mathbb{)}$ is the external
electromagnetic potential calculated with respect to a fixed laboratory
reference frame $\mathcal{K}.$ To take into account the self-interaction of
this particle we will make use \ of a spatially distributed charge density
\rho :M^{4}\rightarrow \mathbb{R},$ satisfying the conditio
\begin{equation}
\xi =\int_{\mathbb{R}^{3}}\rho (t,r)d^{3}r  \label{R5}
\end{equation
for all $t\in \mathbb{R}$ subject to this laboratory reference frame
\mathcal{K}$ with coordinates $(t,r)\in M^{4}.$ Then, owing to \ref{R4} and
the reasonings from Section \ref{Sec_1}, the self-interacting force of this
spatially structured charge $\xi \in \mathbb{R}$ can be expressed with
respect to this laboratory reference frame $\mathcal{K}$ in the following
equivalent form
\begin{equation}
\begin{array}{c}
dp/dt=-\ \frac{1}{c}\frac{d}{dt}\ [\int_{\mathbb{E}^{3}}d^{3}r\rho
(t,r)A_{s}(t,r)]- \\
-\int_{\mathbb{E}^{3}}d^{3}r\rho (t,r)\nabla \varphi _{s}(t,r)\
(1-|u/c|^{2})\
\end{array}
\label{R6}
\end{equation
where we denoted by
\begin{equation}
\varphi _{s}(t,r)=\int_{\mathbb{E}^{3}}\frac{\rho (t^{\prime },r^{\prime
})|_{ret}d^{3}r^{\prime }}{|r-r^{\prime }|},\text{ \ }A_{s}(t,r)=\frac{1}{c
\int_{\mathbb{E}^{3}}\frac{u(t^{\prime })\rho (t^{\prime },r^{\prime
})|_{ret}d^{3}r^{\prime }}{|r-r^{\prime }|},  \label{R7}
\end{equation
the well-known retarded Lienard-Wiechert potentials, which should be
calculated at the retarded time parameter $t^{\prime }:=t-\left\vert
r-r^{\prime }\right\vert /c\in \mathbb{R}.$ Taking additionally into account
the continuity relationshi
\begin{equation}
\partial \rho /\partial t+<\nabla ,J>=0\text{\ \ \ }  \label{R8}
\end{equation
for the spatially distributed charge density $\rho :M^{4}\rightarrow \mathbb
R}$ and current $J=\rho u:M^{4}\rightarrow \mathbb{E}^{3}$ and the Taylor
expansions for retarded potentials \ (\ref{R7})
\begin{eqnarray}
\varphi _{s}(t,r) &=&\sum_{n\in \mathbb{Z}_{+}}\frac{\partial ^{n}}{\partial
t^{n}}\int_{\mathbb{E}^{3}}\frac{(-|r-r^{\prime }|)^{n}}{c^{n}n!}\frac{\rho
(t,r^{\prime })d^{3}r^{\prime }}{|r-r^{\prime }|},  \label{R8a} \\
A_{s}(t,r) &=&\sum_{n\in \mathbb{Z}_{+}}\frac{\partial ^{n}}{\partial t^{n}
\int_{\mathbb{E}^{3}}\frac{(-|r-r^{\prime }|)^{n}}{c^{n}n!}\frac
J(t,r^{\prime })d^{3}r^{\prime }}{|r-r^{\prime }|},  \notag
\end{eqnarray
from (\ref{R6}) and (\ref{R8a}), assuming for brevity the spherical charge
distribution, small enough value $|u|/c\ll 1$ and, respectively, \ slow
acceleration, followed by calculations similar to those of \cite{Jack,MarPin
, \ \ one can obtain tha
\begin{equation}
\begin{array}{c}
F_{s}=\sum_{n\in \mathbb{Z}_{+}}\frac{(-1)^{n+1}}{n!c^{n}}(1-|u/c|^{2})\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
\partial ^{n}}{\partial t^{n}}\rho (t,r^{\prime })\nabla |r-r^{\prime
}|^{n-1}+ \\
\\
+\frac{d}{dt}[\sum_{n\in \mathbb{Z}_{+}}\frac{(-1)^{n+1}}{n!c^{n}}\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\frac{|r-r^{\prime }|^{n-1}}{c^{2}}\frac
\partial ^{n}}{\partial t^{n}}J(t,r^{\prime }]= \\
\\
=\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{(m+2)!c^{m+2}
(1-|u/c|^{2})\int_{\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E
^{3}}d^{3}r^{\prime }\frac{\partial ^{m+2}}{\partial t^{m+2}}\rho
(t,r^{\prime })\nabla |r-r^{\prime }|^{m+1}+ \\
\\
+\frac{d}{dt}[\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{m!c^{m}}\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
\partial ^{m}}{\partial t^{m}}\left( \frac{|r-r^{\prime }|^{m+1}}{c^{2}
J(t,r^{\prime })\right) ]= \\
\end{array}
\label{R9}
\end{equation
\begin{equation*}
\begin{array}{c}
=\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{m!c^{m+2}}\int_{\mathbb{E
^{3}}d^{3}r\rho (t,r)(1-|u/c|^{2})\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
\partial ^{m+1}}{\partial t^{m+1}}<J(t,r^{\prime }),\nabla >|r-r^{\prime
}|^{m+1}= \\
\\
+\frac{d}{dt}[\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{m!c^{m}}\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
|r-r^{\prime }|^{m+1}}{c^{2}}\frac{\partial ^{m}}{\partial t^{m}}\left(
J(t,r^{\prime })\right) ]= \\
\\
=\sum_{n\in \mathbb{Z}_{+}}\frac{(-1)^{n+1}}{n!c^{n+2}}(1-|u|^{2})\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\ \int_{\mathbb{E}^{3}}d^{3}r^{\prime
}|r-r^{\prime }|^{n-1}\frac{\partial ^{n+1}}{\partial t^{n+1}}[\frac
\partial \rho (t,r^{\prime })}{\partial t}\frac{\nabla |r-r^{\prime }|^{n+1
}{(n+2)(n+1)|r-r^{\prime }|^{n-1}}+J(t,r^{\prime })]+ \\
\\
+\frac{d}{dt}[\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{m!c^{m}}\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
|r-r^{\prime }|^{m+1}}{c^{2}}\frac{\partial ^{m}}{\partial t^{m}
J(t,r^{\prime })]
\end{array
\end{equation*
\begin{equation*}
\begin{array}{c}
\\
\\
=\sum_{n\in \mathbb{Z}_{+}}\frac{(-1)^{n+1}}{n!c^{n+2}}(1-|u|^{2})\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)(1-|u|^{2})\int_{\mathbb{E
^{3}}d^{3}r^{\prime }\frac{\partial ^{n+1}}{\partial t^{n+1}}[-<\nabla
,J(t,r^{\prime })>\frac{|r-r^{\prime }|^{n-1}(r-r^{\prime })}{(n+2)}]+ \\
\\
+\frac{d}{dt}[\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{m!c^{m}}\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
|r-r^{\prime }|^{m+1}}{c^{2}}\frac{\partial ^{m}}{\partial t^{m}
J(t,r^{\prime })]= \\
\\
=\sum_{n\in \mathbb{Z}_{+}}\frac{(-1)^{n\ }}{n!c^{n+2}}(1-|u/c|^{2})\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime
}|r-r^{\prime }|^{n-1}\frac{\partial ^{n+1}}{\partial t^{n+1}}\left( \frac{\
J(t,r^{\prime })}{n+2}+\frac{n-1}{n+2}\frac{<r-r^{\prime },J(t,r^{\prime })>
}{|r-r^{\prime }|^{2}}]\right) + \\
+\frac{d}{dt}[\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{m!c^{m}}\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
|r-r^{\prime }|^{m+1}}{c^{2}}\frac{\partial ^{m}}{\partial t^{m}
J(t,r^{\prime })]. \\
\end{array
\end{equation*
The relationship above can be rewritten, owing to the charge continuity
equation \ (\ref{R8}), gives rise to the radiation force expression
\begin{equation}
\begin{array}{c}
\\
F_{s}=\sum_{n\in \mathbb{Z}_{+}}\frac{(-1)^{n\ }}{n!c^{n+2}
(1-|u/c|^{2})\int_{\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E
^{3}}d^{3}r^{\prime }|r-r^{\prime }|^{n-1}\frac{\partial ^{n+1}}{\partial
t^{n+1}}\left( \frac{\ J(t,r^{\prime })}{n+2}+\frac{n-1}{n+2}\frac
<r-r^{\prime },J(t,r^{\prime })>(r-r^{\prime })}{|r-r^{\prime }|^{2}}\right)
+ \\
\\
+\frac{d}{dt}[\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{m!c^{m}}\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
|r-r^{\prime }|^{m+1}}{c^{2}}\frac{\partial ^{m}}{\partial t^{m}
J(t,r^{\prime })]= \\
\\
=\sum_{n\in \mathbb{Z}_{+}}\frac{(-1)^{n+1}}{n!c^{n+2}}(1-|u/c|^{2})\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime
}|r-r^{\prime }|^{n-1}\frac{\partial ^{n+1}}{\partial t^{n+1}}\left( \frac{\
J(t,r^{\prime })}{n+2}+\frac{n-1}{n+2}\frac{|r-r^{\prime
},u|^{2}J(t,r^{\prime })}{|r-r^{\prime }|^{2}|u|^{2}}\right) + \\
\\
+\frac{d}{dt}[\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{m!c^{m}}\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
|r-r^{\prime }|^{m+1}}{c^{2}}\frac{\partial ^{m}}{\partial t^{m}
J(t,r^{\prime })]. \\
\end{array}
\label{R9a}
\end{equation
Now, having applied to \ (\ref{R9a}) the rotational symmetry property for
calculation of the internal integral, one easily obtains that
\begin{equation*}
\begin{array}{c}
F_{s}=\sum_{n\in \mathbb{Z}_{+}}\frac{(-1)^{n}}{n!c^{n+2}}(1-|u/c|^{2})\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime
}|r-r^{\prime }|^{n-1}\frac{\partial ^{n+1}}{\partial t^{n+1}}\left( \frac{\
J(t,r^{\prime })}{n+2}+\frac{(n-1)J(t,r^{\prime })}{3(n+2)}\ \right) + \\
\\
+\frac{d}{dt}[\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{m!c^{m}}\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
|r-r^{\prime }|^{m+1}}{c^{2}}\frac{\partial ^{m}}{\partial t^{m}
J(t,r^{\prime })]= \\
\\
\end{array
\end{equation*
\begin{equation}
\begin{array}{c}
=\sum_{n\in \mathbb{Z}_{+}}\frac{(-1)^{n}}{3n!c^{n+2}}(1-|u/c|^{2})\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime
}|r-r^{\prime }|^{n-1}\frac{\partial ^{n+1}}{\partial t^{n+1}}J(t,r^{\prime
})+ \\
\\
+\frac{d}{dt}[\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{m!c^{m}}\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
|r-r^{\prime }|^{m+1}}{c^{2}}\frac{\partial ^{m}}{\partial t^{m}
J(t,r^{\prime })]= \\
\\
=\frac{d}{dt}[\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{m!c^{m}}\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
|r-r^{\prime }|^{m+1}}{c^{2}}\frac{\partial ^{m}}{\partial t^{m}
[J(t,r^{\prime })-\frac{1}{3}J(t,r^{\prime })]- \\
\\
-\sum_{n\in \mathbb{Z}_{+}}\frac{(-1)^{m\ }}{3m!c^{m+2}}(1-|u/c|^{2})\int_
\mathbb{E}^{3}}d^{3}r\frac{\partial \rho (t,r)}{\partial t}\int_{\mathbb{E
^{3}}d^{3}r^{\prime }|r-r^{\prime }|^{m-1}\frac{\partial ^{m+1}}{\partial
t^{m+1}}J(t,r^{\prime })
\end{array}
\label{R9b}
\end{equation
\begin{equation*}
\begin{array}{c}
=\frac{d}{dt}[\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{m!c^{m}}\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
|r-r^{\prime }|^{m+1}}{c^{2}}\frac{\partial ^{m}}{\partial t^{m}
[J(t,r^{\prime })-\frac{1}{3}J(t,r^{\prime })]- \\
\\
-\sum_{n\in \mathbb{Z}_{+}}\frac{(-1)^{m\ }}{3m!c^{m+2}}(1-|u/c|^{2})\int_
\mathbb{E}^{3}}d^{3}r\frac{\partial \rho (t,r)}{\partial t}\int_{\mathbb{E
^{3}}d^{3}r^{\prime }|r-r^{\prime }|^{m-1}\frac{\partial ^{m+1}}{\partial
t^{m+1}}J(t,r^{\prime })= \\
\\
=\frac{d}{dt}[\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{m!c^{m}}\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
|r-r^{\prime }|^{m+1}}{c^{2}}\frac{\partial ^{m}}{\partial t^{m}
[J(t,r^{\prime })-\frac{1}{3}J(t,r^{\prime })]+ \\
\\
+\sum_{n\in \mathbb{Z}_{+}}\frac{(-1)^{m\ }}{3m!c^{m+2}}(1-|u/c|^{2})\int_
\mathbb{E}^{3}}d^{3}r<\nabla ,J(t,r^{\prime })>\int_{\mathbb{E
^{3}}d^{3}r^{\prime }|r-r^{\prime }|^{m-1}\frac{\partial ^{m+1}}{\partial
t^{m+1}}J(t,r^{\prime })= \\
\\
=\frac{d}{dt}[\sum_{m\in \mathbb{Z}_{+}}\frac{(-1)^{m+1}}{m!c^{m}}\int_
\mathbb{E}^{3}}d^{3}r\rho (t,r)\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\frac
|r-r^{\prime }|^{m+1}}{c^{2}}\frac{\partial ^{m}}{\partial t^{m}
[J(t,r^{\prime })-\frac{1}{3}J(t,r^{\prime })]
\end{array
\end{equation*
where we took into account \cite{Jack} that in case of the spherical charge
distribution the following equalities \
\begin{equation}
\begin{array}{c}
\int_{\mathbb{E}^{3}}d^{3}r\int_{\mathbb{E}^{3}}d^{3}r^{\prime }\rho
(t,r)\rho (t,r^{\prime })\frac{|<r-r^{\prime },u(t)>|^{2}}{|r-r^{\prime
}|^{2}|u(t)|^{2}}=\frac{1}{3}e^{2}, \\
\\
\int_{\mathbb{E}^{3}}d^{3}r<\nabla ,J(t,r^{\prime })>\int_{\mathbb{E
^{3}}d^{3}r^{\prime }|r-r^{\prime }|^{m-1}\frac{\partial ^{m+1}}{\partial
t^{m+1}}J(t,r^{\prime })=0, \\
\\
\int_{\mathbb{E}^{3}}d^{3}r\int_{\mathbb{E}^{3}}d^{3}r\rho (t,r)\rho
(t,r^{\prime })\frac{(r-r^{\prime })}{|r-r^{\prime }|^{3}}=
\end{array}
\label{R10}
\end{equation
hold. Thus, from \ (\ref{R9b}) one easily finds up to the $O(1/c^{4})$
accuracy the following radiation reaction force expression:
\begin{eqnarray}
dp/dt &=&F_{s}=-\frac{d}{dt}\left( \frac{4\mathcal{E}_{es}}{3c^{2}
u(t)\right) -\frac{d}{dt}\left( \frac{2\mathcal{E}_{es}}{3c^{2}}\ |u/c|^{2}\
u(t)\right) +\frac{2\xi ^{2}}{3c^{3}}\frac{d^{2}u}{dt^{2}}+O(1/c^{4})=
\label{R11} \\
&&  \notag \\
&=&-\frac{d}{dt}\left( \frac{4}{3}m_{0,es}(1+\frac{|u/c|^{2}}{2})u(t)\right)
+\frac{2\xi ^{2}}{3c^{3}}\frac{d^{2}u}{dt^{2}}+O(1/c^{4})=  \notag \\
&&  \notag \\
&=&-\frac{d}{dt}\left( \frac{4}{3}\frac{m_{0,es}u(t)}{(1-|u/c|^{2})^{1/2}
\right) \ \ +\frac{2\xi ^{2}}{3c^{3}}\frac{d^{2}u}{dt^{2}}+O(1/c^{4})=
\notag \\
&&  \notag \\
&=&-\frac{d}{dt}\left( \frac{4}{3}m_{es}u(t)\right) +\frac{2\xi ^{2}}{3c^{3}
\frac{d^{2}u}{dt^{2}}+O(1/c^{4}),  \notag
\end{eqnarray
where we defined, respectively, the electrostatic self-interaction repulsive
energy a
\begin{equation}
\mathcal{E}_{es}:=\frac{1}{2}\int_{\mathbb{R}^{3}}d^{3}r\int_{\mathbb{R
^{3}}d^{3}r^{\prime }\frac{\rho (t,r)\rho (t,r^{\prime })}{\left\vert
r-r^{\prime }\right\vert },  \label{R13}
\end{equation
the electromagnetic charged particle rest and inertial masses as
\begin{equation}
m_{0,es}:=\frac{\mathcal{E}_{es}}{c^{2}},\text{ \ \ \ }m_{es}:=\frac{m_{0,es
}{(1-|u/c|^{2})^{1/2}}.  \label{R13a}
\end{equation
Now from (\ref{R4}) one obtains that
\begin{equation}
\frac{d}{dt}\left[ (m_{g}+\frac{4}{3}m_{es})u\right] =\frac{2\xi ^{2}}{3c^{3
}\frac{d^{2}u}{dt^{2}}+O(1/c^{4}),  \label{R14}
\end{equation
where we made use of the inertial mass definition
\begin{equation}
m_{g}:=-\bar{W}_{g}/c^{2},\text{ \ \ }\nabla \bar{W}_{g}\simeq 0,\text{\ }
\label{R15}
\end{equation
following from the vacuum field theory approach, where the $m_{g}\in \mathbb
R}$ is the corresponding gravitational mass of the charged particle $\xi ,$
generated by the vacuum field potential $\bar{W}_{g}.$ The corresponding
radiation force
\begin{equation}
F_{r}=\frac{2\xi ^{2}}{3c^{3}}\frac{d^{2}u}{dt^{2}}+O(1/c^{4}),  \label{R16}
\end{equation
coinciding exactly with the classical Abraham-Lorentz-Dirac results. From
\ref{R14}) one follows that the observable physical charged particle mass
m_{ph}\simeq m_{g}+\frac{4}{3}m_{es}$ consists of two impacts: the
electromagnetic and gravitational components, giving rise to the final force
expression
\begin{equation}
\frac{d}{dt}(m_{ph}u)=\frac{2\xi ^{2}}{3c^{3}}\frac{d^{2}u}{dt^{2}
+O(1/c^{4}).  \label{R17}
\end{equation
It means, in particular, that the real physically observed "inertial" mass
m_{ph}$\ of a real electron strongly depends on the external \ physical
interaction with the ambient vacuum medium, as it was recently demonstrated
within completely different approaches in \cite{Sidh,Mor}, based on the
vacuum Casimir effect considerations. Moreover, the assumed above
boundedness of the electrostatic self-energy $\mathcal{E}_{es}$ appears to
be completely equivalent to the existence of so-called intrinsic Poincar\'{e}
type "\textit{tensions}", analyzed in \cite{BSt,Mor,Gal}, and to the
existence of a special compensating Coulomb "\textit{pressure}", suggested
in \cite{Sidh}, guaranteeing the observable electron stability.

\subsection{Comments}

The charged particle radiation problem, revisited in this Section, allows to
conceive the following explanation of the charged particle mass as that of a
compact and stable object which should be exerted by a vacuum field
interaction energy potential $\bar{W}:\ M^{4}\rightarrow \mathbb{R}\ $ of
negative sign as follows from (\ref{R15}). The latter can be satisfied iff
the expression (\ref{R14}) holds, thereby imposing on the intrinsic charged
particle structure \cite{Med} some nontrivial geometrical constraints.
Moreover, as follows from the physically observed particle mass expressions
\ref{R15}) the electrostatic potential energy, being of the \ \ \ repulsive
force origin, does contribute into the full mass as its main energy
component.

There exist different relativistic generalizations of the force expression \
(\ref{R14}), which suffer the same common physical inconsistency related
with the no radiation effect \ of \ a charged particle at uniform motion.

Another deeply related \ problem to the radiation reaction force analyzed
above is the search for explanation to the Wheeler and Feynman reaction
radiation mechanism, called the absorption radiation theory, strongly based
on the Mach type interaction of a charged particle with the ambient vacuum
electromagnetic medium. Concerning this problem, one can also observe some
its relationships with the one devised here within the vacuum field theory
approach, but this question needs a more detailed and extended analysis.

\section{A charged point particle dynamics and a hadronic string model
analysis}

\subsection{The classical least action principle in electrodynamics\labe
{Sec_1}}

The classical least action principle is considered in modern physics as a
fundamental tool for deriving true and physically sound equations governing
the dynamics of the corresponding physical objects. By means of this
principle there has been described many physical models \cite{LL,Fe,Fe1,Jack}
including those of classical mechanics, electrodynamics and Einsteinian
gravity theory. As it was mentioned in many classical manuals \cit
{Jack,LL,Bar,BS,BS1}, a suitable and physically motivated method of choosing
the corresponding Lagrangian functions appears nowadays to be open for
studying. \ Application of the least action principle is strongly
complicated by inconsistencies often accompanying the derived physical
statements which are well understood and checked by means of other physical
theories. In particular, \ in modern electrodynamics of a charged point
particle moving under influence of an external electromagnetic field, there
is well known misreading \cite{LL,Fe1,Jack} related with the charged point
particle energy expression. Namely, the latter being obtained by means of
the classical least action principle, gives rise to the charged particle
\textquotedblright \textit{dynamical}" mass expression not depending on the
external potential energy. This fact was also discussed in the literature,
for instance in \cite{Br}, where there are also described other physically
reasonable examples. Taking this into account\ and being motivated by R.
Feynman's considerations of the problem in \cite{Fe1,Dy1,Dy2,Lee} and a
recently devised vacuum field theory approach \cite{BP,BPT1,BPS} to
physically reasonable formulating \ the corresponding least action principle
for describing the charged point particle electrodynamics, we have revisited
in \cite{BP-Feynm} the Feynman's approach to describing the electrodynamics
of a charged point particle and stated its complete legacy. Based on this
experience we apply, within the present work, a slightly corrected vacuum
field theory approach to describing the electrodynamics both of a charged
point particle and a charged one-dimensional string under influence of an
external electromagnetic filed, generated by a moving charged point particle
with respect to some laboratory reference frame.

It is well known \cite{BN,Fe1,LL,Pa} that the relativistic least action
principle for a charged point particle $\xi $ in the Minkowski space-time
M^{4}\simeq \mathbb{R}\times \mathbb{E}^{3}$ can be formulated on a time
interval $[t_{1},t_{2}]\subset \mathbb{R}$ (in the light speed units) as
\begin{eqnarray}
\delta S^{(t)} &=&0,\text{ \ \ }S^{(t)}:=\int_{\tau (t_{1})}^{\tau
(t_{2})}(-m_{0}d\tau -<\xi \mathcal{A},dx>_{M^{4}})=  \notag \\
&=&\int_{s(t_{1})}^{s(t_{2})}(-m_{0}<\dot{x},\dot{x}>_{M^{4}}^{1/2}-<\xi
\mathcal{A},\dot{x}>_{M^{4}})ds.  \label{A1.1}
\end{eqnarray
Here $\delta x(s(t_{1}))=0=\delta x(s(t_{2}))$ are the boundary constraints,
$m_{0}\in \mathbb{R}_{+}$ is the so-called particle rest mass, the 4-vector
x:=(t,r)\in M^{4}$ is the particle location in $M^{4},$ $\dot{x}:=dx/ds\in
T(M^{4})$ is the particle 4-vector velocity with respect to a laboratory
reference frame $\mathcal{K},$ parameterized by means of the Minkowski
space-time parameters $(s(t),r)\in M^{4}$ and related to each other by means
of the infinitesimal Lorentz interval relationship
\begin{equation}
d\tau :=<dx,dx>_{M^{4}}^{1/2}:=ds<\dot{x},\dot{x}>_{M^{4}}^{1/2},
\label{A1.1a}
\end{equation
$\mathcal{A}\in $ $T^{\ast }(M^{4})$ is an external electromagnetic 4-vector
potential, satisfying the classical Maxwell equations \cite{Pa,LL,Fe1}, the
sign $<\cdot ,\cdot >_{\mathcal{H}}$ \ means, in general, the corresponding
scalar product in a finite-dimensional vector space $\mathcal{H}$ and
T(M^{4}),T^{\ast }(M^{4})$ \ are, respectively, the tangent and cotangent
spaces \cite{AM,Ar,Th,BPS} to the Minkowski space $M^{4}.$ In particular,
<x,x>_{M^{4}}:=t^{2}-<r,r>_{\mathbb{E}^{3}}$ for any $x:=(t,r)\in M^{4}.$

The subintegral expression in (\ref{A1.1})
\begin{equation}
\mathcal{L}^{(t)}:=-m_{0}<\dot{x},\dot{x}>_{M^{4}}^{1/2}-\ <\xi \mathcal{A}
\dot{x}>_{M^{4}}  \label{A1.2}
\end{equation
is the related Lagrangian function, whose first part is proportional to the
particle world line length with respect to the proper rest reference frame
\mathcal{K}_{r}$ and the second part is proportional to the pure
electromagnetic particle-field interaction with respect to the Minkowski
laboratory reference frame $\mathcal{K}.$ Moreover, the positive rest mass
parameter \ $m_{0}\in \mathbb{R}_{+}$ is introduced into (\ref{A1.2}) as an
external physical ingredient, also describing the point particle\ \ with
respect to the proper rest reference frame $\mathcal{K}_{r}.$ \ The
electromagnetic 4-vector potential $\mathcal{A}\in T^{\ast }(M^{4})$ $\ $is
at the same time expressed as a 4-vector, constructed and measured with
respect to the Minkowski laboratory reference frame $\mathcal{K}$ that
looks, from physical point of view, enough controversial, since the action
functional (\ref{A1.1}) is forced to be extremal \cit
{Br,Gr,t-Ho,Ne,Me,Me1,Tr,We} with respect to the laboratory reference frame
\mathcal{K}.$ This, in particular, means that the real physical motion of
our charged point particle, being realized with respect to the proper rest
reference frame $\mathcal{K}_{r},$ somehow depends on an external
observation data with respect to the occasionally chosen laboratory
reference frame $\mathcal{K}.$ \ This aspect was never discussed in the
physical literature except for very interesting reasonings by R. Feynman in
\cite{Fe1}, who argued that the relativistic expression for the classical
Lorentz force has a physical sense only with respect to \ the Euclidean rest
\ reference frame $\mathcal{K}_{r}$ variables $(\tau ,r)\in \mathbb{E}^{4}$
related to the Minkowski laboratory reference frame $\mathcal{K}$ parameters
$(r,t)\in M^{4}$ by means of the infinitesimal relationship
\begin{equation}
d\tau :=<dx,dx>_{M^{4}}^{1/2}=dt(1-|u|^{2})^{1/2},  \label{A1.2a}
\end{equation
where $u:=dr/dt\in T(\mathbb{R}^{3})$ is the point particle velocity with
respect to the reference frame $\mathcal{K}.$

It is worth to point out here that to be correct, it would be necessary to
include into the action functional the additional part describing the
electromagnetic field itself. But this part is not taken into account, since
there is generally assumed \cite{Kl,Kle,LL,Fe1,B-W,Wi,B-B} that the charged
particle influence on the electromagnetic field is negligible. This is true,
if the particle charge value $\ \xi $ is very small but the support $sup
\mathcal{A\subset }$ $M^{4}$ of the electromagnetic 4-vector potential is
compact. It is clear that in the case of two interacting with each other
charged particles the above assumption cannot be applied, as it is necessary
to take into account the relative motion of two particles and the
respectively changing interaction energy. This aspect of \ the action
functional choice problem appears to be very important when one tries to
analyze the related Lorentz type \ forces exerted by charged particles on
themselves. We will return to this problem in a separate section below.

Having calculated the least action condition (\ref{A1.1}), we easily obtain
from (\ref{A1.2}) the classical relativistic dynamical equations
\begin{eqnarray}
dP/ds &:&=-\partial \mathcal{L}^{(t)}/\partial x=-\ \nabla _{x}<\xi \mathcal
A},\dot{x}>_{M^{4}},  \label{A1.3} \\
P &:&=-\partial \mathcal{L}^{(t)}/\partial \dot{x}=m_{0}\dot{x}<\dot{x},\dot
x}>_{M^{4}}^{-1/2}+\xi \mathcal{A},  \notag
\end{eqnarray
where we denoted by $\nabla _{x}$ the gradient operator with respect to the
variable $x\in M^{4}$ and by $P\in T^{\ast }(M^{4})$ the common
particle-field momentum of the interacting system.

Now at $s=t\in \mathbb{R}$ by means of the standard infinitesimal change of
variables (\ref{A1.2a}), \ we can easily obtain from (\ref{A1.3}) the
classical Lorentz force expressio
\begin{equation}
dp/dt=\xi E+\xi u\times B  \label{A1.6}
\end{equation
with the relativistic particle momentum and mass
\begin{equation}
p:=mu,\text{ \ }\ m:=m_{0}(1-|u|^{2})^{-1/2},\text{ \ }|u|^{2}:=<u,u>_
\mathbb{E}^{3}},  \label{A1.7}
\end{equation
respectively, the electric field
\begin{equation}
E:=-\partial A/\partial t-\nabla \varphi  \label{A1.8}
\end{equation
and the magnetic field
\begin{equation}
B:=\nabla \times A,  \label{A1.9}
\end{equation
where we have expressed the electromagnetic 4-vector potential as $\mathcal{
}=(\varphi ,A)\in T^{\ast }(M^{4}).$

The Lorentz force (\ref{A1.6}), owing to our preceding assumption, means the
force exerted by the external electromagnetic field on our charged point
particle, whose charge $\xi $ is so negligible that it does not exert the
influence on the field. This fact becomes very important if we try to make
use of the Lorentz force expression (\ref{A1.6}) for the case of two charged
interacting with each other particles, since then one cannot assume that our
charge $\xi $ exerts negligible influence on other charged particle. Thus,
the corresponding Lorentz force between two charged particles should be
suitably altered. Nonetheless, the modern physics \cit
{BS,BS1,Di,LL,BD,DJ,JP,BPZZ,Sl,Ba} did not make this naturally needed \
Lorentz force modification and there is used the classical expression (\re
{A1.6}). This situation was observed and analyzed concerning the related
physical aspects in \cite{Rep}, and there was shown that the electromagnetic
Lorentz force between two moving charged particles can be modified in such a
way that it ceases to be dependent on their relative motion contrary to the
classical relativistic case.

To our regret, the least action principle approach to analyze the Lorentz
force structure was in \cite{Rep} completely ignored and that gave rise to
some incorrect and physically not motivated statements concerning
mathematical physics backgrounds of the modern electrodynamics. To make the
problem more transparent we will analyze it in the section below from the
vacuum field theory approach recently devised in \cite{BP0,BP,BPT,BPT1}.

\subsection{The classical least action principle revisited}

Consider the least action principle (\ref{A1.1}) and observe that the
extremality condition
\begin{equation}
\delta S^{(t)}=0,\text{ \ \ \ \ }\delta x(s(t_{1}))=0=\delta x(s(t_{2})),
\label{A2.1}
\end{equation
is calculated with respect to the laboratory reference frame $\mathcal{K},$
whose point particle coordinates $(t,r)\in M^{4}$ are parameterized by means
\ of an arbitrary parameter $s\in \mathbb{R}$ owing to expression (\re
{A1.1a}). Recalling now the definition of the invariant proper rest
reference frame $\mathcal{K}_{r}$ time parameter (\ref{A1.2a}), we obtain
that at the critical parameter value $s=\tau \in \mathbb{R}$ the action
functional (\ref{A1.1}) on the fixed interval $[\tau _{1},\tau _{2}]\subset
\mathbb{R}$ turns into
\begin{equation}
S^{(t)}=\int\limits_{\tau _{1}}^{\tau _{2}}(-m_{0}-\ <\xi \mathcal{A},\dot{x
>_{M^{4}})d\tau  \label{A2.2}
\end{equation
under the \textit{additional constraint}
\begin{equation}
<\dot{x},\dot{x}>_{M^{4}}^{1/2}=1,  \label{A2.2a}
\end{equation
where, by definition, $\ \dot{x}:=dx/d\tau ,$ $\tau \in \mathbb{R}.$

The expressions (\ref{A2.2}) and (\ref{A2.2a}) need some comments since the
corresponding (\ref{A2.2}) Lagrangian function
\begin{equation}
\mathcal{L}^{(t)}:=-m_{0}-\ <\xi \mathcal{A},\dot{x}>_{M^{4}}  \label{A2.3}
\end{equation
depends virtually only on the unobservable rest mass parameter $m_{0}\in
\mathbb{R}_{+}$ and, evidently, it has no \ direct impact on the resulting
particle dynamical equations following from the condition $\delta S^{(t)}=0.$
\ Nonetheless, the rest mass springs up as a suitable Lagrangian multiplier
owing to the imposed constraint (\ref{A2.2a}). To demonstrate this, consider
the extended Lagrangian function (\ref{A2.3}) in the form
\begin{equation}
\mathcal{L}_{\lambda }^{(t)}:=-m_{0}-\ <\xi \mathcal{A},\dot{x
>_{M^{4}}-\lambda (<\dot{x},\dot{x}>_{M^{4}}^{1/2}-1),  \label{A2.3.1}
\end{equation
where $\lambda \in \mathbb{R}$ is a suitable Lagrangian multiplier. The
resulting Euler equations look as follow
\begin{eqnarray}
P_{r} &:&=\partial \mathcal{L}_{\lambda }^{(t)}/\partial \dot{r}=\xi
A+\lambda \dot{r},\text{ \ }P_{t}:=\partial \mathcal{L}_{\lambda
}^{(t)}/\partial \dot{t}=-\xi \varphi -\lambda \dot{t},\text{ }  \notag \\
\text{\ }\partial \mathcal{L}_{\lambda }^{(t)}/\partial \lambda &=&<\dot{x}
\dot{x}>_{M^{4}}^{1/2}-1=0,\text{ \ }dP_{r}/d\tau =\xi \ \nabla <A,\dot{r}>_
\mathbb{E}^{3}}-\xi \dot{t}\ \nabla \varphi ,  \notag \\
\text{ \ }dP_{t}/d\tau &=&\xi <\partial A/\partial t,\dot{r}>_{\mathbb{E
^{3}}-\xi \dot{t}\partial \varphi /\partial t,  \label{A2.3.2}
\end{eqnarray
giving rise, owing to relationship (\ref{A1.2a}), to the following dynamical
equations:

\begin{equation}
\frac{d}{dt}(\lambda u\dot{t})=\xi E+\xi u\times B,\text{ \ }\frac{d}{dt
(\lambda \dot{t})=\xi <E,u>_{\mathbb{E}^{3}},  \label{A2.3.3}
\end{equation
where we denoted by
\begin{equation}
E:=-\partial A/\partial t-\nabla \varphi ,\text{ \ }B=\nabla \times A
\label{A2.3.4}
\end{equation
the corresponding electric and magnetic fields. As a simple consequence of
\ref{A2.3.3}) one obtains
\begin{equation}
\frac{d}{dt}\ln (\lambda \dot{t})+\frac{d}{dt}\ln (1-|u|^{2})^{1/2}=0,
\label{A2.3.5}
\end{equation
being equivalent for all $t\in \mathbb{R},$ owing to relationship (\re
{A1.2a}), \bigskip to the relationship
\begin{equation}
\lambda \dot{t}(1-|u|^{2})^{1/2}=\lambda :=m_{0,}  \label{A2.3.6}
\end{equation
where $m_{0}\in \mathbb{R}_{+}$ is a constant, which could be interpreted as
the rest mass of our charged point particle $\xi .$ Really, the first
equation of (\ref{A2.3.3}) can be rewritten as
\begin{equation}
dp/dt=\xi E+\xi u\times B,  \label{A2.3.7}
\end{equation
where we denoted
\begin{equation}
p:=mu,\text{ }m:=\lambda \dot{t}=m_{0}(1-|u|^{2})^{-1/2},  \label{A2.3.8}
\end{equation
coinciding exactly with that of (\ref{A1.2a}).

