GRAVITATION AND
QUANTUM INTERACTION WITHOUT HYPOTHESES
(ELECTRODYNAMICS-BASED
APPROACH IN HANDLING FORCE-FIELD PROBLEMS)
B.F. Poltoratsky
Variants of elementary particles’ force interaction
behavior through variable electromagnetic fields are
considered below. It
is established that Maxwell’s electrodynamics can be of help in identifying the
fundamental mechanisms of gravitation and quantum interaction.
Introduction
Natural
sciences generally and physics specifically can be based on the two opposite
conceptions, which differ by their different attitude to the role of an
experiment (experience) in the cognitive process.
The
first conception is the traditional one (Aristotle,
Foma Akvinsky, Newton, Maxwell and others). Priority
of an experience is its basis. All
elements of induction and deduction, including mathematics, belong to means
arsenal (or to "craft"), which help to reach the understanding of the
Nature.
Here
all variety of revelations (afflations)
belongs to religion. Hypotheses are admitted only «in sense of prompting to
experience» (Newton).
The
second conception is modern. In the modern conception the right of the
initiative belongs to a revelation (afflation), which takes the form of a
"principle"
or "model". Its
components are all sorts of hypotheses and postulates (It is assumed,
that
the Nature and its laws can be postulated). Here the deduction from the
hypothetical beginning obviously prevails. The experiment role is reduced only
to an illustration of the advantage of this or that revelation (accordingly of
the « scientific school »).
Below
we shall try to return to bottoms of a field theory and substance and to look at
them from position of the first conception.
First, we recognize the connection of
structure of all fundamental particles with an electrodynamics. It is known,
that else Henry
Poincare wrote: «Abraham’s
calculations and Kaufman’s experiments have shown that mechanical mass is equal
to zero and the mass of electrons, at least the mass of negative electrons, is
of an exclusively electrodynamics origin» [1]. However
Poincare's resume can seem to someone a hypothesis. Therefore we shall be based
on such compromise statement, which has absolutely irreproachable experimental
base.
Second, we believe
the following assertion to be also well-grounded, namely - all the fundamental
particles generate a variable electromagnetic field whose base frequency (or
harmonic, designated as ν) is
obtained for all the particles using the equation of the form hν = mc2.
The validity of this assertion can be corroborated by the phenomenon of
particles diffraction and by the capability of any spatial inhomogeneities (or
dislocations - in terms of solid body properties) to radiate and reradiate
fluxes of electromagnetic energy. It is necessary to consider only resonant
character of interaction of particles with a field. The findings
of analytical and
numerical efforts [2] relative to the behavior of local streams of
electromagnetic energy in
dielectric wave-guides and nonlinear mediums testify to the same effect in more details.
It has to be
emphasized, however, that this assertion specifically concerns only variable
fields. Static electrical and magnetic fields caused by charges and moments of
particles can exist too. But those are basically beyond the
interest of this our research.
As will be
shown below, these two known properties of fundamental particles make it possible to
state and solve the force fields problem solely within the framework of
Maxwell’s electrodynamic theory [3].
1. First we shall
consider a single fundamental particle with which middle we shall connect the
beginning of coordinates. Suppose the particle represents for us a certain nonlinear
riddle. We shall surround a particle with a spherical transitive layer, which
thickness is sufficient to define amplitudes of fields and their spatial
derivatives. Be under such circumstances any field configuration in a zone of a
transitive layer can be presented by a finite or infinite set of spherical
harmonics. And the space outside of this layer is obviously usual linear space,
which remains in sphere of action of known mathematical methods. In it bindingly there are unambiguous solution of Maxwell equations for
electromagnetic spherical wave systems [4], which always can be joint
(«cross-linked») with field in a transitive layer. Most important in our
reasoning that we mathematical strictly can give the reason of that fact, that
fundamental particles, as well as any inhomogeneities of the propagation medium
(dislocation), can radiate or reradiate only the limited type of
electromagnetic waves. Other types of waves simply cannot exist.
As an example, we shall write out the electrical part of the available general solution for E-type waves in a spherical wave system [4]. Within the accuracy constraints of a constant magnitude factor, the equations will be of the form as follows:
(1)
where ρ, θ, φ – are spherical
coordinates, 
, 
and 
– are, respectively, radial (dilatational) and angular
(transverse) components of the variable electric field, m, n – are integers,
k – is the wave number, 
- are associated
Legendre functions, ω – is the radian frequency,
t – is time. And
functions 
and 
will be obtained
using expressions given below:
(2)
(3)
where Jn+1/2 and Yn+1/2
– are first and second order Bessel’s functions with a
half-integer index; an, bn,
and 
– are constant values.
Magnetic
components can be calculated with the help of symmetrical formulas.
Transverse
electric modes (H-type waves) can also exist in the same system. The manner of
analytical treatment thereof will not differ much from the procedure applied
herein.
