effects by stariya


									              Interaction of a charge with vacuum.
                                  Nonlinear Effects.

                                             The super–high electric field and electron
                                             clusters are "blank spaces" of modern physics.
                                             To describe them the modern nonlinear phys-
                                             ics is attracted.

According to conceptions of modern physics the charged substance being in va-
cuum interacts with him as with the physically active material medium. For micro-
systems (atoms, nuclei, elementary particles) this interaction is usually described by
the traditional quantum theory and quantum electrodynamics, in particular.

1. The small spatial dimensions, weight and large concentration of charged par-
ticles are typical for microsystems. The last results in the super-high electric field
acting within the area of their localization. In one's turn, the interaction of similar
field with vacuum [1] results in nonlinear effects (polarization, a birth of particles,
self-action of the field) that the modern quantum theory tries to describe.

The classical Maxwell-Lorenz theory does not work in the region of existence of simi-
lar (quantum) systems. It does not take into account the nonlinear effects a role of
which grows with an increase of the created field (i.e. with a growth of the charge
concentration). In fact it does not take into account the interaction of charges with
vacuum at all, because the latter is interpreted by it as an inert containing medium

2. Nevertheless, the super-high electric field is no prerogative only microsystems. In
the main it is possible existence of macrosystems creating the field of the order of
quantum one (1010 v/cm and more). Really, the classical physics laws permit, hav-
ing spent the work (energy), to localize any final charge (electrons) inside the suffi-
ciently small spatial volume. For example, one can localize the charge Q0=1C within
spherical volume of the radius R0=0.1 m.

It is easy can be convinced that the intensity E of similar system (absolute value), if
it is calculated according to the Coulomb law [2],

                E(r) = Q0 / (4  0 r 2) ,           r  R0                             (1)

reaches of the value 1010 v/cm, when r=R0, i.e. of quantum one. Here 0 is the abso-
lute dielectric penetration of vacuum in SI, r is a radial variable.

The super-high field of the localized electron system (we shall call this object as an
electron cluster) is a consequence of the huge concentration of charged particles
about 1021 p/m3 which is typical for quantum systems. All this makes essential an

account of interaction of the electron cluster (EC) with vacuum, i.e. an account of
the nonlinear quantum effects in application to macrosystems.

As to the Coulomb law, it does not work in the region of super–high fields in its
classical form (1). Here one can use it only for the approximate estimation. This law
does already not describe the EC behavior as an account of nonlinear effects
changes the situation in a root. Remaining within the framework of the macroscopic
approach to the problem, one can attempt to correct the Maxwell–Lorentz equations
(Coulomb law) in order to those began to work in the region of super–high fields and
the quantum concentration of charges.

3. In general case the problem of interaction with vacuum for the charged macro-
system creating the field of the order of the quantum one is little investigated in the
modern physics. In addition, a correct formulation of this problem is absent. Never-
theless, the author has risked to offer his point of view on the problem [3]. Not stop-
ping on details, we shall be limited with statement of some results concerning of the
behavior features of macrosystems in this case.

The offered approach is based on the postulates of modern relativistic theory of the
material continuum. At the derivation of the Maxwell–Lorentz equations the attempt
is made to take into account some quantum effects such as the vacuum polarization
and the field self-action.

The derivation itself is based on the integral conservation equations of energy and
charge for the relativistic continuum. No special assumptions going beyond the
scope of modern physics are made. As a result a system of differential quasi-
Maxwell equations is obtained, which differs from the traditional Maxwell-Lorentz
system in nonlinear additions tending to zero for weak fields and low charge densi-

Later on we shall be limited with the special case of the new system which is suffi-
cient for a description of the electrostatic interaction with vacuum of a charged ma-
crosystem. For electrostatics the received quasi–Maxwell equations are essentially
simplified and assume the form [3]:
                                              
            div D =    E D / 2 ,         rot E  0 .                       (2)

            
Here E and D are traditional vectors of the electric intensity and induction,  is a
charge density,  is a certain function [3].

