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Journal of Fundamental and Applied Sciences, Vol. 9, No. 1, pp. 411-467 (2017). http://dx.doi.org/10.4314/jfas.v9i1.25

 

 

The substantial model of the photon

 

Sergey G. Fedosin

Sviazeva Str. 22-79, Perm 614088, Perm Krai, Russia

e-mail intelli@list.ru

 

It is shown that the angular frequency of the photon is nothing else than the averaged angular frequency of revolution of the electron cloud’s center during emission and quantum transition between two energy levels in an atom. On assumption that the photon consists of charged particles of the vacuum field (of praons), the substantial model of a photon is constructed. Praons move inside the photon in the same way as they must move in the electromagnetic field of the emitting electron, while internal periodic wave structure is formed inside the photon. The properties of praons, including their mass, charge and speed, are derived in the framework of the theory of infinite nesting of matter. At the same time, praons are part of nucleons and leptons just as nucleons are the basis of neutron stars and the matter of ordinary stars and planets. With the help of the Lorentz transformations, which correlate the laboratory reference frame and the reference frame, co-moving with the praons inside the photon, transformation of the electromagnetic field components is performed. This allows us to calculate the longitudinal magnetic field and magnetic dipole moment of the photon, and to understand the relation between the transverse components of the electric and magnetic fields, connected by a coefficient in the form of the speed of light. The total rest mass of the particles making up the photon is found, it turns out to be inversely proportional to the nuclear charge number of the hydrogen-like atom, which emits the photon. In the presented picture the photon composed of praons moves at a speed less than the speed of light, and it loses the right to be called an elementary particle due to its complex structure.

Keywords:  matter waves; quantum gravity; electromagnetic interaction; magnetic moments; properties of photon.

PACS: 03.65.-w, 11, 12.10.Dm, 42.25.-p

 

1. Introduction

As is known, the more elementary the particle is, the less we know about it. The photon, the concept of which appeared more than a hundred years ago in the writings of Albert Einstein, is not an exception. What seems surprising about this particle is the absence of the rest mass, but at the same time the presence of wave and corpuscle properties, high stability and the ability to travel over cosmic distances with low energy losses, the indissoluble connection between photons and charged particles in the processes of absorption and emission.

One of the modern methods of studying the photon structure is experiments on colliding photons with each other, with protons and electrons. These experiments show that at small distances a photon can be modelled in the form of fluxes of quarks and gluons [1]. These fluxes should participate in interactions as is prescribed in quantum electrodynamics.

In the oscillating model [2] a photon is regarded as an object periodically changing its volume, the speed of which is less than the speed of light. In this model, it is assumed that the rest mass of the photon with the greatest wavelength can be related to the initial conditions of the early Universe. Based on this assumption the estimate of the mass of the photon’s inner part is made: kg. In contrast, in [3] it is considered that a photon has no proper mass, however under the influence of the vacuum field the effective mass appears.

In [4] the photon diameter is deemed equal to the wavelength  on the ground that this dimension is the limit for the wave diffraction. The soliton model of the photon is constructed in [5], where the equation for the vector potential is used, which is similar to the generalized Schrödinger equation. In [6] it is indicated that the drawback of the soliton model is the difficulty to explain the origin of the soliton, which usually requires a nonlinear medium. The photon diameter according to [7] is equal to , and outside of the photon its field strength must decrease in inverse proportion to the distance to the photon’s axis. This allows the photon to undergo interference in the Young's interference experiment. Description of a photon as a rotating particle in the framework of quantum electrodynamics is presented in [8].

Due to the lack of key information about the internal parameters of electromagnetic quanta, the existing models still require further development and specification, because they do not allow us to define concretely the actual structure of a photon, to relate it to the source of emission at the atomic level and to the experimental data. The purpose of this article is to fill this gap and to provide a more detailed and well-grounded substantial model of the photon. We will do it based on the theory of infinite nesting of matter and the substantial model of the electron [9].

We will start with considering the basic conditions of emission from a hydrogen-like atom and estimating the duration of emission, which is necessary to determine the photon’s length in space and then to calculate its energy density. In Section 3, we will present the main components of the electric and magnetic fields that are created by the charge rotating around the nucleus in the near and wave zones. The energy flux of these fields leads to a standard formula for the charge emission. Our goal is to use certain electromagnetic field components of the rotating charge to find the equations of motion for the smallest charged particles of the vacuum field in Section 4. We consider these particles, called praons, as construction material not only for photons but also for any other elementary particles, including nucleons and leptons. Praons have mass and we use the Lorentz factor to describe their motion at relativistic velocities. This allows us to turn with the help of Lorentz transformations to the reference frame, co-moving with praons, and to understand their motion from the standpoint of a fixed photon.

In Section 5, based on the motion of praons in the electromagnetic field of the emitting electron, periodically changing in space and time, we construct the substantial model of the photon. Section 6 concerns the structure of the electromagnetic field and the strong gravitational field inside the photon and their interaction with praons, which ensures the photon’s stability. In Sections 7, 8, 9 we derive the Lorentz factor for praons and the energy fluxes within the photon, the magnetic dipole moment, and the rest mass of the particles that make up the photon, respectively.

 

2. Emission of a photon from a hydrogen-like atom

According to the Bohr relation, the energy of a photon as an electromagnetic quantum, emitted during the electron’s transition from some energy level  to a lower level , equals the difference between the total energies of the electron at these levels:

 

,                                                    (1)

 

here  is the Dirac constant,

 is the angular frequency of the photon.

 

But how could we describe more clearly what is happening in the atom during emission of the quantum? For simplicity, let us assume that one electron is located in the central-type field of the hydrogen-like atom. If the electron matter rotates totally symmetrically relative to the nucleus, then the electron would not emit. This is due to the fact that for each charge element of the electron matter in an axisymmetric configuration there is a similar charge element on the opposite side of the axis, which is moving in the opposite direction. At large distances, the contribution of the nucleus and of these charge elements into the total electric field strength  and the magnetic induction  will be compensated, and the resulting energy flux will be close to zero.