Thereby, we retrieved here all the results obtained in the section above,
making use of the action functional (\ref{A2.2}), expressed with respect to
the rest reference frame $\mathcal{K}_{r}$ under constraint \ (\ref{A2.2a}).
\ During these derivations, we were faced with a very delicate inconsistency
property of definition \ of the action functional $S^{(t)},$ defined with
respect to the rest reference frame $\mathcal{K}_{r},$ but depending on the
external electromagnetic potential function $\mathcal{A}:M^{4}\rightarrow
T^{\ast }(M^{4}),$ constructed exceptionally with respect to the laboratory
reference frame $\mathcal{K}.$ Namely, this potential function, as a
physical observable quantity, is defined and, \ \ respectively, measurable
only with respect to the fixed laboratory reference frame $\mathcal{K}.$
This, in particular, means that a physically reasonable action functional
should be constructed by means of an expression strongly calculated within
the laboratory reference frame $\mathcal{K}$ by means of coordinates
(t,r)\in M^{4}$ and later suitably transformed subject to the rest reference
frame $\mathcal{K}_{r}$ coordinates $(\tau ,r)\in \mathbb{E}^{4},$
respective for the real charged point particle $\xi $ motion. Thus, the
corresponding dual action functional, in reality, should be from the very
beginning written as
\begin{equation}
S^{(\tau )}=\int\limits_{t(\tau _{1})}^{t(\tau _{2})}(-<\xi \mathcal{A},\dot
x}>_{M^{4}})dt,  \label{A2.4}
\end{equation
where $\dot{x}:=dx/dt,$ $t\in \mathbb{R},$ being calculated on some time
interval $[t(\tau _{1}),t(\tau _{2})]\subset \mathbb{R},$ suitably related
with the proper motion of the charged point particle $\xi $ on the true time
interval $[\tau _{1},\tau _{2}]\subset \mathbb{R}$ with respect to the rest
reference frame $\mathcal{K}_{r}$ and whose charge value is assumed so
negligible that it exerts no influence on the external electromagnetic
field. Now the problem arises: how to compute correctly the variation
\delta S^{(\tau )}=0$ of the action functional (\ref{A2.4})?

To reply to this question we will turn to the R. Feynman reasonings from
\cite{Fe1}, where he argued, when deriving the relativistic Lorentz force
expression, that the real charged particle dynamics can be determined
physically not ambiguously only with respect to the rest reference frame
time parameter. Namely, \ R. Feynman wrote: "...\textit{we calculate a
growth }$\Delta x$\textit{\ for a small time interval }$\Delta t.$\textit{\
But in the other reference frame the interval }$\Delta t$\textit{\ may
correspond to changing both }$t^{\prime }$\textit{\ and }$x^{\prime },
\textit{\ thereby at the change of the only }$t^{\prime }$\textit{\ the
suitable change of }$x$\textit{\ will be\ other... Making use of the
quantity }$d\tau $\textit{\ one can determine a good differential operator \
}$d/d\tau ,$\textit{\ as it is invariant with respect to the Lorentz
reference frames of transformations"}. This means that if our charged
particle $\xi $ moves in the Minkowski space $M^{4}$ during the time
interval $[t_{1},t_{2}]\subset \mathbb{R}$ with respect to the laboratory
reference frame $\mathcal{K},$ its proper real and invariant time of motion
with respect to the rest reference frame $\mathcal{K}_{r}$ will be
respectively $[\tau _{1},\tau _{2}]\subset \mathbb{R}.$

As a corollary of the R. Feynman reasonings, we arrive at the necessity to
rewrite the action functional (\ref{A2.4}) a
\begin{equation}
S^{(\tau )}=\int\limits_{\tau _{1}}^{\tau _{2}}(-<\xi \mathcal{A},\dot{x
>_{M^{4}})d\tau ,\text{ \ }\delta x(\tau _{1})=0=\delta x(\tau _{2}),
\label{A2.5}
\end{equation
where $\dot{x}:=dx/d\tau ,$ $\tau \in \mathbb{R},$ under the \textit
additional constraint}
\begin{equation}
<\dot{x},\dot{x}>_{M^{4}}^{1/2}=1,  \label{A2.5a}
\end{equation
being equivalent to the infinitesimal transformation (\ref{A1.2a}). \ Since
the functional \ (\ref{A2.5}) is still computed in the laboratory reference
frame $\mathcal{K}$ variables $x\in M^{4},$ it is necessary to represent it
with respect to the rest reference frame $\mathcal{K}_{r}.$ Simultaneously
the proper time interval $[\tau _{1},\tau _{2}]\subset \mathbb{R}$ \ is
mapped on the time interval $[t_{1},t_{2}]\subset \mathbb{R}$ by means of
the infinitesimal transformation
\begin{equation}
dt=d\tau (1+|\dot{r}|^{2})^{1/2},  \label{A2.6}
\end{equation
where $\dot{r}:=dr/d\tau ,$ $\tau \in $\ $\mathbb{R}.$ Thus, we can now pose
the \textit{true least action problem} equivalent to (\ref{A2.5}) and (\re
{A2.5a}) as
\begin{equation}
\delta S^{(\tau )}=0,\text{ \ \ \ \ \ }\delta r(\tau _{1})=0=\delta r(\tau
_{2}),  \label{A2.6a}
\end{equation
where the functional
\begin{equation}
S^{(\tau )}=\int\limits_{\tau _{1}}^{\tau _{2}}[-\bar{W}(1+|\dot{r
|^{2})^{1/2}+<\xi A,\dot{r}>_{\mathbb{E}^{3}}]d\tau  \label{A2.7}
\end{equation
is characterized by the Lagrangian function
\begin{equation}
\mathcal{L}^{(\tau )}:=-\bar{W}(1+|\dot{r}|^{2})^{1/2}+<\xi A,\dot{r}>_
\mathbb{E}^{3}}.  \label{A2.8}
\end{equation
Here we denoted, for further convenience, $\bar{W}:=\xi \varphi ,$ \ being
the suitable vacuum field \cite{BPT,BPT1,BP,Rep} potential function. The
resulting Euler equation gives rise to the following relationship
\begin{eqnarray}
P &:&=\partial \mathcal{L}^{(\tau )}/\partial \dot{r}=-\bar{W}\dot{r}(1+
\dot{r}|^{2})^{-1/2}+\xi A,  \label{A2.9} \\
dP/d\tau &:&=\partial \mathcal{L}^{(\tau )}/\partial r=-\nabla \bar{W}(1+
\dot{r}|^{2})^{1/2}+\xi \nabla <A,\dot{r}>_{\mathbb{E}^{3}}.  \notag
\end{eqnarray
Now making use once more of the infinitesimal transformation (\ref{A2.6})
and the crucial \textit{dynamical particle mass} definition \cit
{BPT,BP,Rep,Tr} (in the light speed units )
\begin{equation}
m:=-\bar{W},  \label{A2.9a}
\end{equation
we can easily rewrite equations (\ref{A2.9}) with respect to the parameter
t\in \mathbb{R}$ as the \ classical relativistic Lorentz force
\begin{equation}
dp/dt=\xi E+\xi u\times B,  \label{A2.10}
\end{equation
where we denoted
\begin{eqnarray}
p &:&=mu,\text{ \ \ \ \ \ \ }u:=dr/dt,\text{ }  \label{A2.11} \\
B &:&=\nabla \times A,\text{ \ \ }E:=-\xi ^{-1}\nabla \bar{W}-\partial
A/\partial t.  \notag
\end{eqnarray
Thus, we obtained once more the relativistic Lorentz force expression (\re
{A2.10}), but slightly different from (\ref{A1.6}), since the classical
relativistic momentum $\ $of (\ref{A1.7}) \ does not completely coincide
with our modified relativistic momentum expression
\begin{equation}
p=-\bar{W}u,  \label{A2.12}
\end{equation
depending strongly on the scalar vacuum field potential function $\bar{W
:M^{4}\rightarrow \mathbb{R}.$ But if we recall here that our action
functional (\ref{A2.5}) was written under the assumption that \ the particle
charge value $\xi $ is negligible and \ not exerting the essential influence
on the electromagnetic field source, we can make use of the result before
obtained in \cite{BP0,BP,BPT,BPT1,Rep}, that the vacuum field potential
function $\bar{W}:M^{4}\rightarrow \mathbb{R},$ owing to (\ref{A2.10})-(\re
{A2.12}), satisfies as $\xi \rightarrow 0$ the dynamical equation
\begin{equation}
d(-\bar{W}u)/dt=-\nabla \bar{W},  \label{A2.13}
\end{equation
whose solution at the stationary condition $\partial \bar{W}/\partial t=0$
will be exactly the expressio
\begin{equation}
-\bar{W}=m_{0}(1-|u|^{2})^{-1/2},\text{ }m_{0}=-\left. \bar{W}\right\vert
_{u=0}.  \label{A2.14}
\end{equation
Thereby, we have arrived, owing to (\ref{A2.14}) and (\ref{A2.12}), to the
almost full \textit{coincidence }of our result (\ref{A2.10}) for the
relativistic \ Lorentz force with that of (\ref{A1.6}) under the condition
\xi \rightarrow 0.$

The results obtained above and related inferences we will formulate as the
following proposition.

\begin{proposition}
\label{Pr_A1.2} Under the assumption of the negligible influence of a
charged point particle $\xi $ on an external electromagnetic field source a
true physically reasonable action functional can be represented by
expression (\ref{A2.4}), being equivalently defined with respect to the rest
reference frame $\mathcal{K}_{r}$ in the form (\ref{A2.5}),(\ref{A2.5a}).
The resulting relativistic Lorentz force (\ref{A2.10}) coincides almost
exactly with that of (\ref{A1.6}), obtained from the classical Einstein type
action functional (\ref{A1.1}), but the momentum expression (\ref{A2.12})
differs from the classical expression (\ref{A1.7}), taking into account the
related vacuum field potential interaction energy impact.
\end{proposition}

\bigskip As an important corollary we make the following.

\begin{corollary}
\label{Cor_A2.2}The Lorentz force expression (\ref{A2.10}) \ should be, in
due course, corrected in the case when the weak charge $e$ influence
assumption made above does not hold.
\end{corollary}

\begin{remark}
\label{Rem_A2.3}Concerning the infinitesimal relationship (\ref{A2.6}) one
can observe that it reflects the Euclidean nature of transformations
\mathbb{R\ni }$ $t\rightleftharpoons \tau \in $ \bigskip $\mathbb{R}.$
\end{remark}

In spite of the results obtained above by means of two different least
action principles (\ref{A1.1}) and (\ref{A2.5}), we must claim here that the
first one possesses some logical controversies, which may give rise to
unpredictable, unexplainable and even nonphysical effects. Amongst these
controversies we mention:

$i)$ the definition of Lagrangian function (\ref{A1.2}) as an expression,
depending on the external and undefined rest mass parameter with respect to
the rest \ reference frame $\mathcal{K}_{r}$ time $\tau \in \mathbb{R},$ but
serving as a variational integrand with respect to the laboratory \
reference frame $\mathcal{K}$ time $t\in \mathbb{R};$

$ii)$ the least action condition (\ref{A1.1}) is calculated with respect to
the fixed boundary conditions at the ends of a time interval
[t_{1},t_{2}]\subset \mathbb{R},$ thereby the resulting dynamics becomes
strongly \ dependent on the chosen laboratory reference frame $\mathcal{K},$
which is, following the Feynman arguments \cite{Fe1}, physically
unreasonable;

$iii)$ the resulting relativistic particle mass and its energy depend only
on the particle velocity in the laboratory reference frame $\mathcal{K},$
not taking into account the present vacuum field potential energy of a
charged point particle, exerting not trivial action on the particle motion;

$iv)$ the assumption concerning the negligible influence of a charged point
particle on the external electromagnetic field source is also physically
inconsistent.

\section{\protect\bigskip A charged point particle electrodynamics within
the vacuum field theory approach}

\subsection{A free charged point particle in the vacuum medium}

\bigskip We start now from the following action functional for a charged
point particle $\xi $ moving with velocity $u:=dr/dt\in \mathbb{E}^{3}$
under an external stationary vacuum field potential $\bar{W
:M^{4}\rightarrow \mathbb{R}$ with respect to a laboratory reference frame
\mathcal{K}:
\begin{equation}
S^{(\tau )}:=-\int_{t(\tau _{2})}^{t(\tau _{1})}\bar{W}dt,\text{ \ \ \ \ \ }
\label{A3.1}
\end{equation
being defined on the time interval $[t(\tau _{1}),t(\tau _{2})]\subset
\mathbb{R}.$ The vacuum field potential function $\bar{W}:M^{4}\rightarrow
\mathbb{R}\ $ satisfies the condition $\partial \bar{W}/\partial t=0\ $ and
characterizes the intrinsic properties of the vacuum medium and its
interaction with a charged point particle $\xi ,$ \ jointly with the
Euclidean rest reference frame $\mathcal{K}_{r}$ \textit{relationship}
\begin{equation}
dt=<\dot{\eta},\dot{\eta}>_{\mathbb{E}^{4}}^{1/2}d\tau ,\text{ \ }
\label{A3.1a}
\end{equation
where $\eta :=(\tau ,r)\in \mathbb{E}^{4}$ is a charged point particle
position 4-vector with respect to the proper rest reference frame $\mathcal{
}_{r},$ $\dot{\eta}:=d\eta /d\tau ,$ $\tau \in \mathbb{R}.$ As the real
dynamics of our charged point particle $e$ depends strongly only on the time
interval $[\tau _{1},\tau _{2}]\subset \mathbb{R}$ of its own motion subject
to the rest reference frame $\mathcal{K}_{r},$ we need to calculate the
extremality condition
\begin{equation}
\delta S^{(\tau )}=0,\text{ \ \ \ }\delta r(\tau _{1})=0=\delta r(\tau _{2}).
\label{A3.2}
\end{equation
As action functional (\ref{A3.1}) is equivalent, owing to (\ref{A3.1a}) or
\ref{A2.6}), to the following
\begin{equation}
S^{(\tau )}:=-\int_{\tau _{2}}^{\tau _{1}}\bar{W}(1+|\dot{r
|^{2})^{1/2}d\tau ,  \label{A3.3}
\end{equation
where, by definition, $\dot{r}:=dr/d\tau ,$ $\ |\dot{r}|^{2}:=<\dot{r},\dot{
}>_{\mathbb{E}^{3}},$ $\tau \in \mathbb{R},$ from (\ref{A3.3}) and (\re
{A3.2}) one easily obtains that
\begin{equation}
p:=-\bar{W}\dot{r}(1+|\dot{r}|^{2})^{-1/2},\text{ }dp/d\tau =-\nabla \bar{W
(1+|\dot{r}|^{2})^{1/2}.  \label{A3.4}
\end{equation
Taking into account once more \ relationship (\ref{A2.6}) we can rewrite
\ref{A3.4}) equivalently as
\begin{equation}
dp/dt=-\nabla \bar{W},\text{\ \ \ \ }p:=-\bar{W}u.  \label{A3.5}
\end{equation
If to take into account the dynamic mass definition (\ref{A2.9a}), equation
\ref{A3.5}) turns into the Newton dynamical expression
\begin{equation}
dp/dt=-\nabla \bar{W},\text{ }p=mu.  \label{A3.6}
\end{equation
Having observed now that equation (\ref{A3.6}) is completely equivalent to
equation (\ref{A2.13}), we obtain right away from (\ref{A2.14}) that the
particle mass
\begin{equation}
m=m_{0}(1-|u|^{2})^{-1/2},\text{ }  \label{A3.7}
\end{equation
where
\begin{equation}
m_{0}:=-\left. \bar{W}\right\vert _{u=0}  \label{A3.7a}
\end{equation
is the so-called particle rest mass. Moreover, since the corresponding (\re
{A3.3}) Lagrangian function
\begin{equation}
\mathcal{L}^{(\tau )}:=-\bar{W}(1+|\dot{r}|^{2})^{1/2}  \label{A3.8}
\end{equation
is not degenerate, we can easily construct \cite{AM,Ar,BPS,BP} the related
conservative Hamiltonian function
\begin{equation}
\mathcal{H}^{(\tau )}=-(\bar{W}^{2}-|p|^{2})^{1/2},  \label{A3.9}
\end{equation
where $|p|^{2}:=<p,p>_{\mathbb{E}^{3}},$ \ satisfying the canonical
Hamiltonian equation
\begin{equation}
dr/d\tau =\partial \mathcal{H}^{(\tau )}/\partial p,\text{ }dp/d\tau
=-\partial \mathcal{H}^{(\tau )}/\partial r  \label{A3.10}
\end{equation
and conservation conditions
\begin{equation}
d\mathcal{H}^{(\tau )}/dt=0=d\mathcal{H}^{(\tau )}/d\tau  \label{A3.11}
\end{equation
for all $\tau ,t\in \mathbb{R}.$ Thereby, the quantity
\begin{equation}
\mathcal{E}:=(\bar{W}^{2}-|p|^{2})^{1/2}  \label{A3.12}
\end{equation
can be naturally interpreted as the point particle total energy.

It is important to note here that energy expression (\ref{A3.12}) takes into
account both kinetic and potential energies, but the particle dynamic mass
\ref{A3.7}) depends only on its velocity, reflecting its free motion in \
vacuum. Moreover, since the vacuum potential function $\bar{W
:M^{4}\rightarrow \mathbb{R}$ is not, in general, constant, we claim that
the motion of our particle $e$ with respect to the laboratory reference
frame $\mathcal{K}$ is not, in general, linear and is with not constant
velocity, - the situation, which was already discussed before by R. Feynman
in \cite{Fe1}. Thus, we obtained the classical relativistic mass dependence
on the freely moving particle velocity (\ref{A3.7}), taking into account
both the nonconstant vacuum potential function $\bar{W}:M^{4}\rightarrow
\mathbb{R}$ and the particle velocity $u\in \mathbb{E}^{3}.$

We would also like to mention here that the vacuum potential function $\bar{
}:M^{4}\rightarrow \mathbb{R}$ itself should be simultaneously found by
means of a suitable solution to the Maxwell equation $\partial
^{2}W/\partial t^{2}-\Delta W=\rho ,$ where $\rho \in \mathbb{R}$ is an
ambient charge density and, by definition, $\ \bar{W}(t,r(t))$ $:=$
\lim_{r\rightarrow r(t)}\left. W(r,t)\right\vert ,$ with $r(t)\in \mathbb{E
^{3},$ being the position of the charged point particle at a time moment
t\in \mathbb{R}.$ A more detailed description \cite{BPT} of the vacuum field
potential $W:M^{4}\rightarrow \mathbb{R},$ developing the ideas of \cite{Tr}
and characterizing the vacuum medium \ structure, is given in the Supplement.

We return now to expression (\ref{A3.1}) and rewrite it in the following
invariant form
\begin{equation}
S^{(\tau )}=-\int_{s(\tau _{1})}^{s(\tau _{2})}\bar{W}<\dot{\eta},\dot{\eta
>_{\mathbb{E}^{4}}^{1/2}ds,  \label{A3.13}
\end{equation
where, by definition, $s\in \mathbb{R}$ parameterizes the particle world
line related with the laboratory reference frame $\mathcal{K}$ time
parameter $t\in \mathbb{R}$ by means of the Euclidean infinitesimal \textit
relationship}
\begin{equation}
dt:=<\dot{\eta},\dot{\eta}>_{\mathbb{E}^{4}}^{1/2}ds.  \label{A3.14}
\end{equation
It is easy to observe that at $s=t\in \mathbb{R}$ the functional (\ref{A3.13
) turns into (\ref{A2.2}) and the \textit{constraint} (\ref{A2.2a}), at
s=\tau $ $\in \mathbb{R}$ it transforms into (\ref{A3.1}) and the \textit
relationship} (\ref{A3.1a}). The action functional (\ref{A3.13}) is to be
supplemented naturally with the boundary conditions
\begin{equation}
\delta \eta (s(\tau _{1}))=0=\delta \eta (s(\tau _{2})),  \label{A3.15}
\end{equation
which are, obviously, completely equivalent to those of (\ref{A3.2}), since
the mapping $\mathbb{R}$ $\ni s\rightleftharpoons t\in \mathbb{R},$ owing to
definition (\ref{A3.14}) is one-to-one.

Having calculated the least \ action condition $\delta S^{(\tau )}=0$ \
under constraints (\ref{A3.15}), one easily obtains the same equation (\re
{A3.5}) and relationships (\ref{A3.7}), (\ref{A3.12}) for the particle
dynamical mass and its conservative energy, respectively.

\subsection{A charged point particle electrodynamics under external
electromagnetic field}

We would like to generalize the results obtained above for a free point
particle in the vacuum medium for the case of a charged point particle $\xi $
interacting with an external charged point particle $\xi _{f},$ moving with
respect to a laboratory reference frame $\mathcal{K}$ with the velocity
u_{f}\in \mathbb{E}^{3}.$ Within the vacuum field theory approach, devised
in \cite{BPT,BPT1,BP}, it is natural to reduce the formulated problem to
that considered above, having introduced the reference frame $\mathcal{K
_{f}^{\prime }$ moving with respect to the reference frame $\mathcal{K
_{f}^{\prime }$ \ with the same velocity as that of the external charged
point particle $\xi _{f}.$ Thus, if the external charged particle $\xi _{f},$
considered with respect to the laboratory reference $\mathcal{K}_{f}^{\prime
},$ will be in rest, the test charged point particle $\xi $ \ will be moving
with the resulting velocity $u^{\prime }-u_{f}^{\prime }\in T(\mathbb{R
^{3}),$ where, by definition, $u^{\prime }:=dr/dt^{\prime },$ $u_{f}^{\prime
}:=dr_{f}/dt^{\prime }$ are the corresponding velocities of these charged
point particles $\xi $ and $\xi _{f}$ with respect to the moving reference
frame $\mathcal{K}_{f}^{\prime },$ \ \ parametrized by the Euclidean
variables $(t^{\prime },r_{f})\in \mathbb{E}^{4},$ and which are under the
influence of the corresponding external vacuum field potential $\bar{W
^{\prime }\in \mathbb{R}\ $ related via the Lorentz transform
\begin{equation}
\bar{W}^{\prime }(1-|u_{f}|^{2})^{1/2}=\bar{W}\   \label{A4.0}
\end{equation
to that with respect to the laboratory reference frame $\mathcal{K}.$ As a
result of these reasonings we can write down the following action functional
expressio
\begin{equation}
S^{(\tau )}=-\int_{t^{\prime }(\tau _{1})}^{t^{\prime }(\tau _{2})}\bar{W
^{\prime }dt^{\prime }=\int_{s(\tau _{1})}^{s(\tau _{2})}\bar{W}^{\prime
}<\eta _{f}^{\prime },\eta _{f}^{\prime }>_{\mathbb{E}^{4}}^{1/2}ds,
\label{A4.1}
\end{equation
where, by definition, $\dot{\eta}_{f}:=(\dot{\tau},\dot{r}-\dot{r}_{f})\in
\mathbb{(E}^{4})$ is the charged point particle $\xi $ four-vector velocity
with respect to the rest reference frame $\mathcal{K}_{r}$ and calculated
subject to the corresponding particle world line parameter $s\in \mathbb{R},$
being infinitesimally related to the moving reference frame $\mathcal{K
_{f}^{\prime }$ \ time parameter $t^{\prime }\in \mathbb{R}$ as
\begin{equation}
dt^{\prime }:=<\dot{\eta}_{f},\dot{\eta}_{f}>_{\mathbb{E}^{4}}^{1/2}ds.
\label{A4.2}
\end{equation
The boundary conditions for functional (\ref{A4.1}) are taken naturally in
the for
\begin{equation}
\delta \eta (s(\tau _{1}))=0=\delta \eta (s(\tau _{2}),  \label{A4.3}
\end{equation
where $\eta =(\tau ,r)\in \mathbb{E}^{4}.$ The least action condition
\delta S^{(\tau )}=0$ jointly with (\ref{A4.3}) gives rise to the \bigskip
following equations
\begin{eqnarray}
P &:&=\partial \mathcal{L}^{(\tau )}/\partial \dot{\eta}=-\bar{W}^{\prime
\dot{\eta}_{f}<\dot{\eta}_{f},\dot{\eta}_{f}>_{\mathbb{E}^{4}}^{-1/2},
\notag \\
dP/ds &:&=\partial \mathcal{L}^{(\tau )}/\partial \eta =-\nabla _{\eta }\bar
W}^{\prime }<\dot{\eta}_{f},\dot{\eta}_{f}>_{\mathbb{E}^{4}}^{1/2},
\label{A4.4}
\end{eqnarray
where the Lagrangian function equals
\begin{equation}
\mathcal{L}^{(\tau )}:=-\bar{W}^{\prime }<\dot{\eta}_{f},\dot{\eta}_{f}>_
\mathbb{E}^{4}}^{1/2}.  \label{A4.5}
\end{equation
Having now defined, owing to the relationship (\ref{A4.0}), the charged
point particle $\xi $ momentum $p\in T^{\ast }(\mathbb{E}^{3})$ as
\begin{equation}
p:=-\bar{W}^{\prime }\dot{r}<\dot{\eta}_{f},\dot{\eta}_{f}>_{\mathbb{E
^{4}}^{-1/2}=-\bar{W}^{\prime }u^{\prime }=-\bar{W}u  \label{A4.6}
\end{equation
and the induced external magnetic vector potential $A\in T^{\ast }(\mathbb{E
^{3})$ as
\begin{equation}
\xi A:=\bar{W}^{\prime }\dot{r}_{f}<\dot{\eta}_{f},\dot{\eta}_{f}>_{\mathbb{
}^{4}}^{-1/2}=\bar{W}^{\prime }u_{f}^{\prime }=\bar{W}u_{f},  \label{A4.7}
\end{equation
we obtain, owing to relationship (\ref{A4.2}), the equality
\begin{equation}
d(p+\xi A\ )/dt=-\ \nabla W(1-|u_{f}|_{_{\mathbb{E}^{3}}}^{2}),
\label{A4.7a}
\end{equation
or the final relativistic Lorentz type force expressio
\begin{equation}
dp/dt=\xi E+\xi u\times B-\xi \ \nabla <u-u_{f},A>_{\mathbb{E}^{3}},
\label{A4.8}
\end{equation
where we denoted, by definition,
\begin{equation}
E:=-\xi ^{-1}\ \nabla \bar{W}-\partial A/\partial t,\text{ \ \ \ }B=\ \nabla
\times A,  \label{A4.9}
\end{equation
being, respectively, the external electric and magnetic fields, acting on
the charged point particle $\xi .$

The result (\ref{A4.8}) contains the additional Lorentz force component
\begin{equation}
F_{c}:=-\xi \ \nabla <u-u_{f},A>_{\mathbb{E}^{3}},  \label{A4.10}
\end{equation
not present in the classical relativistic Lorentz force expressions (\re
{A1.6}) and (\ref{A2.10}), obtained before.

\begin{remark}
It is worthy to remark here that just recently a gradient like term, similar
to \ that of (\ref{A4.10}), was obtained by R.T. Hammond in a very
intersting work \cite{Ha}. Yet the exact relationships between these two
representations is stil not established as that from \cite{Ha} contains not
completely specified functional parameters.
\end{remark}

The considered above alternative classical relativistic electrodynamic
model, based on the action functional (\ref{A4.1}) with respect to the rest
\ reference frame $\mathcal{K}_{r},$ where the Lagrangian function is given
by expression (\ref{A4.5}), is strongly supported by physical arguments. \
In particular, it demonstrates a well known physical phenomenon that an
external charged particle $\xi _{f},$ moving with a velocity $u_{f}$ $\in
E^{3}$ tending to the light velocity $c,|c|=1,$ exerts no influence on the
charged test particle $\xi $ and equivalently expressed by the right-hand
side of expression (\ref{A4.7a}). The resulting Lorentz type force
expression (\ref{A4.8}) , being modified by the additional force component
F_{c}:=-$ $\nabla <\xi A,u-u_{f}>_{\mathbb{E}^{3}},$ is important for
explanation \cite{AB,Boy,Tr} of the well known Aharonov-Bohm effect.
Moreover, from (\ref{A4.6}) one obtains that the point particle $\xi $
momentum
\begin{equation}
p=-\bar{W}u:=mu,  \label{A4.11}
\end{equation
where the particle mass
\begin{equation}
m:=-\bar{W}  \label{A4.12}
\end{equation
does not already coincide with the corresponding classical relativistic \
relationship of (\ref{A1.7}).

Consider \ now the least action condition for functional (\ref{A4.1}) at the
critical parameter $s=\tau \in \mathbb{R}:$
\begin{eqnarray}
\delta S^{(\tau )} &=&0,\text{ \ \ \ \ \ }\delta r(\tau _{1})=0=\delta
r(\tau _{2}),  \label{A4.13} \\
S^{(\tau )} &:&=-\int_{\tau _{1}}^{\tau _{2}}\bar{W}(1+|\dot{r}-\dot{r
_{f}|_{\mathbb{E}^{3}}^{2})^{1/2}d\tau .  \notag
\end{eqnarray
The resulting Lagrangian function
\begin{equation}
\mathcal{L}^{(\tau )}:=-\bar{W}(1+|\dot{r}-\dot{r}_{f}|_{\mathbb{E
^{3}}^{2})^{1/2}  \label{A4.14}
\end{equation
gives rise to the generalized momentum expressio
\begin{equation}
P:=\partial \mathcal{L}^{(\tau )}/\partial \dot{r}=-\bar{W}(\dot{r}-\dot{r
_{f})(1+|\dot{r}-\dot{r}_{f}|_{\mathbb{E}^{3}}^{2})^{-1/2}:=p+\xi A,
\label{A4.15}
\end{equation
which makes it possible to construct \cite{AM,Ar,PM,BPS} the corresponding
Hamiltonian function as
\begin{eqnarray}
\mathcal{H} &:&=<P,\dot{r}>_{\mathbb{E}^{3}}-\mathcal{L}^{(\tau )}=-(\bar{W
^{2}-|p+\xi A|_{\mathbb{E}^{3}}^{2})^{1/2}-  \notag \\
- &<&p+\xi A,\xi A>_{\mathbb{E}^{3}}(\bar{W}^{2}-|p+\xi A|_{\mathbb{E
^{3}}^{2})^{-1/2},  \label{A4.16}
\end{eqnarray
satisfying the canonical Hamiltonian equations
\begin{equation}
dP/d\tau :=\partial \mathcal{H}/\partial r,\text{ \ \ }dr/d\tau :=-\partial
\mathcal{H}/\partial r,  \label{A4.16a}
\end{equation
evolving with respect to the proper rest reference frame time $\tau \in
\mathbb{R}$ parameter. When deriving (\ref{A4.16}) we made use of
relationship (\ref{A4.2}) at $s=\tau \in \mathbb{R}$ jointly with
definitions (\ref{A4.6}) and (\ref{A4.7}). Since the Hamiltonian function
\ref{A4.16}) is conservative with respect to the evolution parameter $\tau
\in \mathbb{R},$ owing to relationship (\ref{A4.2}) at $s=\tau \in \mathbb{R}
$ one obtains that
\begin{equation}
d\mathcal{H}/d\tau =0=d\mathcal{H}/dt  \label{A4.17}
\end{equation
for all $t,\tau \in \mathbb{R}.$ The obtained results can be formulated as
the following proposition.

\begin{proposition}
The charged point particle electrodynamics, related with the least action
principle (\ref{A4.1}) and (\ref{A4.3}), reduces to the modified Lorentz
type force equation (\ref{A4.8}), and is equivalent to the canonical
Hamilton frame (\ref{A4.16a}) with respect to the proper rest reference
frame $\mathcal{K}_{\tau }$ time parameter $\tau \in \mathbb{R}.$ The
corresponding Hamiltonian function (\ref{A4.16}) is a conservation law for
the Lorentz type dynamics (\ref{A4.8}), satisfying the conditions (\re
{A4.17}) with respect to both reference frames $\mathcal{K}$ and $\mathcal{K
_{\tau }$ parameters $t,\tau \in \mathbb{R},$ respectively.
\end{proposition}

\bigskip As a corollary, the corresponding energy expression for
electrodynamical model (\ref{A4.8}) can be defined a
\begin{equation}
\mathcal{E}:=(\bar{W}^{2}-|p+\xi A|_{\mathbb{E}^{3}}^{2})^{1/2}+<p+\xi A,\xi
A>_{\mathbb{E}^{3}}(\bar{W}^{2}-|p+\xi A|_{\mathbb{E}^{3}}^{2})^{-1/2}.
\label{A4.18}
\end{equation
The energy expression (\ref{A4.18}) obtained above is a necessary ingredient
for quantizing the relativistic electrodynamics (\ref{A4.8}) of our charged
point particle $\xi $ under the external electromagnetic field influence.