Within the framework of the equations (2) it was formulated the model tesk for the
macrosystem localized in the spherical volume of the radius R0 with the appropriate
mass M and charge Q. It was required to find the external electrostatic field for the
spherically symmetrical distribution of charges. At that, as is well-know [2], the
structure itself of the charge distribution within the area rR0 does not play the part
of. Therefore, it was chosen one of the simplest variants with the charge Q distri-
buted over the spherical conducting shell of the radius R0.

4. Taking account of the model features (symmetries), the equations (2) can reduce

to one nonlinear ordinary differential equation:

               1 d(r 2 E)    Q         Q (r  R 0)
                                 E2               0                         (3)
              r 2  dr       2Mc 2
                                        4r 2

for the radial component of the vector E ={E(r),0,0}. Here (r–R0) is a delta-function,
=0 is a dielectric penetration of vacuum,  is a relative dielectric penetration of
vacuum, r  R0.

For interpretation of the equations (3) it is necessary to take into account what, ac-
cording to [3], M is the complete relativistic mass of the macrosystem carrying the
charge Q. In our case it also contains the mass of shell, and in common case this is
the mass of all the matter concentrated within the area of the charge Q localization.

Besides, the M and Q are not experimentally observed quantities, but “naked” ones,
according to the quantum field theory terminology [1,4]. Such a situation is typical
for the theories trying to take into account polarization of the medium containing a
charge. As to the connection between naked (Q,M) and observed (Q0, M0) quantities,
it should be fixed inside of the theory itself what will be shown later on.

It is easy to be convinced that the solution of the nonlinear equation (3) satisfying to
the attenuation condition E(r)  0 , r   has for r  R0 the shape:

                      Q                 1             ,                        (4)
            E(r) =
                     4r   2           1  1 
                                1  b(Q) 
                                        r  R 0 

                 b(Q) = Q2 / (8Mc2 ) .

5. In accordance with its nature the solution (4) determines only the local proper-
ties of the physical system under investigation. However, this is not enough. The
charge-field macrosystem itself is global as the electric field should occupy the en-
tire space outside of the radius R0. This means that a charge Q, through the field,
interacts with all the Universe vacuum (but not locally) and the global properties of
the Universe must influence on the behavior of the macrosystem.

At a consistent approach, taking into account all this, one must at once formulate
the initial quasi-Maxwell equations in the curved space of the optimum cosmological
model of the Universe (it is coordinated with the observant data in the best way).
However, it was not made and in the main for that reason what all these models are
locally equivalent to the Minkowsky space (the spetial relativity theory space), where
the quasi-Maxwell equations were formulated.

As to the global characteristics the situation can correct in some measure. The pa-
rameters of the optimum cosmological model one must introduce into the behavior
conditions of the charge–field system on the cosmological distances. One of the
ways to make this consists in a use of the integrated equivalent of the appropriate

nonlinear differential equation from (2). The speech is going about the following in-
tegrated relationship:

                                                          
                      ( D n)dS =
                                    r 
                                         d          ( E D) / 2d ,
                                                   ( r )

where (r) is any spherical volume containing the charge localization area, S(r) is a
surface with the unit normal n limiting the volume (r).

In fact the relationship (5) represents the modification of the Gauss theorem [2] in
application to our case. With consideration for well-known relations of the vector
analysis, the equations (3) and (5) for the model task are completely equivalent.
However, the integrated form of the last is adapted more for modeling of the global
properties of the charge–field system.

Let us remind that the system is in vacuum of the real Universe and its global cha-
racteristics such as:

           the field behavior on the border of the Universe,
           the amount of field energy within the Universe volume,

depend essentially from properties of the last (the optimum cosmological model). In
the capacity of such a model was chosen [3,5] the closed model of the uniform iso-
tropic Universe with inflation. It is coordinated with the modern astrophysical data
very well and it is maximumly close by its space–time symmetries to the Minkowski

6. Now we make use of (5) and choose the flat cosmological model (Minkowski
space) in the capacity of the optimum one. Then, integrating (5) all over the 3-
dimensional space at r  , we shall receive:

                       Q  = Q  Q W  ,
                                       
                                   Mc2 


                                              
                        Q = lim             D n  dS ,                  r  ,
                                                

                        Q = lim        d ,                             r  ,

                                            
                                           0 E E     / 2d ,            r
                        W = lim                
                                   (r)         

and instead of the function  it had been put in the quantity Q/Mc2 determined by
the task model.