Therefore, in order to produce emission the electron must move so that its center of inertia is sufficiently removed from the nucleus. Let us assume that the center of the electron cloud rotates at a distance  from the nucleus and is held in relative equilibrium by a force directed towards the nucleus. If the velocity of the cloud’s center is equal to , then for the equality of the central and centripetal forces we can write:

 

,                                    ,                                 (2)

 

where  is the number of protons in the nucleus,  is the elementary charge,  is the vacuum permittivity,  is the electron mass, so that  is the electric force between the positively charged nucleus with the charge  and the negatively charged electron.

 

In (2) we used the approximately equal symbol, since in case of emission the distance  will slowly decrease and the velocity will increase. Besides, we do not take into account the change in the intrinsic electromagnetic energy of the cloud due to the change in the radius and volume of the cloud as it approaches the nucleus. Then we use the standard formula for the power of the total electromagnetic emission from the elementary charge, rotating around a certain center [10]. If we consider the emitted energy per time  up to the sign equal to the change in the total energy  of the electron cloud, then we can write:

 

,                                                      (3)

 

here  is the speed of light, and a small coefficient  reflects the fact that the emission from the electron cloud as a dimensional figure should differ from the emission from of the rotating electron as a point.

 

Assuming that , where  is the angular velocity of rotation of the electron cloud’s center around the nucleus, from the ratio for the power   and (3) we will find the magnitude of the force, decelerating the cloud’s rotation:

 

.                                                         (4)

 

For the angular momentum of the cloud’s center of mass and its rate of change under the influence of the force moment  we can write:

 

,                                .                                  (5)

 

In addition, we obtain the following:

 

,                       ,                             (6)

 

i.e. the change in the electron cloud’s energy with the change in the angular momentum of the cloud’s center is proportional to the angular frequency of rotation.

 

Expressing from (2) the rotation speed in the form   and substituting in (5), in view of (4) we arrive at the differential equation for the dependence of the distance  on the time:

 

,              ,                (7)

 

.

 

here  is the distance to the cloud’s center at the initial time.

 

Expression (7) approximately describes the small changes in the distance  over time for the rotational motion of the electron cloud. Besides the following condition must hold: , , where  is the time of the cloud’s center of mass falling onto the attracting center.

For example, if we assume that the distance changes from  to , where  denotes the Bohr radius, then for the time of transition from the level  to the level  at  and   from (7) we obtain the value of the order of s, that is the typical time of the electromagnetic quantum emission by the electron in atomic transitions.

If we substitute the distance (7) in (3), taking into account , and integrate it over the time, we will find the total energy of the electron cloud:

 

.                                                     (8)

 

From (8) we see that if the electron moves from the energy level  at the energy level , then the energy of the emitted electromagnetic quantum will amount to the value equal to the difference between the electron’s energy levels in the atom: . This relation fully coincides with the Bohr condition for energies. This should have been expected, because from (2) it follows that the electrostatic energy of the electron at the level  is equal to the value , and the kinetic energy of the electron is . The total energy at the level  is supposed to be equal to the sum of these energies: .

From the condition  and (3) it follows that the power of the energy emission is strongly dependent on the current distance :

 

.                                                  (9)

 

According to (9) we can assume that the basic energy of the electromagnetic quantum during the transition from the energy level  to the energy level  is emitted by the electron cloud near the level , where the radius of rotation  of the electron cloud’s center is less. In this case, we find explanation for the fact that the frequency of electromagnetic quanta  in (1) is close, but always less than the frequency of the electron cloud’s rotation near the energy level . If we consider at some time point the emitted electromagnetic quantum along its length in space, then its oscillation frequency should increase when moving from the front part of the quantum to its rear part, and the quantum energy density must reach the maximum closer to the rear part of the quantum.

The constant  was introduced by Planck in 1900 while establishing the law of energy distribution in the blackbody spectrum. This constant turned out to be a universal quantity at the level of elementary particles and atoms, with the dimensionality of a quantum of action. Its role in determining the electromagnetic energy of quanta, despite the fact that the wave oscillation frequency inside these quanta in our opinion cannot be strictly constant, is quite similar to that of the Boltzmann constant in determining the average thermal energy of a set of particles through the temperature, with the energy spread of individual particles, which is always present.

We will show that the angular frequency  of the quantum is the averaged angular frequency  of rotation of the electron cloud’s center at transition between the energy levels  and . For  in view of (6) we have:

 

.

 

At  the energy electromagnetic quantum is:

 

.                                                       (10)

 

From comparison of (10) and (1) we see that . However, if for some reason , the equality  would not exist.

Since during the emission of quanta the electron’s angular momentum changes, the change in the angular momentum should be carried away by the electromagnetic quantum. Photons or electromagnetic quanta, are attributed the angular momentum, equal to . Therefore, during emission the electron loses the angular momentum of the order of  and the same angular momentum is acquired by the photon; the electron loses the energy of the order of , where  is the average rotation frequency of the electron’s center of mass near the nucleus for the period of emission, and the photon acquires this energy. The electron acts in this case as a carrier particle that transfers its kinetic energy and angular momentum into the energy and angular momentum of the electromagnetic wave that are concentrated in the emitted photon.

 

3. The emission from the rotating point charge

Let us assume that a charged point particle with the charge  rotates by a circle of radius  with the angular velocity  and the orbital velocity . We will place a spherical reference frame at the center of this circle and will seek for the components of the electromagnetic field strength from the rotating charge at some remote point with the radius-vector . The current position of the charge is given by the vector , so that the circle of rotation lies in the plane .

In order to determine the electric field strength  and the magnetic field induction  in the first approximation we will use the formulas that take into account any motion of the charge in the special theory of relativity:

 

,                     ,                   (11)

 

here  is the vector from the charge to the remote point at the early time point ,

,

 is the unit vector, directed from the charge to the remote point, taken for the case of rotation of this charge by a circle at an early time point .

 

The formula (11) was first published by Oliver Heaviside in 1902. It was independently discovered by R. P. Feynman, in about 1950, and given in some lectures as a good way of thinking about synchrotron radiation [11].

From the definitions of  and  we see that they are mutually dependent. We will take for them the derivatives with respect to time:

 

,           ,

 

and then we will express these derivatives independently from each other:

 

,              .

 

If the orbital velocity  is significantly less than the speed of light , as is the case for the electron in the atom, we see that:

 

,                               .                                        (12)

 

Taking the first time derivative of the unit vector of the original direction, we find:

 

,                                   (13)

 

,              .

