## 1. Introduction

The basis of modern conception of the world consists of two phenomenological theories (theory of quantum mechanics and theory of relativity), both largely inconsistent, but, in a number of cases, suitable for evaluation of experimental data. Both of these theories have one thing in common: their authors are convicted in limitations of laws and equations of classical mechanics, in absolute validity of Maxwell equations and in essential distinction of laws and mechanisms of the device of macrocosm and microcosm. Nevertheless, such assurance, being dominant in physics in the last hundred years, hasn’t resulted in creation of unifying fundamental physical theory, nor in essential understanding of principal physical conceptions, such as: electric, magnetic and gravitational fields, matter and antimatter, velocity of light, electron, photon and other elementary particles, internal energy, mass, charge, spin, quantum properties, Planck constant, fine structure constant and many others. All laws and the equations of modern physics are attempts to approximate description of the results of natural experiments, rather than strict theoretical (mathematical) findings from the general and uniform laws and mechanisms of the device of the world surrounding us. Moreover, some conclusions from modern physics equations contradict experimental data such as infinite energy or mass of point charge.

In papers (Magnitskii, 2010a, 2011a) bases of the unifying fundamental physical theory which a single postulate is the postulate on existence of physical vacuum (ether) are briefly stated. It is shown, that all basic equations of classical electrodynamics, quantum mechanics and gravitation theory can be derived from two nonlinear equations, which define dynamics of physical vacuum in three-dimensional Euclidean space and, in turn, are derived from equations of Newtonian mechanics. Furthermore, clear and sane definitions are given to all principal physical conceptions from above through the parameters of physical vacuum, namely its density and propagation velocity of various density’s perturbations. Thereby, it is shown that a set of generally unrelated geometric, algebraic and stochastic linear theories of modern physics, which are fudged to agree with experimental data and operating with concepts of multidimensional spaces and space-time continuums, can be replaced with one nonlinear theory of physical vacuum in ordinary three-dimensional Euclidean space, based exclusively on laws of classical mechanics.

In the present paper research of system of equations of physical vacuum is continued with the purpose of studying and the description of processes of a birth of elementary particles and their properties. A system of equations of electrodynamics of the physical vacuum, generalizing classical system of Maxwell's equations and invariant under Galilean transformations is deduced. Definition of the photon is given and process of its curling and a birth from the curled photon of a pair of elementary particles possessing charge, mass and spin are described. The model of an elementary particle is constructed, definitions of its electric and gravitational fields are given and absence of a magnetic field is proved. Coulomb's law and Schrodinger’s and Dirac’s equations for electric field and also the law of universal gravitation for gravitational field are deduced. Definitions of electron, positron, proton, antiproton and neutron are given, and absence of graviton is proved. The elementary model of atom of hydrogen is constructed.

Postulate. All fields and material objects in the Universe are various perturbations of physical vacuum, which is dense compressible inviscid medium in three-dimensional Euclidean space with coordinates
*t* density

With such problem definition, it’s natural to consider that no external forces apply any tension on elements of physical vacuum. Therefore, in compliance with Newtonian mechanics equations of physical vacuum dynamics in the neighborhood of homogeneous stationary state of its density

where first equation is an equation of continuity, and second is the momentum equation. Let's notice, that the physical vacuum has no mass and in this connection dimension of its density does not coincide with dimension of substance (matter).

## 2. Electrodynamics of physical vacuum

Let’s consider a case in which perturbation propagation velocity

Note that the vector of perturbation propagation velocity in physical vacuum can have both transverse and longitudinal components in relation to the direction of propagation of perturbations. Substituting expression for the vector

one can obtain a system of the equations for functions

which we call the *system of the equations of electrodynamics of physical vacuum*

### 2.1. Plane electromagnetic waves. Photon structure

In the particular case of transverse fluctuations of physical vacuum of constant density

Let's introduce into consideration vectors of electric

In the general case of propagation of perturbations in compressible physical vacuum of variable density the vector of electric field intensity has both transverse and longitudinal components, and its divergence is not zero and can be interpreted as linear density of a charge (see item. 2.3). In the considered case of propagation of perturbations in physical vacuum of constant density with constant velocity only transverse component of a vector of electric field intensity is not zero, and its divergence is equal to zero. It is also clear that so defined vector of magnetic field intensity has only a transverse component, divergence of which also is equal to zero, and the vector

Applying to the equation (4) consistently the operators

we shall obtain the classical system of Maxwell's equations describing the propagation of electromagnetic waves in the so-called empty space (vacuum):

The system of equations (6) has a solution in the form

It is considered to be, that the real parts of complex expressions (7) have physical sense. They determine an in-phase plane transverse electromagnetic wave, propagating with a speed of light

propagating with velocity

And since in this case the vectors

Suppose, for example, the transverse wave is propagated in physical vacuum in the direction of the axis