As regards the physical interpretation of the quantity W, it is obvious. This is the
complete field energy within the Universe volume. The charge Q is interpreted by
analogy. The charge Q  corresponds to the charge which figures in the classical
Gauss theorem. It does not coincide with Q only because the model (5) is nonlinear.

One can formally take into account the curvature of the 3-dimensional space in
large scales and the condition Ru< for the Universe radius Ru (the optimum mod-
el). For this one must only demand that the dependence (6) is carried out not when
r  , and when

                       r  R  Ru .                                             (7)

The physical interpretation of all the quantities in (6) remains without any altera-
tions. Nevertheless, when the formula for W is calculated, one must take account of
what the volumes with radius R in the Minkowski space and in the curvature space
of the cosmological model do not coincide by a magnitude [3,5,6]. Using these con-
siderations and introducing the designation

                 = (Q – Q  )/Q ,                 0     1,                  (8)

the following relations were received in [3]:

               DQ 2         (1  R 0 / R)                      W / (Mc2 ) .    (9)
         W                                     ,
              8 R 0          1       1 
                       1  b(Q) 
                                      R 0 

Here the constant D is defined by the topology and geometry of the chosen cosmo-
logical model (RP3 and elliptic, accordingly).

Let us define the charge Q0 observed in experiment as:

                                      Q              ,             r  R.
              Q 0  lim
                                    1  1 
                           1  b(Q)     
                                     r R 0 

This guarantees the Coulomb limit in the solution (4) when r  R. In one’s turn,
taking account of (9), one can construct the system of two nonlinear algebraic equa-
tions for definition of the quantities Q, R as functions from (Q0, R0). This system
looks as follows:

                A(R)(D   )Q2    0 ,
                A(R) Q0 Q2  Q  Q0   0 ,

where, by definition,

                        (R0 / R  1)        
              A(R)                                .
                         8R0Mc2        (D   )Q2

The solution of the nonlinear system is:

               D Q 0                        (D   ) R 0 b(Q)
         Q=              ,         R=                                       (10)
              (D    )                 (D   ) b(Q)   R 
                                                          0

                (D    )Q                    (D   ) R b(Q)
         Q0 =               ,       R0 =                                ,   (11)
                   D                                              
                                           (D    ) b(Q) +   R 
                                                                   

where the quantity   (0     1) is considered as the independent variable.

7. In the relations received above the function b (Q) figures. As it turned out, by
analogy with the quantum theory, in that function one can single out a certain ma-
cro-quantity 0,

                   0  Q2 /(40R00c2 ) ,

where k is a binding factor between masses M0 and M (observed and “naked”). It is
determined later on.

The function b(Q) is defined with the quantity 0 as follows:

                           D2 R 0 0k
                   b(Q) =               .
                          2( D   ) 2

In fact, 0 is a macro-analog of the quantum fine structure constant . The last has
the following shape (in SI):

                      e2 / (2 0 hc) .

Here e0 is an electron charge, h is a quantum of action.

The quantity 0 turns into  if put down

                  Q0 =e0,       M0 =m0,       2R 0 m 0c= h ,                (12)

where m0 is a observed mass of any elementary particle with the observed charge
e0. In doing so the condition for R0 only defines a connection between the particle
size R0 and the length of de Broglie wave :

                    R0 = /2,                  = h/(m0c.

The transition (12) is not only formal, but also as a matter of fact. This means that
the electron cluster (EC) containing only one electron is also described within
framework of the above-mentioned considerations. Leaning against on this and on
the same nonlinear quasi-Maxwell equations, one can write down the system of two

nonlinear algebraic equations for quantities  and e. Here e = e(e0) is a naked charge
of electron, and the vacuum relative inductivity  is unknown,  1.