 

Let us substitute in the right-hand side of (13) the time derivatives by their maximum values according to (12) and calculate the second time derivatives of the unit vector:

 

,

 

,

 

.                                                    (14)

 

For the components of the derivative  in (11), in view of (12) and (13), we obtain:

 

,

 

,

 

.                                           (15)

 

Substituting (14) and (15) in (11), we find the component of the electric field strength :

 

      (16)

 

According to (12), , and then in (16) the third and seventh terms are less than the first term, since . Similarly, taking into account the relation , the fifth and sixth terms in (16) are always less than the second term. As a result, leaving the greatest terms in (16) and (11), for the electromagnetic field components we have the following:

 

.

 

.

 

.                           (17)

 

,         ,

 

.

 

At large distances, when , the last terms containing  in the denominator start predominating in the electric field components (17), and the terms containing  in the denominator start predominating in the magnetic field components.

The Poynting vector or the electromagnetic energy flux equals:

 

.

 

If in (17) we take into account only the last terms that remain at a great distance, then the Poynting vector components are as follows:

 

,

 

,

 

.

 

We must first average the components  and  for one period of the charge’s rotation, when the phase  varies from 0 to . Given that , we have:

 

,           .

 

Now, integrating  over the surface of the remote sphere we find the rate of the electromagnetic energy flux, averaged over the period:

 

.                 (18)

 

here  is a unit vector perpendicular to the sphere surface,  is an area unit, for the spherical coordinates ,  and we assumed that .

 

The emission rate in (18) coincides with the result in (3), while for calculation instead of full expressions for the field we used only the field components from (17) that remain in the remote area.

 

4. Photon formation

4.1. The near zone

We consider an electron in an atom as a flat disk, the center of which is shifted relative to the nucleus and is rotating around the nucleus during emission of a photon. After formation, a photon becomes an independent object and no longer depends on the fields generated by the emitting electron and the atomic nucleus. Now we need to build a model of a photon, to understand what it consists of, how it maintains the perpendicular structure of the electromagnetic field and why a photon is a stable object. For this purpose, we will turn to the results of [12-13], where the photon is regarded as an object consisting of tightly bound charged particles.

In [14] we assume the positively and negatively charged praons as the charged particles that permeate entire space in different directions and create the interaction forces between the electric charges. These particles are one of the components of the vacuum field, along with the graviton field, responsible for the occurrence of gravitational forces [15] in Le Sage’s model. The mass to charge ratio found for praons turns out to be such as it follows from the coefficients of similarity between different levels of matter and from the theory of dimensions. According to the theory of infinite nesting of matter, praons make up the matter of nucleons just as nucleons make up the matter of neutron stars. Besides, the fluxes of charged praons are the cause of the Coulomb force, and inside of photons praons come into a state of steady and orderly rotation.

In the substantial model of electron [9], in a hydrogen atom in its ground state the average radius of the electron disk is assumed to be equal to the Bohr radius , the minimum radius of the disk is  and the maximum radius reaches . These radii correlate with the electron density distribution according to the electron wave function and the solutions of Schrödinger equation. Close to the nucleus, at a radius less than , the electron matter density decreases rapidly. We suppose that the fluxes of praons pass here along the axis , perpendicular to the plane of the electron disk, without direct contact with the electron matter, interacting with the nucleus and electron only by means of the field. From the symmetry of fields of the nucleus and electron disk it follows that near the axis  the praon fluxes mostly move linearly, creating the basis of the emitted photon. Other praons that pass through the electron disk, after interaction with the charged matter of the disk, get into the photon shell with a cross-section of the order of the electron disk’s size. The same pattern holds for the hydrogen-like atom.

For example, in [14] at a first approximation a photon is considered as a long, thin cylinder, rotating at the angular frequency , where  is the wavelength of the photon. For a photon with the wavelength m and the angular frequency  s-1, which emerges in the hydrogen atom at the transition of the electron from the second to the first level in the Lyman series, we assume the average radius of the electron disk  as the photon radius. The total length of the photon is given by the expression , where  is the duration of the photon emission by the atom, according to (7).

Let us analyze the electromagnetic field components at an arbitrary point in space , the coordinates  and  of which do not exceed much the orbit radius  of the rotating emitting charge, and the coordinate  by its absolute value is much larger than the orbit radius:. In this area, the condition  holds, so that in (17) the first terms predominate. Turning to the hydrogen-like atom, in (17) we will also replace  with the negative charge of the electron , where  is the elementary charge, and will add to (17) the static electric field components from the charge  of the atomic nucleus, located in the center of the coordinate system. The result for the field components can be written as follows:

 

,        ,        .     (19)

 

,       ,       .

 

The magnetic field component  in (19) oscillates in a complicated way. If we restrict ourselves to an area, where , that is outside the atom, but with the near zone condition , then we can assume  and . In this case, the component  can be neglected, since it would be  times less than the components  and .

The electric field components, depending on the multiplier , determine the constant field from the effective charge  of the hydrogen-like atom, decreasing with the distance according to the Coulomb's law. This field should accelerate the charged praons, changing their energy. However, the emerging photon has the almost same number of positive and negative praons, which ensures the electroneutrality of the photon. These praons also interact strongly with each other and are in a bound state. Then the component  at sufficiently large distances  will not influence the motion of particles in a neutral average photon.

As a result, the pattern of the moving field will be formed mainly by those components in , ,  and  that are time-dependent. Introducing the transverse vectors  and , from (19) we find for those components the following:

 

,                  ,                      (20)

 

where the unit vector  determines the position of the rotating electron in the plane  at the early time point .

 

From (20) we see that the transverse components of the electric  and magnetic  fields rotate around the axis  at the angular frequency  synchronously with the rotation of the vector  and with the rotation of the electron in the atom. At the same time the component  is directed in space the same way as the radius vector of the electron at the early time, and the magnetic field  is directed oppositely.

We will write down the equation of motion of the negatively charged praons in the external electromagnetic field and will find the mode of their motion, using the general equations of motion in the same way as in [16-17]. The equation of motion with respect to the early time  is determined by the Lorentz force:

 

,                                         (21)

 

here  is the invariant mass of a negatively charged praon,

 is the Lorentz factor,

 is the elementary charge for praon level of matter,

 is the particle velocity vector.