\section{A new hadronic string model: the least action principle and
relativistic electrodynamics analysis within the vacuum field theory approac
}

\subsection{ A new hadronic string model least action formulation}

A classical relativistic hadronic string model was first proposed in \cit
{BC,Na,Go} and deeply studied in \cite{BN}, making use of the least action
principle and related Lagrangian and Hamiltonian formalisms. We will not
discuss here this classical string model and will not comment the physical
problems accompanying it, especially those related to its diverse
quantization versions, but proceed to formulating a new relativistic
hadronic string model, constructed by means of the vacuum field theory
approach, devised in \cite{BPT,BPT1,BP}. The corresponding least action
principle is, following \cite{BN}, formulated as
\begin{equation}
\delta S^{(\tau )}=0,\text{ \ \ }S^{(\tau )}:=\int_{s(\tau _{1})}^{s(\tau
_{2})}ds\int_{\sigma _{1}(s)}^{\sigma _{2}(s)}\bar{W}(x(\eta ))(|\dot{\eta
|^{2}|\eta ^{\prime }|^{2}-<\dot{\eta},\eta ^{\prime }>_{\mathbb{E
^{4}}^{2})^{1/2}d\sigma \wedge ds,  \label{A5.1}
\end{equation
where $\bar{W}:M^{4}\rightarrow \mathbb{R}$ is a vacuum field potential
function, characterizing the interaction of the vacuum medium with our
charged string object, the differential 2-form $d\Sigma ^{(2)}:=(|\dot{\eta
|^{2}|\eta ^{\prime }|^{2}-<\dot{\eta},\eta ^{\prime }>_{\mathbb{E
^{4}}^{2})^{1/2}d\sigma \wedge ds$ $=\sqrt[2]{g(\eta )}d\sigma \wedge ds,$
g(\eta ):=$ $\det (\left. g_{ij}(\eta )\right\vert _{i,j=\overline{1,2}}),$
|\dot{\eta}|^{2}:=$ $<\dot{\eta},\dot{\eta}>_{\mathbb{E}^{4}},$ \ $|\eta
^{\prime }|^{2}:=$ $\ <\eta ^{\prime },\eta ^{\prime }>_{\mathbb{E}^{4}},$
being related with the induced positive define Riemannian infinitesimal
metrics $dz^{2}:=<d\eta ,d\eta >_{\mathbb{E}^{4}}=g_{11}(\eta )d\sigma
^{2}+g_{12}(\eta )d\sigma ds+g_{21}(\eta )dsd\sigma $ $+g_{22}(\eta )ds^{2}$
on the string, means \cite{AM,BN,Th} the infinitesimal two-dimensional world
surface element, parameterized by variables $(s,\sigma )\in \mathbb{R}^{2}$
and embedded into the 4-dimensional Euclidean space-time $\mathbb{E}^{4}$
with coordinates $\eta :=(\tau (s,\sigma ),r(s,\sigma ))\in \mathbb{E}^{4}$
subject to the proper rest reference frame $\mathcal{K}_{r},$ $\dot{\eta
:=\partial \eta /\partial s,$ $\eta ^{\prime }:=\partial \eta /\partial
\sigma $ are the corresponding partial derivatives. The related boundary
conditions are chosen as
\begin{equation}
\delta \eta (s,\sigma (s))=0  \label{A5.2}
\end{equation
at string parameter $\sigma (s)\in \mathbb{R}$ for all $s\in \mathbb{R}.$
The action functional expression is strongly motivated by that constructed
for the point particle action functional (\ref{A3.1}):
\begin{equation}
S^{(\tau )}:=-\int_{\sigma _{1}}^{\sigma _{2}}dl(\sigma )\int_{t(\sigma
,\tau _{1})}^{t(\sigma ,\tau _{2})}\bar{W}dt(\tau ,\sigma ),  \label{A5.3}
\end{equation
where the laboratory reference time parameter $t(\tau ,\sigma )\in \mathbb{R}
$ is related to the proper rest string reference frame time parameter $\tau
\in \mathbb{R}$ by means of the standard Euclidean infinitesimal
relationship
\begin{equation}
dt(\tau ,\sigma ):=(1+|\dot{r}_{\perp }|^{2}(\tau ,\sigma ))^{1/2}d\tau
\text{ \ \ \ }|\dot{r}_{\perp }|^{2}:=<\dot{r}_{\bot },\dot{r}_{\bot }>_
\mathbb{E}^{3}},  \label{A5.4}
\end{equation
with $\sigma \in \lbrack \sigma _{1},\sigma _{2}]\subset \mathbb{R},$ being
a spatial variable \ parameterizing the string length measure $dl(\sigma )$
on the real axis $\mathbb{R},$ $\dot{r}_{\perp }(\tau ,\sigma ):=\hat{N}$
\dot{r}(\tau ,\sigma )\in \mathbb{E}^{3}$ being the orthogonal to the string
velocity component, and
\begin{equation}
\text{\ }\ \hat{N}:=(1-|r^{\prime }|^{-2}r^{\prime }\otimes r^{\prime })
\text{ \ \ }|r^{\prime }|^{-2}:=<r^{\prime },r^{\prime }>_{\mathbb{E
^{3}}^{-1},  \label{A5.5}
\end{equation
being the corresponding projector operator in $\mathbb{E}^{3}$ on the
orthogonal to the string direction, expressed for brevity by means of the
standard tensor product $"\otimes "$ in the Euclidean space $\mathbb{E}^{3}.$
The result of calculation of (\ref{A5.3}) gives rise to the following
expression
\begin{equation}
S^{(\tau )}=-\int_{\tau _{1}}^{\tau _{2}}d\tau \int_{\sigma _{1}(\tau
)}^{\sigma _{2}(\tau )}\bar{W}((|r^{\prime }|^{2}(1+|\dot{r}|^{2})-<\dot{r
,r^{\prime }>_{\mathbb{E}^{3}}^{2})^{1/2}d\ \sigma ,  \label{A5.6}
\end{equation
where we made use of the infinitesimal measure representation $dl(\sigma
)=<r^{\prime },r^{\prime }>_{\mathbb{E}^{3}}^{1/2}d\sigma ,$ $\sigma \in
\lbrack \sigma _{1},\sigma _{2}].$ If now to introduce on the string world
surface local coordinates $(s(\tau ,\sigma ),\sigma )\in \mathbb{E}^{2}$ and
the related Euclidean string position vector $\eta :=(\tau ,r(s,\sigma ))\in
\mathbb{E}^{4},$ the string action functional reduces equivalently to that
of (\ref{A5.1}).

Below we will proceed to Lagrangian and Hamiltonian analyzing the least
action conditions for expressions (\ref{A5.1}) and (\ref{A5.6}).

\subsection{ Lagrangian and Hamiltonian analysis}

First we will obtain the corresponding to (\ref{A5.1}) Euler equations with
respect to the special \cite{BN} internal conformal variables $(s,\sigma
)\in \mathbb{E}^{2}$ on the world string surface, with respect to which the
metrics on it becomes equal to $dz^{2}=|\eta ^{\prime }|^{2}d\sigma ^{2}+
\dot{\eta}|^{2}ds^{2},$ where $<$ $\eta ^{\prime },\dot{\eta}$ $>_{\mathbb{E
^{4}}=0=|\eta ^{\prime }|^{2}-|\dot{\eta}|^{2}$ are the imposed constraints,
and the corresponding infinitesimal world surface measure $d\Sigma ^{(2)}$
becomes $d\Sigma ^{(2)}=|\eta ^{\prime }|\dot{\eta}|d\sigma \wedge ds.$ As a
result of simple calculations one finds the linear second order partial
differential equation
\begin{equation}
\partial (\bar{W}\dot{\eta})/\partial s+\partial (\bar{W}\eta ^{\prime
})/\partial \sigma =\partial (|\eta ^{\prime }|\text{ }|\dot{\eta}|\bar{W
)/\partial \sigma  \label{A6.1}
\end{equation
under the suitably chosen boundary condition
\begin{equation}
\eta ^{\prime }-\dot{\eta}\text{ }\dot{\sigma}=0  \label{A6.2}
\end{equation
for all $s\in \mathbb{R}.$ It is interesting to mention that equation \ (\re
{A6.1}) \ is of \textit{elliptic type}, contrary to the case considered
before in \cite{BN}. This is, evidently, owing to the fact that the
resulting metrics on the string world surface is Euclidean, as we took into
account that the real string motion is, in reality, realized with respect to
its proper rest reference frame $\mathcal{K}_{r},$ being not dependent on
the string motion observation data, measured with respect to any external
laboratory reference frame $\mathcal{K}.$ The latter can be used for
physically motivated evidence of the dynamical stability of the relativistic
charged string object, modeling a charged hadronic particle \cit
{Ba,Gr,Na,Wi} with non-trivial internal structure.

The differential equation (\ref{A6.1}) strongly depends on the vacuum field
potential function $\bar{W}:M^{4}\rightarrow \mathbb{R},$ which, as a
function of the Minkowski 4-vector variable $x:=(t(s,\sigma ),r)\in M^{4}$
of the laboratory reference frame $\mathcal{K},$ should be expressed as a
function of the variables $(s,\sigma )\in \mathbb{E}^{2},$ making use of the
infinitesimal relationship (\ref{A5.4}) in the following form
\begin{equation}
dt=<\hat{N}\partial \dot{\eta}/\partial \tau ,\hat{N}\partial \dot{\eta
/\partial \tau >_{\mathbb{E}^{3}}^{1/2}(\frac{\partial \tau }{\partial s}ds
\frac{\partial \tau }{\partial \sigma }d\sigma ),  \label{A6.3}
\end{equation
defined on the string world surface. The function $\bar{W}:M^{4}\rightarrow
\mathbb{R}$ itself should be simultaneously found, following ideas of \cit
{Br,Tr} and recent results of \cite{BP0,BP,BPT,BPT1}, by means of a suitable
solution to the Maxwell equation $\partial ^{2}W/\partial t^{2}-\Delta
W=\rho ,$ where $\rho \in \mathbb{R}$ is an ambient charge density and, by
definition, $\ \bar{W}(r(t))$ $:=$ $\lim_{r\rightarrow r(t)}\left.
W(r,t)\right\vert ,$ with $r(t)\in \mathbb{E}^{3}$ being the position of the
string element with a proper rest reference coordinates $(\tau ,\sigma )\in
\mathbb{E}^{2}$ at the time moment $t=t(\tau ,\sigma )\in \mathbb{R}.$

We proceed now to constructing the dynamical Euler equations for our string
model, making use of the general action functional (\ref{A5.6}) in the
following form
\begin{equation}
S^{(\tau )}=-\int_{\tau _{1}}^{\tau _{2}}d\tau \int_{\sigma _{1}(\tau
)}^{\sigma _{2}(\tau )}\bar{W}|r^{\prime }|(1+|\dot{r}|^{2}-<r^{\prime
}|r^{\prime }|^{-1},\dot{r}>_{\mathbb{E}^{3}}^{2})^{1/2}d\ \sigma ,
\label{A6.3a}
\end{equation
It is easy to calculate that the generalized momentum density
\begin{eqnarray}
p &:&=\partial \mathcal{L}^{(\tau )}/\partial \dot{r}=\frac{-\bar{W
|r^{\prime }|(\dot{r}-r^{\prime }|r^{\prime }|^{-2}<r^{\prime },\dot{r}>_
\mathbb{E}^{3}})}{(|\dot{r}|^{2}+1-<r^{\prime }|r^{\prime }|^{-1},\dot{r}>_
\mathbb{E}^{3}}^{2})^{1/2}}=  \notag \\
&=&\frac{-\bar{W}|r^{\prime }|\hat{N}dr\ }{d\tau (|\dot{r}|^{2}+1-<r^{\prime
}|r^{\prime }|^{-1},\dot{r}>_{\mathbb{E}^{3}}^{2})^{1/2}}=-|r^{\prime }|\bar
W}\hat{N}dr/dt\ =-|r^{\prime }|\hat{N}(\bar{W}u)  \label{A6.4}
\end{eqnarray
satisfies the dynamical equation
\begin{eqnarray}
dp/d\tau &:&=\delta \mathcal{L}^{(\tau )}/\delta r=-(|r^{\prime }|^{2}(|\dot
r}|^{2}+1)-<r^{\prime },\dot{r}>_{\mathbb{E}^{3}}^{2})^{1/2}\ \nabla \bar{W}+
\label{A6.5} \\
&&+\frac{\partial }{\partial \sigma }\left\{ \frac{\bar{W}(1+|\dot{r}|^{2
\hat{T})r^{\prime }/r^{\prime }}{(1+<|\dot{r}|^{2}\hat{T}r^{\prime
}|r^{\prime }|^{-1},r^{\prime }|r^{\prime }|^{-1}>_{\mathbb{E}^{3}})^{1/2}
\right\} ,  \notag
\end{eqnarray
where we denoted by
\begin{equation}
\mathcal{L}^{(\tau )}:=-\bar{W}(|r^{\prime }|^{2}(1+|\dot{r}|^{2})-<\dot{r
,r^{\prime }>_{\mathbb{E}^{3}}^{2})^{1/2}=-\bar{W}(|r^{\prime }|^{2}+|\dot{r
|^{2}<r^{\prime },\hat{T}_{\dot{r}}r^{\prime }>_{\mathbb{E}^{3}})^{1/2}
\label{A6.6}
\end{equation
the corresponding Lagrangian function, and for any vector $w\in \mathbb{E
^{3}$
\begin{equation}
\hat{T}_{w}:=1-|w|^{-2}\text{ }w\otimes w,\text{ \ \ \ }|w|^{2}:=<w,w>_
\mathbb{E}^{3}}^{2}\   \label{A6.7}
\end{equation
the usual projector operator in $\mathbb{E}^{3}.$ As a result of (\ref{A6.5
) one finds tha
\begin{eqnarray}
dp/dt &=&-|r^{\prime }|\ \nabla \bar{W}+  \label{A6.7a} \\
+(1-|u|^{2}) &<&u,r^{\prime }>_{\mathbb{E}^{3}}^{2})^{-1/2}\frac{\partial }
\partial \sigma }\left\{ \frac{\bar{W}(1-|u|^{2}+<u,r^{\prime }>_{\mathbb{E
^{3}}^{2}+|u|^{2}\hat{T}_{u})r^{\prime }/r^{\prime }}{(1-|u|^{2}+<u,r^
\prime }>_{\mathbb{E}^{3}}^{2})^{1/2}}\right\} ,  \notag
\end{eqnarray
where we took into account that owing to (\ref{A5.4})
\begin{equation}
\dot{r}=dr/d\tau =dr/dt\cdot (dt/d\tau )=u(1-|u|^{2}+<u,r^{\prime }>_
\mathbb{E}^{3}}^{2})^{-1/2}.  \label{A6.7b}
\end{equation}

The Lagrangian function is degenerate \cite{BN}, satisfying the obvious
identity
\begin{equation}
<p,r^{\prime }>_{\mathbb{E}^{3}}=0  \label{A6.8}
\end{equation
for all $\tau \in \mathbb{R}.$ Concerning the Hamiltonian formulation of the
dynamics (\ref{A6.5}) we construct the corresponding Hamiltonian functional
a
\begin{equation}
\begin{array}{c}
\mathcal{H}:=\int_{\sigma _{1}}^{\sigma _{2}}(<p,\dot{r}>_{\mathbb{E}^{3}}
\mathcal{L}^{(\tau )})d\sigma = \\
\\
=\int_{\sigma _{1}}^{\sigma _{2}}\left( \frac{-\bar{W}|r^{\prime }|(|\dot{r
|^{2}-r^{\prime }|r^{\prime }|^{-2}<r^{\prime },\dot{r}>_{\mathbb{E
^{3}}^{2})}{(|\dot{r}|^{2}+1-<r^{\prime }|r^{\prime }|^{-1},\dot{r}>_
\mathbb{E}^{3}}^{2})^{1/2}}+\frac{\bar{W}|r^{\prime }|(|\dot{r
|^{2}+1-<r^{\prime }|r^{\prime }|^{-1},\dot{r}>_{\mathbb{E}^{3}}^{2})\ }{(
\dot{r}|^{2}+1-<r^{\prime }|r^{\prime }|^{-1},\dot{r}>_{\mathbb{E
^{3}}^{2})^{1/2}}\ \right) d\sigma = \\
\\
=\int_{\sigma _{1}}^{\sigma _{2}}\left( \frac{\ \bar{W}|r^{\prime }|}{(|\dot
r}|^{2}+1-<r^{\prime }|r^{\prime }|^{-1},\dot{r}>_{\mathbb{E}^{3}}^{2})^{1/2
}\right) d\sigma =-\int_{\sigma _{1}}^{\sigma _{2}}(\bar{W}^{2}|r^{\prime
}|^{2}-|p|^{2})^{1/2}d\sigma
\end{array}
\label{A6.9}
\end{equation}
satisfying the canonical Hamiltonian equations
\begin{equation}
dr/d\tau :=\delta \mathcal{H}/\delta p,\text{ \ }dp/d\tau :=-\delta \mathcal
H}/\delta r,  \label{A6.10}
\end{equation
where
\begin{equation}
d\mathcal{H}/d\tau =0,  \label{A6.10a}
\end{equation
holding only with respect to the proper rest reference frame \ $\mathcal{K
_{r}$ time parameter $\tau \in \mathbb{R}.$ Now making use of identity (\re
{A6.8}) the Hamiltonian functional (\ref{A6.9}) can be equivalently
represented \cite{BN} in the symbolic form as
\begin{equation}
\mathcal{H}=\int_{\sigma _{1}}^{\sigma _{2}}|\bar{W}r^{\prime }\pm ip|_
\mathbb{C}}d\sigma ,\text{ }  \label{A6.11}
\end{equation
where \ $i:=\sqrt{-1}.$ Moreover, concerning the result obtained above we
need to mention here that one can not construct the suitable Hamiltonian
function expression and relationship of type (\ref{A6.10a}) with respect to
the laboratory reference frame $\mathcal{K},$ since expression (\ref{A6.11})
is not defined on the whole for a separate laboratory time parameter $t\in
\mathbb{R}$ \ locally dependent both on the spatial parameter $\sigma \in
\mathbb{R}$ and the proper rest reference frame time parameter $\tau \in
\mathbb{R}.$

Thereby, one can formulate the following proposition.

\begin{proposition}
The hadronic string model (\ref{A5.1}) allows, on the related world surface,
the conformal local coordinates, with respect to which the resulting
dynamics is described by means of the linear second order elliptic equation
\ref{A6.1}). Subject to the proper Euclidean rest reference frame $\mathcal{
}_{r}$ coordinates the corresponding dynamics is equivalent to the canonical
Hamiltonian equations (\ref{A6.10}) with Hamiltonian functional (\ref{A6.9}).
\end{proposition}

We proceed now to construct the action functional expression for a charged
string under an external electromagnetic field, generated by a point
velocity charged particle $\xi _{f},$ moving with some velocity
u_{f}:=dr_{f}/dt\in $ $\mathbb{E}^{3}$\ subject to a laboratory reference
frame $\mathcal{K}.$ To solve this problem we make use of the trick of
Section \ref{Sec_1}, passing to the string, \ considered with respect to the
proper rest reference frame $\mathcal{K}_{r}$ moving under the external
vacuum field potential $\bar{W}^{\prime }$ with respect to the relative
reference frame $\mathcal{K}_{f}^{\prime },$ specified by its own Euclidean
coordinates $(t^{\prime },r_{f})\in \mathbb{E}^{4},$ which simultaneously
moves with velocity $u_{f}\in $ $\mathbb{E}^{3}$ with respect to the
laboratory reference frame $\mathcal{K}.$ As a result of this reasoning we
can write down the action functional:
\begin{equation}
S^{(\tau )}=-\int_{\tau _{1}}^{\tau _{2}}d\tau \int_{\sigma _{1}(\tau
)}^{\sigma _{2}(\tau )}\bar{W}^{\prime }(|r^{\prime }|^{2}(1+|\dot{r}-\dot{r
_{f}|^{2})-<\dot{r}-\dot{r}_{f},r^{\prime }>_{\mathbb{E}^{3}}^{2})^{1/2}d\
\sigma ,  \label{A6.12}
\end{equation
giving rise to the following dynamical equation
\begin{eqnarray}
dP/d\tau &:&=\delta \mathcal{L}^{(\tau )}/\delta r=-(|r^{\prime }|^{2}(1+
\dot{r}-\dot{r}_{f}|^{2})-<\dot{r}-\dot{r}_{f},r^{\prime }>_{\mathbb{E
^{3}}^{2})^{1/2}\ \nabla \bar{W}^{\prime }+  \label{A6.13} \\
&&+\frac{\partial }{\partial \sigma }\left\{ \frac{\bar{W}^{\prime }(1+|\dot
r}-\dot{r}_{f}|^{2}\hat{T}_{\dot{r}-\dot{r}_{f}})r^{\prime }}{(|r^{\prime
}|^{2}(1+|\dot{r}-\dot{r}_{f}|^{2})-<\dot{r}-\dot{r}_{f},r^{\prime }>_
\mathbb{E}^{3}}^{2})^{1/2}}\right\} ,  \notag
\end{eqnarray
where the generalized momentum density
\begin{equation}
P:=\frac{-\bar{W}^{\prime }|r^{\prime }|^{2}\hat{N}(\dot{r}-\dot{r}_{f})}
(|r^{\prime }|^{2}(1+|\dot{r}-\dot{r}_{f}|^{2})-<\dot{r}-\dot{r
_{f},r^{\prime }>_{\mathbb{E}^{3}\mathbb{E}^{3}}^{2})^{1/2}}.  \label{A6.14}
\end{equation
Owing to (\ref{A4.0}), one can define
\begin{eqnarray}
p &:&=\frac{-\bar{W}^{\prime }|r^{\prime }|^{2}\hat{N}\dot{r}}{(|r^{\prime
}|^{2}(1+|\dot{r}-\dot{r}_{f}|^{2})-<\dot{r}-\dot{r}_{f},r^{\prime }>_
\mathbb{E}^{3}}^{2})^{1/2}}=  \label{A6.16} \\
&=&-\frac{-\bar{W}^{\prime }|r^{\prime }|\hat{N}dr}{d\tau (1+|\dot{r}-\dot{r
_{f}|^{2}-<\dot{r}-\dot{r}_{f},r^{\prime }/|r^{\prime }|>_{\mathbb{E
^{3}}^{2})^{1/2}}=-\bar{W}^{\prime }|r^{\prime }|\hat{N}u^{\prime }=-\bar{W
|r^{\prime }|\hat{N}u  \notag
\end{eqnarray
as the local string momentum density and
\begin{eqnarray}
\xi |r^{\prime }|A &:&=\frac{\bar{W}^{\prime }|r^{\prime }|^{2}\hat{N}\dot{r
_{f}}{(|r^{\prime }|^{2}(1+|\dot{r}-\dot{r}_{f}|^{2})-<\dot{r}-\dot{r
_{f},r^{\prime }>_{\mathbb{E}^{3}}^{2})^{1/2}}=  \label{A6.17} \\
&=&\frac{\bar{W}^{\prime }|r^{\prime }|\hat{N}dr_{f}}{d\tau (1+|\dot{r}-\dot
r}_{f}|^{2}-<\dot{r}-\dot{r}_{f},r^{\prime }/|r^{\prime }|>_{\mathbb{E
^{3}}^{2})^{1/2}}=\bar{W}^{\prime }|r^{\prime }|\hat{N}u_{f}^{\prime }=\bar{
}|r^{\prime }|\hat{N}u_{f}  \notag
\end{eqnarray
as the external vector magnetic potential density, where $\xi \in \mathbb{R}$
is a uniform charge density, distributed along the string length. Thus,
equation (\ref{A6.13}) reduces t
\begin{equation}
\begin{array}{c}
d(p+\xi |r^{\prime }|A)/dt^{\prime }=-|r^{\prime }|\ \nabla \bar{W}^{\prime
}+(1-|u^{\prime }-u_{f}^{\prime }|^{2}+<u^{\prime }-u_{f}^{\prime
},r^{\prime }>_{\mathbb{E}^{3}}^{2})^{-1/2}\times \\
\end{array}
\label{A6.17a}
\end{equation
\begin{equation*}
\times \frac{\partial }{\partial \sigma }\left\{ \frac{\bar{W}^{\prime
}(1-|u^{\prime }-u_{f}^{\prime }|^{2}+<u^{\prime }-u_{f}^{\prime },r^{\prime
}>_{\mathbb{E}^{3}}^{2}+|u^{\prime }-u_{f}^{\prime }|^{2}\hat{T}_{u^{\prime
}-u_{f}^{\prime }}r^{\prime }/r^{\prime })}{(1-|u^{\prime }-u_{f}^{\prime
}|^{2}+<u^{\prime }-u_{f}^{\prime },r^{\prime }>_{\mathbb{E}^{3}}^{2})^{1/2}
\right\}
\end{equation*
with respect to the moving reference frame $\mathcal{K}_{f}^{\prime },$ or
equivalently, t
\begin{equation}
\begin{tabular}{l}
$d(p+\xi |r^{\prime }|A)/dt=-|r^{\prime }|$ $\nabla \bar{W}(1-|u_{f}|^{2})+\
\ $ \\
\\
$+(1-|u_{f}|^{2})(1-|u_{f}|^{2}-|u\ -u_{f}|^{2}+<u-u_{f},r^{\prime }>_
\mathbb{E}^{3}}^{2})^{-1/2}\times $ \\
\\
$\times \frac{\partial }{\partial \sigma }\left\{ \frac{\bar{W
(1-|u_{f}|^{2}-|u-u_{f}|^{2}+<u-u_{f},r^{\prime }>_{\mathbb{E
^{3}}^{2}+|u-u_{f}|^{2}\hat{T}_{u-u_{f}}r^{\prime }/r^{\prime })}
(1-|u_{f}|^{2}-|u-u_{f}|^{2}+<u-u_{f},r^{\prime }>_{\mathbb{E
^{3}}^{2})^{1/2}}\right\}
\end{tabular}
\label{A6.17b}
\end{equation
with respect to the moving laboratory frame $\mathcal{K}.$ \ The latter can
be easily rewritten also as the Lorentz type force expression

\begin{equation}
\begin{array}{c}
dp/dt=\xi |r^{\prime }|E+\xi |r^{\prime }|u\times B-\xi |r^{\prime }|\
\nabla <u-u_{f},A>_{\mathbb{E}^{3}}+ \\
\\
+(1-|u_{f}|^{2})(1-|u_{f}|^{2}-|u\ -u_{f}|^{2}+<u-u_{f},r^{\prime }>_
\mathbb{E}^{3}}^{2})^{-1/2}\times \\
\\
\times \frac{\partial }{\partial \sigma }\left\{ \frac{\bar{W
(1-|u_{f}|^{2}-|u-u_{f}|^{2}+<u-u_{f},r^{\prime }>_{\mathbb{E
^{3}}^{2}+|u-u_{f}|^{2}\hat{T}_{u-u_{f}}r^{\prime }/r^{\prime })}
(1-|u_{f}|^{2}-|u-u_{f}|^{2}+<u-u_{f},r^{\prime }>_{\mathbb{E
^{3}}^{2})^{1/2}}\right\}
\end{array}
\label{A6.18}
\end{equation
where $B=\ \nabla \times A$ means, as usual, the external magnetic field and
\begin{equation}
E\ =\ \partial A\ /\partial t-\ \nabla \bar{W}\ \   \label{A6.19a}
\end{equation
means the corresponding electric field, acting on the string. Making use of
the standard scheme, one can derive, as above, the Hamiltonian
interpretation of dynamical equations \ (\ref{A6.13}), but which will not be
here discussed.

\subsection{Comments}

Based on the vacuum field theory approach, devised recently in \cit
{BPT,BPT1,BP}, we revisited the alternative charged point particle and
hadronic string electrodynamics models, having succeeded in treating their
Lagrangian and Hamiltonian properties. The obtained results were compared
with classical ones, owing to which a physically motivated choice of a true
model was argued. Another important aspect of the developed vacuum field
theory approach consists in singling out the decisive role of the related
rest \ reference frame $\mathcal{K}_{r},$ with respect to which the
relativistic object motion, in reality, realizes. Namely, with respect to
the proper rest reference \ frame evolution parameter $\tau \in \mathbb{R}$
all of our electrodynamics models allow both the Lagrangian and Hamiltonian
physically reasonable formulations, suitable for the canonical procedure.
The deeper physical nature of this fact remains, up today, as we assume, not
enough understood. We would like to recall here only very interesting
reasonings by R. Feynman, F.A. Berezin and M.S. Marinov, who argued in \cit
{Fe1,BM} that the relativistic expressions obtain physical sense only with
respect to \ the proper rest reference frames. In a sequel of our work we
plan to analyze our relativistic electrodynamic models subject to their
quantization and make a step toward the related vacuum quantum field theory
of infinite many particle systems.

\subsection{Supplement: The Maxwell electromagnetism within the vacuum field
theory}

We start from the following field theoretical model \cite{BP0} of the
microscopic vacuum medium structure, considered as some physical reality
imbedded into the standard three-dimensional Euclidean space reference frame
marked by three spatial coordinates $r\in \mathbb{E}^{3},$ endowed, as
before, with the standard scalar product $\ <\cdot ,\cdot >_{\mathbb{E
^{3}}, $ and parameterized by means of the scalar temporal parameter $t\in
\mathbb{R}.$ \ First we will describe the physical vacuum medium endowing it
with an everywhere smooth enough four-vector potential function
(W,A):M^{4}\rightarrow T^{\ast }\mathbb{(}M^{4}),$ \ defined in the
Minkowski space $M^{4}$ and naturally related to light propagation
properties. The material objects, imbedded into the vacuum medium, we will
model \ (classically here) by means of the scalar charge density function $\
\rho :M^{4}\rightarrow \mathbb{R}$ and the vector current density $\
J:M^{4}\rightarrow \mathbb{E}^{3},$ being also everywhere smooth enough
functions.

\begin{enumerate}
\item The \textit{first} field theory principle regarding the vacuum we
accept is formulated as follows: the four-vector function
(W,A):M^{4}\rightarrow T^{\ast }\mathbb{(}M^{4})$ satisfies the standard
Lorentz type continuity relationshi
\begin{equation}
\frac{1}{c}\frac{\partial W}{\partial t}+<\nabla ,A>_{\mathbb{E}^{3}}=0,
\label{M1.1}
\end{equation
where, by definition, $\nabla :=\partial /\partial r$ is the usual gradient
operator$\ $with respect to the spatial variable $r\in \mathbb{E}^{3}$ and
c $ is the light velocity in the vacuum.

\item The \textit{second} field theory principle we accept is a dynamical
wave relationship on the scalar potential component $W:M^{4}\rightarrow
\mathbb{R}:
\begin{equation}
\frac{1}{c^{2}}\frac{\partial ^{2}W}{\partial t^{2}}-\nabla ^{2}W=\rho ,
\label{M1.2}
\end{equation
assuming the linear law \ of the small vacuum uniform and isotropic
perturbation propagations in the space-time, understood here, evidently, as
a first (linear) approximation in the case of weak enough fields.

\item The \textit{third }principle is similar to the first one and means
simply the \ continuity condition for the density and current density
functions
\begin{equation}
\partial \rho /\partial t+<\nabla ,j>_{\mathbb{E}^{3}}=0.  \label{M1.3}
\end{equation}
\end{enumerate}

We need to note here that the vacuum field perturbations velocity parameter
c>0,$ used above, coincides with the vacuum light velocity, as we are trying
to derive successfully from these first principles the well-known Maxwell
electromagnetism field equations, to analyze the related Lorentz forces and
\ special relativity relationships. To do this, we first combine equations
\ref{M1.1}) and (\ref{M1.2})
\begin{equation*}
\frac{1}{c^{2}}\frac{\partial ^{2}W}{\partial t^{2}}=-<\nabla ,\frac{1}{c
\frac{\partial A}{\partial t}>_{\mathbb{E}^{3}}=<\nabla ,\nabla W>_{\mathbb{
}^{3}}+\rho ,
\end{equation*
whence
\begin{equation}
<\nabla ,-\frac{1}{c}\frac{\partial A}{\partial t}-\nabla W>_{\mathbb{E
^{3}}=\rho .  \label{M1.4}
\end{equation
Having put, by definition
\begin{equation}
E:=-\frac{1}{c}\frac{\partial A}{\partial t}-\nabla W,  \label{M1.5}
\end{equation
we obtain the first material Maxwell equatio
\begin{equation}
<\nabla ,E>_{\mathbb{E}^{3}}=\rho  \label{M1.6}
\end{equation
for the electric field $E:M^{4}\rightarrow \mathbb{E}^{3}.$ Having now
applied the rotor-operation $\ \ \nabla \times $ \ to expression (\ref{M1.5
) we obtain the first Maxwell field equatio
\begin{equation}
\frac{1}{c}\frac{\partial B}{\partial t}-\nabla \times E=0  \label{M1.7}
\end{equation
on the magnetic field vector function $B:M^{4}\rightarrow \mathbb{E}^{3},$
defined as \ \
\begin{equation}
B:=\nabla \times A.  \label{M1.8}
\end{equation}

\begin{remark}
\label{Rem_M1}It is useful to remark that \emph{the second field theory
principle} is exactly equivalent to the experimentally stated physical
relationships \ (\ref{M1.5}) and \ (\ref{M1.6}) for the electric field
E:M^{4}\rightarrow \mathbb{E}^{3}.$ Really, having applied the operator
\nabla \times $ to the left-hand side of \ (\ref{M1.5}), one obtains the
wave relationship \ (\ref{M1.2}).
\end{remark}

To derive the second Maxwell field equation we will make use of (\ref{M1.8
), (\ref{M1.1}) and (\ref{M1.5})
\begin{eqnarray}
\nabla \times B &=&\nabla \times (\nabla \times A)=\nabla <\nabla ,A>_
\mathbb{E}^{3}}-\nabla ^{2}A=  \notag \\
&=&\nabla (-\frac{1}{c}\frac{\partial W}{\partial t})-\nabla ^{2}A=\frac{1}{
}\frac{\partial }{\partial t}(-\nabla W-\frac{1}{c}\frac{\partial A}
\partial t}+\frac{1}{c}\frac{\partial A}{\partial t})-\nabla ^{2}A=  \notag
\\
&=&\frac{1}{c}\frac{\partial E}{\partial t}+(\frac{1}{c^{2}}\frac{\partial
^{2}A}{\partial t^{2}}-\nabla ^{2}A).  \label{M1.9}
\end{eqnarray
We have from (\ref{M1.5}), (\ref{M1.6}) and (\ref{M1.3}) that
\begin{equation*}
<\nabla ,\frac{1}{c}\frac{\partial E}{\partial t}>_{\mathbb{E}^{3}}=\frac{1}
c}\frac{\partial \rho }{\partial t}=-\frac{1}{c}<\nabla ,j>_{\mathbb{E}^{3}},
\end{equation*
or
\begin{equation}
<\nabla ,-\frac{1}{c^{2}}\frac{\partial ^{2}A}{\partial t^{2}}-\nabla (\frac
1}{c}\frac{\partial W}{\partial t})+\frac{1}{c}j>_{\mathbb{E}^{3}}=0.
\label{M1.10}
\end{equation
Now making use of (\ref{M1.1}), from (\ref{M1.10}) we obtain that \
\begin{equation}
\frac{1}{c^{2}}\frac{\partial ^{2}A}{\partial t^{2}}-\nabla ^{2}A=\frac{1}{c
(j+\nabla \times S)  \label{M1.12}
\end{equation
for some smooth vector function $S:M^{4}\rightarrow \mathbb{E}^{3}.$ Here we
need to note that continuity equation (\ref{M1.3}) is defined, concerning
the current density vector $j:M^{4}\rightarrow \mathbb{R}^{3},$ up to a
vorticity expression, that is $j\simeq j+\nabla \times S$ and equation (\re
{M1.12}) can finally be rewritten as
\begin{equation}
\frac{1}{c^{2}}\frac{\partial ^{2}A}{\partial t^{2}}-\nabla ^{2}A=\frac{1}{c
j.  \label{M1.13}
\end{equation
Having substituted (\ref{M1.13}) into (\ref{M1.9}) we obtain the second
Maxwell field equation

\begin{equation}
\nabla \times B-\frac{1}{c}\frac{\partial E}{\partial t}=\frac{1}{c}j.
\label{M1.14}
\end{equation
In addition, from (\ref{M1.8}) one also finds the magnetic no-charge
relationshi
\begin{equation}
<\nabla ,B>_{\mathbb{E}^{3}}=0.  \label{M1.15}
\end{equation}

Thus, we have derived all the Maxwell electromagnetic field equations from
our three main principles (\ref{M1.1}), (\ref{M1.2}) and (\ref{M1.3}). The
success of our undertaking will be more impressive if we adapt our results
to those following from the well-known relativity theory in the case of
point charges or masses. Below we will try to demonstrate the corresponding
derivations based on some completely new physical conceptions of the vacuum
medium first discussed in \cite{Rep,BP}.