This system is:

            a(r0)e0e2 + e – e 0 = 0,
             a(r0) (D –      1)e2   +  = 0.

                           (r0 / Ru  1) ,                  h ,              r0  R0 .
                a(r0 )                             r0 =
                           80m0c2r0                      2m0c

Solving the system (13), we shall receive the following analytical solution:

                 2R u e0 ,                       2 D  1R u .                        (14)
           e                                
                D R u  r0                     D2  R u  r0 

With consideration for (14) the dependence е = е(е0) may also write down in the

                   e 0 D                           (D  1)e
           e             ,                  e0             .                           (15)
                  (D  1)                             D

The last is coordinated with (10) and (12) when   = 1,              R = Ru .

Remark 1. If R u  , then the vacuum dielectric penetration  from (14) will be
the same for all the elementary particles,   1/. However, if Ru < , then = (r0)
and depends on a choice of an elementary particle. In this case the interaction of
particles with vacuum differs from on the small amount of the order of r0/R0. All
that allows to speak (formally) about existence of the own vacuum for each particle
(the electron-nositron vacuum, proton-antiproton vacuum and so on). Such an ap-
proach is typical also for the quantum theory [1,4].    

8. To establish the connection between the naked mass M and the observed mass
M0, it is necessary to attract, in addition to the quasi-Maxwell equations (2), the
modified equations of gravistatics. The last are analogues of the Einstein equations
in the Minkowski space in the appropriate approximation [5]. Moreover, in the
common form, these equations generalize the analogous nonlinear equations offered
by Xevisaid, Karstua, Brillouin [7]. However, in our case the nonlinear additions
connected with those equations were not taken into account by force of the gravita-
tion field weakness.

Let us write out the nonlinear gravistatics equations necessary for us in the form:

                                                               
            div( 0 G)  c2  (E D) / 2 ,                   rot G = 0                   (16)

Here G is a vector of the gravitation field intensity,  is a relativistic mass density
(the field source),
                       0   c2 / (4 G 0) ,
G0 is a gravity constant.

Remark 2. For a complete account of the gravitational and electromagnetic interac-
tions with vacuum the mathematical models (2), (16) must formulate and solve
jointly. However, in the considered approximation taking account of the weakness of
the object gravity field, the equations (2) and (16) are divided. That has allowed to
solve theirs successively.      

The solution of system (16) for the model task is easily to find:

                            G 0M          1                                   (17)
                G(r)         2 
                             r            1  1 
                                 1  b(Q)     
                                          r R0  

                              G = {G (r), 0, 0},                R0  r.

Let us define the observed mass M0 by analogy with the charge Q0 as:

                   M 0  lim
                                       M                             r  R.
                                       1  1 
                               1  b(Q)     
                                       r  R0  

That guarantees the Nouton limit on cosmological distances in the solution (17).

Now let us allow for the quantity Q0 definition and connections (10), (11). This al-
lows to establish the relations between quantities M0 and M in the shape:

                 DM 0 ,                           (D   ) M
           M                             M0                                 (18)
                (D   )                              D
           M0 =kM,                         k=(D–   )/D.

In the relatively weak fields, when the Maxwell theory still works, R=Ru,              0
and accordingly Q  Q0, M  M0,                  k  1.

In the other extreme case, when a field is very strong and the Maxwell theory does
already not work in a classical form, then R  R u,   =1 and
                    DQ 0 ,            (D  1)Q ,                       (19)
              Q                  Q0 
                   (D  1)               D

                          ,                     (D 1)M .                     (20)
              M                       M0 
                   (D 1)                          D

9. With consideration for the connections between quantities Q, Q0 and M, M0,
the quantity   (0     1) introduced in (8 – 9) assumes the shape:

                      DQ2  1    
                      0     1,                                         (21)
                    80M0c  0 R 

           R0  R  Ru and   = W/(Mc2).