 

After substitution of the fields (20) into (21), we obtain for the motion of particles the following:

 

,

 

,

 

.                          (22)

 

We will take into account that the Lorentz force components in (22), containing  in the denominator, can be excluded from calculation because they are much less than the components without the speed of light. From the ratio of these components the condition follows: . The velocity of praons  along the axis  reaches almost the speed of light, so that the condition has the form , which corresponds to the previously accepted expression  for the near zone. Consequently, an approximate solution to the equations (22) for the particles’ velocity has the form:

 

,      ,     .

 

What would change, if at these velocities we will take into account that the electric field components (19) also contain constant terms, containing the multiplier ? If the action of the field component  can be neglected, then taking into account the constant terms in the components  and  leads to emerging of an additional centripetal force. This force influences the negative praons and changes their velocity on the stable rotation trajectory up to the following values:

 

,      ,     .

(23)

 

For the hydrogen atom  and the expressions for velocities are simplified. In this case, from (23) we see that in the near zone at the time , which does not exceed the period of the electron’s rotation around the nucleus, the negative praons rotate around the axis  at velocity  following the electron’s rotation in the atom. For the positive praons, at the same angular velocity vector, the linear velocity components in (23) will be in opposite direction relative to the velocity components of the negative praons, due to a different sign of charge. For such motion, it is enough for the negative praons to rotate on the same side as the electron at the early time and for the positive praons to be on the opposite side relative to the axis , at equal common rotation. We can also take into account that the negative praons at their matter level are the analogues of electrons and therefore the mass ratio of positive and negative praons is equal to the mass ratio of proton and electron: . Substituting  instead of  in (23) leads to the fact that the velocities and radii of rotation of the positive praons will be significantly less than those of the negative praons.

Figure 1 shows a surface perpendicular to the axis  and shifted along this axis for a certain distance from the atom, on which we can see the directions of the electric and magnetic fields (20) and the velocities of praons according to (23) for the hydrogen atom at . All vectors correspond to the time point , at which the condition  is met, where . The vector  stands for the velocity of the positive praons.

 

 

Надпись: Fig. 1. The fields (20) and velocities (23) of praons close to the axis   in the near zone at   for the hydrogen atom.

 

4.2. The wave zone

Let us now consider another extreme case, when the coordinate  is in the remote wave zone. Here the properties of the emerging photon should reveal themselves to the full extent. If we consider the electric field strength components in (17), we see that among all the terms those terms become the maximum terms, which contain the square of the speed of light. These terms slowly decrease with the distance, because they contain the distance  to the first power in the denominator. In the magnetic components, the largest terms also slowly decrease with the distance, since they are proportional to the multiplier .

Turning again to the hydrogen atom, we will replace  in (17) with the electron’s negative charge  and will add to (17) the components of the static electric field from the charge  of the atomic nucleus, located in the center of the coordinate system. As a result, the field components in the wave zone can be expressed as follows:

 

,           ,          .

(24)

 

,     ,     .

 

We will consider sufficiently long distances , when the conditions ,  are met. Then in the components  and  in (24) we can neglect the constant terms from the nuclear field. As for component , it should have little effect on the photon’s motion also due to its electrical neutrality.

The magnetic field component  as compared with the components  and  is small. For example, the amplitude ratio of the components  and  at small  and  is estimated as . Further on we will consider that the component  is close to zero and in the wave zone it is not involved in the processes inside the photon. Then the electric and magnetic fields that remain in (24) would be perpendicular to each other and to the axis , besides the magnetic field components would be shifted forward relative to the electric field components at an angle  and rotate at the same frequency . In addition, the relation  appears. In a photon the same conditions are met, and it is expected that the fields in the form of (24) should form a circularly polarized photon, that is, with rotation of the electric vector relative to the photon’s axis.

In (24) we will turn from the earlier time  to the current time  in the laboratory reference frame, taking into account the definition: . We will also introduce the wave vector with the amplitude , which is directed along the axis , so that at any sign of the coordinate  and velocity of praons   the following relations are satisfied: , . Then , and for periodically varying fields we can write the following:

 

,   ,

 

,           ,

 

,                          .                           (25)

 

As we can see, at any constant value , the fields in (25) depend on the time according to the sine law. In addition, as the coordinate  increases at the points, where the condition  is satisfied and , the fields (25) rotate synchronously with each other along the axis . Thus, the field acquires a periodic spatial structure, repeated after a minimum distance equal to the wavelength. In the previous case, when equation (21) was solved, the spatial structure was not considered, as we were considering the near zone, the size of which is of the order of less than one wavelength.

Similarly to (21), we will write the equation of motion for the negative praons, but with respect to the current time :

 

.

 

After substituting the fields (25) into this equation we obtain:

 

,            (26)

 

,

 

.

 

In the right side of (26) the Lorentz force depends on two variables – the time  and the coordinate  that define the distance to the emitting atom. Therefore, during the motion of the charged particles in the electromagnetic field, the acceleration and velocity of the particles also become the functions of  and . Due to this, we presented the time derivatives in the left side of (26) as material derivatives.

The change of  has a more significant impact on the argument of sines and cosines than the change of  in the amplitude’s denominator. If we consider  as a variable only in the sines and cosines, then the approximate solution of equations (26) for the velocity of the particles has the form:

 

,      ,     .

(27)

 

If the time  is fixed, then in case of changing the position of the coordinate from  to , the velocity vector in (27) will make complete revolution around the axis , while at large  the decrease of the velocity amplitude due to the change of  will be little. This proves our approximate solution of (27), though solving the equations we have not taken into account the change of  in  in the denominator of the Lorentz force’s amplitude.

Figure 2 shows a surface perpendicular to the axis , where the directions of the electric and magnetic fields (25) and praons’ velocities are shown according to (27). The vectors  and  denote the velocities of the positive and negative praons, respectively.

 

 

Надпись: Fig. 2. The fields (25) and velocities (27) of the praons near the axis   in the remote wave zone at  .