It is interesting to analyze a particular case of the first field theory
vacuum principle (\ref{M1.1}) when the following local conservation law for
the scalar potential field function $W:M^{4}\rightarrow \mathbb{R}$ holds
\begin{equation}
\frac{d}{dt}\int_{\Omega _{t}}W(t,r^{\prime })d^{3}r^{\prime }=0,
\label{M1.16}
\end{equation
where $\Omega _{t}\subset \mathbb{E}^{3}$ is any open domain in space
\mathbb{E}^{3}$ with the smooth boundary $\partial \Omega _{t}$ \ for all
t\in \mathbb{R}$ and $d^{3}r^{\prime }$ \ is the standard volume measure in
\mathbb{E}^{3}$ in a vicinity of the point $r^{\prime }(t):=r_{f}(t)-r\in
\Omega _{t}.$ Having calculated expression (\ref{M1.16}) we obtain the
following equivalent continuity equation
\begin{equation}
\frac{1}{c}\frac{\partial W}{\partial t}+<\nabla ,\frac{u}{c}W>_{\mathbb{E
^{3}}=0,  \label{M1.17}
\end{equation
where $\nabla :=$ $\partial /\partial r^{\prime }$ is, as above, the
gradient operator with respect to the spatial variable $r^{\prime }\in
\mathbb{E}^{3}$ and $u:=dr_{f}(t)/dt$ is the velocity vector of a vacuum
medium perturbation influenced by the charge $\xi ,$ located at point
r_{f}(t)\in \mathbb{E}^{3}$ and carrying the field potential quantity $W.$
Comparing now equations (\ref{M1.1}), (\ref{M1.17}) and using equation (\re
{M1.3}) we can make the suitable very important identifications:
\begin{equation}
A=\frac{u}{c}W,\text{ \ \ \ \ }j=\rho u,  \label{M1.18}
\end{equation
well known from the classical electrodynamics \cite{LL} and
superconductivity theory \cite{Fe1,Kle}. Thus, we are faced with a new
physical interpretation of the conservative electromagnetic field theory
when the vector \ potential $A:M^{4}\rightarrow \mathbb{E}^{3}$ is
completely determined via expression (\ref{M1.18}) by the scalar field
potential function $W:M^{4}\rightarrow \mathbb{R}.$ It is also evident that
all the Maxwell electromagnetism filed equations derived above hold as well
in the case

Consider now the conservation equation (\ref{M1.16}) jointly with the
related integral "vacuum momentum" conservation condition
\begin{equation}
\frac{d}{dt}\int_{\Omega _{t}}(\frac{Wv}{c^{2}})d^{3}r^{\prime }=0,~~~\Omega
_{t}|_{t=0}=\Omega _{0},  \label{M1.19}
\end{equation
where, as above, $\Omega _{t}\subset \mathbb{E}^{3}$ is for any time $t\in
\mathbb{R}$ an open domain with the smooth boundary $\partial {\Omega _{t}},$
whose evolution is governed by the equation
\begin{equation}
dr^{\prime }/dt=u(r^{\prime },t)  \label{M1.20}
\end{equation
for all $r^{\prime }\in \Omega _{t}$ and $t\in \mathbb{R},$ as well as by
the initial state of the boundary $\partial {\Omega }_{0}.$ As a result of
relation (\ref{M1.19}) one obtains the new continuity equation
\begin{equation}
\frac{d(uW)}{dt}+uW<\nabla ,u>_{\mathbb{E}^{3}}=0.  \label{M1.21}
\end{equation
Now making use of (\ref{M1.17}) in the equivalent form
\begin{equation*}
\frac{dW}{dt}+W<\nabla ,u>_{\mathbb{E}^{3}}=0,
\end{equation*
we finally obtain a very interesting local conservation relationship \
\begin{equation}
du/dt=0\   \label{M1.22}
\end{equation
on the vacuum matter perturbations velocity $u=dr^{\prime }/dt,$ which holds
for all values of the time parameter $t\in \mathbb{R}.$ As it is easy to
observe, the obtained relationship completely coincides with the well-known
hydrodynamic equation \cite{MC} of ideal compressible liquid without any
external exertion, that is, any external forces and field "pressure" \ are
equally identical to zero. We received a natural enough result where the
propagation velocity of the vacuum field matter is constant and equals
exactly $u=c,$ that is the light velocity in the vacuum, if to take into
account the starting wave equation (\ref{M1.2}) owing to which the small
vacuum field matter perturbations propagate in the space with the light
velocity.

\section{The electromagnetic Dirac-Fock-Podolsky problem and symplectic
properties of the Maxwell and Yang-Mills type dynamical systems}

\subsection{Introduction}

When investigating different dynamical systems on canonical symplectic
manifolds, invariant under action of certain symmetry groups, additional
mathematical structures often appear, the analysis of which shows their
importance for understanding many related problems under study. Amongst them
\ we here mention the Cartan type connection on an associated principal
fiber bundle, which enables one to study in more detail the properties of
the investigated dynamical system in the case of its reduction upon the
corresponding invariant submanifolds and quotient spaces, associated with
them.

Problems related to the investigation of properties of reduced dynamical
systems on symplectic manifolds were studied, e.g., in \cite{AM,PM,BPS,Pr,PS
, where the relationship between a symplectic structure on the reduced space
and the available connection on a principal fiber bundle was formulated in
explicit form. Other aspects of dynamical systems related to properties of
reduced symplectic structures were studied in \cite{Ku,Ra,HK,HKup}, where,
in particular, the reduced symplectic structure was explicitly described
within the framework of the classical Dirac scheme, and several applications
to nonlinear (including celestial) dynamics were given.

It is well-known \cite{BS,Th,Di,BPT,BP,Pa} that the Hamiltonian theory of
electromagnetic Maxwell equations faces a very important classical problem
of introducing into the unique formalism the well known Lorenz conditions,
ensuring both the wave structure of propagating quanta and the positivity of
energy. Regretfully, in spite of classical studies on this problem given by
Dirac, Fock and Podolsky \cite{DFP}, the problem remains open, and the
Lorenz condition is imposed within the modern electrodynamics as the
external constraint not entering a priori the initial Hamiltonian (or
Lagrangian) theory. Moreover, when trying to quantize the electromagnetic
theory, as it was shown by Pauli, Dirac, Bogolubov and Shirkov and others
\cite{BS,Pa,Di,BjD}, within the existing approaches the quantum Lorenz
condition could not be satisfied, except in the average sense, since it
becomes not compatible with the related quantum dynamics. This problem
stimulated us to study this problem from the so called symplectic reduction
theory \cite{MW,PS,Pr}, which allows the systematic introduction into the
Hamiltonian formalism the external charge and current conditions, giving
rise to a partial solution to the Lorenz condition problem mentioned above.
Some applications of the method to Yang-Mills type equations interacting
with a point charged particle, are presented in detail. In particular, based
on analysis of reduced geometric structures on fibered manifolds, invariant
under the action of a symmetry group, we construct the symplectic structures
associated with connection forms on suitable principal fiber bundles. We
present suitable mathematical preliminaries of the related Poissonian
structures on the corresponding reduced symplectic manifolds, which are
often used \cite{AM,MW,Kup} in various problems of dynamics in modern
mathematical physics, and apply them to study the non-standard Hamiltonian
properties of the Maxwell and Yang-Mills type dynamical systems. We
formulate a symplectic analysis of the important Lorenz type constraints,
which describe the electrodynamic vacuum properties.

We formulate a symplectic reduction theory of the classical Maxwell
electromagnetic field equations and prove \cite{BPTP} that the important
Lorenz condition, ensuring the existence of electromagnetic waves \cit
{BS,BjD,Fe1,LL}, can be naturally included into the Hamiltonian picture,
thereby solving the Dirac, Fock and Podolsky problem \cite{DFP} mentioned
above. We also study from the symplectic reduction theory the Poissonian
structures and the classical minimal interaction principle related with
Yang-Mills type equations.

\subsection{The symplectic reduction on cotangent fiber bundles with symmetr
}

Consider an $n$-dimensional smooth manifold $M$ and the cotangent vector
fiber bundle $T^{\ast }(M).$ We equip (see \cite{Go}, Chapter VII; \cit
{DNFo}) the cotangent space $T^{\ast }(M)$ with the canonical Liouville
1-form $\lambda (\alpha ^{(1)}):=$ $pr_{M}^{\ast }\alpha ^{(1)}\in \Lambda
^{1}(T^{\ast }(M)),$ where $pr_{M}:T^{\ast }(M)\rightarrow M$ is the
canonical projection and, by definition
\begin{equation}
\alpha ^{(1)}(u)=\sum_{j=1}^{n}v_{j}du^{j},  \label{1}
\end{equation
where $(u,v)\in T^{\ast }(M)$ are the corresponding canonical local
coordinates on $T^{\ast }(M).$ Thus, any group of diffeomorphisms of the
manifold $M$ naturally lifted to the fiber bundle $T^{\ast }(M)$ preserves
the invariance of the canonical 1-form $\lambda (\alpha ^{(1)})\in \Lambda
^{1}(T^{\ast }(M)).$ In particular, if a smooth action of a Lie group $G$ is
given on the manifold $M,$ then every element $a\in \mathcal{G},$ where
\mathcal{G}$ is the Lie algebra of the Lie group $G,$ generates the vector
field $k_{a}\in T(M)$ in a natural manner. Furthermore, since the group
action on $M,$ i.e.,
\begin{equation}
\varphi :G\times M\rightarrow M,  \label{2}
\end{equation
generates a diffeomorphism $\ \varphi _{g}\in Diff$ $M$ for every element
g\in G,$ this diffeomorphism is naturally lifted to the corresponding
diffeomorphism $\varphi _{g}^{\ast }$ $\in Diff$ $T^{\ast }(M)$ of the
cotangent fiber bundle $T^{\ast }(M),$ which also leaves \ the canonical
1-form $pr_{M}^{\ast }\alpha ^{(1)}\in \Lambda ^{1}(T^{\ast }(M))$
invariant. Namely, the equality

\begin{equation}
\varphi _{g}^{\ast }\lambda (\alpha ^{(1)})=\lambda (\alpha ^{(1)})
\label{3}
\end{equation
holds \cite{AM,Go,PM} for every 1-form $\alpha ^{(1)}\in \Lambda ^{1}(M).$
Thus, we can define on $T^{\ast }(M)$ the corresponding vector field
K_{a}:T^{\ast }(M)\rightarrow T(T^{\ast }(M))$ for every element $a\in
\mathcal{G}.$ Then condition (\ref{3}) can be rewritten in the following
form for all $a\in \mathcal{G}:$

\begin{equation*}
L_{K_{a}}\cdot pr_{M}^{\ast }\alpha ^{(1)}=pr_{M}^{\ast }\cdot
L_{k_{a}}\alpha ^{(1)}=0,
\end{equation*
where $L_{K_{a}}$ and $L_{k_{a}}$ are the ordinary Lie derivatives on
\Lambda ^{1}(T^{\ast }(M))$ and $\Lambda ^{1}(M),$ respectively.

The canonical symplectic structure on $T^{\ast }(M)$ is defined as

\begin{equation}
\omega ^{(2)}:=d\lambda (\alpha ^{(1)})  \label{4}
\end{equation
and is invariant, i.e., $L_{K_{a}}\omega ^{(2)}=0$ for all $a\in \mathcal{G
. $

For any smooth function $H\in D(T^{\ast }(M)),$ \ a Hamiltonian vector field
$K_{H}:T^{\ast }(M)\rightarrow T(T^{\ast }(M))$ \ such that

\begin{equation}
i_{K_{H}}\omega ^{(2)}=-dH  \label{5}
\end{equation
is defined, and vice versa, because the symplectic 2-form (\ref{4}) is
non-degenerate. Using (\ref{5}) and (\ref{4}), we easily establish that the
Hamiltonian function $H:=H_{K}\in D(T^{\ast }(M))$ is given by the
expression $H_{K}=pr_{M}^{\ast }\alpha ^{(1)}(K_{H})=\alpha
^{(1)}(pr_{M}^{\ast }K_{H})=\alpha ^{(1)}(k_{H}),$ where $k_{H}\in T(M)$ is
the corresponding vector field on the manifold $M,$ whose lifting to the
fiber bundle $T^{\ast }(M)$ coincides with the vector field $K_{H}:T^{\ast
}(M)\rightarrow T(T^{\ast }(M)).$ For $K_{a}:T^{\ast }(M)\rightarrow
T(T^{\ast }(M)),$ where $a\in \mathcal{G},$ \ it is easy to establish that
the corresponding Hamiltonian function $H_{a}=\alpha
^{(1)}(k_{a})=pr_{M}^{\ast }$ $\alpha ^{(1)}(K_{a})$ for $a\in \mathcal{G}$
defines \cite{AM,PM,BPS} a linear momentum \ mapping $l:T^{\ast
}(M)\rightarrow \mathcal{G}^{\ast }$ according to the rule

\begin{equation}
H_{a}:=<l,a>,  \label{6}
\end{equation
where $<$\textperiodcentered $,$\textperiodcentered $\ >$ \ is the
corresponding convolution on $\mathcal{G}^{\mathcal{\ast }}$ $\times
\mathcal{G}.$ By virtue of definition (\ref{6}), the momentum mapping
l:T^{\ast }(M)\rightarrow \mathcal{G}^{\ast }$ is invariant under the action
of any invariant Hamiltonian vector field $K_{b}:T^{\ast }(M)\rightarrow
T(T^{\ast }(M))$ for any $b\in \mathcal{G}.$ Indeed,
L_{K_{b}}<l,a>=L_{K_{b}}H_{a}=-L_{K_{a}}H_{b}=0,$ because, by definition,
the Hamiltonian function $H_{b}\in D(T^{\ast }(M))$ is invariant under the
action of any vector field $K_{a}:T^{\ast }(M)\rightarrow T(T^{\ast }(M)),$
a\in \mathcal{G}.$

We now fix a regular value of the momentum mapping $l(u,v)=\xi \in \mathcal{
}^{\ast }$ and consider the corresponding submanifold $\mathcal{M}_{\xi
}:=\{(u,v)\in T^{\ast }(M):l(u,v)=\xi \in \mathcal{G}^{\ast }\}.$ On the
basis of definition (\ref{1}) and the invariance of the 1-form $pr_{M}^{\ast
}$ $\alpha ^{(1)}\in \Lambda ^{1}(T^{\ast }(M))$ under the action of the Lie
group $G$ on $T^{\ast }(M),$ we can write the equalities

\begin{eqnarray}
&<&l(g\circ (u,v)),a>=pr_{M}^{\ast }\alpha ^{(1)}(K_{a})(g\circ (u,v))=
\notag \\
&=&pr_{M}^{\ast }\alpha ^{(1)}(K_{Ad_{g-1}a})(u,v):=  \label{7} \\
&=&<l(u,v),Ad_{g-1}a>=<Ad_{g-1}^{\ast }l(u,v),a>  \notag
\end{eqnarray
for any $g\in G$ and all $a\in \mathcal{G}$ and $(u,v)\in T^{\ast }(M).$
Using (\ref{7}) we establish that, for every $g\in G$ and all $(u,v)\in
T^{\ast }(M),$ the following relation is true: $l(g\circ
(u,v))=Ad_{g^{-1}}^{\ast }l(u,v).$ This means that the diagra
\begin{equation*}
\begin{array}{ccc}
T^{\ast }(M) & \overset{l}{\rightarrow } & \mathcal{G}^{\mathcal{\ast }} \\
g\downarrow &  & \downarrow Ad_{g-1}^{\ast } \\
T^{\ast }(M) & \overset{l}{\rightarrow } & \mathcal{G}^{\mathcal{\ast }
\end{array
\end{equation*
is commutative for all elements $g\in G;$ the corresponding action
g:T^{\ast }(M)\rightarrow T^{\ast }(M)$ is called equivariant \cite{AM,PM}.

Let $\ G_{\xi }\subset G$ \ denote the stabilizer of a regular element $\xi
\in \mathcal{G}^{\ast }$ with respect to the related coadjoint action. \ It
is obvious that in this case the action of the Lie subgroup $G_{\xi }$ on
the submanifold $\mathcal{M}_{\xi }\subset T^{\ast }(M)$ is naturally
defined; we assume that it is free and proper. According to this action on
\mathcal{M}_{\xi },$ we can define \cite{AM,Ra,HKup,Mo,Kup} a so-called
reduced space $\mathcal{\bar{M}}_{\xi }$ by taking the factor with respect
to the action of the subgroup $G_{\xi }$ on $\mathcal{M}_{\xi },$ i.e.,

\begin{equation}
\mathcal{\bar{M}}_{\xi }:=\mathcal{M}_{\xi }/G_{\xi }.  \label{8}
\end{equation
The quotient space (\ref{8}) induces a symplectic structure $\bar{\omega
_{\xi }^{(2)}\in \Lambda ^{2}(\mathcal{\bar{M}}_{\xi })$ on itself, which is
defined as follows
\begin{equation}
\bar{\omega}_{\xi }^{(2)}(\bar{\eta}_{1},\bar{\eta}_{2})=\omega _{\xi
}^{(2)}(\eta _{1},\eta _{2}),  \label{9}
\end{equation
where $\bar{\eta}_{1},\bar{\eta}_{2}\in T(\mathcal{\bar{M}}_{\xi })$ are
arbitrary vectors onto which vectors $\eta _{1},\eta _{2}\in T(\mathcal{M
_{\xi })$ are projected, taken at any point $(u_{\xi },v_{\xi })\in \mathcal
M}_{\xi },$ being uniquely projected onto the point $\bar{\mu}_{\xi }\in
\mathcal{\bar{M}}_{\xi },$ according to (\ref{8}).

Let $\pi _{\xi }:\mathcal{M}_{\xi }\rightarrow T^{\ast }(M)$ denote the
corresponding imbedding mapping into $T^{\ast }(M)$ and let $r_{\xi }
\mathcal{M}_{\xi }\mathcal{\rightarrow \bar{M}}_{\xi }$ denote the
corresponding reduction to the space $\mathcal{\bar{M}}_{\xi }.$ Then
relation (\ref{9}) can be rewritten equivalently in the form of the equality
\begin{equation}
r_{\xi }^{\ast }\bar{\omega}_{\xi }^{(2)}=\pi _{\xi }^{\ast }\omega ^{(2)},
\label{10}
\end{equation
defined on vectors on the cotangent space $T^{\ast }(\mathcal{M}_{\xi }).$
To establish the symplecticity of the 2-form $\omega _{\xi }^{(2)}\in
\Lambda ^{2}(\mathcal{\bar{M}}_{\xi }),$ we use the corresponding
non-degeneracy of the Poisson bracket $\{$\textperiodcentered $,
\textperiodcentered $\}_{\xi }^{r}$ on $\mathcal{\bar{M}}_{\xi }.$ To
calculate it, we use a Dirac type construction, defining functions on $\
\mathcal{\bar{M}}_{\xi }$ as certain $G_{\xi }$-invariant functions on the
submanifold $\mathcal{M}_{\xi }.$ Then one can calculate the Poisson bracket
$\{$\textperiodcentered $,$\textperiodcentered $\}_{\xi }$ of such functions
that corresponds to symplectic structure (\ref{4}) as an ordinary Poisson
bracket on $T^{\ast }(M),$ arbitrarily extending these functions from the
submanifold $\mathcal{M}_{\xi }$ $\subset T^{\ast }(M)$ to a certain
neighborhood $U(\mathcal{M}_{\xi })\subset T^{\ast }(M).$ It is obvious that
two extensions of a given function to the neighborhood $U(\mathcal{M}_{\xi
}) $ of this type differ by a function that vanishes on the submanifold
\mathcal{M}_{\xi }\subset T^{\ast }(M).$ The difference between the
corresponding Hamiltonian fields of these two different extensions to $U
\mathcal{M}_{\xi })$ is completely controlled by the conditions of the
following lemma (see also \cite{AM,PM,HK,HKup,Ra,Pr}).

\begin{lemma}
\label{Lm_2.1}Suppose that a function $f:U(\mathcal{M}_{\xi })\rightarrow
\mathbb{R}$ is smooth and vanishes on $\mathcal{M}_{\xi }$ $\subset T^{\ast
}(M),$ i.e., $f|_{\mathcal{M}_{\xi }}=0.$ Then, at every point $(u_{\xi
},v_{\xi })\in \mathcal{M}_{\xi }$ the corresponding Hamiltonian vector
field $K_{f}\in T(U(\mathcal{M}_{\xi }))$ is tangent to the orbit
Or(G;(u_{\xi },v_{\xi })).$
\end{lemma}

\begin{proof}
It is obvious that the submanifold $M_{\xi }\subset T^{\ast }(M)$ is defined
by a certain collection of relations of the typ
\begin{equation}
H_{a_{s}}=\xi _{s},\text{ \ \ \ \ \ \ \ \ }\xi _{s}:=<\xi ,a_{s}>,
\label{11}
\end{equation
where $a_{s}\in \mathcal{G},s=\overline{1,dimG},$ is a certain basis of the
Lie algebra $\mathcal{G},$ which follows from definition (\ref{6}). Since a
function $f:U(\mathcal{M}_{\xi })\rightarrow \mathbb{R}$ vanishes on
\mathcal{M}_{\xi },$ we can write the following equality
\begin{equation*}
f=\sum_{s=1}^{\dim \mathcal{G}}(H_{a_{s}}-\xi _{s})f_{s},
\end{equation*
where $\ f_{s}:U(\mathcal{M}_{\xi })\rightarrow \mathbb{R},$ $\ \ s
\overline{1,dimG},$ is a certain collection of functions in the neighborhood
$U(\mathcal{M}_{\xi })$. We take an arbitrary tangent vector $\eta \in T(U
\mathcal{M}_{\xi }))$ at the point $(u_{\xi },v_{\xi })\in \mathcal{M}_{\xi
} $ \ and calculate the expressio
\begin{eqnarray}
&<&df(u_{\xi },v_{\xi }),\eta (u_{\xi },v_{\xi })>=\sum_{s=1}^{\dim \mathcal
G}}<dH_{a_{s}}(u_{\xi },v_{\xi }),\eta (u_{\xi },v_{\xi })>f_{s}(u_{\xi
},v_{\xi })=  \notag \\
&=&-\sum_{s=1}^{\dim \mathcal{G}}\omega ^{(2)}(K_{a_{i}}(u_{\xi },v_{\xi
}),\eta (u_{\xi },v_{\xi }))f_{s}(u_{\xi },v_{\xi })=  \notag \\
&=&-\omega ^{(2)}(\sum_{s=1}^{\dim \mathcal{G}}K_{a_{s}}(u_{\xi },v_{\xi
})f_{s}(u_{\xi },v_{\xi }),\eta (u_{\xi },v_{\xi }))=  \notag \\
&=&-<i_{\left( \sum_{s=1}^{\dim \mathcal{G}}K_{a_{s}}(u_{\xi },v_{\xi
})f_{s}(u_{\xi },v_{\xi })\right) }\omega ^{(2)},\eta (u_{\xi },v_{\xi })>.
\label{12}
\end{eqnarray
It follows from the arbitrariness of the vector $\eta \in T(\mathcal{M}_{\xi
})$ at the point $(u_{\xi },v_{\xi })\in \mathcal{M}_{\xi }$ and relation
\ref{12}) tha
\begin{equation*}
K_{f}=\sum_{s=1}^{\dim \mathcal{G}}K_{a_{s}}f_{s},
\end{equation*
i.e., $K_{f}:\mathcal{M}_{\xi }\rightarrow T(Or(G)),$ which was to be proved.
\end{proof}

As a corollary of Lemma \ref{Lm_2.1}, we obtain an algorithm for the
determination of the reduced Poisson bracket $\{$\textperiodcentered $,
\textperiodcentered $\}_{\xi }^{r}$ on the space $\mathcal{\bar{M}}_
\mathcal{\xi }}$ according to definition (\ref{10}). Namely, we choose two
functions defined on $\mathcal{M}_{\xi }$ and invariant under the action of
the subgroup $G_{\xi }$ and arbitrarily smoothly extend them to a certain
open domain $U(\mathcal{M}_{\xi })\subset T^{\ast }(M).$ Then we determine
the corresponding Hamiltonian vector fields on $T^{\ast }(M)$ and project
them onto the space tangent to $\mathcal{M}_{\xi },$ adding, if necessary,
the corresponding vectors tangent to the orbit $Or(G).$ It is obvious that
the projections obtained depend on the chosen extensions to the domain $U
\mathcal{M}_{\xi })\subset T^{\ast }(M).$ As a result, we establish that the
reduced Poisson bracket $\{$\textperiodcentered $,$\textperiodcentered
\}_{\xi }^{r}$ is uniquely defined via the restriction of the initial
Poisson bracket upon $\mathcal{M}_{\xi }$ $\subset T^{\ast }(M).$ By virtue
of the non-degeneracy of the latter and the functional independence of the
basis functions (\ref{11}) on the submanifold $U(\mathcal{M}_{\xi })\subset
T^{\ast }(M),$ the reduced Poisson bracket $\{$\textperiodcentered $,
\textperiodcentered $\}_{\xi }^{r}$ appears to be \cite{AM,PM,Ra}
non-degenerate on $\mathcal{\bar{M}}_{\xi }.$ \ As a consequence of the
non-degeneracy, we establish that the dimension of the reduced space
\mathcal{\bar{M}}_{\xi }$ is even. Taking into account that the element $\xi
\in \mathcal{G}^{\mathcal{\ast }}$ is regular and the dimension of the Lie
algebra of the stabilizer $\mathcal{G}_{\xi }$ is equal to $dim$ $G_{\xi },$
we easily establish that $dim$ $\mathcal{\bar{M}}_{\xi }=$ $dim$ $T^{\ast
}(M)-2dim$ $\mathcal{G}_{\xi }.$ Since, by construction, $dim$ $T^{\ast
}(M)=2n,$ we conclude that the dimension of the reduced space $\mathcal{\bar
M}}_{\xi }$ is necessarily even.

For the correctness of the algorithm, it is necessary to establish the
existence of the corresponding projections of Hamiltonian vector fields onto
the tangent space $T(\mathcal{M}_{\xi }).$ The following statement is true.

\begin{theorem}
\label{Tm_2.1}At every point $(u_{\xi },v_{\xi })\in \mathcal{M}_{\xi },$
one can choose a vector $V_{f}\in T(Or(G))$ such that $K_{f}(u_{\xi },v_{\xi
})$ $+V_{f}(u_{\xi },v_{\xi })\in T_{(u_{\xi },v_{\xi })}(\mathcal{M}_{\xi
}).$ Furthermore, the vector $V_{f}\in T(Or(G))$ is determined uniquely up
to a vector tangent to the orbit $Or(G_{\xi }).$
\end{theorem}

\begin{proof}
Note that the orbit $Or(G;(u_{\xi },v_{\xi }))$ passing through the point
(u_{\xi },v_{\xi })\in \mathcal{M}_{\xi }$ is always symplectically
orthogonal to the tangent space $T_{(u_{\xi },v_{\xi })}(\mathcal{M}_{\xi
}). $ Indeed, for any vector $\eta \in T(\mathcal{M}_{\xi })$ and $a\in
\mathcal{G},$ we have $\omega ^{(2)}(\eta ,K_{a})=-i_{K_{a}}\omega
^{(2)}(\eta )=dH_{a}(\eta )=0,$ because the submanifold $\mathcal{M}_{\xi }$
$\subset T^{\ast }(M)$ is defined by the equality $<\xi ,a>=H_{a}$ for all
a\in \mathcal{G},$ i.e., $dH_{a}=0$ on $\mathcal{M}_{\xi }.$ Thus, $T
\mathcal{M}_{\xi })\cap T(Or(G))=T(Or(G))$ because $H_{a}\circ g_{\xi
}=H_{a} $ for all $g_{\xi }\in G_{\xi },$ which follows from the invariance
of the element $\xi \in \mathcal{G}^{\ast }$ under the action of the Lie
group $G\xi .$\bigskip\ We now solve the imbedding condition $\
K_{f}+V_{f}\in T(\mathcal{M}_{\xi }), $ or the equation
\begin{equation}
\omega ^{(2)}(K_{f}+V_{f},K_{a})=0  \label{13}
\end{equation
on the manifold $\mathcal{M}_{\xi }$ $\subset T^{\ast }(M)$ for all $a\in
\mathcal{G}.$ We rewrite equality (\ref{13}) in the form
\begin{equation}
K_{a}f=\omega ^{(2)}(V_{f},K_{a})  \label{14}
\end{equation
on $\mathcal{M}_{\xi }$ for all $a\in \mathcal{G};$ it is obvious that the
2-form on the right-hand side of (\ref{14}) depends only on the element $\xi
\in G^{\ast }.$ Taking into account the equivariance of the group action on
T^{\ast }(M)$ and the obvious equalit
\begin{equation*}
\omega ^{(2)}(K_{a},K_{b})=pr_{M}^{\ast }\alpha
^{(1)}([K_{a},K_{b}])=-pr_{M}^{\ast }\alpha ^{(1)}(K_{[a,b]})
\end{equation*
for all $a,b\in \mathcal{G},$ we establish that there exists an element
a_{f}\in \mathcal{G}$ such that $V_{f}=K_{a_{f}}\in T(Or(G))$ an
\begin{eqnarray}
\omega ^{(2)}(V_{f},K_{a}) &=&\omega ^{(2)}(K_{a_{f}},K_{a})=pr_{M}^{\ast
}\alpha ^{(1)}([K_{a},K_{a_{f}}])=  \notag \\
&=&pr_{M}^{\ast }\alpha ^{(1)}(K_{[a_{f},a]})=H_{[a_{f},a]}=<l,[a_{f},a]>=
\notag \\
&=&<\xi ,[a_{f},a]>=<ad_{a_{f}}^{\ast }\xi ,a>  \label{15}
\end{eqnarray
on $\mathcal{M}_{\xi }$ for all $a\in \mathcal{G}.$ Since $ad_{a_{f}}^{\ast
}\xi =0$ for any $a_{f}\in \mathcal{G}_{\xi },$ we conclude that, on the
quotient space $\mathcal{G}/\mathcal{G}_{\xi }$ the right-hand side of (\re
{15}) defines a non-degenerate skew-symmetric form associated with the
canonical isomorphism $\hat{\xi}:\mathcal{G}/\mathcal{G}_{\xi }\rightarrow
\mathcal{G}/\mathcal{G}_{\xi })^{\ast },$ where, by definition
\begin{equation}
<\hat{\xi}(\tilde{a}),\tilde{b}>:=<\xi ,[a,b]>  \label{16}
\end{equation
for any $\tilde{a}$ and $\tilde{b}$ $\in $ $\mathcal{G}/\mathcal{G}_{\xi }$
with the corresponding representatives $a$ and $b{\in }\mathcal{G}.$
Further, since the function $f:\mathcal{M}_{\xi }\rightarrow \mathbb{R}$ is
G_{\xi }$-invariant on $\mathcal{M}_{\xi }$ $\subset T^{\ast }(M),$ the
right-hand side of (\ref{14}) defines an element $\mu _{f}\in (\mathcal{G}
\mathcal{G}_{\xi })^{\ast }$ by the equality
\begin{equation*}
\mu _{f}:\tilde{a}:=-K_{a}f
\end{equation*
for all $a\in G.$ Using relations (\ref{15}) ) and (\ref{16}), we establish
that there exists the element
\begin{equation*}
\tilde{a}_{f}=\hat{\xi}^{-1}\circ \mu _{f}\in \mathcal{G}/\mathcal{G}_{\xi }.
\end{equation*
Since the element $\tilde{a}_{f}\in \mathcal{G}/\mathcal{G}_{\xi }$ is
associated with the element $a_{f}$ $(mod$ $\mathcal{G}_{\xi })$ $\in
\mathcal{G},$ which uniquely generates a locally defined vector field
K_{a_{f}}:Or(G)\rightarrow T(Or(G)),$ using the fact that $V_{f}=K_{a_{f}}$
on $\mathcal{M}_{\xi },$ we complete the proof of the theorem.
\end{proof}

Now assume that two functions $f_{1},f_{2}\in D(\mathcal{M}_{\xi })$ are
G_{\xi }$-invariant. Then their reduced Poisson bracket $\{f_{1},f_{2}\}_
\xi }^{r}$ on $\mathcal{\bar{M}}_{\xi }$ is defined according to the rule:
\begin{equation}
\{f_{1},f_{2}\}_{\xi }^{r}:=-\omega
^{(2)}(K_{f_{1}}+V_{f_{1}},K_{f_{2}}+V_{f_{2}})=\{f_{1},f_{2}\}+\omega
^{(2)}(V_{f_{1}},V_{f_{2}}),  \label{17}
\end{equation
where we have used the following identities on $\mathcal{M}_{\xi }$ $\subset
T^{\ast }(M)$:
\begin{equation*}
\omega ^{(2)}(K_{f_{1}}+V_{f_{1}},V_{f_{2}})=0=\omega
^{(2)}(K_{f_{2}}+V_{f_{2}},V_{f_{1}}),
\end{equation*
being simple consequences of equality (\ref{13}) on $\mathcal{M}_{\xi }.$
Regarding (\ref{15}), relation (\ref{17}) takes the form
\begin{equation}
\{f_{1},f_{2}\}_{\xi }^{r}=\{f_{1},f_{2}\}+\frac{1}{2
(V_{f_{1}}f_{2}-V_{f_{2}}f_{1}),  \label{18}
\end{equation
where $f_{1},f_{2}\in D(\mathcal{M}_{\xi })$ are arbitrary smooth extensions
of the $G_{\xi }$-invariant functions defined earlier on the domain $U
\mathcal{M}_{\xi }).$ Thus, the following theorem holds.

\begin{theorem}
\label{Tm_2.2}The reduced Poisson bracket of two functions on the quotient
space $\mathcal{\bar{M}}_{\xi }$ $=$ $\mathcal{M}_{\xi }/G_{\xi }$ is
determined with the use of their arbitrary smooth extensions to functions on
an open neighborhood $U(\mathcal{M}_{\xi })$ according to the Dirac-type
formula (\ref{18}).
\end{theorem}

\subsubsection{The symplectic reduction on principal fiber bundles with
connection}

We begin by reviewing the backgrounds of the reduction theory subject to
Hamiltonian systems \ with symmetry on principle fiber bundles. The material
is partly available in \cite{GS,Ku}, so here it will be only sketched in
notations suitable for us.