We remind that here W is the complete energy of the electrostatic field created
when r < R by the naked charge Q and W is the part of this energy not involved in
the vacuum polarization. It is measured directly in the experiment.

Meaning the formula (21), let us return to the base definition of the parameter   in
(8) and give a physical interpretation of the area 0    1. For this purpose we
make use of the modified Gauss theorem (5). With a help (6) we present it in the
                Q  =Q –   Q.                                         (22)

When in (22) the nonlinear addition   Q  0 at the expense of    0, that is
characteristic for the Maxwell theory, then Q   Q and we come to the classical
Gauss theorem with the appropriate physical interpretation.

In common case the polarization addition   Q can only decrease Q. At that in the
Gauss theorem a certain effective charge Q  = Q (1–   ) will be involved instead Q.
Obviously, the maximum vacuum polarization corresponds to the complete neutra-
lization of the charge Q. In this case the effective charge Q  vanishes. All that puts
the limitation 0  Q  /Q  1 on the limits of the charge Q  alteration. It is fully
equivalent to the limitation (8) for   .

10. Analysing the relation (21) for   , one can easily be convinced that all the vari-
able EC macro-characteristics (Q0, M0, R0, R) are contained there. The first three of
them (Q0, M0, R0,) define the EC core charge, mass and size observed experimental-
ly. These determine just the electron cluster with substance accompanying to it. As
regards the fourth quantity R, it is the global characteristic of all the charge–field

At the physical strict definition of the quantities   and R it is necessary to interp-
ret theirs as functions:

                    =   (Q0, M0, R0, R),         R  (R0, Ru],

                   R = R (Q0, M0, R0,   ),            (0, 1].

The last, with consideration for (10) and (21), can write down as:

                     1,                                                   R0  R  Ru
                   D Q 0
                                2        1   1
                                               1,                      R  Ru
                                             
                                    2    R 0 R
                      8   M 0 c

                     R u,                                                 0    1
                R   (D  1)bR 0
                     (D  1)b  R  R u ,
                                0                                         1 .

All that, including the physical interpretation of the quantities R,   , allows to di-
vide the area of their definition ( R0 < R  Ru,  0 <    1 ) as follows.

A. The incomplete compensation Area. Inside this area the charge Q is com-
pensated with the polarization addition   Q only in part. In doing so the following
conditions are carried out:

               R=Ru               0 <  < 1                 ( Q   0).                    (24)

This case is typical for the weak (Maxwell’s) and moderately strong fields. At that, by
force (23),   is a function of the quantities Q0, M0, R0 only,

                   DQ2  1
                       0        1                              W =   Mc2.               (25)
                       2R  R 
                 8 0M 0c  0   u

It is obvious, the area A is characterized by its set of meanings for macro characte-
ristics Q0, M, R0 not leading out the quantity   out of the definition interval.

Provided the relations (24) and (25) are carried out, then the less   (or Q0 and
Q0/M0) the less the EC behavior will be differing from the Maxwell’s one dictated by
the Coulomb law. In this case, as is well-known, the similar system of the same
charges is not stable and disintegrates under action of the Coulomb repulsion. 

В. The complete compensation Area. In this area the charge Q is compensated
with the polarization addition   Q in full. At that the effective charge Q  becomes
zero and it is fulfilled:

              R=Ru,                   =1              ( Q  = 0).                          (26)

Here the compensation is realized in entire 3-dimensional space of the Universe
when r  R,      R=Ru. The transition the area A to B is reached in (23) with an in-
crease of the values Q0 and Q0/M0 (when R0 is fixed). That results in intensifica-
tion of the interaction of the naked charge with vacuum.

This case is characteristic for fields of the order of the quantum ones. It singles out
within the area of a change of quantities R,   a certain critical point (Ru,1). In this

point the relations (23) assume the shape:

                  DQ2  1
                      0          1                   Mc2 = W.                 (27)
                         2  R  R   1,
                8 0M 0c  0     u

Obviously, there is always a set of the macro-characteristics (Q0, M0, R0) not leading
out the EC out of the region of action of the conditions (27) and (26). 