That is, in full accordance with classical electrodynamics, vectors

Now we can compare the spiral wave in the physical vacuum, obtained as the solution of the equation (4), and the classical electromagnetic wave, obtained as the solution of system of Maxwell's equations (6). Both waves have an arbitrary frequencies and corresponding wavelengths, so the two solutions describe all plane transverse electromagnetic waves existing in nature. However, it is easy to see from the above analysis, that the vectors of classical electric and magnetic fields are artificial vectors, namely, the derivatives of the same true vector of the velocity perturbations propagation in the physical vacuum. Furthermore, a classical electromagnetic wave (Fig. 1a) does not allow to correctly define the concept of a quantum of electromagnetic waves (photon), because it except for wavelength

### 2.2. Galileo transformations of electrodynamics equations

Consider an inertial rest reference frame

Galilean transformation implies the same time in all frames of reference (absolute time). It is known also that all equations of classical mechanics are written the same in any inertial reference system, i.e. they are invariant under Galilean transformations. Let's show that any law, mathematical notation of which represents the full time derivative of any function

(15) |

From this assertion follows immediately that the physical vacuum equations (1) are invariant under the Galilean transformations, since

Also the system of equations of electrodynamics of physical vacuum (3) is invariant under the Galilean transformation that follows from the system of equations (1).

Now consider in reference frames

As it shown above, to this solution of system of equations (1) with the function

Expanding now the vector

Solution (11) is the solution of equations (1) and (3) in the reference frame

From the above it follows that, in contrast to the equations of a spiral wave (3) which are invariant under Galilean transformations, Maxwell's equations (6) describe the propagation of plane electromagnetic waves in moving inertial reference frames only approximately for small

either to admit that Maxwell's equations are not absolutely correct and are need to be changed so that they should satisfy the Galilean transformations;

or to recognize that equations of classical mechanics are not quite correct and should be considered only as an approximation to the true equations, satisfying the Lorentz transformations.

Unfortunately, world science has chosen the second option, despite the reasoned objections of many outstanding scientists of the last century, among which the first is the name of Nikola Tesla ( Tesla, 2003). The way chosen by world science has led to an absolutization of speed of light and Maxwell's equations and has led to full termination of researches in the field of search more general equations of electrodynamics satisfying the principle of Galilean relativity. The present research proves that the correct way to exit from the crisis of science in early twentieth century was not in updating the equations of classical mechanics with the use of relativistic additives but, on the contrary, in finding the equations generalizing Maxwell's equations and satisfying the Galilean transformations.

### 2.3. Longitudinal electromagnetic waves. Currents

Consider the general case of propagation of spiral waves (2) in physical vacuum of variable density. As shown in Sec. 2.1, these waves are solutions of the equations of electrodynamics of physical vacuum (3). Applying to the sum of the second and the third equations of system (3) consistently the operators

Note that in this case the electric field intensity vector

Let's introduce into consideration the linear charge density

Then from (12) we shall obtain the system of equations

The system of equations (13) at

As it is already mentioned above, the classical system of Maxwell's equations describing propagation of electromagnetic waves in presence of charges and currents can be obtained from (13) at

can be obtained by applying the operator

## 3. Elementary particles of a matter

We show in this section that processes of a birth of elementary particles of matter and antimatter from the physical vacuum (ether), as well as all basic quantum-mechanical properties of elementary particles can be obtained from the system of equations (1) written in spherical system of coordinates:

(24) |

where

### 3.1. Birth of elementary particles from physical vacuum

Let’s consider a spiral wave of photon (8)

propagating with the velocity

So, we shall put in (14)

(26) |

The solution for the curled photon we shall find from the system (15), putting in it

Let’s show now that equation system (15) has solutions, which possess all known properties of elementary particles when

and small-amplitude oscillations of function

Substitution of assumed form of solution of (16) into equation system (15), with a drop of second infinitesimal order terms and multiplications of small terms, will result in the following system of equations

It is necessary to notice that at such approximation nonlinear term of second infinitesimal order

It’s not difficult to get the solutions of equation system (17) in the following form

However, not every solution in form (16), (18) is an elementary particle. Such solution has to possess properties of charge conservation and universality, as well as quantum properties of mass, momentum and energy. Moreover, over the time of full roundabout way of the wave along the sphere equator, electric field intensity must conserve its sign. Such classical and quantum mechanical terms as electric and magnetic field of elementary particle, its charge, mass, energy, momentum, spin also need correct definitions through the characteristics of physical vacuum.

First, let’s give the definition of electric field and electric charge of elementary particle similarly to the case of plane electromagnetic waves propagation, examined above.