Thus and
so, electromagnetic fields around fundamental particles possess commonly known
properties or properties which can readily be determined using available
formulas. This circumstance compels us to imply by the "particle"
word a system comprising both the particle itself and the related field.
However, no separate and isolated particles can exist in a real world
environment. Therefore, not only does a particle have its intrinsic field, but
it is also interacts with the fields of other adjacent particles. Hence, it
makes no sense to investigate the behavior of a single isolated particle - we
will never be able to validate such analytical findings empirically. Even two
particles’ wave interference is of qualitative interest at most. That is why,
it is the problem of a interacting of many particles that is
only worth considering, which is actually the subject-matter of this
publication.
Specific
reference ought to be made to the particles’ stability properties – and it is
only stable particles that we consider herein – such a steady-state condition
evidently resulting in the formation of fields, which present a system of
standing waves. We are dealing here with system of such waves, which
have separated spatially maximums of electrical and magnetic components having a π/2 phase lag
of related fields. Besides, maximums of
electrical and magnetic fields, alternating in space and time, describe integral energy behavior in
a similar manner.
2. Here we will analyze
field interaction phenomena only in a two-particle system. We shall begin by
considering the radial dependence of particle-induced electric fields. This
dependence in equation (1) will be set by the functions 
or 
. It is
evident that at small distances (kρ) the first function does
prevail, i.e. the one related to the radial component of
the alternating
electric field 
in (1). At greater ranges (kρ),
this component tends to rapidly decrease, the angular components 
and 
become the main contributors. That is
why in further reasoning we shall mostly deal with angular components, due
allowance being also made for a capability to analyze the short-range
near-field interaction via dilatational components.
Electrical
fields E1 and E2 of fundamental particles
pair can be represented schematically in space Ω as it is shown on
the figure Fig.1.
Fig.
1
The shaded circles in the figure shown above represent
the fundamental particles’ internal domains having volumes V1 and V2
respectively. The intrinsic electric field’s transverse components are
designated by symbols E1i and E2i,
L standing for the distance between them.
Below
Fig.2 presents a schematic view of the one-dimensional field distribution
pattern.
Fig.2
This
figure features particles whose central points are designated with two gray
strips in a manner having x1 and x2
coordinates on the x-axis, whereby x2 - x1
= L. The electric field’s respective transverse components E1
and E2 are presented in a graphic form depending on the
actual distance. The curve for the left-hand particle presents a solid line,
whereas the graph for the right-hand one – a dashed line.
Maximums of own
electric fields in the middle of each particle are co-phased in our example
with a field induced by the adjacent particle. It means that
the variable electric field, which features an unconditional inter-period
stability, for instance that of the left-hand particle E1,
will now present a vectorial sum of the particle’s intrinsic field and the
field induced by the nearby particle E2. Besides, under the
conditions stated for this problem, the particle is assumed to be in a steady
state. And the mechanism of stabilization has in the
bottom a negative feedback on energy [2]. The law of matter preservation leads
us to the same conclusion. I.e. quantity 
is stabilized inside V1. A symmetric behavior
pattern
is also peculiar to the field of the second particle. Such an interference
pattern results in that the transverse waves’ total electrical energy in a
two-particle system will be defined by the formula given below:
(4)
where 
, 
, ε0 – is a constant, ε1 and ε2 – are the respective fundamental particles’
dielectric permeability coefficients which, outside the V1 and V2 spheres, are assumed to be equal to unity.
It
naturally follows that the 
component does not depend on the distance between the particles and,
hence, it makes no impact on the force fields. Therefore, the other summand in
the equation (4) deserves greater attention. Fig.3 features a typical
dependence of two identical particles’ 
component and the 
force acting between them
on the distance kL between the particles’ centers. The averaged Ff force has been obtained by smoothing the F(kL) curve using a standard low-frequency filter.
Fig.3
The numerical
experiment was executed by means of PC using initial data as
follows: space Ω presents a parallelepiped containing 572õ260õ260 points; the particles feature a spherical shape of a kρ0
= 5 radius, the particles being herewith placed symmetrically relative to the
central point of the Ω space. As
the waves’ phase and absolute amplitude value were of minor interest to us, we
used the formula 
at n = 1. It is easily validated that
neither any other linear combinations in (2), nor any other values of n
can change the overall pattern in Fig.3.
It follows from Fig.3
that, within the accuracy constraints of this experiment, no permanently acting
long-range forces were identified for the separate particles’ pair-wise
interaction scenario. Short-range forces are prevailing obviously. They are of
exceptionally periodical nature, featuring only stepwise interaction or
quantum-type behavior.
3. We
will now consider a system of electric fields created by two arrays of
fundamental particles. An adequate description of such a system will require
the use of statistical methods and, in particular, the Gi
distribution function of the particles’ ri coordinates and ρ0i
radii. This distribution function will be represented using the expression of a
form given below:
(5)
Then
the formula (4) can be adapted for a system of consecutively numbered N
particles located in the Ω space in a manner as follows:
. (6)
Evidently, we have thus obtained a formula, which
shows that the total energy of the system is not equal to the sum total of its
constituent particles’ energies. The first summand represents a simple sum of
individual particles’ stable internal energies, such a sum not depending on
inter-particle distances. The second part of the system’s total energy (6)
presents an implicit function of their relative position and, consequently,
forms the basis for calculating any forces acting within the system. To this
end, it will suffice to differentiate the right-hand side of the equation (6)
or only the second summand thereof to a particular interaction distance.