С. The space-limited compensation Area. Inside this area the charge Q is also
compensated in full at the expense of the vacuum polarization, but that is reached
within a finite volume of the Universe where r < R . In doing so it is carried out:

             R 0< R<Ru,               = 1        ( Q  = 0).                  (28)

In the region (28) the relations (23) assume the shape:

                             1  1                 Mc2 =  W.                   (29)
                                  1,
               8 0M 0c2    R0 R 

The transition B to C is realized by a further increase Q0 and Q0/M0 (when R0 is
fixed). In doing so the condition (29) is fulfilled at the expense of a change R, R R0.

Having presented R as the function R = R(Q0, M0, R0) or having substituted   =1
in the formula (10), one can solve (29) regarding R. Eventually, it gives:

                       (D  1)b(Q)R0 ,             R0< R< Ru.                     (30)
                     (D  1)b(Q)  R0 
From the definition of the radius R as the size of the complete compensation area of
the charge Q it follows that the electric field is fully absent when r > R. It is con-
centrated within the layer    R0< r< R, where it changes according to the law (4). At
that, in accordance with the relationship W=Mc2 from (29), if the mass M is fixed,
then the total field energy within this layer remains without the changes in spite of
a growth Q (or Q0). However, the field energy density will be increasing at the ex-
pense of R  R0 or a decrease of the layer thickness.

Remark 3. One can try to make an analogy. In all the cases (A – C) the examined
charge-field system reminds with its structure the spherical condenser with the ca-
pacitor plate radii R0 and R. In doing so the charge Q is placed at r = R0 and the
charge QR= –   Q is placed at r = R. Formally, this means that one can introduce
the electrocapacities Сn and С0 for a naked charge and for a observed one:

               4  0 D  1  1                   4 0 (D   )  1 1  1     (31)
          Cn =               
               (D   )  R 0 R              C0                       .
                       *                                D         R0 R 

In doing so the appropriate electrostatic field energy is determined as:

                   D Q2 ,                       D Q2 .                        (32)
            W =                         W       0
                   2C n                         2C0

That is coordinated with a definition W and with the similar formulas from the clas-
sical electrostatics [2]. These formulas are only distinguished by the factor D=const
taking into account a difference of 3-dimensional volume in the Minkowski space
(the model task) and 3-dimensional volume in the curved space of the real Universe
(the optimum cosmology model). The factor is introduced from the same considera-
tions that are also used by us for the energy W definition (see above) and from the
global properties of the charge-field system.            

 Remark 4. In spite of all its evidence, the analogy with a spherical condenser is
superficial enough. An analysis of the differential-geometric structure of the field in-
side the layer R0< r < R taking into account the space curvature of the cosmologi-
cal model shows [3] that for cases B and C the geometry and topology of the charge-
field system coincide with analogous characteristics of the optimum cosmological
model (topology is RP3, geometry is elliptic).

In a language of the force lines this means, leaving the charge distribution area at
r R0, those do not go in infinity as in the Maxwell case in the space R3. The force
lines bend when r = R and turn back. In the idealized one-dimensional case this pic-
ture looks as follows (Fig. 1).

                 Q                                          Q
                                    R1                                         R1
                                                                   RP 1
             Weak field
             Topology R 1
                                                         Strong field
                                                         Topology RP 1

             Strong field
             Topology S
                                          Fig. 1

Here the transition the topology S1 to RP1 is realized as usually [8,9].
       Formally, just this situation is presented by the formulas (31) and (32) in the
classical electrostatics terms, i.e. by the spherical condenser model.        

Obviously, the real curvature of the electric field lines is a consequence of the elec-
tric field interaction with physical vacuum, vacuum polarization and the field self-
action. All that is displayed in the geometry and topology of the charge-field system
under investigation. Those cease to be Euclidean and the field is self-locked within
a finite 3-dimensional volume. As to the quasi-Maxwell case A, especially when
  0, then the behavior of the charge-field system and its geometry differ from the

classical not very much. In particular, if the distance r is slight in comparison with
the cosmological one (r << R u).