 

Let us pay attention to the difference between the solutions (23) and (27), which consists in the fact that the praons’ velocities in them for the hydrogen atom at  have different signs. In this case, at the boundary between the near and wave zones, which is reflected by the condition , a change of the field action takes place. Specifically, the total field of the electron changes its phase to the opposite, due to the increased field components (25) in comparison with the field components (20). As a result, when the electron is rotating on the one side from the axis , the negative praons are located and rotating under the field action on the opposite side of the axis . As for the positive praons, they are now located on the side, where the electron is moving, and are rotating at lower speed and with a smaller radius of rotation, due to their large mass.

Additionally, the photon obtains spatial structure in the wave zone. How can it be explained from the standpoint of physics? Assume that the electromagnetic field of the electron, which is periodically varying in the course of rotation, achieves a certain cross section of the photon at a distance  from the atom and sets its particles into motion. Then, the electron makes a revolution inside the atom, and at this point new particles come to the cross section at  from the side of the atom. The electron exerts influence on them by its field, as in the previous case, and everything is repeated. The same holds true for the points with coordinates , where the field comes from the electron in the same phase and respectively it was emitted by the electron at the earlier time points. Since the beginning of the photon’s emission, as the time was passing and the number of the electron’s revolutions was increasing, the number of single-phase points with coordinates  was increasing until the rotating field of the electron would not cover the entire area that should be occupied by the photon. Besides, if the motion of particles inside the photon occurs in a certain way and synchronously with the electron’s motion, as in (27), then it creates the necessary conditions for the wave structure inside the photon, which is periodically varying in space and time.

The velocity components  in (27) are recorded in the reference frame , associated with the atom emitting the photon. Let us now turn to the reference frame , which is moving along the axis  at the velocity  almost reaching the speed of light. For this purpose we will use the direct Lorentz transformations as follows:

 

,         ,         ,         .

 

.                                                   (28)

 

In  we denoted the proper time with , to avoid confusion with the earlier time  in (20). We also would need to transform the velocities, that is to establish relation between the velocities in both reference frames. In this case, we obtain the following:

 

,          ,          ,

 

.                               (29)

 

The velocity  denotes the full velocity of the particle in , and  is the Lorentz factor of the particle in . Let us substitute into (27) the Lorentz transformation for the wave phase (28) and the transformation (29) of the velocities and the Lorentz factor, leaving  in the velocity’s amplitude constant and expressed in terms of the coordinates in :

 

,         .      (30)

 

As is known, the role of the Lorentz transformations reduces to establishing the relation between the clock values and the coordinates of events in the inertial reference frames. It follows from them that in the moving reference frames the rate of clock slows down. In (30), in the reference frame , in view of the relation  and the inverse Lorentz transformations in the wave phase (28), the role of the angular rate of rotation of the velocity vector is played by the quantity . The angular velocity  is less than the angular velocity of rotation  of the electron in the atom and the angular frequency of the photon due to the time dilation effect. At the same time, in  the wave vector  becomes smaller and the wavelength  becomes larger, which is due to the effect of reduction of the longitudinal dimensions of the moving bodies in .

According to (30), in the reference frame  we observe rotation of the negative praons at the angular velocity  in the plane , and in case of instantaneous motion of the observer along the axis  with changing of  we discover displacement of the rotation phase by the value . For this to happen the particles inside the photon must be arranged as if they are located on the surface of the right-threaded screw with the pitch , while the screw is rotating to the right at the angular velocity , without moving along the axis . If we consider the positive praons as the rotating particle inside the photon, then due to their increased mass, their rotation velocity in (30) would be less. The positive praons can be placed on the surface of the screw, the radius of which is  times less than the radius of the screw for the negative praons. At each time point the positive praons would be on the same side as the electron at a corresponding delayed time , while the negative praons would be located on the other side of the axis .

 

4.3. The second field component

Let us consider the action of the second field component in (17) on the motion of the particles inside the photon in the wave zone. The field of this component at  in view of the electric field of the nucleus is as follows:

 

,          ,           .

(31)

 

,       ,       .

 

Under conditions , , in the electric field components in (31) we can neglect the constant terms from the nuclear field including . Additionally, we can also neglect the magnetic field component , since it would be  times less than the components  and . In (31) let us turn from the early time point  to the current time point , taking into account the definition: . At  we find:

 

,      ,

 

,          ,

 

,                          .                           (32)

 

Doing the same as in the previous section, similarly to (26) we obtain the following:

 

,

 

,

 

.

 

The approximate solution of these equations for the velocity of particles has the form:

 

,      ,     .

(33)

 

Figure 3 shows the surface perpendicular to the axis , on which the directions of the electric and magnetic fields (32) and the velocities of praons are shown, according to (33). All the vectors correspond to the time point , at which the condition  is met. The vectors  and  denote the velocities of the positive and negative praons, respectively.

 

 

Надпись: Fig. 3. The fields (32) and velocities (33) of praons near the axis   in the wave zone at  .

 

5. The photon structure

The presence of  in the denominator of the velocity in (23) for the near zone and in (27) for the wave zone leads to decreasing of the velocity amplitude while the distance from the emitting atom is increasing. Obviously, for the photon to exist independently at a certain distance from the atom, the amplitude of the rotation velocity of the negatively charged praons in the photon must stop being dependent on . By analogy with (20) and (25), in which we will replace  with a certain constant distance , we will assume the following expressions for the amplitude of the electric field inside the photon in the near zone and in the wave zone, respectively:

 

,                         .

 

We have an opportunity to estimate the value of  using the data from [14] for the photon with the angular frequency  s-1, which emerges in the hydrogen atom in the electron’s transition from the second to the first level in the Lyman series, setting the photon radius equal to . Based on the photon energy and its volume, with equality of the density of this energy and the electromagnetic energy density, we determine the amplitude of the electric field inside the photon: V/m.

If we equate  and , we obtain , where  is the Bohr radius. However, if we equate  and , then we should obtain . In the near zone the field  is substantially smaller than the field , and therefore, only at a small distance from the nucleus of the order of , the field  could set the photon’s praons in motion so that it could have a field of the order of . At the boundary between the near and wave zones, which is reflected by the condition , the value . Consequently, the internal electromagnetic energy of the photon, associated with the motion of the charged particles in it, appears in it already in the near zone. Here, the particles of the emerging photon are influenced by the electric field (17), consisting of three main components, which get aligned with each other at  and the value .