Let $G$ denote a given Lie group with the unity element $e\in G$ and the
corresponding Lie algebra $\mathcal{G}$ $\simeq T_{e}(G).$ Consider a
principal fiber bundle $p:(M,\varphi )\rightarrow N$ with the structure
group $G$ and base manifold $N,$ on which the Lie group $G$ acts by means of
a mapping $\ \varphi :M\times G\rightarrow M.$ Namely, for each $g\in G$
there is a group diffeomorphism $\varphi _{g}:M\rightarrow M,$ generating
for any fixed $u\in M$ the following induced mapping: $\hat{u}:G\rightarrow
M,$ where
\begin{equation}
\hat{u}(g)=\varphi _{g}(u).  \label{0.1}
\end{equation}

On the principal fiber bundle $p:(M,\varphi )\rightarrow N$ \ a connection
\Gamma (${$\mathcal{A}$}$)$ is assigned by means of such a morphism {
\mathcal{A}$}$:(T(M),\varphi _{g\ast })\rightarrow (\mathcal{G},Ad_{g^{-1}})$
that for each $u\in M$ a mapping $\mathcal{A}(u):T_{u}(M)\rightarrow
\mathcal{G}$ is a left inverse one to the mapping \ $\hat{u}_{\ast }(e)
\mathcal{G}\rightarrow T_{u}(M),$ that is

\begin{equation}
\mathit{\mathcal{A}}(u)\hat{u}_{\ast }(e)=1.  \label{0.2}
\end{equation}

As usual, denote by $\varphi _{g}^{\ast }:T^{\ast }(M)\rightarrow T^{\ast
}(M)$ the corresponding lift of the mapping \ $\varphi _{g}:M\rightarrow M$
\ at any $g\in G.$ If $\alpha ^{(1)}\in \Lambda ^{1}(M)$ is the canonical $G$
- invariant 1-form on $\ M,$ the canonical symplectic structure $\omega
^{(2)}\in \Lambda ^{2}(T^{\ast }(M))$ given by
\begin{equation}
\omega ^{(2)}:=d\text{ }pr^{\ast }\alpha ^{(1)}  \label{0.3}
\end{equation
generates the corresponding momentum mapping $l:T^{\ast }(M)\rightarrow
\mathcal{G}^{\ast },$ where
\begin{equation}
l(\alpha ^{(1)})(u)=\hat{u}^{\ast }(e)\alpha ^{(1)}(u)  \label{0.4}
\end{equation
for all $u\in M.$ Remark here that the principal fiber \ bundle structure
p:(M,\varphi )\rightarrow N$ \ means in part the exactness of the following
sequences of mappings:
\begin{equation}
0\rightarrow \mathcal{G}\overset{\hat{u}_{\ast }(e)}{\rightarrow }T_{u}(M
\overset{p_{\ast }(u)}{\rightarrow }T_{p(u)}(N)\rightarrow 0,  \label{0.5}
\end{equation
that is
\begin{equation}
p_{\ast }(u)\hat{u}_{\ast }(e)=0=\hat{u}^{\ast }(e)p^{\ast }(u)  \label{0.6}
\end{equation
for all $u\in M.$ Combining (\ref{0.6}) with (\ref{0.2}) and (\ref{0.4}),
one obtains such an embedding:
\begin{equation}
\lbrack 1-\mathcal{A}^{\ast }(u)\hat{u}^{\ast }(e)]\alpha ^{(1)}(u)\in \text
range }p^{\ast }(u)  \label{0.7}
\end{equation
for the canonical 1-form $\alpha ^{(1)}\in \Lambda ^{1}(M)$ at $u\in M.$ The
expression (\ref{0.7}) means of course, that
\begin{equation}
\hat{u}^{\ast }(e)[1-\mathcal{A}^{\ast }(u)\hat{u}^{\ast }(e)]\alpha
^{(1)}(u)=0  \label{0.8}
\end{equation
for all $u\in M.$ Now taking into account that the mapping \ $p^{\ast
}(u):T^{\ast }(N)\rightarrow T^{\ast }(M)$ \ is for each $u\in M$ injective,
it has the unique inverse mapping \ $\ (p^{\ast }(u))^{-1}$ upon its image \
$p^{\ast }(u)T_{p(u)}^{\ast }(N)\subset T_{u}^{\ast }(M).$ Thereby \ for
each $u\in M$ one can define a morphism $p_{\mathcal{A}}:(T^{\ast
}(M),\varphi _{g}^{\ast })\rightarrow T^{\ast }(N)$ as
\begin{equation}
p_{\mathcal{A}}(u):\alpha ^{(1)}(u)\rightarrow (p^{\ast }(u))^{-1}[1
\mathcal{A}^{\ast }(u)\hat{u}^{\ast }(e)]\alpha ^{(1)}(u).  \label{0.9}
\end{equation
Based on the definition (\ref{0.9}) one can easily check that the diagram
\begin{equation}
\begin{array}{ccc}
T^{\ast }(M) & \overset{p_{\mathcal{A}}}{\rightarrow } & T^{\ast }(N) \\
\left. pr_{M}\right\downarrow &  & \left\downarrow pr_{N}\right. \\
M & \overset{p}{\rightarrow } &
\end{array}
\label{0.10}
\end{equation
is commutative.

Let an element $\xi \in \mathcal{G}^{\ast }$ be $G$-invariant, that is $\ \
Ad_{g^{-1}}^{\ast }\xi =\xi $ for all $\ g\in G.$ Denote also by \ $p_
\mathcal{A}}^{\xi }$ \ the restriction of the mapping (\ref{0.9}) upon the
subset $\mathcal{M}_{\xi }:=l^{-1}(\xi )\in T^{\ast }(M),$ that is \ \ $p_
\mathcal{A}}^{\xi }:\mathcal{M}_{\xi }\rightarrow T^{\ast }(N),$ where for
all $u\in M
\begin{equation}
p_{\mathcal{A}}^{\xi }(u):l^{-1}(\xi )\rightarrow (p^{\ast }(u))^{-1}[1
\mathcal{A}^{\ast }(u)\hat{u}^{\ast }(e)]l^{-1}(\xi ).  \label{0.11}
\end{equation
Now one can characterize the structure of the reduced phase space {$\mathcal
\bar{M}}_{\xi }:=$}$l^{-1}(\xi )/G$ \ by means of the following lemma.

\begin{lemma}
{\label{lem_01} {The mapping }$p_{\mathcal{A}}^{\xi }(u):\mathcal{M}_{\xi
}\rightarrow T^{\ast }(N),$ where \ }$\mathcal{M}_{\xi }:=${\ }$l^{-1}(\xi
), ${\ {\ is a principal fiber }$G$ {-bundle with the reduced space \ }
\mathcal{\bar{M}}_{\xi },$ {\ being diffeomorphic to }$T^{\ast }(N).$}
\end{lemma}

Denote by $<.,.>_{\mathcal{G}}$ the standard $Ad$-invariant non-degenerate
scalar product on $\mathcal{G}\times \mathcal{G}.$\ \ Based on Lemma \re
{lem_01} one derives the following \ characteristic theorem.

\begin{theorem}
{\ \label{th_02}\ {Given a principal fiber \ }$G${-bundle with a connection
$\Gamma (\mathcal{A})$ {\ and a }$G${-invariant element }$\ \xi \in \mathcal
G}^{\ast },$ {\ then every such connection }$\Gamma (\mathcal{A})$ {\
defines a symplectomorphism }$\nu _{\xi }:{\mathcal{\bar{M}}_{\xi }
\rightarrow T^{\ast }(N)$ {\ between the reduced phase space }$\mathcal{\bar
M}}_{\xi }$ {\ and \ cotangent bundle \ }$T^{\ast }(N),$ {\ where }
l:T^{\ast }(M)\rightarrow \mathcal{G}^{\ast }$ {\ is the naturally
associated momentum mapping for the group }$G${-action on }$M.$ {\ Moreover,
the following equality }
\begin{equation}
(p_{\mathcal{A}}^{\xi })(d\text{ }pr^{\ast }\beta ^{(1)}+pr^{\ast }\text{
\Omega _{\xi }^{(2)})=\left. d\text{ }pr^{\ast }\alpha ^{(1)}\right\vert
_{l^{-1}(\xi )}  \label{0.12}
\end{equation
{holds for the canonical 1-forms \ }$\beta ^{(1)}\in \Lambda ^{1}(N)$ {\ and
\ }$\alpha ^{(1)}\in \Lambda ^{1}(M),$ {\ where \ }$\Omega _{\xi
}^{(2)}:=<\xi ,\Omega ^{(2)}\ >_{\mathcal{G}}$ {\ is the }$\xi ${-component
of the corresponding curvature form }$\Omega ^{(2)}\in \Lambda
^{(2)}(N)\otimes \mathcal{G}.$}
\end{theorem}

\begin{proof}
One has that on $l^{-1}(\xi )\subset M$ the following expression, due to
\ref{0.9}), holds:
\begin{equation*}
p^{\ast }(u)p_{\mathcal{A}}^{\xi }(\alpha ^{(1)}(u))=p^{\ast }(u)\beta
^{(1)}(pr_{N}(u))=\alpha ^{(1)}(u)-\mathcal{A}^{\ast }(u)\hat{u}^{\ast
}(e)\alpha ^{(1)}(u)
\end{equation*
for \ any $\beta ^{(1)}\in T^{\ast }(N)$ and all $u\in M_{\xi
}:=p_{M}l^{-1}(\xi )\subset M.$ Thus we easily get that
\begin{equation*}
\alpha ^{(1)}(u)=(p_{\mathcal{A}}^{\xi })^{-1}\beta ^{(1)}(p_{N}(u))=p^{\ast
}(u)\beta ^{(1)})(pr_{N}(u))+<\xi ,\mathcal{A}(u)>_{\mathcal{G}}
\end{equation*
for all $u\in M_{\xi }.$ Recall now that in virtue of (\ref{0.10}) on the
manifold $M_{\xi }$ there hold relationships
\begin{equation*}
p\text{ }\circ pr_{M_{\xi }}=pr_{N}\text{ }\circ p_{\mathcal{A}}^{\xi }
\text{ \ }pr_{M_{\xi }}^{\ast }\text{\ }\circ p^{\ast }\text{ }=\text{ }(p_
\mathcal{A}}^{\xi })^{\ast }\text{\ }\circ pr_{N}^{\ast }\text{ .\ }
\end{equation*
Therefore we can now write down that
\begin{eqnarray*}
pr_{M_{\xi }}^{\ast }\alpha ^{(1)}(u) &=&pr_{M_{\xi }}^{\ast }\beta
^{(1)}(p_{N}(u))+pr_{M_{\xi }}^{\ast }<\xi ,\mathcal{A}(u)>_{\mathcal{G}} \\
&=&(p_{\mathcal{A}}^{\xi })^{\ast }(pr_{N}^{\ast }\beta
^{(1)})(u)+pr_{M_{\xi }}^{\ast }<\xi ,\mathcal{A}(u)>_{\mathcal{G}},
\end{eqnarray*
whence taking the external \ differential, one arrives at the following
equalities:
\begin{eqnarray*}
d\text{ }pr_{M_{\xi }}^{\ast }\alpha ^{(1)}(u) &=&(p_{\mathcal{A}}^{\xi
})^{\ast }d(pr_{N}^{\ast }\beta ^{(1)})(u)+pr_{M_{\xi }}^{\ast }<\xi ,d\text{
}\mathcal{A}(u)\ >_{\mathcal{G}} \\
&=&(p_{\mathcal{A}}^{\xi })^{\ast }d(pr_{N}^{\ast }\beta
^{(1)})(u)+pr_{M_{\xi }}^{\ast }<\xi ,\Omega (p(u))\ >_{\mathcal{G}} \\
&=&(p_{\mathcal{A}}^{\xi })^{\ast }d(pr_{N}^{\ast }\beta
^{(1)})(u)+pr_{M_{\xi }}^{\ast }p^{\ast }<\xi ,\Omega ,>_{\mathcal{G}}(u) \\
&=&(p_{\mathcal{A}}^{\xi })^{\ast }d(pr_{N}^{\ast }\beta ^{(1)})(u)+(p_
\mathcal{A}}^{\xi })^{\ast }pr_{N}^{\ast }<\xi ,\Omega >_{\mathcal{G}}(u) \\
&=&(p_{\mathcal{A}}^{\xi })^{\ast }[d(pr_{N}^{\ast }\beta
^{(1)})(u)+pr_{N}^{\ast }<\xi ,\Omega \ >_{\mathcal{G}}(u)].
\end{eqnarray*
When deriving the above expression we made use of the following property
satisfied by the curvature 2-form $\Omega \in \Lambda ^{2}(M)\otimes
\mathcal{G}:
\begin{eqnarray*}
&<&\xi ,d\mathcal{A(}u)\ \mathcal{>_{\mathcal{G}}=<}\xi ,d\mathcal{A(}u)
\mathcal{A(}u)\wedge \mathcal{A(}u)\ >_{\mathcal{G}}-<\xi ,\mathcal{A(
u)\wedge \mathcal{A(}u)\ >_{\mathcal{G}} \\
&=&<\xi ,\Omega (p_{N}(u))\ >_{\mathcal{G}}=<\xi ,p_{N}^{\ast }\Omega \ >_
\mathcal{G}}(u)
\end{eqnarray*
at any $u\in M_{\xi },$ since for any $A,B\in \mathcal{G}$ there holds $<\xi
,[A,B]\ >_{\mathcal{G}}=<Ad^{\ast }A\xi ,B>_{\mathcal{G}}=0$ in virtue of
the invariance condition $Ad_{G}^{\ast }\xi =\xi .$ Thereby the proof is
finished.
\end{proof}

\begin{remark}
\label{rem_03} {As the canonical 2-form \ \ \ }$d$ {\ }$pr^{\ast }\alpha
^{(1)}\in $ {\ }$\Lambda ^{(2)}(T^{\ast }(M))$ {\ \ \ is }$G$ {-invariant on
\ \ \ }$T^{\ast }(M)$ {\ due to construction, it is evident that its
restriction upon the \ }$G$ {-invariant submanifold \ \ \ $\mathcal{M}_{\xi
} $}$\subset T^{\ast }(M)$ {\ \ will be effectively defined only on the
reduced space \ \ $\mathcal{\bar{M}}_{\xi }$}$,$ {\ that ensures the
validity of the equality sign in (\ref{0.12}). \ }
\end{remark}

As a consequence of Theorem \ref{th_02} one can formulate the following
useful for applications theorems.

\begin{theorem}
\label{th_04} {Let an element }$\xi \in \mathcal{G}^{\ast }$ {\ have the
isotropy group }$G_{\xi }$ {\ \ \ acting on the subset \ \ $\mathcal{M}_{\xi
}$}$\subset T^{\ast }(M)$ {\ freely and properly, so that the reduced phase
space \ }$({\mathcal{\bar{M}}_{\xi }}\ ,\sigma _{\xi }^{(2)})$ where, by
definition, ${\mathcal{\bar{M}}_{\xi }:=}l^{-1}(\xi )/G_{\xi },$ {\ is
symplectic whose symplectic structure is defined as }
\begin{equation}
\sigma _{\xi }^{(2)}:=\left. d\text{ }pr^{\ast }\alpha ^{(1)}\right\vert
_{l^{-1}(\xi )}.  \label{0.13}
\end{equation
{If a principal fiber bundle \ \ }$p:(M,\varphi )\rightarrow N$ {\ has a
structure group coinciding with }$G_{\xi },${\ then\ \ the reduced
symplectic space \ \ \ \ }$({\mathcal{\bar{M}}_{\xi }},\sigma _{\xi }^{(2)})$
{\ is symplectomorphic to the cotangent symplectic space }$(T^{\ast
}(N),\omega _{\xi }^{(2)}),${\ where }
\begin{equation}
\omega _{\xi }^{(2)}=d\text{ }pr^{\ast }\beta ^{(1)}+pr^{\ast }\Omega _{\xi
}^{(2)},  \label{0.14}
\end{equation
{and the corresponding symplectomorphism \ is given by a relation like (\re
{0.12}).}
\end{theorem}

\begin{theorem}
{\label{th_05} {In order that two symplectic spaces \ \ \ }$({\mathcal{\bar{
}}_{\xi }},\sigma _{\xi }^{(2)})$ {\ \ \ \ and \ \ \ }$(T^{\ast
}(N),dpr^{\ast }\beta ^{(1)})$ were {symplectomorphic, it is necessary and
sufficient that the element \ }$\xi \in \ker ${\ }}${h,}${\ where for }${G}$
{-invariant element \ \ }$\xi \in \mathcal{G}^{\ast }$ {\ the mapping }
h:\xi \rightarrow \lbrack \Omega _{\xi }^{(2)}]\in H^{2}(N;\mathbb{Z}),$ {\
with }$H^{2}(N;\mathbb{Z})$\ {\ being the cohomology class of 2-forms \ on
the manifold }$N.$}
\end{theorem}

\subsection{The Hamiltonian analysis of the Maxwell electromagnetic
dynamical systems}

We take the Maxwell electromagnetic equations to b
\begin{eqnarray}
\partial E/\partial t &=&\nabla \times B-j,\text{ \ \ \ }\partial B/\partial
t=-\nabla \times E,  \label{1.1} \\
&<&\nabla ,E>=\rho ,\text{ \ \ \ \ \ \ \ }<\nabla ,B>=0,\text{\ }  \notag
\end{eqnarray
on the cotangent phase space $T^{\ast }(N)$ \ to $N\subset T(D;\mathbb{E
^{3}),$ \ being the smooth manifold of smooth vector fields on an open
domain $D\subset \mathbb{R}^{3},$ all expressed in the light speed units.
Here $(E,B)\in T^{\ast }(N)$ \ \ is a vector of electric and magnetic
fields, $\rho :D\rightarrow \mathbb{R}$ and $j:D\rightarrow \mathbb{E}^{3}$\
are, simultaneously, fixed charge and current densities in the domain $D,$
satisfying the equation of continuity
\begin{equation}
\partial \rho /\partial t+<\nabla ,j>=0,  \label{1.1a}
\end{equation
holding for all $t\in \mathbb{R},$ where we denoted by the sign $"\nabla "$
the gradient operation with respect to a variable $x\in $ $D,$ by the sign
"\times "$ the usual vector product in $\mathbb{E}^{3}:=(\mathbb{R
^{3},<\cdot ,\cdot >),$ \ being the standard three-dimensional Euclidean
vector space $\ \mathbb{R}^{3}$ endowed with the usual scalar product
<\cdot ,\cdot >.$

Aiming to represent equations (\ref{1.1}) as those on reduced symplectic
space, we define an appropriate configuration space $M$ $\subset \mathcal{T
(D;\mathbb{E}^{3})$ with a vector potential field coordinate \ $A\in M.$ The
cotangent space $T^{\ast }(M)$\ \ may be identified with pairs $(A;Y)\in
T^{\ast }(M),$ where $Y\in \mathcal{T}^{\ast }(D;\mathbb{E}^{3})$ is a
suitable vector field density in $D.$ On the space $T^{\ast }(M)$ there
exists the canonical symplectic form $\omega ^{(2)}\in \Lambda ^{2}(T^{\ast
}(M)),$ allowing, $\ $owing to the definition$\ $of\ the\ Liouville\ from$\
\ \ \ \ \ \ \ \ \ \ \ \
\begin{equation}
\lambda (\alpha ^{(1)})(A;Y)=\int_{D}d^{3}x(<Y,dA>:=(Y,dA),  \label{1.2}
\end{equation
the canonical expression$\ \ \
\begin{equation}
\omega ^{(2)}:=d\lambda (\alpha ^{(1)})=(dY,\wedge dA).  \label{1.2b}
\end{equation
Here we denoted by $"\wedge "$ the usual external differentiation, by
d^{3}x,$ $x\in D,$ \ the Lebesgue measure in the domain $D$ and by
pr:T^{\ast }(M)\rightarrow M$ \ the standard projection upon the base space
M.$ Define now a Hamiltonian function $\tilde{H}\in \mathcal{D}(T^{\ast
}(M)) $ a
\begin{equation}
H(A,Y)=1/2[(Y,Y)+(\nabla \times A,\nabla \times A)+(<\nabla ,A>,<\nabla
,A>)],  \label{1.2c}
\end{equation
describing the well-known Maxwell equations in vacuum, if the densities
\rho =0$ and $j=0.$ Really, owing to (\ref{1.2b}) one easily obtains from
\ref{1.2c}) that
\begin{eqnarray}
\partial A/\partial t &:&=\delta H/\delta Y=Y,  \label{1.2d} \\
\partial Y/\partial t &:&=-\delta H/\delta A=-\nabla \times B+\nabla <\nabla
,A>,  \notag
\end{eqnarray
being true wave equations in vacuum, where we put, by definition,
\begin{equation}
B:=\nabla \times A,  \label{1.2e}
\end{equation
being the corresponding magnetic field. \ Now defining
\begin{equation}
E:=-Y-\nabla W  \label{1.1f}
\end{equation
for some function $W:M\rightarrow \mathbb{R}$ as the corresponding electric
field, the system of equations (\ref{1.2d}) will become, owing to definition
(\ref{1.2e}),
\begin{equation}
\partial B/\partial t=-\nabla \times E,\text{ \ }\partial E/\partial
t=\nabla \times B,  \label{1.3}
\end{equation
exactly coinciding with the Maxwell equations in vacuum, if the Lorenz
condition
\begin{equation}
\partial W/\partial t+<\nabla ,A>=0  \label{1.3a}
\end{equation
is involved.

Since definition (\ref{1.1f}) was essentially imposed rather than arising
naturally from the Hamiltonian approach and our equations are valid only for
a vacuum, we shall try to improve upon these matters by employing the
reduction approach devised above. Namely, we start with the Hamiltonian (\re
{1.2c}) and observe that it is invariant with respect to the following
abelian symmetry group $G:=\exp \mathcal{G},$ where $\mathcal{G}\simeq
C^{(1)}(D;\mathbb{R}),$ acting on the base manifold $M$ naturally lifted to
T^{\ast }(M):$ for any $\psi \in \mathcal{G}$ and $(A,Y)\in T^{\ast }(M)
\begin{equation}
\varphi _{\psi }(A):=A+\nabla \psi ,\ \ \ \ \varphi _{\psi }(Y)=Y.
\label{1.4}
\end{equation
The 1-form (\ref{1.2}) under transformation (\ref{1.4}) also is invariant
since
\begin{equation}
\begin{array}{c}
\varphi _{\psi }^{\ast }\lambda (\alpha ^{(1)})(A,Y)=(Y,dA+\nabla d\psi )=
\\
=(Y,dA)-(<\nabla ,Y>,d\psi )=\lambda (\alpha ^{(1)})(A,Y)
\end{array}
\label{1.5}
\end{equation
where we made use of the condition $d\psi \simeq 0$ in $\Lambda ^{1}(T^{\ast
}(M))$ for any $\psi \in \mathcal{G}.$ Thus, the corresponding momentum
mapping (\ref{0.4}) is given as
\begin{equation}
l(A,Y)=-<\nabla ,Y>  \label{1.6}
\end{equation
for all $(A,Y)\in T^{\ast }(M).$ If $\rho \in \mathcal{G}^{\ast }$is fixed,
one can define the reduced phase space $\mathcal{\bar{M}}_{\rho
}:=l^{-1}(\rho )/G,$ since evidently, the isotropy group $G_{\rho }=G,$
owing to its commutativity and the condition (\ref{1.4}). Consider \ \ now a
principal fiber bundle $p:M\rightarrow N$ with the abelian structure group
G $ and a base manifold $N$ taken as
\begin{equation}
N:=\{B\in \mathcal{T}(D;\mathbb{E}^{3}):\text{ \ }<\nabla ,\text{ }B>=0
\text{ \ }<\nabla ,E(S)>=\rho \},  \label{1.7}
\end{equation
where, by definition,
\begin{equation}
p(A)=B=\nabla \times A.  \label{1.8}
\end{equation
We can construct a connection 1-form $\ \ \mathcal{A}\in \Lambda ^{1}(M
\mathbb{\otimes }\mathcal{G}$ on this bundle, where for all $A\in M$
\begin{equation}
\mathcal{A}(A)\cdot \hat{A}_{\ast }(l)=1,\text{ \ \ }d<\mathcal{A}(A),\rho
>_{\mathcal{G}}=\Omega _{\rho }^{(2)}(A)\in H^{2}(M;\mathbb{Z}),  \label{1.9}
\end{equation
where $\mathcal{A}(A)\in \Lambda ^{1}(M)$ is some differential 1-form, which
we choose in the following form
\begin{equation}
\mathcal{A}(A):=-(W,d<\nabla ,A>),  \label{1.9a}
\end{equation
where $W\in C^{(1)}(D;\mathbb{R})$ is some scalar function, still not
defined. As a result, the Liouville form (\ref{1.2}) transforms into
\begin{equation}
\lambda (\tilde{\alpha}_{\rho }^{(1)}):=(Y,dA)-(W,d<\nabla ,A>)=(Y+\nabla
W,dA):=(\tilde{Y},\text{ }dA),\ \tilde{Y}:=Y+\nabla W,  \label{1.9aa}
\end{equation
giving rise to the corresponding canonical symplectic structure on $T^{\ast
}(M)$ as
\begin{equation}
\tilde{\omega}_{\rho }^{(2)}:=d\lambda (\tilde{\alpha}_{\rho }^{(1)})=(
\tilde{Y},\wedge dA).  \label{1.9aaa}
\end{equation
Respectively, the Hamiltonian function (\ref{1.2c}), as a function\ on $\
T^{\ast }(M),$ transforms into
\begin{equation}
\tilde{H}_{\rho }(A,\tilde{Y})=1/2[(\tilde{Y},\tilde{Y})+(\nabla \times
A,\nabla \times A)+(<\nabla ,A>,<\nabla ,A>)],  \label{1.9ab}
\end{equation
coinciding with the well-known Dirac-Fock-Podolsky \cite{BS,DFP} Hamiltonian
expression. The corresponding Hamiltonian equations on the cotangent space
\ \ T^{\ast }(M)$
\begin{eqnarray*}
\partial A/\partial t &:&=\delta \tilde{H}/\delta \tilde{Y}=\tilde{Y},\text{
\ \ }\tilde{Y}:=-E-\nabla W, \\
\partial \tilde{Y}/\partial t &:&=-\delta \tilde{H}/\delta A=-\nabla \times
(\nabla \times A)+\nabla <\nabla ,A>,
\end{eqnarray*
describe true wave processes related to the Maxwell equations in vacuum,
which do not take into account boundary charge and current densities
conditions. Really, from (\ref{1.9ab}) we obtain that
\begin{equation}
\partial ^{2}A/\partial t^{2}-\nabla ^{2}A=0\Longrightarrow \partial
E/\partial t+\nabla (\partial W/\partial t\text{\ }+\text{ }<\nabla
,A>)=-\nabla \times B,\text{\ }  \label{1.9abb}
\end{equation
giving rise to the true vector potential wave equation, but the
electromagnetic Faraday induction law is satisfied if one to impose
additionally the Lorenz condition (\ref{1.3a}).

To remedy this situation, we will apply to this symplectic space \ the
reduction technique devised in Section 1. Namely, owing to Theorem \re
{th_04}, the constructed above cotangent manifold $T^{\ast }(N)\ $ is
symplectomorphic to the corresponding reduced phase space $\mathcal{\bar{M}
_{\rho },$ that is
\begin{equation}
\mathcal{\bar{M}}_{\rho }\simeq \{(B;S)\in T^{\ast }(N):\ <\nabla
,E(S)>=\rho ,\text{ \ \ }<\nabla ,B>=0\}  \label{1.9b}
\end{equation
with the reduced canonical symplectic 2-form
\begin{equation}
\omega _{\rho }^{(2)}(B,S)=(dB,\wedge dS=d\lambda (\alpha _{\rho
}^{(1)})(B,S),\text{ \ \ \ }\lambda (\alpha _{\rho }^{(1)})(B,S):=-(S,dB),
\label{1.10}
\end{equation
\ where we put, by definition,

\begin{equation}
\nabla \times S+F+\nabla W=-\tilde{Y}:=E+\nabla W,\text{ \ \ }<\nabla
,F>:=\rho ,  \label{1.10a}
\end{equation
for some fixed vector mapping $F\in C^{(1)}(D;\mathbb{E}^{3}),$ depending on
the imposed boundary conditions. The result (\ref{1.10}) follows right away
upon substituting the expression for the electric field \ $E=\nabla \times
S+F$ into the symplectic structure (\ref{1.9aaa}), and taking into account
that $dF=0$ in $\Lambda ^{1}(M).$ \ The Hamiltonian function (\ref{1.9ab})
reduces, respectively, to the following symbolic form
\begin{eqnarray}
H_{\rho }(B,S) &=&1/2[(B,B)+(\nabla \times S+F+\nabla W,\nabla \times
S+F+\nabla W)+  \notag \\
+( &<&\nabla ,(\nabla \times )^{-1}B>,<\nabla ,(\nabla \times )^{-1}B>)],
\label{1.11}
\end{eqnarray
where $"(\nabla \times )^{-1}"$ means, by definition, the corresponding
inverse curl-operation, mapping \cite{MW} the divergence-free subspace $C_{
div}}^{(1)}(D;\mathbb{E}^{3})\subset C^{(1)}(D;\mathbb{E}^{3})$ into itself.
As a result from (\ref{1.11}), \ the Maxwell equations (\ref{1.1}) become a
canonical Hamiltonian system upon the reduced phase space $T^{\ast }(N),$
endowed with the canonical symplectic structure (\ref{1.10}) and \ the
modified Hamiltonian function (\ref{1.11}). Really, one easily obtains tha
\begin{eqnarray}
\partial S/\partial t &:&=\delta H/\delta B=B-(\nabla \times )^{-1}\nabla
<\nabla ,(\nabla \times )^{-1}B>,  \label{1.11a} \\
\text{\ \ }\partial B/\partial t &:&=-\delta H/\delta S=-\nabla \times
(\nabla \times S+F+\nabla W):=-\nabla \times E,  \notag
\end{eqnarray
where we make use of the definition $E=\nabla \times S+F$ and the elementary
identity $\nabla \times \nabla =0.$ Thus, the second equation of (\ref{1.11a
) coincides with \ the second Maxwell equation of (\ref{1.1}) in the
classical form
\begin{equation*}
\partial B/\partial t=-\nabla \times E.
\end{equation*
Moreover, from (\ref{1.10a}), owing to (\ref{1.11a}), one obtains via the
differentiation with respect to $t\in \mathbb{R}$ that
\begin{eqnarray}
\partial E/\partial t &=&\partial F/\partial t+\nabla \times \partial
S/\partial t=  \label{1.11b} \\
&=&\partial F/\partial t+\nabla \times B,  \notag
\end{eqnarray
as well as, owing to (\ref{1.1a}),
\begin{equation}
<\nabla ,\partial F/\partial t>=\partial \rho /\partial t=-<\nabla ,j>.
\label{1.11c}
\end{equation
So, we can find from (\ref{1.11c}) that, up to non-essential curl-terms
\nabla \times (\cdot ),$ the following relationshi
\begin{equation}
\partial F/\partial t=-j  \label{1.11d}
\end{equation
holds. Really, the current density vector $j\in C^{(1)}(D;\mathbb{E}^{3}),$
owing to the equation of continuity (\ref{1.1a}), is defined up to
curl-terms $\nabla \times (\cdot )$ which can be included into the
right-hand side of (\ref{1.11d}). Having now substituted (\ref{1.11d}) into
\ref{1.11b}), we obtain exactly the first Maxwell equation of (\ref{1.1}):
\begin{equation}
\partial E/\partial t=\nabla \times B-j,  \label{1.11cd}
\end{equation
being supplemented, naturally, with the external boundary constraint
conditions
\begin{equation}
\begin{array}{c}
<\nabla ,B>=0,\text{ \ \ }<\nabla ,E>=\rho , \\
\partial \rho /\partial t+<\nabla ,j>=0
\end{array}
\label{1.11cdd}
\end{equation
owing to the continuity relationship (\ref{1.1a}) and definition (\ref{1.9b
).

Concerning the wave equations, related to the Hamiltonian system (\ref{1.11a
), \ we obtain the following: the electric field $E$ is recovered from the
second equation as
\begin{equation}
E:=-\partial A/\partial t-\nabla W,  \label{1.11dd}
\end{equation
where $W\in C^{(1)}(D;\mathbb{R})$ is some smooth function, depending on the
vector field $A\in M.$ To retrieve this dependence, we substitute (\re
{1.11d}) \ into equation (\ref{1.11cd}), having taken into account that
B=\nabla \times A:
\begin{equation}
\partial ^{2}A/\partial t^{2}-\nabla (\partial W/\partial t+<\nabla
,A>)=\nabla ^{2}A+j.  \label{1.11de}
\end{equation
\ With the above, if we now impose the Lorenz condition (\ref{1.3a}), we
obtain from (\ref{1.11de}) \ the corresponding true wave equations in the
space-time, taking into account the external charge and current density
conditions (\ref{1.11cdd}).