11. In the offered area division, where the quantities R and   are changed, the
area B is critical both formally and as a matter of fact. Indeed, formally, the signific-
ances of quantities R and   reached here are maximum and fixed. Let us
represent it as follows:

                ( R = Ru ,        < 1 )  A,
                ( R = Ru ,        = 1 )  B,                                (33)
                ( R < Ru ,        = 1 )  C.

Here, as one can see, the transition out of B into one of areas A or C is connected
with the appropriate decrease values   and R. The physical interpretation of that
will be more transparent, if we attempt to determine the binding energy between the
charged particles forming objects a kind of the EC.

For the binding energy definition we consider one each object submitted by follow-
ing parameters in each of areas A – C:

                 ( Q0, Ma, R0, Ru )  A,
                 ( Q0, Mb, R0, Ru )  B,                                        (34)
                 ( Q0, Mc, R0, R )  C.

Here the size each of the macro-objects and its charge (R0 и Q0 ) are fixed and the
same for everything, and the mass M0 assumes a meaning M0  (Ma , Mb , Mc ).

Formally, the transition out of the classical area A, where the object is stable, into
the area B and then into C is realized by the mass M0 change:

            Ma  Mb  Mc,                   Ma > Mb > Mc.

On the other hand, by definition in (25–29),

        Ma c2  Wa k,
                             Mb c2  Wb k,                                    (35)

                                                  Mc c2  Wc k.
At that Wa= Wb and

                             D2 Q 2  1        1 ,
                 Wa               0            
                           8 0 (D  1)  R 0 R u 
                                                  

                         D2 Q 2  1       1
                 Wc           0         ,
                       8 0 (D  1)  R 0 R 
                                            

where k=(D–1)/D is defined in (18), and each of quantities Wa, Wb, Wc is the
complete energy of field created by the same charge Q0(R0) at r  R, but in the dif-
ferent areas A – C.

The change (decrease) of this energy at the transition А  B  C signifies that its
appropriate part is absorbed by vacuum and it means it is used at formation of the
binding energy between particles of a cluster. All that allows to define the "naked"
binding energy for one of N particles as follows:

               ES = (Wc – Wb)/N.                                         (36)

As to the observed binding energy E0, with consideration for relation   M=M0/k
between naked and observed mass from (18), it is defined

               E0= k ES.                                                  (37)

Using expressions (35), the binding energy ES can present in the unwrapped shape:

                          D2 Q 2  1
                                  0       1 .                            (38)
              ES                          
                       8 0 (D  1)N  R R u 

Having excluded the radius R from the last formula with a help of the relation (30),
we receive ES as the function E S = E S (Q 0 ,M 0 ,R 0 ,N),

                   M 0 c2 D (Q0)2  1   1
              ES                       .                              (39)
                    kN     8 k N  R0 Ru 
                                            

12. Properly from the binding energy definition, there is one-to-one correspondence
between quantities ES and R. In doing so

             ES(R) < 0,                         R < Ru,

             ES(R) = 0,                         R = Ru.

That means, the quantity ES can use instead R for the definition of the areas A – C
(see above (33)). The last makes the physical interpretation of these areas more
transparent and habitual. However, to be more consistent one must introduce a di-
mensionless quantity also for the energy ES. Let us mark it as  S and write out to-
gether with   for a convenience of comparison them:

                      W          D Q 0)2  1 1  ,
                                               
                      Mc 2
                                 8 M 0 c  R 0 R 
                      ES          D Q 0)2  1         1 .
               S                                     
                      Mc 2        8 M  c2  R u       R

Here W  (Wa, Wb, Wc).

The new dimensionless parameter  S replaces ES uniquely and it is connected with R
by conditions similar to (40):

                                   S (R) < 0,             R< Ru ,

                                   S (R) = 0,             R = Ru .