The amplitude of the transverse electric field of the second component in (17), according to (32), at  is equal to . From the equality  we obtain the estimate: , so that in the near zone the second component in (17) has lower degree of influence on the particles inside the photon than the first component, but its influence is stronger than that of the third field component.

Analyzing the field directions and the velocities of particles in Figures 1-3, resulting from the field components (17), in a first approximation we can develop the photon model, which is symmetric in its form. The photon’s cross section at  in this model is presented in Figure 4.

 

 

 

Надпись: Fig. 4. The photon’s cross section at  .  The positive charge near the axis   is indicated with  , the negative charges of the lobes are indicated with  .

 

When the rotation phase of the electron in the atom satisfies the relation , where  is the earlier time point, then for any time point  we can choose such , with which the pattern of events will be repeated in the same way as in Figure 4. In this case in Figure 4 the lobes along the axis  are formed by the negative praons under action of the fields of the form (20) and (25), as in Figures 1 and 2, respectively. In Figure 4, we also added the lobes of the negative praons that are likely to occur under action of the fields (32), as in Figure 3. In the center, near the axis  the positive praons are concentrated. The whole lobes’ construction is rotating around the axis  at the angular velocity  and it is also moving along this axis at the velocity , which is almost equal to the speed of light. At a given time point  with the change  the phase  would change and other lobes would appear in the new cross section of the photon, which would be shifted relative to the lobes in Figure 4 at a corresponding angle. This means that the entire set of lobes of the negative praons form continuous helical lines in space at a pitch equal to the wavelength and with the length along the axis  equal to the photon’s length.

In this case we can expect that at  formation of the two corresponding lobes stops and they become independent of the field , which is decreasing in amplitude. For the rest of the lobes the same is true at  for the field  and at  for the field . Although these estimates are not entirely accurate, as we assumed the constancy of  in the denominator of the solutions for the praons’ velocities (23), (27) and (33), the general result remains the same: the electric field of the rotating electron cloud is able to create the lobe structure of the emerging photon. The lobes arising from praons are then fixed by the forces acting between the praons.

Theoretically, inside each lobe in Figure 4 there should be sufficiently smooth distribution of the charge, from the positive charge at the center – to the prevalence of the negative charge at the edges of the lobes. This should also be accompanied by smooth change of the mass density along the lobes. In this case, the lobes contain not only the negative praons but also a significant number of positive praons. We compare the positive praons with protons and the negative praons with electrons, and admit the existence of neutral praons as the analogues of neutrons.

From the solutions for the velocities of praons (23), (27) and (33) we see that these velocities in a first approximation do not depend on whether the photon is propagating in the positive or negative direction of the axis . These solutions do not contradict the fact that in quantum transition two photons will be simultaneously emitted from the atom in opposite directions. These photons must have oppositely directed circular polarization, that is, have different rotation directions of the electric vector with respect to the velocity of the photon.

However, as a rule, except for the special cases of excitation, a multielectron atom emits one photon. To explain this, let us turn to the results in [9], where the substantial model of electron was presented, which was understood not as a charged point, but as a three-dimensional structure in the form of an electron disk. It is assumed that the matter in the disk rotates differentially around the disk’s center at the angular velocity , which depends on the current radius  of the matter unit’s location. In addition, the disk’s center can be located at a distance  from the nucleus and rotate around it at the angular velocity . The latter allows us to explain the electron spin as the result of rotation of the electron disk as a whole around the nucleus, as well as to ensure the possibility of electromagnetic emission from the atom, while there is rotation at the angular velocity . In [9] we take into account that the negatively charged matter of the electron cloud is attracted to the positively charged nucleus and at the same time repels from itself by electrical forces. In this model, the matter is also under influence of the strong gravitation from the nucleus with the strong gravitational constant m3∙kg-1∙s-2. The sum of all these forces makes the electron matter rotate around the nucleus. The stable quantized states of the atom’s energy are explained by the fact that in these states the equality is achieved of the kinetic energy flux of the electron matter and of the sum of the fluxes of the electromagnetic and gravitational energies in this matter. Besides, the emission from the atom tends to zero and the state of the electron disk’s rotation remains unchanged for a long time and the field momentum is not transferred to the matter.

In the substantial model of the electron it is important that in the multielectron atom the electron disks are located in the atomic shells approximately parallel to each other and the number of electrons in each filled shell is even. This leads to the fact that the magnetic energy of the atom tends to a minimum, since the magnetic moments of the respective paired electrons are directed oppositely (the Pauli exclusion principle). To explain the Pauli principle the well-known Lenz rule is used: as the magnetic field in the conductor increases, the magnetic field is formed which opposes the initial magnetic field. If an atom has an unpaired electron and is combined with a free electron, the latter will have such a rotation of the disk’s matter as to create a magnetic field, according to the Lenz rule and to the Pauli principle. This explanation does not require any reference to the quantum spins of the electrons and is based on the known electromagnetic phenomena.

For the maximum number of electrons in the electron shell of the atom there is a quadratic dependence . As for the multiplier 2 before , we believe that this is a consequence of the Pauli principle, as well as of the symmetry in the arrangement of electrons in the form of rings-disks. As for the quadratic dependence , we can see that as the shell’s number  increases the distance to this shell increases too. The shell’s area like the sphere’s area varies in proportion to the squared distance. Therefore, in case of a corresponding change of the distance to the shell with the number , the relative density of electrons as the number of electrons per unit area of the shell, remains unchanged. Only in this case, the entire matter of each electron will be located on the shell with the same density, and the total charges and masses of the electrons on the shells will be integer-valued.

Now let us consider the simplest case, when there are two paired electrons in the form of two parallel disks, one of them is an excited electron in the state of quantum transition, the momentum of which decreases with emission of a photon. We see that the situation for the two opposite fluxes of praons passing through the disk of the excited electron is asymmetric: one flux of praons passes through the first electron and then through the second excited electron. The other flux of praons first passes through the second electron and then through the first electron. Since the first electron is not excited, the center of its disk is not shifted with respect to the nucleus and does not rotate, the electron does not emit, and it influences the flux of praons only by its stationary fields. This results in a slight shift of the flux of praons, which then interacts with the matter and fields of the excited electron, and becomes involved in the formation of a photon in the propagation direction of the flux of praons. The opposite flux of praons first passes through the disk of the excited electron, and transverse rotation of praons emerges in this flux, which is necessary for a photon. But then the flux passes through the disk of the second electron, where the transverse rotation of praons is suppressed by the action of the electron’s rotating charged matter, which hinders the formation of a photon in this direction. For the atomic shell, where the number of electrons is more than two, the situation gets more complicated, but does not change fundamentally – in order to emit the photon from the atom mainly in one direction, asymmetry of the excited electron’s position with respect to other electrons is needed.