Now we consider an example with two groups of 8
identical particles distributed in a pattern shown in Fig.4 below:
Fig.4
Here in a
position a) - is the scheme, and b) - is the Gi image;
parameters b and d are constants, L
being a variable.
Based on Fig.4 the
16-particle distribution function has the shape of a 3D numerical array. Its
substitution into the right-hand side of the equation (6) makes it possible to
calculate the variable part of the particles’ energy W~ as per the procedure
used previously for a two-particle system described above. The findings of such
a numerical experiment are presented in Fig.5, where the Ω space
contains 572õ260õ260 points, while b = d
= k20. The software example
on MatLab can be taken in a file - "Soft".
Fig.5
A detailed analysis of the Fig.5 data shows
that the particles’ collective interaction via a variable electric field
significantly differs from that within an isolated pair of particles, which was
demonstrated above in Fig.3. Now total force of interaction - F contains not only the strong
variable component, which is responsible for quantum or a strong interaction. But it has a constant component - Ff , which
possesses
long-range interaction properties. Herewith, this component has
positive values at all points, i.e. the two systems of particles feature mutual
attraction properties. However the flex point of Ff curve
in
a kL zone from 250 up to 300 testifies that the curve is not a
hyperbole, which is characteristic for the standard gravitation. Or gravitation
is not limited by hyperbolic dependence in all range of distances. This basic
result is the fact, which can be confirmed and updated by use of more powerful
computing means. But it requires as well more intent analytical
research of collective processes in systems of particles. Here the main task is
stated so: the constant
force should be determined and identified from the point of view of
mechanics.
Analytical estimation of gravitation
The more formal analysis of a gravitation problem
can be begun from equivalent transformation of the equation (6), which can be presented as follows:
(7)
where 
, 
– is the distribution function of the
overall particles system.
We assume that the
system consists of two separate subsystems N and M which centers
are apart form each other on distance L. Then, differentiating equation
(7) to L is possible to find the force FNM
between subsystems N and M:
(8)
Let's consider more
closely subintegral expressions in equation (8). The first two components contain the components responsible at
first approximation only for pair interactions inside each system. But they are
equal to zero in a distant zone, as we have found out earlier by means of
example Fig.1 and Fig.3. Last components are expressed through coefficients of
mutual coherence 
according to logic of
van-Zittert - Zernike theorem [5]:
, (9)
where WN
and WM are separate energies of systems N and M, and 
is linear combination from coefficients of a
mutual coherence 
, which always has the positive constant component.
It is
obvious, that the second component in equation (8) will contain r in the third
degree in the denominator. And it means, that
(10)
Now we should specify the function of distribution - G. Basically it should satisfy to a condition of the stationary equilibrium state of the overall system with environment or a thermostat. If this condition is met, for such state it is possible to write down:
, (11)
where 
is the equilibrium energy of the system taking into
consideration aprioristic data, W0
– thermostat energy, 
is the constant or the statistical sum.
Differentiation (11) to L leads us to the equation:
. (12)
Substitution of the
formula (12) in the equation (10) allows calculating force of gravitation
between ensembles of particles N and M.
Let's note two
important features of the right part of equation (12). At first, the
exponential function contains squares of functions Ei and Ep,
which prevail on average quantity. It means, that the exponential function
weakly depends on distance between subsystems N and M. Secondly,
derivative of the sum in preexponential member is approximately equal to the
quantity of the sum owing to sine-like dependence of Ei and Ep from distance.
Hence,
we get the following estimation of the force FNM, acting
between systems N and M:
, (13)
where m1
and m2 are masses of N and M
subsystems (as is known m=W/c2).
It is easy to notice the obvious similarity of the force FNM expressed by equation (13), and the force of gravitation.
Conclusion
The arguments presented
above testify that the conventional electrodynamics, constructed
on the known results of great number of the most convincing experiments,
can give answers to many principle questions of a quantum
mechanics and the theory of gravitation.
The software example on MatLab can be taken in a file - "Soft".
BIBLIOGRAPHY
1. Henri Poincare. Bulletin
des Scientist Mathematiques, series 2, 1904, XXVIII, 302-324.
2.
B. F. Poltoratsky. Fundamental particles in pictures without hypothesis. Moscow, «Sputnik+», 2007.
3.
James Clerk Maxwell. Royal Society Transactions, v. CLV, 1864.
4. Andre Angot. Complements de Mathematiques. Paris, Chapter VII, 1957.
5. Born, E. Wolf. Principles of optics. N-Y, Pergamon press, Chapters 10 and 11, 1964.