Notwithstanding our progress so far, the problem of fulfilling the Lorenz
constraint (\ref{1.3a}) naturally within the canonical Hamiltonian formalism
still remains to be completely solved. To this end, we are compelled to
analyze the structure of the Liouville 1-form\ (\ref{1.9aa}) for Maxwell
equations in vacuum on a slightly extended functional manifold $M\times L.$
As a first step, we rewrite 1-form (\ref{1.9aa}) as
\begin{eqnarray}
\lambda (\tilde{\alpha}_{\rho }^{(1)}) &:&=(\tilde{Y},dA)=(Y+\nabla
W,dA)=(Y,dA)+  \notag \\
+(W,-d &<&\nabla ,A>):=(Y,dA)+(W,d\chi ),  \label{1.11e}
\end{eqnarray
where we put, by definition,
\begin{equation}
\chi :=-<\nabla ,A>.  \label{1.11f}
\end{equation
Considering now the elements $(Y,A;\chi ,W)$ $\in T^{\ast }(M\times L)$ as
new canonical variables on the extended cotangent phase space $T^{\ast
}(M\times L),$ where $L:=C^{(1)}(D;\mathbb{R}),$ we can rewrite the
symplectic structure (\ref{1.9aaa}) in the following canonical for
\begin{equation}
\tilde{\omega}_{\rho }^{(2)}:=d\lambda (\tilde{\alpha}_{\rho
}^{(1)})=(dY,\wedge dA)+(dW,\wedge d\chi ).  \label{1.11g}
\end{equation
Subject to the Hamiltonian function (\ref{1.9ab}) we obtain the expression
\begin{equation}
H(A,Y;\chi ,W)=1/2[(Y-\nabla W,Y-\nabla W)+(\nabla \times A,\nabla \times
A)+(\chi ,\chi )],  \label{1.11h}
\end{equation
with respect to which the corresponding Hamiltonian equations take the form
\begin{eqnarray}
\partial A/\partial t &:&=\delta H/\delta Y=Y-\nabla W,\text{ \ \ }Y:=-E,
\notag \\
\partial Y/\partial t &:&=-\delta H/\delta A=-\nabla \times (\nabla \times
A),  \notag \\
\partial \chi /\partial t &:&=\delta H/\delta W=<\nabla ,Y-\nabla W>,  \notag
\\
\partial W/\partial t &:&=-\delta H/\delta \chi =-\chi .  \label{1.11hh}
\end{eqnarray
From (\ref{1.11hh}) \ we obtain, owing to external boundary conditions (\re
{1.11cdd}), \ successively tha
\begin{eqnarray}
\partial B/\partial t+\nabla \times E &=&0,\text{ \ }\partial ^{2}W/\partial
t^{2}-\nabla ^{2}W=\rho ,  \label{1.11hi} \\
\partial E/\partial t-\nabla \times B &=&0,\text{ \ \ }\partial
^{2}A/\partial t^{2}-\nabla ^{2}A=-\nabla (\partial W/\partial t+<\nabla
,A>).  \notag
\end{eqnarray
As is seen, these equations describe electromagnetic Maxwell equations in
vacuum, but without the Lorenz condition (\ref{1.3a}). Thereby, as above, we
will apply to the symplectic structure (\ref{1.11g}) the reduction technique
devised in Section 1. We obtain that under transformations (\ref{1.10a}) the
corresponding reduced manifold $\mathcal{\bar{M}}_{\rho }$ becomes endowed
with the symplectic structure
\begin{equation}
\bar{\omega}_{\rho }^{(2)}:=(dB,\wedge dS)+(dW,\wedge d\chi ),  \label{1.11i}
\end{equation
and the Hamiltonian (\ref{1.11h}) assumes the form
\begin{equation}
H(S,B;\chi ,W)=1/2[(\nabla \times S+F+\nabla W,\nabla \times S+F+\nabla
W)+(B,B)+(\chi ,\chi )],  \label{1.11j}
\end{equation
whose Hamiltonian equations
\begin{eqnarray}
\partial S/\partial t &:&=\delta H/\delta B=B,\text{ \ \ \ \ \ \ }\partial
W/\partial t:=-\delta H/\delta \chi =-\chi ,  \label{1.11k} \\
\partial B/\partial t &:&=-\delta H/\delta S=-\nabla \times (\nabla \times
S+F+\nabla W)=-\nabla \times E,  \notag \\
\partial \chi /\partial t &:&=\delta H/\delta W=-<\nabla ,\nabla \times
S+F+\nabla W>=-<\nabla ,E>-\Delta W,  \notag
\end{eqnarray
coincide completely with Maxwell equations (\ref{1.1}) under conditions (\re
{1.10a}), describing true wave processes in vacuum, as well as the
electromagnetic Maxwell equations, taking into account \textit{a priori}
both the imposed external boundary conditions (\ref{1.11cdd}) and the Lorenz
condition (\ref{1.3a}), solving the problem mentioned in \cite{BS,DFP}.
Really, it is easy to obtain from (\ref{1.11k}) that
\begin{eqnarray}
\partial ^{2}W/\partial t^{2}-\Delta W &=&\rho ,\text{ \ \ \ \ \ \ \ \ \
\partial W/\partial t+<\nabla ,A>=0,  \label{1.11l} \\
\nabla \times B &=&j+\partial E/\partial t,\text{ \ \ \ \ \ \ \ }\partial
B/\partial t=-\nabla \times E,  \notag
\end{eqnarray
Based now on (\ref{1.11l}) and (\ref{1.11cdd}) one can easily calculate \cit
{BP,BPT} the magnetic wave equatio
\begin{equation}
\partial ^{2}A/\partial t^{2}-\Delta A=j,  \label{1.11m}
\end{equation
supplementing the suitable wave equation on the scalar potential $W\in L,$
finishing the calculations. Thus, we can formulate the following proposition.

\begin{proposition}
The electromagnetic Maxwell equations (\ref{1.1}) jointly with Lorenz
condition (\ref{1.3a}) are equivalent to the Hamiltonian system (\ref{1.11k
) \ with respect to the canonical symplectic structure \ (\ref{1.11i}) \ and
Hamiltonian function (\ref{1.11j}), which correspondingly reduce to
electromagnetic equations (\ref{1.11l}) and (\ref{1.11m}) under external
boundary conditions (\ref{1.11cdd}).
\end{proposition}

The obtained above result can be, eventually, used\ for developing an \
alternative quantization procedure\ of Maxwell \ electromagnetic equations,
being free of some quantum operator problems, discussed in detail in \cit
{BS}. We hope to consider this aspect of quantization problem in a specially
devoted study.

\begin{remark}
{\ If one considers a motion of a charged point particle under a Maxwell
field, it is convenient to introduce a trivial fiber bundle structure $\pi
:M\rightarrow N,$ \ such that $M\simeq N\times G$}${,}${\ $\ N:=D\subset
\mathbb{R}^{3},$ with $G:=\mathbb{R}\backslash \{0${$\}$}\ being the
corresponding (abelian) structure Lie group. An analysis similar to the
above gives rise to the reduced (on the reduced space }$\mathcal{\bar{M}
_{\xi }{:=l}${$^{-1}(\xi )/G\simeq T^{\ast }(N),$ $\xi \in \mathcal{G)}$\
symplectic structure
\begin{equation*}
{\omega ^{(2)}(q,p)=<dp,\wedge dq>+d<\xi ,\mathcal{A}(q,g)>_{\mathcal{G}},}
\end{equation*
{\ where $\mathcal{A}(q,g):=<A(q),dq>+g^{-1}dg$ is a suitable connection
1-form on phase\ space $\ M,$ with $(q,p)\in T^{\ast }(N)$ and $g\in G.$ The
corresponding canonical Poisson brackets on $T^{\ast }(N)$ are easily found
to be
\begin{equation}
\{q^{i},q^{j}\}=0,\text{ \ \ }\{p_{j},q^{i}\}=\delta _{j}^{i},\text{ \ \ \ \
\ \ }\{p_{i},p_{j}\}=F_{ji}(q)  \label{1.21}
\end{equation
for all $(q,p)\in T^{\ast }(N).$ If one introduces a new momentum variable
\tilde{p}:=p+A(q)$ on $T^{\ast }(N)\ni (q,p),$ it is easy to verify that \
\omega _{\xi }^{(2)}\rightarrow \tilde{\omega}_{\xi }^{(2)}:=<d\tilde{p
,\wedge dq>$}${,}${\ giving rise to the following Poisson brackets \cit
{Kup,Pr,PS}:
\begin{equation}
\{q^{i},q^{j}\}=0,\text{ \ \ \ \ }\{\tilde{p}_{j},q^{i}\}=\delta _{j}^{i}
\text{ \ \ \ \ \ \ }\{\tilde{p}_{i},\tilde{p}_{j}\}=0,  \label{1.22}
\end{equation
where $i,j=\overline{1,3},$ iff \ \ the standard Maxwell field equations
\begin{equation}
\partial F_{ij}/\partial q_{k}+\partial F_{jk}/\partial q_{i}+\partial
F_{ki}/\partial q_{j}=0  \label{1.23}
\end{equation
are satisfied on $N$ for all $i,j,k=\overline{1,3}$ with the curvature
tensor $F_{ij}(q):=\partial A_{j}/\partial q^{i}-\partial A_{i}/\partial
q^{j},$ \ $i,j=\overline{1,3},$ $q\in N.$}
\end{remark}

Such a construction permits a natural generalization to the case of
non-abelian structure Lie group yielding a description of Yang-Mills field
equations within the reduction approach, to which we proceed below.

\subsection{The Hamiltonian analysis of the Yang-Mills type dynamical system
}

As above, we start with defining a phase space $M$ of a particle \ \ \ under
a Yang-Mills field in a region $D\subset \mathbb{R}^{3}$ as $M:=D\mathbb
\times }G,$ where $G$ is a (not in general semi-simple) Lie group, acting on
$M$ \ from the right. Over the space $M$ one can define quite naturally a
connection $\Gamma (\mathcal{A})$ \ if we consider the following trivial
principal fiber bundle $p:M\rightarrow N,$ where $N:=D,$ with the structure
group $G.$ Namely, if $g\in G,$ $\ q\in N,$ then a connection 1-form on
M\ni (q,g)$ can be expressed \cite{GS,PM,BPS,Mo} as
\begin{equation}
\mathcal{A}(q;g):=g^{-1}(d+\sum_{i=1}^{n}a_{i}A^{(i)}(q))g,  \label{2.1}
\end{equation
where $\{a_{i}\in \mathcal{G}:i=\overline{1,n}\}$ is a basis of the Lie
algebra $\mathcal{G}$ \ of the Lie group $G$, and $A_{i}:D\rightarrow
\Lambda ^{1}(D),$ $i=\overline{1,n},$ are the Yang-Mills fields on the
physical space \ \ $D\subset \mathbb{R}^{3}.$

Now one defines the natural left invariant Liouville form on $M$ \ as
\begin{equation}
\alpha ^{(1)}(q;g):=<p,dq>+<y,g^{-1}dg>_{\mathcal{G}},  \label{2.2}
\end{equation
where $y\in T^{\ast }(G)$ and $\ <\cdot ,\cdot >_{\mathcal{G}}$ denotes, \
as before, the usual Ad-invariant non-degenerate bilinear form on $\mathcal{
}^{\ast }\times \mathcal{G},$ as, evidently, $g^{-1}dg\in \Lambda
^{1}(G)\otimes \mathcal{G}\mathbf{.}$ The main assumption we need to proceed
is that the connection 1-form is compatible with the Lie group $G$ \ action
on $M.$ The latter means that the condition
\begin{equation}
R_{h}^{\ast }\mathcal{A}(q;g)=Ad_{h^{-1}}\mathcal{A}(q;g)  \label{2.3}
\end{equation
is satisfied for all \ \ $(q,g)\in M$ and $h\in G,$ where
R_{h}:G\rightarrow G$ means the right translation by an element $h\in G$ on
the Lie group $\ G.$

Having stated all preliminary conditions needed for the reduction Theorem
\ref{th_04} to be applied to our model, suppose that the Lie group $G$
canonical action on $M$ is naturally lifted to that on the cotangent space
T^{\ast }(M)$ endowed due to (endowed owing to ({\ref{1.2}) with the
following $G$-invariant canonical symplectic structure: \ \ \ \
\begin{eqnarray}
\omega ^{(2)}(q,p;g,y) &:&=d\text{ }pr^{\ast }\alpha
^{(1)}(q,p;g,y)=<dp,\wedge dq>+  \label{2.4} \\
+ &<&dy,\wedge g^{-1}dg>_{\mathcal{G}}+<ydg^{-1},\wedge dg>_{\mathcal{G}}
\notag
\end{eqnarray
for all $(q,p;g,y)\in T^{\ast }(M).$ Take now an element $\xi \in \mathcal{G
^{\ast }$ and assume that its isotropy subgroup $G_{\xi }=G,$ that is
Ad_{h}^{\ast }\xi =\xi $ for all $h\in G.$ In the general case such an
element $\xi \in \mathcal{G}^{\ast }$ cannot exist but trivial $\xi =0,$ as
it happens, for instance, in the case of the Lie group $G=SL_{2}(\mathbb{R
). $ Then one can construct the reduced phase space $l^{-1}(\xi )/G$
symplectomorphic to $(T^{\ast }(N),\omega _{\xi }^{(2)}),$ where owing to
\ref{0.12}) for any $(q,p)\in T^{\ast }(N)
\begin{eqnarray}
\omega _{\xi }^{(2)}(q,p) &=&<dp,\wedge dq>+<\Omega ^{(2)}(q),\xi >_
\mathcal{G}}=  \label{2.5} \\
&=&<dp,\wedge
dq>+\sum_{s=1}^{n}\sum_{i,j=1}^{3}e_{s}F_{ij}^{(s)}(q)dq^{i}\wedge dq^{j}.
\notag
\end{eqnarray
In the above we have expanded the element $\xi =\sum_{i=1}^{n}e_{i}a^{i}$ }
\in \mathcal{G}^{\ast }${{\ }with respect to the bi-orthogonal basis
\{a^{i}\in \mathcal{G}^{\ast },a_{j}\in \mathcal{G}:$ $\ <a^{i},a_{j}>_
\mathcal{G}}=\delta _{j}^{i},$ $i,j=\overline{1,n}\},$ \ with $e_{i}\in
\mathbb{R},$ $i=\overline{1,3},$ being some constants, and we, as well,
denoted by $F_{ij}^{(s)}(q),$ $i,j=\overline{1,3},$ $s=\overline{1,n},$ the
corresponding curvature 2-form $\Omega ^{(2)}\in \Lambda ^{2}(N)\otimes
\mathcal{G}$ components, that is
\begin{equation}
\Omega ^{(2)}(q):=\sum_{s=1}^{n}\sum_{i,j=1}^{3}a_{s\text{
}F_{ij}^{(s)}(q)dq^{i}\wedge dq^{j}  \label{2.6}
\end{equation
for any point $q\in N.$ Summarizing the calculations accomplished above, we
can formulate the following result. }

\begin{theorem}
\label{th_2.1} {\ Suppose the Yang-Mills field (\ref{2.1}) on the fiber
bundle }$p:M\rightarrow N$ {\ with }$M=D\times G$ {\ is invariant with
respect to the Lie group }$G$ {\ action }$G\times M\rightarrow M.$ {\
Suppose also that an element }$\xi \in G^{\ast }$ {\ is chosen so that }
Ad_{G}^{\ast }\xi =\xi .$ {\ Then for the naturally constructed momentum
mapping }$l:T^{\ast }(M)\rightarrow G^{\ast }$ {\ (being equivariant) the
reduced phase space }$l^{-1}(\xi )/G\simeq T^{\ast }(N)$ {\ is endowed with
the symplectic structure (\ref{2.5}), having the following component-wise
Poisson brackets form:}
\begin{equation}
\{p_{i},q^{j}\}_{\xi }=\delta _{i}^{j},\text{ \ \ }\{q^{i},q^{j}\}_{\xi
}=0,\ \ \{p_{i},p_{j}\}_{\xi }=\sum_{s=1}^{n}e_{s}F_{ji}^{(s)}(q)
\label{2.7}
\end{equation
{for all }$i,j=\overline{1,3}$ {\ and }$(q,p)\in T^{\ast }(N).$
\end{theorem}

The respectively extended Poisson bracket on the whole cotangent space
T^{\ast }(M)$ amounts owing to (\ref{1.4}) into the following set of Poisson
relationships:
\begin{eqnarray}
\{y_{s},y_{k}\}_{\xi } &=&\sum_{r=1}^{n}c_{sk\text{ }}^{r}y_{r},\text{ \ \ \
\ \ \ \ \ \ \ \ \ }\ \ \{p_{i},q^{j}\}_{\xi }=\ \delta _{i}^{j}\ \ ,\text{ }
\label{2.8} \\
\text{\ }\{y_{s},p_{j}\}_{\xi } &=&0=\{q^{i},q^{j}\},\text{\ \
\{p_{i},p_{j}\}_{\xi }=\sum_{s=1}^{n}y_{s\text{ }}F_{ji}^{(s)}(q),  \notag
\end{eqnarray
where $i,j=\overline{1,n},$ $\ c_{sk}^{r}\in \mathbb{R},$ \ $s,k,r=\overline
1,m},$ are the structure constants of the Lie algebra $\mathcal{G},$ and we
made use of the expansion \ $A^{(s)}(q)=\sum_{j=1}^{n}A_{j}^{(s)}(q)$
dq^{j} $ as well we introduced alternative fixed values $e_{i}:=y_{i},$ $i
\overline{1,n}.$ The result (\ref{2.8}) can be easily seen if one one makes
a shift within the expression (\ref{2.4}) as $\sigma ^{(2)}\rightarrow
\sigma _{ext}^{(2)},$ where $\sigma _{ext}^{(2)}:=\left. \sigma
^{(2)}\right\vert _{\mathcal{A}_{0}\rightarrow \mathcal{A}}$ $,$ $\mathcal{A
_{0}(g):=g^{-1}dg,$ $g\in G.$ Thereby one can obtain in virtue of the
invariance properties of the connection $\Gamma (\mathcal{A})$ that
\begin{equation*}
\sigma _{ext}^{(2)}(q,p;u,y)=<dp,\wedge dq>+d<y(g),Ad_{g^{-1}}\mathcal{A
(q;e)>_{\mathcal{G}}=
\end{equation*
\begin{equation*}
=<dp,\wedge dq>+<d\text{ }Ad_{g^{-1}}^{\ast }y(g),\wedge \mathcal{A}(q;e)>_
\mathcal{G}}=<dp,\wedge dq>+\sum_{s=1}^{m}dy_{s}\wedge du^{s}+
\end{equation*}

\begin{equation*}
+\sum_{j=1}^{n}\sum_{s=1}^{m}A_{j}^{(s)}(q)dy_{s}\wedge
dq-<Ad_{g^{-1}}^{\ast }y(g),\mathcal{A}(q,e)\wedge \mathcal{A}(q,e)>_
\mathcal{G}}+
\end{equation*
\begin{equation}
+\sum_{k\geq s=1}^{m}\sum_{l=1}^{m}y_{l}\text{ }c_{sk}^{l}\text{
du^{k}\wedge du^{s}+\sum_{s=1}^{n}\sum_{i\geq
j=1}^{3}y_{s}F_{ij}^{(s)}(q)dq^{i}\wedge dq^{j},  \label{2.9}
\end{equation
where coordinate points $(q,p;u,y)\in T^{\ast }(M)$ \ are defined as
follows: $\mathcal{A}_{0}(e):=\sum_{s=1}^{m}du^{i}$ $a_{i},$
Ad_{g^{-1}}^{\ast }y(g)=y(e):=\sum_{s=1}^{m}y_{s}$ $a^{s}$ for any element
g\in G.$ Hence one gets straightaway the Poisson brackets (2.8) plus
additional brackets connected with conjugated sets of variables $\{u^{s}\in
\mathbb{R}:$\texttt{\ }$s=\overline{1,m}\}$ $\in \mathcal{G}^{\ast }$\ and
\{y_{s}\in \mathbb{R}:$\texttt{\ }$s=\overline{1,m}\}\in \mathcal{G}:$

\begin{equation}
\{y_{s},u^{k}\}_{\xi }=\delta _{s}^{k},\text{ \ }\{u^{k},q^{j}\}_{\xi }=0
\text{ \ }\{p_{j},u^{s}\}_{\xi }=A_{j}^{(s)}(q),\text{ \
\{u^{s},u^{k}\}_{\xi }=0,  \label{2.10}
\end{equation
where $j=\overline{1,n},$ \ $k,s=\overline{1,m},\ $and $\ \ q\in N.$

Note here that the transition suggested above from the symplectic structure
\sigma ^{(2)}$ \ on $T^{\ast }(N)$ to its extension $\sigma _{ext}^{(2)}$ on
\ $T^{\ast }(M)$ just consists formally in adding to the symplectic
structure $\sigma ^{(2)}$ \ an exact part, which transforms it into an
equivalent one. Looking now at the expressions (\ref{2.9}), one can infer
immediately that an element \ $\xi :=\sum_{s=1}^{m}e_{s}a^{s}\in \mathcal{G
^{\ast }$ \ will be invariant with respect to the $Ad^{\ast }$-action of the
Lie group $\ G$ \ iff
\begin{equation}
\left. \{y_{s},y_{k}\}_{\xi }\right\vert
_{y_{s}=e_{s}}=\sum_{r=1}^{m}c_{sk}^{r}\text{ }e_{r}\text{ }=0  \label{2.11}
\end{equation
identically for all $s,k=\overline{1,m},$ \ $j=\overline{1,n}$ \ and $\ q\in
N.$ In this, and only this case, the reduction scheme elaborated above will
go through.

Returning our attention to expression (\ref{2.10}), one can easily write the
following exact expression:
\begin{equation}
\omega _{ext}^{(2)}(q,p;u,y)=\omega ^{(2)}(q,p+\sum_{s=1}^{n}y_{s}\text{
A^{(s)}(q)\text{ };u,y),  \label{2.12}
\end{equation
on the phase space $T^{\ast }(M)\ni (q,p;u,y),$ where we abbreviated $\
<A^{(s)}(q),dq>$ as $\sum_{j=1}^{n}A_{j}^{(s)}(q)$ $dq^{j}.$ The
transformation like (\ref{2.12}) was discussed within somewhat different
contexts in articles \cite{Kup,Pr} containing also a good background for the
infinite dimensional generalization of symplectic structure techniques.
Having observed from (\ref{2.12}) that the simple change of variable
\begin{equation}
\tilde{p}:=p+\sum_{s=1}^{m}y_{s}\text{ }A^{(s)}(q)  \label{2.13}
\end{equation
of the cotangent space $T^{\ast }(N)$ recasts our symplectic structure (\re
{2.9}) into the old canonical form (\ref{2.4}), one obtains that the
following new set of canonical Poisson brackets on $T^{\ast }(M)$ $\ni (q
\tilde{p};u,y):
\begin{eqnarray}
\{y_{s},y_{k}\}_{\xi } &=&\sum_{r=1}^{n}c_{sk}^{r}\text{ }y_{r},\text{ \ \ \
}\{\tilde{p}_{i},\tilde{p}_{j}\}_{\xi }=0,\text{\ \ \ \ \ }\{\tilde{p
_{i},q^{j}\}=\delta _{i}^{j},\text{ }  \label{2.14} \\
\{y_{s},q^{j}\}_{\xi } &=&0\text{ }=\{q^{i},q^{j}\}_{\xi },\text{\
\{u^{s},u^{k}\}_{\xi }=0,\text{ \ \ }\{y_{s},\tilde{p}_{j}\}_{\xi }=0,\text{
\ }  \notag \\
\{u^{s},q^{i}\}_{\xi } &=&0,\text{ \ \ \ \ \ \ \ \ \ }\{y_{s},u^{k}\}_{\xi
}=\delta _{s}^{k},\text{ \ \ \ \ \ \ \ \ \ \ }\{u^{s},\tilde{p}_{j}\}_{\xi
}=0,  \notag
\end{eqnarray
where $\ k,s=\overline{1,m}$ \ and $i,j=\overline{1,n},$ holds iff the
non-abelian Yang-Mills type field equations
\begin{equation}
\partial F_{ij}^{(s)}/\partial q^{l}+\partial F_{jl}^{(s)}/\partial
q^{i}+\partial F_{li}^{(s)}/\partial q^{j}+  \label{2.15}
\end{equation
\begin{equation*}
+\sum_{k,r=1}^{m}c_{kr}^{s}(F_{ij}^{(k)}A_{l}^{(r)}+F_{jl}^{(k)}A_{i}^{(r)}+F_{li}^{(k)}A_{j}^{(r)})=0
\end{equation*
are fulfilled for all $\ s=\overline{1,m}$ \ and $i,j,l=\overline{1,n}$ on
the base manifold $\ N.$ This effect of complete reduction of gauge
Yang-Mills type variables from the symplectic structure (\ref{2.9}) is known
in literature \cite{Kup} as the principle of "\emph{minimal interaction}"
and appeared to be useful enough for studying different interacting systems
as in \cite{MW,PZ}. We plan to continue further the study of the geometric
properties of reduced symplectic structures connected with such interesting
infinite-dimensional coupled dynamical systems of Yang-Mills-Vlasov,
Yang-Mills-Bogolubov and Yang-Mills-Josephson types \cite{MW,PZ} as well as
their relationships with associated principal fiber bundles endowed with
canonical connection structures.

\section{\label{Sec_2}The Maxwell electromagnetic equations and the Lorentz
type force derivation - the Feynman's approach legacy}

\subsection{ Problem setting}

Still in 1948 R. \ Feynman presented but not published \cite{Dy1,Dy2} a very
interesting, in some aspect \textquotedblleft heretical\textquotedblright ,
quantum-mechanical derivation of the classical Lorentz force acting on a
charged particle under influence of an external electromagnetic field. His
result was analyzed by many authors \cite{Lee,Bre,Dom,Sil,And,Far,VF,Hug,HS}
from different points of view, including its relativistic generalization
\cite{Tan}. As this problem is completely classical, we reanalyze the
Feynman's derivation from the classical Hamiltonian dynamics point of view
on the coadjoint space $T^{\ast }(N),N\subset \mathbb{R}^{3},$ and construct
its nontrivial generalization compatible with results \cite{BPT,BP0,BP-Feynm}
of Section 1, based on a recently devised vacuum field theory approach \cit
{BPT0,BPT1}. Having further obtained the classical Maxwell electromagnetic
equations we supply the complete legacy of the Feynman's approach to the
Lorentz force derivation and demonstrate its compatibility with the
relativistic generalization, presented in Section 1.

Consider a motion of a charged point particle under a electromagnetic field.
For its description, following Section 3, it is convenient to introduce a
trivial fiber bundle structure ${\pi }${$:M\rightarrow N,M=N\times G,$ }
N\subset \mathbb{R}^{3},$ {with the abelian structure group $G:=\mathbb{R
\backslash \{0${$\},$} \ equivariantly acting on the canonically symplectic
coadjoint space }$T^{\ast }(M),\ $\ {and to endow it with some connection
one-form $\mathcal{A}:M$}${\mathcal{\rightarrow }T^{\ast }(M)\times \mathcal
G}}\ \ $a
\begin{equation}
{\mathcal{A}(q,g):=\ g^{-1}(d+\ \alpha ^{(1)}(q))\ \ g}  \label{Qeq3.19}
\end{equation
on the phase\ space $\ M$, where $d:\Lambda (M)\rightarrow \Lambda (M)$ is
the usual exterior differentiation, $\alpha ^{(1)}{:M\rightarrow \ }\Lambda
^{1}(N)\mathcal{\otimes }\mathcal{G}$ is some smooth mapping, $\ q\in N$ and
$g\in G.$ If $l:T^{\ast }(M)\rightarrow \mathcal{G}^{\ast }$ is the related
momentum mapping, one can respectively construct the reduced phase space
\mathcal{\bar{M}}_{\xi }{:=l}${$^{-1}(\xi )/G\simeq T^{\ast }(N),$ where
\xi \in \mathcal{G}^{\ast }\mathcal{\simeq }$}$\mathbb{R}\ $ is taken to be
fixed,\ \ possessing the reduced symplectic structure
\begin{equation}
{\bar{\omega}_{\xi }^{(2)}(q,p)=<dp,\wedge dq>+d<\xi ,\alpha ^{(1)}(q)}>_
\mathcal{G}}{.\ }  \label{Qeq3.20}
\end{equation
{From (\ref{Qeq3.20}) one finds easily the corresponding Poisson brackets on
$T^{\ast }(N)$}${:}$
\begin{equation}
\{q^{i},q^{j}\}_{{\omega _{\xi }^{(2)}}}=0,\text{ \ \ }\{p_{j},q^{i}\}_{
\omega _{\xi }^{(2)}}}=\delta _{j}^{i},\text{ \ \ \ \ \ \ }\{p_{i},p_{j}\}_{
\omega _{\xi }^{(2)}}}=\xi F_{ji}(q)  \label{Qeq3.21}
\end{equation
for $i,j=\overline{1,3}$ with respect to the reference frame }$\mathcal{K(
t,q\mathcal{)},$ characterized by the phase space {coordinates $(q,p)\in
T^{\ast }(N).$ If one introduces a new momentum variable $\tilde{p}:=p+\xi
A(q)$ on $T^{\ast }(N)\ni (q,p),$ where }${\alpha ^{(1)}(q):=<}A(q){,dq>}$ $
\in T}_{q}^{\ast }{(N),}$ {it is easy to verify that \ $\bar{\omega}_{\xi
}^{(2)}\rightarrow \tilde{\omega}_{\xi }^{(2)}:=<d\tilde{p},\wedge dq>$}${,}
{\ giving rise to the following "minimal interaction" canonical \ \ \ \
Poisson brackets \cite{Kup,Pr,PS}:
\begin{equation}
\{q^{i},q^{j}\}_{{\tilde{\omega}_{\xi }^{(2)}}}=0,\text{ \ \ \ \ }\{\tilde{p
_{j},q^{i}\}_{{\tilde{\omega}_{\xi }^{(2)}}}=\delta _{j}^{i},\text{ \ \ \ \
\ \ }\{\tilde{p}_{i},\tilde{p}_{j}\}_{{\tilde{\omega}_{\xi }^{(2)}}}=0
\label{Qeq3.22}
\end{equation
for $i,j=\overline{1,3}$ with respect to the reference frame }$\mathcal{K
_{f}(t,q-q_{f}),$ characterized by the phase space coordinates {$(q,\tilde{p
)\in T^{\ast }(N),$} {iff \ the Maxwell field equations
\begin{equation}
\partial F_{ij}/\partial q_{k}+\partial F_{jk}/\partial q_{i}+\partial
F_{ki}/\partial q_{j}=0  \label{Qeq3.23}
\end{equation
are satisfied on $N$ for all $i,j,k=\overline{1,3}$ with the curvature
tensor $F_{ij}(q):=\partial A_{j}/\partial q^{i}-\partial A_{i}/\partial
q^{j},$ \ $i,j=\overline{1,3},$ $q\in N.$ }

\subsection{The Lorentz type force and Maxwell electromagnetic field
equations - the Lagrangian analysis}

The Poisson structure (\ref{Qeq3.22}) makes it possible to describe a
charged particle $\xi \in \mathbb{R},$ located at point $q\in N\subset $
\mathbb{R}^{3},\ $moving with a velocity $dq/dt:=u\in T_{q}(N)$ with respect
to the laboratory reference frame $\mathcal{K(}t,q\mathcal{)},$ specified by
coordinates $(t,q)\in M^{4},$ being under the electromagnetic influence of
an external charged particle $\xi _{f}\in \mathbb{R}$ \ located at point
q_{f}\in N\subset $ $\mathbb{R}^{3}$ \ and moving with respect to the same
reference frame $\mathcal{K(}t,q\mathcal{)}$ with a velocity
dq_{f}/dt:=u_{f}\in T_{q_{f}}(N).$ \ Really, consider a new shifted
reference frame $\mathcal{K}_{f}^{\prime }\mathcal{(}t^{\prime },q-q_{f
\mathcal{)}$ moving with respect to the reference frame $\mathcal{K(}t,
\mathcal{)}$ with \ the velocity $u_{f}.$ With respect to the reference
frame $\mathcal{K}_{f}^{\prime }\mathcal{(}t^{\prime },q-q_{f}),$ specified
by coordinates $(t^{\prime },q-q_{f})\in M^{4},$ the charged point particle
\xi $ \ moves with the velocity $u^{\prime }-u_{f}^{\prime }:=dr/dt^{\prime
}-dr_{f}/dt^{\prime }\in T_{q-q_{f}}(N)$ and, respectively, the charged
particle $\xi _{f}$ \ stays in rest. Then one can write down the standard
\textit{classical Lagrangian function} of the charged particle $\xi $ with a
mass $m^{\prime }\in \mathbb{R}_{+}$ \ subject to the reference frame
\mathcal{K}_{f}^{\prime }\mathcal{(}t^{\prime },q-q_{f}\mathcal{)}:
\begin{equation}
\mathcal{L}_{f}(q,u^{\prime })=\frac{m^{\prime }}{2}|u^{\prime
}-u_{f}^{\prime }|^{2}-\xi \varphi ^{\prime },  \label{Qeq3.24}
\end{equation
and the suitably Lorentz transformed scalar potential $\varphi ^{\prime
}=\varphi /(1+|u_{f}^{\prime }|^{2})\in C^{2}(N;\mathbb{R})$ is the
corresponding potential energy with respect to the reference frame $\mathcal
K}_{f}^{\prime }\mathcal{(}t^{\prime },q-q_{f}\mathcal{)}.$ On the other
hand, owing to (\ref{Qeq3.24}) and the Poisson brackets (\ref{Qeq3.22}) the
following equality for the charged particle $\xi $ \textit{canonical}
\textit{momentum} with respect to the reference frame $\mathcal{K
_{f}^{\prime }\mathcal{(}t^{\prime },q-q_{f}\mathcal{)}$ holds:
\begin{equation}
\tilde{p}^{\prime }:=p^{\prime }+\xi {A}^{\prime }{(q)=\partial }\mathcal{L
_{f}(q,u^{\prime })/\partial u^{\prime },  \label{Qeq3.25}
\end{equation
or, equivalently,
\begin{equation}
p^{\prime }+\xi {A}^{\prime }{(q)=m}^{\prime }{(u^{\prime }}-u_{f}^{\prime
}),  \label{Qeq3.26}
\end{equation
expressed in the units when the light speed $c=1.$ Taking into account that
the charged particle $\xi $ momentum with respect to the reference frame
\mathcal{K(}t,q\mathcal{)}$ equals $p^{\prime }:=m^{\prime }u^{\prime }\in
T_{q}(N),$ one can easily obtain from (\ref{Qeq3.26}) the important
relationship
\begin{equation}
\xi {A}^{\prime }{(q)=-m}^{\prime }{u_{f}^{\prime }}  \label{Qeq3.27}
\end{equation
for the vector potential $A\in C^{2}(N;\mathbb{E}^{3}),$ which was before
obtained in \cite{BPT0,BPT1,Rep} and described before in Section \ref{Sec_1
. Taking now into account (\ref{Qeq3.24}) and (\ref{Qeq3.27}) one finds the
following Lagrangian equation
\begin{equation}
\frac{d}{dt^{\prime }}[p^{\prime }+\xi {A}^{\prime }{(q)]=}\partial \mathcal
L}_{f}(q,u^{\prime })/\partial q=-\xi \nabla \varphi ^{\prime },
\label{Qeq3.28}
\end{equation
obtained before with respect to the shifted reference frame $\mathcal{K
_{f}^{\prime }\mathcal{(}t^{\prime },q-q_{f}\mathcal{)}$ in \cite{BPT0,BPT1}
and giving rise, as the result of obvious relationships $p^{\prime
}=p,A^{\prime }=A,$ to the following charged point particle $\xi $ dynamics:
\begin{eqnarray}
dp/dt &=&-\xi \partial A/\partial t-\xi \nabla \varphi (1-|u_{f}|^{2})-\xi
<u,\nabla >A=  \notag \\
&&  \notag \\
&=&-\xi \partial A/\partial t-\xi \nabla \varphi -\xi <u,\nabla >A+  \notag
\\
&&  \label{Qeq3.29} \\
+\xi \nabla &<&u,A>-\xi \nabla <u-u_{f},A>=  \notag \\
&&  \notag \\
&=&-\xi (\partial A/\partial t+\nabla \varphi )+\xi u\times (\nabla \times
A)-\xi \nabla <u-u_{f},A>  \notag
\end{eqnarray
with respect to the laboratory reference frame $\mathcal{K(}t,q\mathcal{)}.$
\ \ Based now on (\ref{Qeq3.29}) we obtain the modified Lorentz type force
\begin{equation}
dp/dt=\xi E+\xi u\times B-\xi \nabla <u-u_{f},A>,  \label{Qeq3.30}
\end{equation
where we put, as usually by definition,
\begin{equation}
E:=-\partial A/\partial t-\nabla \varphi ,\text{ \ \ }B:=\nabla \times A,
\label{Qeq3.30a}
\end{equation
and slightly differing from the classical \cite{Jack,Bar,Fe1,LL} Lorentz
force expression
\begin{equation}
dp/dt=\xi E+\xi u\times B  \label{Qeq3.31}
\end{equation
by the gradient component
\begin{equation}
F_{c}:=-\xi \nabla <u-u_{f},A>.  \label{Qeq3.32}
\end{equation}