 

6. The fields inside the photon

We should note that the charged praons themselves cannot generate such an electromagnetic field inside the photon so that this field in turn could lead to the required motion of particles and hold them together. For such motion of praons there should either an additional external force, for example the electromagnetic Lorentz force in (26) from the field of the rotating electron in the atom, or some internal non-electromagnetic force. For the praons inside the photon we suppose the action of strong gravitation [14], which should act between the positively charged praons in the photon’s core and the negatively charged praons in the lobes. The other two forces are the electromagnetic force of attraction between the oppositely charged praons in the photon’s core and in the lobes of the photon and the electromagnetic force of repulsion of the negative praons from each other. The praons in the lobes are in the state of continuous rotation, so that we must also take into account the centripetal force.

Earlier we derived the strong gravitational constant  for the level of atoms, equating the magnitude of all the four above-mentioned forces, acting on the electron’s matter in the hydrogen atom [9], [12]. Let us assume in a first approximation that the same condition of the forces’ equality holds for the praons inside the photon in the reference frame , associated with the photon. Let us represent any of the lobes in the form of a capacitor, one plate of which is positively charged and is located at the axis , and the other plate is negatively charged and is located at the end of lobe. We believe that the photon is neutral in general and the number of positively and negatively charged praons in it is equal. The capacitor plates are attracted to each other by the electric force, as well as by the attraction force of the strong gravitation:

 

,                 ,

 

where  and  are the charge and mass of the positive praons on the capacitor plate, which is adjacent to the axis ; the product  is the total mass of the negative praons on the other capacitor plate at the end of the lobe;  and  set the masses of the positive and negative praons;

 and  are the electric filed strength and the gravitational field strength, respectively, which are acting on the plate with the negative praons,  denotes the area of the capacitor plate.

 

From the equality of the forces  and  , as well as the condition , we find:

 

.                                                    (34)

 

The charge and mass of the positive praon were found in [14], using the theory of dimensions and the coefficients of similarity between the atomic and praon levels of matter: C, kg, here we used the coefficients of similarity in mass , in sizes  and in speed of processes . Substituting the praon’s charge and mass in (34), in view of the relations , , we obtain an estimate of the strong gravitational constant: m3·kg-1·s-2. On the other hand, the gravitational constants at the stellar, atomic and praon levels of matter, according to the theory of dimensions, are related to each other by the similarity coefficients:

 

 m3·kg-1·s-2,              m3·kg-1·s-2,

 

besides, for the strong gravitational constant at the level of atoms, the following relation from [12] holds true, which coincides by its sense with (34):

 

.

 

The strong gravitational constant  allowed us not only to describe the stability of electrons in the atom, but also to explain the nature of the rest energy of elementary particles, to derive a formula for the magnetic moment of the proton [9] and to calculate its radius [18]. The stability of nucleons in the atomic nucleus is also explained, in this case the strong gravitational attraction between the nucleons is opposed by the repulsive force of the torsion fields of the strong gravitation of nucleons (the spin-spin interaction in the gravitational model of strong interaction). The coupling constant of strong gravitation is close in value to the standard coupling constant of strong interaction. In [15] it is shown that the strong gravitation structure is the same as that of the ordinary gravitation, and the range of the strong gravitation’s action in the matter, which has the same mass density as that of the Earth, is not more than 0.7 m [9].

Thus it can be stated that the strong gravitation at the level of praons and the electromagnetic forces are able to keep the positive and negative praons inside the photon near each other and to ensure the photon’s integrity. As for the stability of the positive praons concentrated mainly in the photon’s core, here the main forces are the gravitational forces of attraction and the forces of praons’ repulsion from each other, according to the gravitational model of strong interaction [9]. In this model, the gravitational force is opposed by the spin forces from the gravitational torsion field, which in particular ensures the stability of the atomic nuclei.

In Figure (4) the electric field inside the lobes is directed outwardly from the axis . In the photon in addition to the electric field there must be a magnetic field perpendicular to the electric field. In order to understand how this magnetic field appears, we will consider the rotation of the photon’s lobes in Figure 4 from the perspective of the reference frame , as in Section 4.2. This reference frame is moving along the axis  at the same velocity  as the photon in the reference frame . Taking into account (29), in  in each cross section of the photon the lobes rotate around the axis  at the angular velocity:

 

.                          (35)

 

In (35)  denotes the Lorentz factor close to 1 for the praons in the reference frame , and  is a very large Lorentz factor for the praons in the reference frame , in which the full velocity  and the velocity  of the praons are close to the speed of light. In [14] an estimate of the Lorentz factor for the praons in the photon was made: . The value  is substantially less than , which characterizes the effect of time dilation. Inside the lobes a certain electric field  is acting, which is in equilibrium with the strong gravitational field strength and maintains the form of the lobes in view of their rotation.

Let us consider the transformation of the electromagnetic field of the lobes from the reference frame  into the reference frame . The electromagnetic field components are the components of the electromagnetic field tensor and therefore they can be transformed from one inertial reference frame into another not as the components of a four-vector, but as the components of a four-tensor. In particular, for the transformation of the field components during the motion of the reference frame  along the axis  we can write:

 

,           ,            ,                     (36)

 

,            ,            .

 

The slow rotation of the negatively charged lobes creates a certain common magnetic field . Let us consider in Figure 4 one of the lobes, in which the internal electric field at a given time is directed along the axis  and is equal to . At this time point, other electric field components averaged with respect to the volume of the lobe are zero,  and , and similarly the magnetic field components  and . Then from (36) it follows that for the components of the lobe’s field, which are averaged with respect to the volume, in the reference frame  we should obtain:

 

,   ,   ,   ,   ,   .