Definition.* Electric field intensity distribution
*

It follows from (16) and (18) that inside a particle at

Let’s determine an instant value of the charge

(32) |

What follows from formula (21) is that solution (16),
(18) of the equation system (15) can be interpreted as an elementary particle only in such case, when wave number

Integrating the density distribution of charge over sphere’s volume with radius

It’s important to point out that electric field of elementary particle directed along radius is created by particle’s electric charge, but at the same time the charge is divergence of a completely different inner field of the particle, which is represented by second term in the third equation of equation system (15) and directed by the angle

### 3.2. Other basic properties of elementary particles

Let’s now determine other properties of an elementary particle: internal energy ε, mass m, momentum p and spin σ. Expressions of Planck constant

Here B is the volume of elementary particle sphere of radius

and it has to execute the work over not only electric charge with distribution density

let’s insert it as well as derived expression of internal field

(36) |

Integrating the last equation and taking into account that

Now, to derive the well-known main formulas and correlations of quantum mechanics, it’s suffice to denote the mass of elementary particle and Planck constant as

From this it follows immediately:

Einstein’s formula for internal energy of a particle and formulas of impulse and energy for de Broglie’s waves

formula for spin of a particle

fine structure constant formula

These formulas, derived exclusively by the methods of classical mechanics, are completely identical to the well-known expressions of quantum mechanics as well as clearly reflect the physical essence of charge, mass, energy and spin of elementary particles, allowing to understand the nature of quantum processes in microcosm. It can be seen that the internal energy of the particle is indeed proportional to the square of velocity of light, and proportionality coefficient (mass of the particle) linearly grows with the increase of wave number k, as well as frequency ω of the parental photon. The Plank constant is indeed a constant value depending only on characteristics of physical vacuum and not on the type of the elementary particle. The spin of the particle indeed has a value of either integer or half-integer number of

Note also that the simplest particles with the spin of ½ when n = 1 are double period cycles in relation to the initial cycle defined by the motion of curled photon. That brings another proof of the theory introduced in this research – the interpretation of the Pauli principle, the corollary fact of which is that electron returns to the initial state only after the turn of 720, not 360 degrees. According to R. P. Feynman (Feynman & Weinberg, 1987), particle with topology of Moebius band meets the Pauli principle. But in the Feigenbaum-Sharkovskii-Magnitskii universal theory of dynamical chaos (FSM theory) (Magnitskii, 2008a, 2008b, 2010, 2010b, 2011b; Magnitskii & Sidorov, 2006; Evstigneev & Magnitskii, 2010), results of which valid for every nonlinear differential equation system of macrocosm, the solution’s difficulty increase starts from double period bifurcation of the original singular cycle. Interesting enough, the newborn cycle of doubled period belongs to the Moebius band around the original cycle! In another words, according to the FSM theory electron and proton are initial and simplest double period bifurcations from the infinite bifurcation cascade. Therefore, FSM theory works not only in macrocosm, but also in microcosm, and elementary particles defined by formulas (16), (18), are not a full infinite set of all elementary particles, which can be born as a result of bifurcations in nonlinear equation system (15). Furthermore, more complex nonperiodic solutions of systems (14) and (15) can be foreseen, which are singular attractors in terms of FSM theory. Thus, any attempts of an experimental detection of the simplest (most elementary), as well as the most complex of elementary particles are essentially futile.

### 3.3. Some main classical equations and laws

Another proof of validity of the theory presented in this paper is the possibility of a rigorous mathematical conclusion from its unique postulate on existence of physical vacuum of some important phenomenological equations and laws of the modern physics which are widely used by classical electrodynamics and quantum mechanics and not contradicting to common sense interpretation of variables included in them. We consider here the Coulomb's law and Schrodinger’s and Dirac’s equations.

#### 3.3.1. Coulomb's law

We assume that outside of a particle of radius

The vector of electric field intensity distribution of a particle will become

Then a vector of electric field intensity of an elementary particle

For the particles having a positive charge

#### 3.3.2. Schrodinger's equation

Let's show, that for a free particle of mass

is a scalar function

then averaging the right part of last expression by the angle

Multiplying the last expression on

that coincides with the equation (24). Thus, it becomes clear a physical sense of

#### 3.3.3. Dirac’s equation

It was already shown in (Magnitskii, 2010a, 2011a) that electric field intensity and charge of elementary particle defined above agree with electromagnetic form of Dirac’s equation for electron in bispinor form. Here we shall consider this question in more detail. Dirac’s equation in bispinor form has a kind

that is a consequence of operator equation

where

Solution of the system (27), consistent with a solution of the system (19) in the vicinity of the equatorial areas of the elementary particle, has the following kind:

Then, as it is easy to verify by the direct substitution, the vector

in the vicinity of

Equation (28) differs from the equation (26) those, that in it instead of the electron mass

Therefore, the true physical meaning of wave function ψ from Dirac equation for electron in bispinor form (25) becomes clear – it’s a 4-vector