It is seen that the modified Lorentz type force expression (\ref{Qeq3.30})
can be naturally generalized to the relativistic case if to take into
account that the standard Lorenz condition
\begin{equation}
\partial \varphi /\partial t+<\nabla ,A>=0  \label{Qeq3.33}
\end{equation
is imposed on the electromagnetic potential $(\varphi ,A)\in C^{2}(N;M^{4}).$

Really, from (\ref{Qeq3.30a}) one obtains that the Lorentz invariant field
equation
\begin{equation}
\partial ^{2}\varphi /\partial t^{2}-\Delta \varphi =\rho _{f},
\label{Qeq3.34}
\end{equation
where $\rho _{f}:N\rightarrow \mathcal{D}^{\prime }(N)$ is a generalized
density function of the external charge distribution $\xi _{f}.$ Following
now by the calculations from \cite{BPT0,BPT1} we can easily find from (\re
{Qeq3.34}) and the charge conservation law
\begin{equation}
\partial \rho _{f}/\partial t+<\nabla ,j_{f}>=0  \label{Qeq3.35}
\end{equation
the next \textit{Lorentz invariant} equation on the vector potential $A\in
C^{2}(N;\mathbb{E}^{3}):
\begin{equation}
\partial ^{2}A/\partial t^{2}-\Delta A=j_{f}.  \label{Qeq3.36}
\end{equation
Moreover, relationships (\ref{Qeq3.30a}),(\ref{Qeq3.34}) and (\ref{Qeq3.36})
easily entail the true classical Maxwell equations
\begin{eqnarray}
\nabla \times E &=&-\partial B/\partial t,\text{ }\nabla \times B=\partial
E/\partial t+j_{f},  \label{Qeq3.37} \\
&<&\nabla ,E>=\rho _{f},\text{ \ \ }<\nabla ,B>=0\text{\ }  \notag
\end{eqnarray
on the \ electromagnetic field $(E,B)\in C^{2}(N;\mathbb{E}^{3}\mathbb
\times E}^{3}).$

Consider now the Lorenz condition (\ref{Qeq3.33}) and observe that it is
equivalent to the following local conservation law
\begin{equation}
\frac{d}{dt}\int_{\Omega _{t}}Wd^{3}q=0,  \label{Qeq3.38}
\end{equation
giving rise to the important relationship for the magnetic potential $A\in
C^{2}(N;\mathbb{E}^{3})$ \ \textit{\ }
\begin{equation}
A=u_{f}\varphi  \label{Qeq3.39}
\end{equation
with respect to the laboratory reference frame $\mathcal{K(}t,q\mathcal{)},$
\ where $\Omega _{t}\subset N$ \ is any open domain with the smooth boundary
$\partial \Omega _{t},$ moving jointly with the charge distribution $\xi
_{f} $ in the domain $N\subset \mathbb{R}^{3}$ with the corresponding
velocity $u_{f}^{\prime }.$ Taking into account relationship (\ref{Qeq3.27})
one can find the expression for our charged particle $\xi $
\textquotedblleft \textit{inertial}\textquotedblright\ mass:
\begin{equation}
m=-\bar{W},\text{ \ \ }\bar{W}:=\xi \varphi ,  \label{Qeq3.40}
\end{equation
coinciding with that obtained before in \cite{BPT0,BPT1,Rep}, where we
denoted by $\bar{W}\in C^{2}(N;\mathbb{R})$ the corresponding potential
energy of the charged point particle $\xi .$

\subsection{The modified least action principle and the Hamiltonian analysis}

Based on the representations (\ref{Qeq3.39}) and (\ref{Qeq3.40}) \ one can
rewrite the determining Lagrangian equation (\ref{Qeq3.28}) with respect to
the shifted reference frame $\mathcal{K}_{f}^{\prime }\mathcal{(}t^{\prime
},q_{f}\mathcal{)}$ as follows
\begin{equation}
\frac{d}{dt^{\prime }}[-\bar{W}^{\prime }(u^{\prime }-u_{f}^{\prime
})]=-\nabla \bar{W}^{\prime },  \label{Qeq3.41}
\end{equation
which is reduced to the Lorentz type force expression (\ref{Qeq3.30})
calculated with respect to the reference frame $\mathcal{K(}t,q\mathcal{)}:$

\begin{equation}
dp/dt=\xi E+\xi u\times B-\xi \nabla <u-u_{f},A>,  \label{Qeq3.41a}
\end{equation
where we put, as before,
\begin{equation}
E:=-\partial A/\partial t-\nabla \varphi ,\text{ \ \ }B:=\nabla \times A.
\label{Qeq3.41b}
\end{equation}

\begin{remark}
It is interesting to remark here that equation (\ref{Qeq3.41a}) does not
allow the Lagrangian representation with respect to the reference frame
\mathcal{K(}t,q\mathcal{)}$ in contrast to that of equation (\ref{Qeq3.41})
which is equivalent to \ (\ref{Qeq3.28}).
\end{remark}

The remark above is a challenging source of our further analysis concerning
the direct relativistic generalization of the modified Lorentz type force
\ref{Qeq3.30}). Namely, the following proposition holds.

\begin{proposition}
The Lorentz type force (\ref{Qeq3.30}) in the case when the charged point
particle $\xi $ momentum is defined, owing to (\ref{Qeq3.40}), as $p=-\bar{W
u$ is the exact relativistic expression \ allowing the Lagrangian
representation of the charged particle $\xi $ dynamics with respect to the
rest reference frame $\mathcal{K}_{\tau }(\tau ,q-q_{f}),$ related to the
shifted reference frame $\mathcal{K}_{f}^{\prime }(t^{\prime },q-q_{f})$ by
means of the classical relativistic proper time infinitesimal transformation
\begin{equation}
dt^{\prime }=d\tau (1+|u^{\prime }-u_{f}^{\prime }|^{2})^{1/2},
\label{Qeq3.42}
\end{equation
where $\tau \in \mathbb{R}$ is the proper time parameter in the rest
reference frame $\mathcal{K}_{\tau }(\tau ,q-q_{f}).$
\end{proposition}

\begin{proof}
Take the following action functional with respect to the charged point
particle $\xi $ rest reference frame $\mathcal{K}_{\tau }(\tau ,q-q_{f}):
\begin{equation}
S^{(\tau )}:=-\int_{t_{1}(\tau _{1})}^{t_{2}(\tau _{2})}\bar{W}^{\prime
}dt^{\prime }=\int_{\tau _{1}}^{\tau _{2}}\bar{W}^{\prime }(1+|u^{\prime
}-u_{f}^{\prime }|^{2})^{1/2}d\tau ,  \label{Qeq3.43}
\end{equation
where the proper temporal values $\tau _{1},\tau _{2}\in \mathbb{R}$ are
considered, in a Feynman spirit \cite{Fe1}, to be fixed in contrast to the
temporal parameters $t_{2}(\tau _{2}),t_{2}(\tau _{2})\in \mathbb{R}$ \
depending, owing to (\ref{Qeq3.42}), on the charged particle $\xi $
trajectory in the phase space $M^{4}.$ The least action condition
\begin{equation}
\delta S^{(\tau )}=0,\delta q(\tau _{1})=0=\delta q(\tau _{2}),
\label{Qeq3.43a}
\end{equation
applied to (\ref{Qeq3.43}), entails exactly the dynamical equation (\re
{Qeq3.41}), being simultaneously equivalent to the relativistic Lorentz type
force expression (\ref{Qeq3.30}) with respect to the laboratory reference
frame $\mathcal{K(}t,q).$ The latter proves the proposition.
\end{proof}

Making use of the relationships between the reference frames $\mathcal{K(}t,
\mathcal{)}$ and $\mathcal{K}_{\tau }(\tau ,q-q_{f})$ in the case when the
external charge particle velocity $u_{f}=0,$ we can easily derive the
following corollary.

\begin{corollary}
Let the external charge point   $e_{f}$ \ be in   rest, that is the velocity
$u_{f}=0.$ Then equation (\ref{Qeq3.41}) reduces to
\begin{equation}
\frac{d}{dt}(-\bar{W}u){]=}-\nabla \bar{W},  \label{Qeq3.41aa}
\end{equation
allowing the following conservation law
\begin{equation}
H_{0}=\bar{W}(1-|u|^{2})^{1/2}=-(\bar{W}^{2}-|p|^{2})^{1/2}.  \label{Qeq3.44}
\end{equation
Moreover, equation (\ref{Qeq3.41aa}) is Hamiltonian with respect to the
canonical Poisson structure (\ref{Qeq3.22}), Hamiltonian function (\re
{Qeq3.44}) and the rest reference frame $\mathcal{K}_{r}(\tau ,q):
\begin{equation}
\left.
\begin{array}{c}
dq/d\tau :=\partial H_{0}/\partial p=p(\bar{W}^{2}-|p|^{2})^{-1/2} \\
dp/d\tau :=-\partial H_{0}/\partial q=-\bar{W}(\bar{W}^{2}-|p|^{2})^{-1/2
\nabla \bar{W
\end{array
\right\} \Rightarrow \left.
\begin{array}{c}
dq/dt\text{ }\ =-p\bar{W}^{-1}, \\
dp/dt=-\nabla \bar{W
\end{array
\right\} .  \label{Qeq3.45}
\end{equation
In addition, if to define the rest particle mass $m_{0}:=-H_{0}|_{u=0},$ the
"inertial" particle mass quantity $m\in \mathbb{R}$ obtains the well known
classical relativistic form
\begin{equation}
m=-\bar{W}=m_{0}(1-|u|^{2})^{-1/2},  \label{Qeq3.46}
\end{equation
depending on the particle velocity $u\in \mathbb{R}^{3}.\ $
\end{corollary}

Concerning the general case of equation (\ref{Qeq3.41}) analogous ones to
the above results hold, which were also described in part in Section \ \re
{Sec_1}. We need only to mention that the induced Hamiltonian structure of
the general equation (\ref{Qeq3.41}) results naturally from its least action
representation (\ref{Qeq3.43}) and (\ref{Qeq3.43a}) with respect to the rest
reference frame $\mathcal{K}_{\tau }(\tau ,q).$

\subsubsection{Comments}

Within Section \ \ref{Sec_2} we have demonstrated the complete legacy of the
Feynman's approach to the Lorentz force based derivation of the Maxwell
electromagnetic field equations. Moreover, we have succeeded in finding the
exact relationship between the Feynman's approach and the vacuum field
approach of Section \ref{Sec_1}, devised before in \cite{BPT0,BPT1}. Thus,
the results obtained firmly argue for \ the \ deep physical backgrounds
lying in the vacuum field theory approach, based on which one can
simultaneously describe the physical phenomena both of electromagnetic and
gravity origins. The latter is physically based on the particle "inertial"
mass expression (\ref{Qeq3.40}), naturally following both from the Feynman's
approach to the Lorentz force derivation and from the vacuum field approach.

\section{The generalized Fock spaces, quantum currents algebra
representations and electrodynamics}

\subsection{Preliminaries: Fock space and its realizations}

Let $\Phi $ be a separable Hilbert space, $F$ be a topological real linear
space and $\mathcal{A}:=\left\{ A(f):f\in F\right\} $ a family of commuting
self-adjoint operators in $\Phi $ (i.e. these operators commute in the sense
of their resolutions of the identity). Consider the Gelfand rigging \cit
{Ber1} of the Hilbert space ${\Phi ,}$ i.e., a chain
\begin{equation}
\mathcal{D}\subset {\Phi }_{+}\subset {\Phi }\subset {\Phi }_{-}\subset
\mathcal{D^{^{\prime }}}  \label{FQeq2.1}
\end{equation
in which ${\Phi }_{+}$ and ${\Phi }_{-}$ are further Hilbert spaces, and the
inclusions are dense and continuous, i.e. ${\Phi }_{+}$ is topologically
(densely and continuously) and quasi-nucleus (the inclusion operator $i:
\Phi }_{+}\longrightarrow {\Phi }$ is of the Hilbert - Schmidt type)
embedded into ${\Phi },$ the space $\ {\Phi }_{-}$ is the dual of ${\Phi
_{+}$ with respect to the scalar product $<.,.>_{{\Phi }}$ in ${\Phi },$ and
$\mathcal{D}$ is a separable projective limit of Hilbert spaces,
topologically embedded into ${\Phi }_{+}.$ Then, the following structural
theorem \cite{Ber1,BerK} holds:

\begin{theorem}
\label{FQth2.1} Assume that the family of operators $\mathcal{A}$ \
satisfies the following conditions:

a)$\quad \mathcal{D}\subset DomA(f),\;f\in F,$ and the closure of the
operator $A(f)\uparrow \mathcal{D}$ coincides with $A(f)$ for any $f\in F,$
that is $A(f)\uparrow \mathcal{D}=A(f)$ in ${\Phi };$

b) the Range $A(f)\uparrow \mathcal{D}\subset {\Phi }_{+}$ \ for any $f\in F
;

c) for every $\psi \in \mathcal{D}$ the mapping $F\ni f\longrightarrow
A(f)\psi \in {\Phi }_{+}$ is linear and continuous;

d) there exists a strong cyclic (vacuum) vector $|\Omega \rangle \in
\bigcap_{f\in F}DomA(f),$ such that the set of all vectors $|\Omega \rangle
, $ $\prod_{j=1}^{n}A(f_{j})|\Omega \rangle ,$ $n\in \mathbb{Z}_{+},$ is
total in ${\Phi }_{+}$ (i.e. their linear hull is dense in ${\Phi }_{+}$).

Then there exists a probability measure $\mu $ on $(F^{\prime },C_{\sigma
}(F^{\prime }))$, where $F^{\prime }$ is the dual of $F$ and $C_{\sigma
}(F^{\prime })$ is the $\sigma -$algebra generated by cylinder sets in
F^{\prime }$ such that, for $\mu -$almost every $\eta \in F^{\prime }$ there
is a generalized joint eigenvector $\omega (\eta )\in {\Phi }_{-}$ of the
family $\mathcal{A},$ corresponding to the joint eigenvalue $\eta \in
F^{\prime },$ that is
\begin{equation}
<\omega (\eta ),A(f)\psi >_{{\Phi }}=\eta (f)<\omega (\eta ),\psi >_{{\Phi }}
\label{FQeq2.1a}
\end{equation
with $\eta (f)\in \mathbb{R}$ denoting the pairing between $F$ and
F^{\prime }$.

The mapping
\begin{equation}
{\Phi }_{+}\ni \psi \longrightarrow <\omega (\eta ),\psi >_{{\Phi }}:=\psi
(\eta )\in \mathbb{C}  \label{FQeq2.1b}
\end{equation
for any $\eta \in F^{\prime }$ can be continuously extended to a unitary
surjective operator $\mathcal{F}:{\Phi }\longrightarrow L_{2}^{(\mu
)}(F^{\prime };\mathbb{C}),$ where
\begin{equation}
\mathcal{F}\text{ }\psi (\eta ):=\psi (\eta )  \label{FQeq2.1c}
\end{equation
for any $\eta \in F^{\prime }$ is a generalized Fourier transform,
corresponding to the family $\mathcal{A}$. Moreover, the image of the
operator $A(f)$, $f\in F^{\prime }$, under the $\mathcal{F}-$mapping is the
operator of multiplication by the function $F^{\prime }\ni \eta \rightarrow
\eta (f)\in \mathbb{C}.$
\end{theorem}

We assume additionally that the main Hilbert space $\Phi $ possesses the
standard Fock space (bose)-structure \cite{BogB,Ber1,PrTB}, that is
\begin{equation}
{\Phi }=\oplus _{n\in \mathbb{Z}_{+}}{\Phi }_{(s)}^{\otimes n},
\label{FQeq2.1d}
\end{equation
where subspaces ${\ \Phi }_{(s)}^{\otimes n},$ $n\in \mathbb{Z}_{+}$, are
the symmetrized tensor products of a Hilbert space $\mathcal{H}:=L_{2}
\mathbb{R}^{m};\mathbb{C}).$ If a vector $g:=(g_{0},g_{1},...,g_{n},...)\in
\Phi $, its norm
\begin{equation}
\Vert g\Vert _{\Phi }:=\left( \sum_{n\in \mathbb{Z}_{+}}\Vert g_{n}\Vert
_{n}^{2}\right) ^{1/2},  \label{FQeq2.2}
\end{equation
where $g_{n}\in {\Phi }_{(s)}^{\otimes n}\simeq L_{2,(s)}((\mathbb{R
^{m})^{\otimes n};\mathbb{C})$ and $\parallel ...\parallel _{n}$ is the
corresponding norm in ${\Phi }_{(s)}^{\otimes n}$ for all $n\in \mathbb{Z
_{+}$. Note here that concerning the rigging structure (\ref{FQeq2.1}),
there holds the corresponding rigging for the Hilbert spaces ${\Phi
_{(s)}^{\otimes n}$, $n\in \mathbb{Z}_{+}$, that is
\begin{equation}
\mathcal{D}_{(s)}^{n}\subset {\Phi }_{(s),+}^{\otimes n}\subset {\Phi
_{(s)}^{\otimes n}\subset {\Phi }_{(s),-}^{\otimes n}  \label{FQeq2.3}
\end{equation
with some suitably chosen dense and separable topological spaces of
symmetric functions $\mathcal{D}_{(s)}^{n},$ $n\in \mathbb{Z}_{+}.$
Concerning expansion (\ref{FQeq2.1}) we obtain by means of projective and
inductive limits \cite{Be,BeS,Ber1,BerK} the quasi-nucleus rigging of the
Fock space $\Phi $ in the form (\ref{FQeq2.1})
\begin{equation*}
\mathcal{D}\subset {\Phi }_{+}\subset {\Phi }\subset {\Phi }_{-}\subset
\mathcal{D^{^{\prime }}}.
\end{equation*}

Consider now any vector $|(\alpha )_{n}\rangle \in {\Phi }_{(s)}^{\otimes
n}, $ $n\in \mathbb{Z}_{+},$ which can be written \cite{Be,BogB,KowS} in the
following canonical Dirac ket-form:
\begin{equation}
|(\alpha )_{n}\rangle :=|\alpha _{1},\alpha _{2},...,\alpha _{n}\rangle ,
\label{FQeq2.4}
\end{equation
where, by definition, \
\begin{equation}
|\alpha _{1},\alpha _{2},...,\alpha _{n}\rangle :=\frac{1}{\sqrt{n!}
\sum_{\sigma \in S_{n}}|\alpha _{\sigma (1)}\rangle \otimes |\alpha _{\sigma
(2)}\rangle ...|\alpha _{\sigma (n)}\rangle  \label{FQeq2.5}
\end{equation
and $|\alpha _{j}\rangle \in {\Phi }_{(s)}^{\otimes 1}(\mathbb{R}^{m}
\mathbb{C}):=\mathcal{H}$ for any fixed $j\in \mathbb{Z}_{+}$. The
corresponding scalar product of base vectors as (\ref{FQeq2.5}) is given as
follows:
\begin{equation}
\begin{array}{c}
\langle (\beta )_{n}|(\alpha )_{n}\rangle :=\langle \beta _{n},\beta
_{n-1},...,\beta _{2},\beta _{1}|\alpha _{1},\alpha _{2},...,\alpha
_{n-1},\alpha _{n}\rangle \\[5pt]
=\sum_{\sigma \in S_{n}}\langle \beta _{1}|\alpha _{\sigma (1)}\rangle
...\langle \beta _{n}|\alpha _{\sigma (n)}\rangle :=per\{\langle \beta
_{i}|\alpha _{j}\rangle :i,j=\overline{1,n}\}
\end{array}
\label{FQeq2.6}
\end{equation
where \textquotedblleft $per$\textquotedblright\ denotes the permanent of
matrix and $\langle .|.\rangle $ is the corresponding product in the Hilbert
space $\mathcal{H}$. Based now on representation (\ref{FQeq2.4}) one can
define an operator $a^{+}(\alpha ):{\Phi }_{(s)}^{\otimes n}\longrightarrow
\Phi }_{(s)}^{\otimes (n+1)}$ for any $|\alpha \rangle \in \mathcal{H}$ as
follows:
\begin{equation}
a^{+}(\alpha )|\alpha _{1},\alpha _{2},...,\alpha _{n}\rangle :=|\alpha
,\alpha _{1},\alpha _{2},...,\alpha _{n}\rangle ,  \label{FQeq2.7}
\end{equation
which is called the "\textit{creation}" operator in the Fock space $\Phi $.
The adjoint operator $a(\beta ):=(a^{+}(\beta ))^{\ast }:{\Phi
_{(s)}^{\otimes (n+1)}\longrightarrow {\Phi }_{(s)}^{\otimes n}$ with
respect to the Fock space ${\Phi }$ (\ref{FQeq2.1}) for any $|\beta \rangle
\in \mathcal{H},$ called the "\textit{annihilation}" operator, acts as
follows:
\begin{equation}
a(\beta )|\alpha _{1},\alpha _{2},...,\alpha _{n+1}\rangle :=\sum_{\sigma
\in S_{n}}\langle \beta ,\alpha _{j}\rangle |\alpha _{1},\alpha
_{2},...,\alpha _{j-1},\hat{\alpha}_{j},\alpha _{j+1},...,\alpha
_{n+1}\rangle ,  \label{FQeq2.8}
\end{equation
where the $"hat"$ over a vector denotes that it should be omitted from the
sequence.

It is easy to check that the commutator relationship
\begin{equation}
\lbrack a(\alpha ),a^{+}(\beta )]=\langle \alpha ,\beta \rangle
\label{FQeq2.9}
\end{equation
holds for any vectors $|\alpha \rangle \in \mathcal{H}$ and $|\beta \rangle
\in \mathcal{H}$. Expression (\ref{FQeq2.9}), owing to the rigged structure
\ref{FQeq2.1}), can be naturally extended to the general case, when vectors
\ |\alpha \rangle $ and $|\beta \rangle \in \mathcal{H}_{-}$, conserving its
form. In particular, taking $|\alpha \rangle :=|\alpha (x)\rangle =\frac{1}
\sqrt{2\pi }}e^{i\langle \lambda ,x\rangle }\in \mathcal{H}_{-}:=L_{2,-}
\mathbb{R}^{m};\mathbb{C})$ for any $x\in \mathbb{R}^{m}$, one easily gets
from (\ref{FQeq2.9}) that
\begin{equation}
\lbrack a(x),a^{+}(y)]=\delta (x-y),  \label{FQeq2.10}
\end{equation
where we put, by definition, $a^{+}(x):=a^{+}(\alpha (x))$ and
a(y):=a(\alpha (y))$ for all $x,y\in \mathbb{R}^{m}$ and denoted by $\delta
(\cdot )$ the classical Dirac delta-function.

The construction above makes it possible to observe easily that there exists
a unique vacuum vector $|\Omega \rangle \in \mathcal{H}_{+}$, such that for
any $x\in \mathbb{R}^{m}$
\begin{equation}
a(x)|\Omega \rangle =0,  \label{FQeq2.11}
\end{equation
and the set of vectors
\begin{equation}
\left( \prod_{j=1}^{n}a^{+}(x_{j})\right) |\Omega \rangle \in {\Phi
_{(s)}^{\otimes n}  \label{FQeq2.12}
\end{equation
is total in ${\Phi }_{(s)}^{\otimes n}$, that is their linear integral hull
over the dual functional spaces $\hat{\Phi}_{(s)}^{\otimes n}$ is dense in
the Hilbert space ${\Phi }_{(s)}^{\otimes n}$ for every $n\in \mathbb{Z}_{+}
. This means that for any vector $g\in \Phi $ the following representation
\begin{equation}
g=\oplus _{n\in \mathbb{Z}_{+}}\int_{(\mathbb{R}^{m})^{n}}\hat{g
_{n}(x_{1},...,x_{n})a^{+}(x_{1})a^{+}(x_{2})...a^{+}(x_{n})|\Omega \rangle
\label{FQeq2.13}
\end{equation
holds with the Fourier type coefficients $\hat{g}_{n}\in \hat{\Phi
_{(s)}^{\otimes n}$ for all $n\in \mathbb{Z}_{+}$, with $\hat{\Phi
_{(s)}^{\otimes 1}:=\mathcal{H}\simeq {L}_{2}(\mathbb{R}^{m};\mathbb{C}).$
The latter is naturally endowed with the Gelfand type quasi-nucleus rigging
dual to
\begin{equation}
\mathcal{H}_{+}\subset \mathcal{H}\subset \mathcal{H}_{-},  \label{FQeq2.14}
\end{equation
making it possible to construct a quasi-nucleolus rigging of the dual Fock
space ${\hat{\Phi}}:=\oplus _{n\in \mathbb{Z}_{+}}{\hat{\Phi}
_{(s)}^{\otimes n}.$ Thereby, chain (\ref{FQeq2.14}) generates the dual Fock
space quasi-nucleolus rigging
\begin{equation}
\mathcal{\hat{D}}\subset {\hat{\Phi}}_{+}\subset {\hat{\Phi}}\subset {\hat
\Phi}}_{-}\subset \mathcal{\hat{D}}^{\prime }  \label{FQeq2.15}
\end{equation
with respect to the central Fock type Hilbert space ${\hat{\Phi}},$ where
\mathcal{\hat{D}}\simeq \mathcal{D},$ easily following from (\ref{FQeq2.1})
and (\ref{FQeq2.14}).

Construct now the following self-adjoint operator $\ \ \rho (x):\Phi
\rightarrow \Phi $ \ \ as
\begin{equation}
\rho (x):=a^{+}(x)a(x),  \label{FQeq2.16}
\end{equation
called the density operator at a point $x\in \mathbb{R}^{m},$ satisfying the
commutation properties:
\begin{equation}
\begin{array}{c}
\lbrack \rho (x),\rho (y)]=0, \\[5pt]
\lbrack \rho (x),a(y)]=-a(y)\delta (x-y), \\[5pt]
\lbrack \rho (x),a^{+}(y)]=a^{+}(y)\delta (x-y
\end{array}
\label{FQeq2.17}
\end{equation
for all $y\in \mathbb{R}^{m}$.

Now, if to construct the following self-adjoint family $\mathcal{A}:=\left\{
\int_{\mathbb{R}^{m}}\rho (x)f(x)dx:f\in F\right\} $ of linear operators in
the Fock space $\Phi ,$\ where $F$ $:=\mathcal{S}(\mathbb{R}^{m};\mathbb{R})$
is the Schwartz functional space, one can derive, making use of Theorem \re
{FQth2.1}, that there exists the generalized Fourier transform (\re
{FQeq2.1c}), such that
\begin{equation}
{\Phi }(\mathcal{H})=L_{2}^{(\mu )}(\mathcal{S}^{\prime };\mathbb{C})\simeq
\int_{\mathcal{S}^{\prime }}^{\oplus }\Phi _{\eta }d\mu (\eta )
\label{FQeq2.17a}
\end{equation
for some Hilbert space sets $\Phi _{\eta },$ $\eta \in F^{\prime },$ \ and a
suitable measure $\mu $ on $\mathcal{S}^{\prime },$ \ with respect to which
the corresponding joint eigenvector $\omega (\eta )\in \Phi _{+}$ for any
\eta \in F^{\prime }$ generates the Fourier transformed family $\hat{u
=\left\{ \eta (f)\in \mathbb{R}:\;\;f\in F\right\} $. Moreover, if $\dim
\Phi _{\eta }=1$ for all $\ \eta \in F,$ the Fourier transformed eigenvector
$\hat{\omega}(\eta ):=\Omega (\eta )=1$ for all $\eta \in F^{^{\prime }}.$

Now we will consider the family of self-adjoint operators $\hat{u}$ as
generating a unitary family $\mathcal{U}:=\left\{ U(f):f\in F\right\} =\exp
(i\hat{u}),$ where for any $\rho (f)\in \hat{u},$ $f\in F,$ the operator
\begin{equation}
U(f):=\exp [i\rho (f)]  \label{FQeq2.18}
\end{equation
is unitary, satisfying the abelian commutation condition
\begin{equation}
U(f_{1})U(f_{2})=U(f_{1}+f_{2})  \label{FQeq2.19}
\end{equation
for any $f_{1},f_{2}\in F.$

Since, in general, the unitary family $\mathcal{U}=\exp (i\hat{u})$ is
defined in some Hilbert space $\Phi $, not necessarily being of Fock type,
the important problem of describing its Hilbertian cyclic representation
spaces arises, within which the factorization
\begin{equation}
\rho (f)=\int_{\mathbb{R}^{m}}a^{+}(x)a(x)f(x)dx  \label{FQeq2.20}
\end{equation
jointly with relationships (\ref{FQeq2.17}) hold for any $f\in F$. This
problem can be treated using mathematical tools devised both within the
representation theory of $C^{\ast }$-algebras \cite{Di,Di1} and the
Gelfand--Vilenkin \cite{GV} approach. Below we will describe the main
features of the Gelfand--Vilenkin formalism, being much more suitable for
the task, providing a reasonably unified framework of constructing the
corresponding representations.

\begin{definition}
\label{FQdef2.1} Let $F$ be a locally convex topological vector space,
F_{0}\subset F$ be a finite dimensional subspace of $F$. \ Let
F^{0}\subseteq F^{\prime }$ be defined by
\begin{equation}
F^{0}:=\left\{ \xi \in F^{\prime }:\;\;\xi |_{F_{0}}=0\right\} ,
\label{FQeq2.21}
\end{equation
and called the annihilator of $F_{0}$.
\end{definition}

The quotient space $F^{\prime 0}:=F^{\prime }/F^{0}$ may be identified with
F_{0}^{\prime }\subset F^{\prime }$, the adjoint space of $F_{0}$.

\begin{definition}
\label{FQdef2.2} Let $A\subseteq F^{\prime };$ then the subset

\begin{equation}
X_{F^{0}}^{(A)}:=\left\{ \xi \in F^{\prime }:\xi +F^{0}\subset A\right\}
\label{FQeq2.22}
\end{equation
is called the cylinder set with base $A$ and generating subspace $F^{0}$.
\end{definition}

\begin{definition}
\label{FQdef2.3} Let $n=\dim F_{0}=\dim F_{0}^{\prime }=\dim F^{\prime 0}.$
One says that a cylinder set $X^{(A)}$ has Borel base, if $A$ is Borel, when
regarded as a subset of $\mathbb{R}^{m}$.
\end{definition}

The family of cylinder sets with Borel base forms an algebra of sets.

\begin{definition}
\label{FQdef2.4} The measurable sets in $F^{\prime }$ are the elements of
the $\sigma -$ algebra generated by the cylinder sets with Borel base.
\end{definition}

\begin{definition}
\label{FQdef2.5} A cylindrical measure in $F^{\prime }$ is a real-valued
\sigma -$pre-additive function $\mu $ defined on the algebra of cylinder
sets with Borel base and satisfying the conditions $0\leq \mu (X)\leq 1$ for
any $X,$ $\mu (F^{\prime })=1$ and $\mu \left( \coprod_{j\in \mathbb{Z
_{+}}X_{j}\right) =\sum_{j\in \mathbb{Z}_{+}}\mu (X_{j}),$ if all sets
X_{j}\subset F^{\prime }$, $j\in \mathbb{Z}_{+}$, have a common generating
subspace $F_{0}\subset F$.
\end{definition}

\begin{definition}
\label{FQdef2.6} A cylindrical measure $\mu $ satisfies the commutativity
condition if and only if for any bounded continuous function $\alpha
:F^{n}\longrightarrow \mathbb{R}$ of $n\in \mathbb{Z}_{+}$ real variables
the function
\begin{equation}
\alpha \lbrack f_{1},f_{2},...,f_{n}]:=\int_{F^{\prime }}\alpha (\eta
(f_{1}),\eta (f_{2}),...,\eta (f_{n}))d\mu (\eta )  \label{FQeq2.23}
\end{equation
is sequentially continuous in $f_{j}\in F,$ $j=\overline{1,m}.$ (It is well
known \cite{GV,Gol} that in countably normalized spaces the properties of
sequential and ordinary continuity are equivalent).
\end{definition}

\begin{definition}
\label{FQdef2.7} A cylindrical measure $\mu $ is countably additive if and
only if for any cylinder set $X=\coprod_{j\in \mathbb{Z}_{+}}X_{j}$, which
is the union of countably many mutually disjoints cylinder sets
X_{j}\subset F^{\prime },j\in \mathbb{Z}_{+},$ $\mu (X)=\sum_{j\in \mathbb{Z
_{+}}\mu (X_{j})$.
\end{definition}

The following propositions hold.

\begin{proposition}
\label{FQpr2.8} A countably additive cylindrical measure $\mu $ can be
extended to a countably additive measure on the $\sigma $- algebra,
generated by the cylinder sets with Borel base. Such a measure will also be
called a cylindrical measure.
\end{proposition}

\begin{proposition}
\label{FQpr2.9} Let $F$ be a nuclear space. Then any cylindrical measure
\mu $ on $F^{\prime }$, satisfying the continuity condition, is countably
additive.
\end{proposition}

\begin{definition}
\label{FQdef2.10} Let $\mu $ be a cylindrical measure in $F^{\prime }$. The
Fourier transform of $\mu $ is the nonlinear functional
\begin{equation}
\mathcal{L}(f):=\int_{F^{\prime }}\exp [i\eta (f)]d\mu (\eta ).
\label{FQeq2.24}
\end{equation}
\end{definition}

\begin{definition}
\label{FQdef2.11} The nonlinear functional $\mathcal{L}:F\longrightarrow
\mathbb{C}$ on $F,$ defined by (\ref{FQeq2.24}), is called positive
definite, if and only if for all $f_{j}\in F$ and $\lambda _{j}\in \mathbb{C
,$ $j=\overline{1,n},$ the condition
\begin{equation}
\sum_{j,k=1}^{n}\bar{\lambda}_{j}\mathcal{L}(f_{k}-f_{j})\lambda _{k}\geq 0
\label{FQeq2.25}
\end{equation
holds for any $n\in \mathbb{Z}_{+}$.
\end{definition}

\begin{proposition}
\label{FQpr2.12} The functional $\mathcal{L}:F\longrightarrow\mathbb{C}$ on
F $, defined by (\ref{FQeq2.24}), is the Fourier transform of a cylindrical
measure on $F^{\prime }$, if and only if it is positive definite,
sequentially continuous and satisfying the condition $\mathcal{L}(0)=1$.
\end{proposition}

Suppose now that we have a continuous unitary representation of the unitary
family $\mathcal{U}$ in a Hilbert space $\Phi $ with a cyclic vector
|\Omega \rangle \in \Phi $. Then we can put
\begin{equation}
\mathcal{L}(f):=\langle \Omega |U(f)|\Omega \rangle  \label{FQeq2.26}
\end{equation
for any $f\in F:=\mathcal{S}$, being the Schwartz space on $\mathbb{R}^{m}$,
and observe that functional (\ref{FQeq2.26}) is continuous on $F$ owing to
the continuity of the representation. Therefore, this functional is the
generalized Fourier transform of a cylindrical measure $\mu $ on $\mathcal{S
^{\prime }:$
\begin{equation}
\langle \Omega |U(f)|\Omega \rangle =\int_{\mathcal{S}^{\prime }}\exp [i\eta
(f)]d\mu (\eta ).  \label{FQeq2.27}
\end{equation
From the spectral point of view, based on Theorem \ref{FQth2.1}, there is an
isomorphism between the Hilbert spaces $\Phi $ and $L_{2}^{(\mu )}(\mathcal{
}^{\prime };\mathbb{C})$, defined by $|\Omega \rangle \longrightarrow \Omega
(\eta )=1$ and $U(f)|\Omega \rangle \longrightarrow \exp [i\eta (f)]$ and
next extended by linearity upon the whole Hilbert space $\Phi $.