Using  S in (33), we result in the following definition of the areas A – C:

                   ( S = 0,           < 1 )  A
                   ( S = 0,           = 1 )  B                             (42)
                   ( S < 0,           = 1 )  C

Remark 5. In fact the parameter   determines the normalize of the general energy
W for the field created by a charged macro-object. As one can see from (42) it can-
not surpass the value Mc2.      

Now let us try to represent graphically the correlation of quantities   and  S . For
this we shall be considering (for simplicity) that ( S ,   )  R2. This allows to draw
the following curves

               A                                                A     Ru
      O             1                                O                   R


          s                              Fig 2

On the Fig.2 there are tree characteristic areas of interaction with vacuum of the
charge system. So, in the quasi–Maxwell area A the behavior of the system of the
same charges are near to the classical one, especially for   << 1, and the stable
EC existence is impossible. With a growth of the values Q0 and Q0/M0 (R0 is fixed)
the object gets in some critical point B(0,1).

At the further growth of the values Q0 и Q0/M0 this object goes over from the area B
to C, i. e. it find oneself ("comes down") in the potential well ES = ES (R) connected
with C and represented on the Fig. 2 to the right. Here the EC is already stable.

In the point B (0,1) the situation is interpreted somewhat more complicated. In fact
the point B (0,1) acts the part of a phase transition point between areas A and C,
and the relation (27) is the EC stability condition. With consideration for the defini-
tions made in (10–14), let us solve the condition (27) concerning Q0 and write down
in form:

                         4 0D(Ru  r0 )
                 Q2 
                  C                       R M c2  ,
                         D (Ru  R0 ) 0 0
                           8 0Ru
                 Q2 
                  C                  R M c2  .
                         D(Ru  R0 ) 0 0

Here Q0QC, QC is some critical (minimum) charge with which the macrosystem gets
in the point B(0,1) when M0 и R0 are fixed (see Fig. 2). For any charge Q>QC the sys-
tem already gets inside the EC stability area C.

Obviously, by force of the correlation  = () (see (14)), where  is the fine struc-
ture constant, the relations in (43) are fully equivalent (in doing so R0< R, R=Ru). In
an obvious approximation, when r0 << R u, R0<<Ru, the more compact formulas fol-
low out of (43):

                         4 0D                                8 0
                  Q2 
                   C             R M c2  ,
                          D  0 0
                                                        Q2 
                                                         C      D
                                                                      R0M 0c2 

that are more convenient in the direct accounts.

Summarizing the discussion of areas A – C, we give them the following brief charac-

1. A ( S = 0,  0 <   < 1) – This is a quasi-Maxwell, quasi-classical area, the EC in-
   stability area.
2. B ( S =0,   = 1) – This is a quasi-quantum area (elementary particles, critical
   macro-objects), the phase transition point B (0,1).
3. C ( S < 0,   = 1) – This is a post-quantum area (macroscopic electron clusters),
   the EC stability area.

The area A have been studied by modern physics regarding well, especially when
  << 1. However, this may not affirm about other areas. The consecutive investi-
gate of laws working here will yet have to.

References (in Russian)
    1. Migdal A. B. Fermions and bosons in the high fields. M., 1978.
    2. Jackson J. Classical Electrodynamics. M., 1965.
    3. Buerakov V. A. The Maxwell equation in physical vacuum and experimental
       consequences. Manuscript, 1991.
    4. Berestetskij V. B. and other. Relativistic quantum theory. Part 1. M., 1968.
    5. Buerakov V. A. Some problems of the gravitation theory and cosmology. Preprint
       AS Ukraine, IPPMM–11–88. Lvov, 1988, 48 p.
    6. Tolman R. Relativity, thermodynamics, cosmology. M., 1974,.
    7. Brillouin L. New point of view on the relativity theory. M., 1972.
    8. Aleksandryan R. A., Mirzaxanyan Ye.A. Common topology. M., 1979.
    9. Wolf J. Spaces of constant curvature. M., 1982.


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