### 3.4. Electron, positron, proton, antiproton, neutron and atom of hydrogen

It’s obvious, that more complex, multi-curled elementary particles correspond to high-frequency perturbation waves with bigger mass and energy. So, it’s natural to imply that the simplest half-curled particles with the spin of ½ when n = 1 are pairs “electron-positron” and “proton-antiproton”. Both pairs of particles have the same mechanism of a birth. The difference is in the values of frequencies of parental photons and, accordingly, in radiuses of their curling

It is obvious that the charges of proton and electron should have different signs. Thus, their combinations can form atoms of substance only in the case when the electric field intensity of a particle of smaller radius (proton) is directed to its center, and, accordingly, the electric field intensity of a particle of the greater radius (electron) is directed from its center. That is, proton should have a negative charge in the sense of expression (20), and electron should have a positive charge. Then for instant density of physical vacuum of proton

Similar expression we shall receive for instant density of physical vacuum of electron

Consequently, proton is compressed, and electron is rarefied areas of physical vacuum with respect to its stationary density

Consider now the possibility of the formation from a pair of proton-electron of the simplest electrically neutral structures, such as neutron and atom of hydrogen. Since the electron has a much larger radius than the radius of a proton, then in the most part of elements of physical vacuum laying inside of the electron, the electric field of the electron directed from its center, less than an electric field of the proton directed to its center. Therefore, an electron having got in area of its capture by an electric field of a proton, should move in its direction until some stable structure in the form of a sphere with a radius of an electron, in which center there is a nucleus as a sphere with a radius of a proton is formed. The electric field intensity outside of an external sphere is equal to zero, as at

We can assume that the simplest atom of hydrogen, as well as arbitrary neutron are arranged in this manner. The neutron can differ from the atom of hydrogen in radius and, accordingly, in frequencies of oscillations of waves of its electron and proton. In Fig. 3 a diagram of a hydrogen atom and also a picture of a real hydrogen atom made in Japan (Podrobnosti, 04.11.2010) are presented.

In this model, the impossibility of formation of atoms of antimatter can be easily explained by the fact that the electric field of the antiproton, which has much smaller radius than the positron, is directed from its center, which prevents the formation of stable structures of antimatter.

## 4. Gravitation and gravitational waves

Let’s demonstrate that the creation of any elementary particle is accompanied by appearance of the gravitation, notably the pressure force in physical vacuum, generated by small periodic perturbations of its density, which in its own turn generate gravitational wave, propagating to the center of newborn particle. It’s natural to propose that gravitation works over any distance from the particle, and that when the distance is large, perturbations of physical vacuum density created by the newborn particle depend only on distance r and are independent of angles θ and φ. Based on such assumption, let’s seek solutions of system (15) when

Equation system (15) will take a form

meaning that of all four fields in the initial system (15) only gravitational field

Furthermore, gravitational field differs from three other previously examined fields since it’s severely nonlinear. It can’t be linearized basing on the form of velocity

Let’s seek the solution of equation system (29) in the form of

In this case the pressure force of gravitational wave (gravitational field intensity) expresses as

and it agrees with the law of universal gravitation. However, the physical essence of gravitation comes in somewhat different light than before. The bodies do not attract each other – each material body creates its own gravitational wave, which propagates from infinity to its center of mass and puts an external pressure on other body with the force, proportional to the mass of the body and inversely proportional to the square of distance between the bodies.

Let’s note another significant difference between gravitational and electromagnetic waves. Electromagnetic wave moving with constant velocity has a wavelength, thus, resulting in the existence of electromagnetic wave quant or photon. Gravitational wave moves with velocity dependant on radius, thus, there can be no gravitational wave quant. Traditional parallel between the gravitational wave and its hypothetical carrier, graviton, is apparently the main obstacle for the real discovery of gravitational waves in nature.

## 5. Conclusion

The theoretical research carried out in the work and its results allow to draw several fundamental conclusions and statements which looks more than plausibly:

all fields and material objects in the Universe are various perturbations of physical vacuum, microcosm and macrocosm are organized by the same laws – laws of classical mechanics, described by nonlinear differential equation systems in tree-dimensional plane Euclidean space and bifurcations in such systems;

electromagnetic fields can exist without mass and gravitation, and electromagnetic waves can propagate in any direction with constant velocity (velocity of light) and arbitrary oscillation frequency, which is defined by oscillation frequency of physical vacuum without changes of its density;

there exist equations, more common than Maxwell equations, deduced from the physical vacuum equations and invariant concerning Galileo transformations, many experimentally established laws of classical and quantum mechanics can be successfully deduced from the physical vacuum equations;

existence of gravitation, mass and charge inseparably linked with the creation of elementary particles in form of curls of a single gravi-electromagnetic field, the attracting force is actually a pressure force in physical vacuum created by gravitational wave, which propagates to the center of the particle with variable velocity and has no wave length;