In the case of the non-cyclic case there exists a finite or countably
infinite family of measures $\left\{ \mu _{k}:k\in \mathbb{Z}_{+}\right\} $
on $\mathcal{S}^{\prime },$ with ${\Phi }\simeq \oplus _{k\in \mathbb{Z
_{+}}L_{2}^{(\mu _{k})}(\mathcal{S}^{\prime };\mathbb{C})$ and the unitary
operator $U(f):{\Phi }\longrightarrow {\Phi }$ for any $f\in \mathcal{S
^{\prime }$ corresponds in all $L_{2}^{(\mu _{k})}(\mathcal{S}^{\prime }
\mathbb{C})$, $k\in \mathbb{Z}_{+}$, to $\exp [i\eta (f)]$. This means that
there exists a single cylindrical measure $\mu $ on $\mathcal{S}^{\prime }$
and a $\mu -$measurable field of Hilbert spaces ${\Phi }_{\eta }$ on
\mathcal{S}^{\prime }$, such that
\begin{equation}
{\Phi }\simeq \int_{\mathcal{S}^{\prime }}^{\oplus }{\Phi }_{\eta }d\mu
(\eta ),  \label{FQeq2.28}
\end{equation
with $U(f):{\Phi }\longrightarrow {\Phi }$, corresponding \cite{GV} to the
operator of multiplication by $\exp [i\eta (f)]$ for any $f\in \mathcal{S}$
and $\eta \in \mathcal{S}^{\prime }$. Thereby, having constructed the
nonlinear functional (\ref{FQeq2.24}) in an exact analytical form, one can
retrieve the representation of the unitary family $\mathcal{U}$ in the
corresponding Hilbert space ${\Phi }$ of the Fock type, making use of the
suitable factorization (\ref{FQeq2.20}) as follows: ${\Phi }=\oplus _{n\in
\mathbb{Z}_{+}}{\Phi }_{n}$, where
\begin{equation}
{\Phi }_{n}=\underset{f_{n}\in L_{2,s}(\mathbb{R}^{m\times }{}^{n};\mathbb{C
)}{span}\left\{ \prod_{j=\overline{1,n}}a^{+}(x_{j})|\Omega \rangle \right\}
,  \label{FQeq2.29}
\end{equation
for all $n\in \mathbb{Z}_{+}$. The cyclic vector $|\Omega \rangle \in {\Phi }
$ can be, in particular, obtained as the ground state vector of some
unbounded self-adjoint positive definite Hamilton operator $\mathbb{H}:{\Phi
}\longrightarrow {\Phi }$, commuting with the self-adjoint particles number
operator
\begin{equation}
\mathbb{N}:=\int_{\mathbb{R}^{m}}\rho (x)dx,  \label{FQeq2.30}
\end{equation
that is $[\mathbb{H},\mathbb{N}]=0$. Moreover, the conditions
\begin{equation}
\mathbb{H}|\Omega \rangle =0  \label{FQeq2.31}
\end{equation
and
\begin{equation}
\inf_{g\in dom\mathbb{H}}\langle g,\mathbb{H}g\rangle =\langle \Omega
\mathbb{H}|\Omega \rangle =0  \label{FQeq2.32}
\end{equation
hold for the operator $\mathbb{H}:{\Phi }\longrightarrow {\Phi },$ where
dom $ $\mathbb{H}$ denotes its domain of definition.

To find the functional (\ref{FQeq2.26}), which is called the generating
Bogolubov type functional for moment distribution functions
\begin{equation}
F_{n}(x_{1},x_{2},...,x_{n}):=\langle \Omega |:\rho (x_{1})\rho
(x_{2})...\rho (x_{n}):|\Omega \rangle ,  \label{FQeq2.33}
\end{equation
where $x_{j}\in \mathbb{R}^{m}$, $j=\overline{1,n}$, and the normal ordering
operation $:\cdot :$ is defined as
\begin{equation}
:\rho (x_{1})\rho (x_{2})...\rho (x_{n}):\;=\prod_{j=1}^{n}\left( \rho
(x_{j})-\sum_{k=1}^{j}\delta (x_{j}-x_{k})\right) ,  \label{FQeq2.34}
\end{equation
it is convenient to choose the Hamilton operator $\mathbb{H}:{\Phi
\longrightarrow {\Phi }$ in the following \cite{GGPS,Gol,BPr} algebraic
form:
\begin{equation}
\mathbb{H}:=\frac{1}{2}\int_{\mathbb{R}^{m}}K^{+}(x)\rho
^{-1}(x)K(x)dx+V(\rho ),  \label{FQeq2.35}
\end{equation
being equivalent in the Hilbert space $\Phi $ to the positive definite
operator expression
\begin{equation}
\mathbb{H}:=\frac{1}{2}\int_{\mathbb{R}^{m}}(K^{+}(x)-A(x;\rho ))\rho
^{-1}(x)(K(x)-\mathcal{A}(x;\rho ))dx,  \label{FQeq2.35a}
\end{equation
satisfying conditions (\ref{FQeq2.31}) and (\ref{FQeq2.32}), where $A(x;\rho
):\Phi \rightarrow \Phi ,$ $\ x\in \mathbb{R}^{m},$ is some specially chosen
linear self-adjoint operator. The \textquotedblleft potential" operator
V(\rho ):{\Phi }\longrightarrow {\Phi }$ \ is, in general, a polynomial (or
analytical) functional of the density operator $\rho (x):{\Phi
\longrightarrow {\Phi }$ and the operator is given as
\begin{equation}
K(x):=\nabla _{x}\rho (x)/2\ +\ iJ(x),  \label{FQeq2.36}
\end{equation
where the self-adjoint \textquotedblleft current" operator $J(x):{\Phi
\longrightarrow {\Phi }$ can be defined (but non-uniquely) from the equality
\begin{equation}
\partial \rho /\partial t=\frac{1}{i}[\mathbb{H},\rho (x)]=-<\nabla
_{x},J(x)>_{,}  \label{FQeq2.37}
\end{equation
holding for all $x\in \mathbb{R}^{m}.$ Such an operator $J(x):{\Phi
\longrightarrow {\Phi },$ $x\in \mathbb{R}^{m},$ can exist owing to the
commutation condition $[\mathbb{H},\mathbb{N}]=0,$ giving rise to the
continuity relationship (\ref{FQeq2.37}), if taking into account that
supports $supp\;\rho $ of the density operator $\rho (x):{\Phi
\longrightarrow {\Phi },$ $x\in \mathbb{R}^{m},$ can be chosen arbitrarily
owing to the independence of (\ref{FQeq2.37}) on the potential operator
V(\rho ):{\Phi }\longrightarrow {\Phi },$ but its strict dependence on the
corresponding representation (\ref{FQeq2.28}). In particular, based on the
Fock space $\Phi ,$ defined by (\ref{FQeq2.1}) and generated by the
creation-annihilation operators (\ref{FQeq2.7}) and (\ref{FQeq2.8}), the
current operator $\ \ J(x):{\Phi }\longrightarrow {\Phi },$ $x\in \mathbb{R
^{m},$ can be constructed as follows:
\begin{equation}
J(x)=\frac{1}{2i}[a^{+}(x)\nabla a(x)-\nabla a^{+}(x)a(x)],
\label{FQeq2.37a}
\end{equation
satisfying jointly with the density operator $\rho (x):{\Phi
\longrightarrow {\Phi },$ $x\in \mathbb{R}^{m},$ defined by \ (\ref{FQeq2.16
), the following quantum current Lie algebra \cite{GMS,BPr,PM} relationships
\begin{eqnarray}
\lbrack J(g_{1}),J(g_{2})] &=&iJ([g_{1,}g_{2}]),\text{\ \ }
\label{FQeq2.37b} \\
\lbrack J(g_{1}),\rho (f_{1})] &=&i\rho (<g_{1},\nabla f_{1}>),  \notag \\
\lbrack \rho (f_{1}),\rho (f_{2})] &=&0,\text{ }  \notag
\end{eqnarray
holding for all $f_{1},f_{1}\in F$ and $g_{1},g_{2}\in F^{3},$ where \ we
put, by definition,
\begin{equation}
\lbrack g_{1,}g_{2}]:=<g_{1},\nabla )g_{2}-<g_{2},\nabla )g_{1},
\label{FQeq2.37c}
\end{equation
being the usual commutator of vector fields in the Euclidean space $\mathbb{
}^{m}.$ It is easy to observe that the current algebra (\ref{FQeq2.37b}) is
\ the Lie algebra $\mathcal{G},$ corresponding to the Banach Lie group
G=Diff$ $\mathbb{E}^{3}\rightthreetimes F,$\ the semidirect product of the \
Banach Lie group of diffeomorphisms \ \ $Diff$ $\mathbb{E}^{3}$ of the
three-dimensional Euclidean space $\mathbb{E}^{3}$ and the abelian subject
to the multiplicative operation Banach group of smooth functions $F$ . We
note also that representation (\ref{FQeq2.35a}) holds only under the
condition that there exists such a self-adjoint operator $\mathcal{A}(x;\rho
):{\Phi }\longrightarrow {\Phi },$ $x\in \mathbb{R}^{m},$ that
\begin{equation}
K(x)|\Omega \rangle =\mathcal{A}(x;\rho )|\Omega \rangle   \label{FQeq2.38}
\end{equation
for all ground states $|\Omega \rangle \in {\Phi },$ correspond to suitably
chosen potential operators $V(\rho ):{\Phi }\longrightarrow {\Phi }.$

The self-adjointness of the operator $\mathcal{A}(x;\rho ):{\Phi
\longrightarrow {\Phi },$ $x\in \mathbb{R}^{m},$ can be stated following
schemes from works \cite{GGPS,BPr}, under the additional condition of the
existence of such a linear anti-unitary mapping $T:{\Phi }\longrightarrow
\Phi }$ that the following invariance conditions hold:
\begin{equation}
T\rho (x)T^{-1}=\rho (x),\qquad T\;J(x)\;T^{-1}=-J(x),\qquad T|\Omega
\rangle =|\Omega \rangle  \label{FQeq2.39}
\end{equation
for any $x\in \mathbb{R}^{m}$. Thereby, owing to conditions (\ref{FQeq2.39
), the following expressions
\begin{equation}
K^{\ast }(x)|\Omega \rangle =\mathcal{A}(x;\rho )|\Omega \rangle
=K(x)|\Omega \rangle  \label{FQeq2.40}
\end{equation
hold for any $x\in \mathbb{R}^{m}$, giving rise to the self-adjointness of
the operator $\mathcal{A}(x;\rho ):{\Phi }\longrightarrow {\Phi },$ $x\in
\mathbb{R}^{m}$.

Based now on the construction above one easily deduces from expression (\re
{FQeq2.37}) that the generating Bogolubov type functional (\ref{FQeq2.26})
obeys for all $x\in \mathbb{R}^{m}$ the following functional-differential
equation:
\begin{equation}
\left[ \nabla _{x}-i\nabla _{x}f\right] \frac{1}{2i}\frac{\delta \mathcal{L
(f)}{\delta f(x)}=\mathcal{A}\left( x;\frac{1}{i}\frac{\delta }{\delta f
\right) \mathcal{L}(f),  \label{FQeq2.41}
\end{equation
whose solutions should satisfy the Fourier transform representation (\re
{FQeq2.27}). In particular, a wide class of special so-called Poissonian
white noise type solutions to the functional-differential equation (\re
{FQeq2.41}) was obtained in \cite{GGPS,BPr} by means of functional-operator
methods in the following generalized form:
\begin{equation}
\mathcal{L}(f)=\exp \left\{ 2\mathcal{A}\left( \frac{1}{i}\frac{\delta }
\delta f}\right) \right\} \exp \left( \bar{\rho}\int_{\mathbb{R}^{m}}\{\exp
[if(x)]-1\}dx\right) ,  \label{FQeq2.41a}
\end{equation
where $\bar{\rho}:=\langle \Omega |\rho |\Omega \rangle \in \mathbb{R}_{+}$
is a Poisson distribution density parameter.

It is worth to remark here that solutions to equation (\ref{FQeq2.41})
realize the suitable physically motivated representations of the abelian
Banach subgroup $F$ of the Banach group $G=Diff$ $\mathbb{E
^{3}\rightthreetimes F,$ mentioned above. In the general case of the Banach
group $\ G=Diff$ $\mathbb{E}^{3}\rightthreetimes F$ one can also construct
\cite{BPr,PK} a generalized Bogolubov type functional equation, whose
solutions give rise to \ suitable physically motivated representations of
the corresponding current Lie algebra $\mathcal{G}.$

\subsection{The quantum current Lie algebra and \ the magnetic Aharonow-Bohm
effect}

\bigskip

In the Section above we could get convinced that different representations
of the equal-time current algebra (\ref{FQeq2.37b})
\begin{eqnarray}
\lbrack \rho (f_{1}),\rho (f_{2})] &=&0,\text{ \ \ }  \label{FQeq3.19} \\
\lbrack J(g_{1}),\rho (f_{1})] &=&i\rho (<g_{1},\nabla f_{1}>),  \notag \\
\lbrack J(g_{1}),J(g_{2})] &=&iJ([g_{2},g_{1}])+i\rho (<B,g_{1}\times
g_{2}>),  \notag
\end{eqnarray
where $f_{1},f_{2}\in F$ and $g_{2},g_{1}\in F^{3},$ acting in the Fock
space $\Phi $ and describing a non-relativistic quasi-stationary system
consisting of a test charged particle $q,$ imbedded into a cylindrical
region $\Gamma \subset \mathbb{E}^{3},$ being under influence of the
magnetic field $B=\nabla \times A.$ \ Here $A\in \mathbb{E}^{3}$ is a
magnetic vector potential, the sign \textquotedblleft $\times
\textquotedblright\ means the vector product in $\mathbb{E}^{3}$ and the
current $\ \ J(x):{\Phi }\longrightarrow {\Phi },$ $x\in \mathbb{R}^{m},$ is
defined, owing to the \textit{minimal interaction} principle (\ref{FQeq2.13
), as
\begin{equation}
J(x):=\frac{1}{2}a^{+}(x)(\frac{1}{i}\nabla -A)a(x)-[(\frac{1}{i}\nabla
+A)a^{+}(x)]a(x).  \label{FQeq3.20}
\end{equation
In particular, it is assumed that $supp$ $B\subset \Gamma $ that gives rise
to the equality $\rho (<B,g_{1}\times g_{2}>=0$ \ for all points $x\in
\mathbb{E}^{3}\backslash \Gamma .$

As the suitable representations of the current algebra $\mathcal{G},$
defined by (\ref{FQeq3.20}), describe the physical quantum states of the
system $\Gamma $ under regards, we consider them following \cite{GMS,GMS1}
as those realized in the Hilbert space $L_{2}(\mathbb{E}^{3};\mathbb{C})$
under the condition that the charged test particle $q$ can penetrate the
boundary $\partial \Gamma $ \ of the region $\Gamma .$ Namely, let for $\psi
\in L_{2}($ $\mathbb{\Gamma };\mathbb{C})$
\begin{eqnarray}
\rho (f)\psi (x) &:&=f(x)\psi (x),  \label{FQeq3.21} \\
J(g)\psi (x) &=&\frac{1}{2i}\left\{ [<g(x),\cdot \nabla >+<\nabla ,\cdot
g(x)>]-<g(x),\int_{\Gamma }d^{3}y\frac{\nabla \times B(y)}{4\pi |x-y|
>\right\} \psi (x),  \notag
\end{eqnarray
for all $x\in \mathbb{E}^{3},$ where the sign \textquotedblleft $\cdot
\textquotedblright\ means that the natural operator composition. When
deriving (\ref{FQeq3.21}) there was imposed the invariant Coulomb gauge
constraint $\ <\nabla ,A>=0$ allowing to determine the vector potential
A\in \mathbb{E}^{3}$ as
\begin{equation}
A=\int_{\Gamma }d^{3}y\frac{\nabla \times B(y)}{4\pi |x-y|}  \label{FQeq3.22}
\end{equation
using the classical Maxwell equations (\ref{L0.6}), since the electric
displacement current component $\ \partial E/\partial t=0.$ The wave
function $\psi \in L_{2}(\mathbb{\Gamma };\mathbb{C})$ satisfies \ in the
cylindrical coordinates $x(r,\theta ,z)\in \Gamma $ \ the natural
quasi-periodical conditio
\begin{equation}
\psi (r,\theta +2\pi n,z)=\exp (i\lambda n)\psi (r,\theta ,z)
\label{FQeq3.23}
\end{equation
for some $\lambda \in \mathbb{R}$ and any $n\in \mathbb{Z},$ which should be
determined from the physically realizable representation (\ref{FQeq3.21}).
To do this, we need preliminarily to define the following \cite{GMS1}
unitary operator in the Hilbert space $L_{2}(\mathbb{E}^{3};\mathbb{C}):
\begin{equation}
Q\psi (x):=\left\{
\begin{array}{c}
\psi (x),\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ }x\in \Gamma ; \\
\exp \left\{ -i\int_{l_{\infty }}^{x}dl(y)\int_{E^{3}\backslash \Gamma
}dy^{\prime }\frac{\nabla \times B(y^{\prime })}{|y-y^{\prime }|}\right\}
\psi (x),\text{ \ \ }x\in \mathbb{E}^{3}\backslash \Gamma ;\text{\
\end{array
\right.  \label{FQeq3.24}
\end{equation
where the path $l_{\infty }\subset \mathbb{E}^{3}\backslash \Gamma $
connects an infinite point $\infty \in \mathbb{E}^{3}$ with the chosen point
$x\in \mathbb{E}^{3}\backslash \Gamma .$ Making use now of the unitary
transformation $\ \tilde{J}(g):=QJ(g)Q^{-1}$ and the fact that the magnetic
field
\begin{equation}
B(x)=\nabla \times \int_{\Gamma }\frac{\nabla \times B(y)d^{3}y}{4\pi |x-y|}
\label{FQeq3.25}
\end{equation
for any $x\in \mathbb{E}^{3},$ one obtains easily that the current operator
\ \tilde{J}(g):L_{2}(\mathbb{E}^{3};\mathbb{C})\longrightarrow L_{2}(\mathbb
E}^{3};\mathbb{C})$ is self-adjoint for any $g\in F^{3}$ and
\begin{equation}
\tilde{J}(g)\psi (x)=\frac{1}{2i}[<g(x),\cdot \nabla >+<\nabla ,\cdot
g(x)>]\psi (x),  \label{FQeq3.26}
\end{equation
and whose domain of definition $dom$ $\tilde{J}(g)\subset $ $L_{2}(\mathbb{E
^{3};\mathbb{C})$ is constrained by the functions $\psi \in L_{2}(\mathbb{E
^{3};\mathbb{C}),$ satisfying the condition
\begin{equation}
\psi (r,\theta +2\pi ,z)=\exp [i\lambda (B)])\psi (r,\theta ,z),
\label{FQeq3.27}
\end{equation
where, owing to (\ref{FQeq3.24}),
\begin{equation}
\lambda (B)=-\int_{\partial \Gamma }<B,dS>.  \label{FQeq3.28}
\end{equation
Thus, the found above representation (\ref{FQeq3.26}) of the current Lie
algebra (\ref{FQeq3.19}) in the Hilbert space $L_{2}(\mathbb{E}^{3};\mathbb{
}),$ \ in the case when $supp$ $B$ $\subset \Gamma ,$ describes the complete
set of observables if \ the charged particle $q$ is not excluded from the
region $\Gamma .$ \ In contrast, if the region $\Gamma $ \ possesses a
potential barrier at the boundary $\partial \Gamma ,$ such that the charged
particle $q$ can not penetrate it and enter the \ region $\Gamma ,$ \ the
value of $\lambda (B)\in \mathbb{R},$ defined by (\ref{FQeq3.28}), remains
constant. This entails that a suitable outside the region $\Gamma $
measurement can certainly indicate the presence of the magnetic field inside
$\Gamma .$ So, as it was mentioned in \cite{GMS}, the constructed above
current algebra representation completely describes our non-relativistic
quasi-stationary system not giving rise to the Aharonov-Bohm \cite{AB}
paradox. \ Moreover, the outside measurements results simply depend on the
representation of the current algebra (\ref{FQeq3.19}), which in turn
depends on the history of the system and the topology of the space outside
the barrier. As a related physical aspect of the explanation above it is
necessary to stress that \ vanishing \ of the magnetic field outside the
region $\Gamma ,$ possessing a nontrivial topology, does not imply the
simultaneous vanishing of the corresponding magnetic potential outside the
region $\Gamma .$ Namely the latter in somewhat obscured form was used in
the analysis of the current algebra representation, suitable for describing
the complete set of physical observables both inside and outside the region
\Gamma .$

\

\subsection{The Fock space embedding method, nonlinear dynamical systems and
their complete linearization}

This final Section is devoted to an interesting application of the

\bigskip

Consider now the case, when the basic Fock space ${\Phi }=\otimes _{j=1}^{s}
\Phi }^{(j)}$, where ${\Phi }^{(j)},$ $j=\overline{1,s}$, are Fock spaces
corresponding to the different types of independent cyclic vectors $|\Omega
_{j}\rangle \in {\Phi }^{(j)},$ $j=\overline{1,s}.$ This, in particular,
means that the suitably constructed creation and annihilation operators
a_{j}(x),a_{k}^{+}(y):{\Phi }\longrightarrow {\Phi },$ $j,k=\overline{1,s},$
satisfy the following commutation relations:
\begin{equation}
\begin{array}{c}
\lbrack a_{j}(x),a_{k}(y)]=0, \\[5pt]
\lbrack a_{j}(x),a_{k}^{+}(y)]=\delta _{jk}\delta (x-y
\end{array}
\label{FQeq2.42}
\end{equation
for any $x,y\in \mathbb{R}^{m}$.

\begin{definition}
\label{FQdef2.13} A vector $|u\rangle \in {\Phi },$ $x\in \mathbb{R}^{m},$
is called coherent \cite{Gl,PrBGT} with respect to a mapping $u\in L_{2}
\mathbb{R}^{m};\mathbb{R}^{s}):=M,$ if it satisfies the eigenfunction
condition
\begin{equation}
a_{j}(x)|u\rangle =u_{j}(x)|u\rangle  \label{FQeq2.43}
\end{equation
for each $j=\overline{1,s}$ and all $x\in \mathbb{R}^{m}$.
\end{definition}

It is easy to check that the coherent vectors $|u\rangle \in {\Phi }$ exist.
Really, the following vector expression
\begin{equation}
|u\rangle :=\exp \{(u,a^{+})\}|\Omega \rangle ,  \label{FQeq2.44}
\end{equation
where the cyclic state $|\Omega \rangle :=\otimes _{j=1}^{m}|\Omega
_{j}\rangle \ $and $\ (.,.)$ is the standard scalar product in the Hilbert
space $M,$ satisfies the defining condition (\ref{FQeq2.43}), and moreover,
its norm
\begin{equation}
\Vert u\Vert _{{\Phi }}:=\langle u|u\rangle ^{1/2}=\exp (\frac{1}{2}\Vert
u\Vert ^{2})<\infty ,  \label{FQeq2.45}
\end{equation
since $u\in M$ and the norm $\Vert u\Vert :=(u,u)^{1/2}$ is bounded.

Take now any function $u\in M:=L_{2}(\mathbb{R}^{m};\mathbb{R}^{s})$ and
observe that the Fock space embedding mapping
\begin{equation}
\phi :M\ni u\longrightarrow |u\rangle \in {\Phi },  \label{FQeq3.1}
\end{equation
defined by means of the coherent vector expression (\ref{FQeq2.44}) realizes
a smooth isomorphism between Hilbert spaces $M$ and ${\Phi }.$ The inverse
mapping $\phi ^{-1}:{\Phi }\longrightarrow M$ is given by the following
exact expression:
\begin{equation}
u(x)=\langle \Omega |a(x)|u\rangle ,  \label{FQeq3.2}
\end{equation
holding for almost all $x\in \mathbb{R}^{m}$. Owing to condition (\re
{FQeq2.45}), one finds from (\ref{FQeq3.2}) that, the corresponding function
$u\in M.$

In the Hilbert space $M,$ let now define a nonlinear dynamical system (which
can, in general, be non-autonomous) in partial derivatives
\begin{equation}
du/dt=K[u],  \label{FQeq3.3}
\end{equation
where $t\in \mathbb{R}_{+}$ is the corresponding evolution parameter,
[u]:=(t,x;u,u_{x},u_{xx},...,u_{rx}),r\in \mathbb{Z}_{+}$, and a mapping
K:M\longrightarrow T(M)$ is Frechet smooth. Assume also that the Cauchy
problem
\begin{equation}
u|_{t=+0}=u_{0}  \label{FQeq3.4}
\end{equation
is solvable for any $u_{0}\in M$ in an interval $[0,T)\subset \mathbb{R
_{+}^{1}$ for some $T>0.$ Thereby, the smooth evolution mapping is defined
\begin{equation}
T_{t}:M\ni u_{0}\longrightarrow u(t|u_{0})\in M,  \label{FQeq3.5}
\end{equation
for all $t\in \lbrack 0,T).$

It is now natural to consider the following commuting diagram
\begin{equation}
\begin{array}{ccc}
M & \overset{\phi }{\longrightarrow } & \Phi \\
T_{t}\downarrow &  & \downarrow {\mathbb{T}_{t}} \\
M & \overset{\phi }{\longrightarrow } & \Phi
\end{array}
\label{FQeq3.6}
\end{equation
where the mapping $\mathbb{T}_{t}:{\Phi }\longrightarrow {\Phi }$, $t\in
\lbrack 0,T)$, is defined from the conjugation relationship
\begin{equation}
\phi \circ T_{t}=\mathbb{T}_{t}.\circ \phi  \label{FQeq3.7}
\end{equation}

Now take coherent vector $|u_{0}\rangle \in {\Phi },$ corresponding to
u_{0}\in M,$ and construct the vector
\begin{equation}
|u\rangle :=\mathbb{T}_{t}\text{ }|u_{0}\rangle  \label{FQeq3.8}
\end{equation
for all $t\in \lbrack 0,T).$ Since vector (\ref{FQeq3.8}) is, by
construction, coherent, that is
\begin{equation}
a_{j}(x)|u\rangle :=u_{j}(x,t|u_{0})|u\rangle  \label{FQeq3.9}
\end{equation
for each $j=\overline{1,s}$, $t\in \lbrack 0,T)$ and almost all $x\in
\mathbb{R}^{m}$, owing to the smoothness of the mapping $\xi
:M\longrightarrow {\Phi }$ with respect to the corresponding norms in the
Hilbert spaces $M$ and $\Phi ,$ we derive that coherent vector (\ref{FQeq3.8
) is differentiable with respect to the evolution parameter $t\in \lbrack
0,T)$. Thus, one can easily find \cite{KowS,Kow,PrBGT,PrBGT1} that
\begin{equation}
\frac{d}{dt}|u\rangle =\hat{K}[a^{+},a]|u\rangle ,  \label{FQeq3.10}
\end{equation
where
\begin{equation}
|u\rangle |_{t=+0}=|u_{0}\rangle  \label{FQeq3.11}
\end{equation
and a mapping $\hat{K}[a^{+},a]:{\Phi }\longrightarrow {\Phi }$ is defined
by the exact analytical expression
\begin{equation}
\hat{K}[a^{+},a]:=(a^{+},K[a]).  \label{FQeq3.12}
\end{equation}

As a result of the consideration above we obtain the following theorem.

\begin{theorem}
\label{FQth3.1} Any smooth nonlinear dynamical system (\ref{FQeq3.3}) in
Hilbert space $M:=L_{2}(\mathbb{R}^{m};\mathbb{R}^{s})$ is representable by
means of the Fock space embedding isomorphism $\phi :M\longrightarrow {\Phi }
$ in the completely linear form (\ref{FQeq3.10}).
\end{theorem}

We now make some comments concerning the solution to the linear equation
\ref{FQeq3.10}) under the Cauchy condition (\ref{FQeq3.11}). Since any
vector $|u\rangle \in {\Phi }$ allows the series representation
\begin{equation}
\begin{array}{l}
|u\rangle =\underset{n=\sum_{j=1}^{s}n_{j}\in \mathbb{Z}_{+}}{\bigoplus
\frac{1}{(n_{1}!n_{2}!...n_{s}!)^{1/2}}\int_{(\mathbb{R
^{m})^{n}}f_{n_{1}n_{2}...n_{s}}^{(n)}(x_{1}^{(1)},x_{2}^{(1)},...,x_{{n
_{1}}^{(1)}; \\[10pt]
\qquad x_{1}^{(2)},x_{2}^{(2)},...,x_{{n
_{2}}^{(2)};...;x_{1}^{(s)},x_{2}^{(s)},...,x_{{n}_{s}}^{(s)}
\prod_{j=1}^{s}\left(
\prod_{k=1}^{n_{j}}dx_{k}^{(j)}a_{j}^{+}(x_{k}^{(j)})\right) |\Omega \rangle

\end{array}
\label{FQeq3.13}
\end{equation
where for any $n=\sum_{j=1}^{s}n_{j}\in \mathbb{Z}_{+}$ functions
\begin{equation}
f_{n_{1}n_{2}...n_{s}}^{(n)}\in \bigotimes_{j=1}^{s}L_{2,s}((\mathbb{R
^{m})^{n_{j}};\mathbb{C})\simeq L_{2,s}(\mathbb{R}^{mn_{1}}\times \mathbb{R
^{mn_{2}}\times ...\mathbb{R}^{mn_{s}};\mathbb{C}),  \label{FQeq3.14}
\end{equation
and the norm
\begin{equation}
\Vert u\Vert _{{\Phi }}^{2}=\sum_{n=\sum_{j=1}^{s}n_{j}\in \mathbb{Z
_{+}}\Vert f_{n_{1}n_{2}...n_{s}}^{(n)}\Vert _{2}^{2}=\exp (\Vert u\Vert
^{2}).  \label{FQeq3.15}
\end{equation
By substituting (\ref{FQeq3.13}) into equation (\ref{FQeq3.10}), reduces
\ref{FQeq3.10}) to an infinite recurrent set of linear evolution equations
in partial derivatives on coefficient functions (\ref{FQeq3.14}). The latter
can often be solved \cite{Kow,PrBGT,PrBGT1} step by step analytically in
exact form, thereby, making it possible to obtain, owing to representation
\ref{FQeq3.2}), the exact solution $u\in M$ to the Cauchy problem (\re
{FQeq3.4}) for our nonlinear dynamical system in partial derivatives (\re
{FQeq3.3}).

\begin{remark}
\label{FQre3.2} Concerning some applications of nonlinear dynamical systems
like (\ref{FQeq3.1}) in mathematical physics problems, it is very important
to construct their so called conservation laws or smooth invariant
functionals $\gamma :M\longrightarrow \mathbb{R}$ on $M$. Making use of the
quantum mathematics technique described above one can suggest an effective
algorithm for constructing these conservation laws in exact form.
\end{remark}

Indeed, consider a vector $|\gamma \rangle \in {\Phi },$ satisfying the
linear equation:
\begin{equation}
\frac{\partial }{\partial t}|\gamma \rangle +\hat{K}^{\ast }[a^{+},a]|\gamma
\rangle =0.  \label{FQeq3.16}
\end{equation}

Then, the following proposition \cite{Kow,PrBGT,PrBGT1} holds.

\begin{proposition}
\label{FQpr3.3} The functional
\begin{equation}
\gamma :=\langle u|\gamma \rangle  \label{FQeq3.17}
\end{equation
is a conservation law for dynamical system (\ref{FQeq3.1}), that is
\begin{equation}
d\gamma /dt|_{K}=0  \label{FQeq3.18}
\end{equation
along any orbit of the evolution mapping (\ref{FQeq3.5}).
\end{proposition}

\subsection{Comments}

Within the scope of this Section we have described general aspects of the
current algebra representations approach to describing \ quantum dynamical
systems in Fock type soaces. The problem of constructing physically
realizable quantum states was analyzed in much detail using the
Gelfand-Vilenkin representation theory \cite{GV,BLOT} of infinite
dimensional groups and the Goldin-Menikoff-Sharp theory \cite{GGPS,Gol,GMS}
of generating Bogolubov type functionals, classifying these representations.
The related problem of constructing Fock type space representations and
retrieving their creation-annihilation generating structure still needs a
deeper investigation within the approach devised. Here we mention only that
some aspects of this problem within the so-called Poissonian White noise
analysis were studied in a series of works \cite{BerK,Ber2,AKS,KSWY,LRS},
based on some generalizations of the Delsarte type characters technique.
Based on the quantum current Lie algebra description of a bounded
non-relativistic quantum system under an external electromagnetic field
within the \textit{minimal interaction} principle, the magnetic
Aharonow-Bohm effect has been interpreted. As it was mentioned in \cite{GMS
, the suitably constructed current algebra representation completely
describes our non-relativistic quasi-stationary system not giving rise to
the Aharonov-Bohm \cite{AB} paradox.

We have also presented main mathematical preliminaries and properties of the
related quantum mathematics techniques suitable for analytical studying of
the important linearization problem for a wide class of nonlinear dynamical
systems in partial derivatives in Hilbert spaces. Concerning this direction
it is worthy to mention the related results also obtained in \cit
{KS1,Kow,KowS,BPT,BPT1,SPTPrB}, devoted to the application of the Fock space
embedding method to studying solutions to a wide nonlinear dynamical systems
and to constructing quantum computing algorithms.

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