 

Since  we arrive at the condition , which holds in the photon and relates its electric field strength and magnetic field. Meanwhile the transverse magnetic field  appears as a consequence of transformation of the electromagnetic field components from the reference frame  into the reference frame , in which the photon is moving at the velocity . At each point of the lobe under consideration in , the magnetic field  appears, which is perpendicular to the electric field  inside this lobe.

If in Figure 4 we take another lobe, then the direction of the magnetic field, which is transverse to this lobe, would change accordingly, but it will be perpendicular to the axis of the lobe and to its internal electric field. In a first approximation, we can assume that the electric fields in the cross section of the photon in Figure 4 rotate with the lobes and are directed radially from the axis, and the magnetic field is located on the segments of circles in the places, where it crosses the lobes, and is directed along the tangents to the circle.

 

7. The Lorentz factor and energy fluxes

We will recall, as in [14] we found the charge to mass ratio and the Lorentz factor  for the praons in the photon. Suppose some praon is located on the radius  and rotates at a certain velocity  around the photon’s axis. For the period of rotation of the particle we can write the following:

 

,                                .

 

The relation between the centripetal force, required to rotate the particle, and the electric force, exerted on the particle with the charge  and the rest mass , is as follows:

 

.                                                (37)

 

For the photon, it is assumed that half of its energy  is the energy of the particles’ rotation, and the other half of its energy is the total energy of all the fields. Besides, in the reference frame  the angular momentum of the photon is equal to the Dirac constant and is given by a formula, which corresponds to a rotating cylinder composed of  particles:

 

 .                                                   (38)

 

Based on (38), we can estimate the energy of rotation: .

Dividing the photon energy by the photon volume, we obtain the energy density, which can be equated to the double density  of the electromagnetic energy inside the photon:

 

.                                                 (39)

 

From (37) and (39), for the photon under consideration in view of (7) with the photon radius  and  s-1 it follows:

 

 C/kg.                                 (40)

 

In Section 6, we have shown how the praon’s charge and mass are calculated from the theory of similarity of matter levels with the use of similarity coefficients. If we substitute  in (40), then we would find the value of the Lorentz factor .

For the case of a hydrogen-like atom we can see that in (40) the following proportions hold true:  according to (7), as well  and . If the principal quantum numbers of the energy states  and  are large enough, and the condition  is met, then we will have . Consequently, the Lorentz factor  increases in proportion to the nuclear charge number  or to the square root of the photon energy: . The highest value  of the photon is expected in the hydrogen-like atom, which has the nucleus with the largest number of protons, and in electron transitions near the smallest orbits. In this case the largest fields of the atom influence the praons of the emerging photon and transfer their energy to them.

From transformations of the electromagnetic field components (36) it was found that  in the photon. Deriving (34) we assumed that in the photon the balance is achieved between the electromagnetic force and the force from the strong gravitation, and in (37) we also took into account the equality of the electromagnetic force and the centripetal force, arising from rotation of the praons inside the photon. The full balance of the forces should also include the fourth force, arising from repulsion of the praons’ charges from each other. All the four forces are approximately equal in magnitude. Let us now consider the ratio of the energy fluxes inside the photon. For the average values of the electromagnetic Poynting vector and accordingly of the gravitational Heaviside vector [12], [19], after averaging over the wave period of the periodically varying field components, we can write:

 

,     ,     ,     ,

 

here  is the vector of the gravitational torsion field as the strong gravitational field component, which is similar in its meaning to the magnetic field induction, and we used the conditions of the form  and  for the amplitudes of the field components.

 

Let us calculate the ratio of the amplitude of the average value of the gravitational energy flux vector to the amplitude of the average value of the electromagnetic energy flux vector. Again, we will consider the model of a lobe inside the photon in the form of a capacitor, as in derivation of (34), and will take into account the expressions for the amplitudes of the field strengths in the form:  and . In view of the relation  and (34) we obtain:

 

.

 

The ratio of the fluxes of gravitational and electromagnetic energies in the photon turns out to be equal to the ratio of the proton mass to the electron mass. This correlates with the fact that, according to [14], for each matter level the ratio of the energy density of the field of gravitons in the vacuum field, responsible for the gravitational forces, to the energy density of the charged particles in the vacuum field, responsible for the electromagnetic forces, is also equal to the ratio of the proton mass to the electron mass.

 

8. The magnetic dipole moment

The non-zero component of the magnetic field  inside the photon leads to the fact that the photon as a whole must have some magnetic dipole moment. Let us consider the photon in the reference frame , where the amplitude of the wave vector and the wavelength, in view of (29), are as follows:

 

,          .       (41)

 

To estimate the magnetic field we will represent the photon in the reference frame  in the form of a solenoid with the length  that contains  turns. In the reference frame , associated with the photon, in each turn there is a current . Here  is the duration of the photon emission from the atom,  denotes the number of particles carrying the praon’s charge   in one turn, and the product  determines the total number of charges, involved in the magnetic field creation.

The mass  in (38) is a certain effective mass and is close to the praon mass ; the difference between these masses is due to the fact that the photon is not a solid cylinder but a helical structure. Besides, the cross section of the photon in Figure 4 has the form of lobes, and the mass density in the lobes must be a function of the distance from the axis . In the reference frame  the angular momentum is written as in (38), and taking into account (35) and (38) we have:

 

.                                    (42)

 

The magnetic field inside a long solenoid depends only on the number of turns per unit length  and on the flowing current :

 

.

 

Substituting here , using (35) and (41), and expressing  in terms of  from (42), we find:

 

.             (43)

 

If we substitute here the data for the photon under consideration: , , s, , C, and if instead of  we use the praon mass kg, we obtain the estimate of the longitudinal magnetic field inside the photon: T.

The magnetic field in the reference frame  is associated with a certain effective current  in the turns of the solenoid, which models the photon:

 

.

 

We introduce the effective current into consideration because the charged praons in  do not just revolve around the axis , but also fly along this axis at the velocity . Taking into account the above-mentioned and the ratios  and (43), on the assumption , we find a dipole magnetic moment of the photon of insignificant value in the reference frame :

 

 A·m2.               (44)

 

If the negative praons move as is shown in Figure 4, and the photon propagates in the positive direction of the axis , then the magnetic moment of the photon would be directed in the negative direction of the axis  and opposite to the photon velocity. If the photon propagates oppositely to the axis , the direction of the magnetic moment and the velocity of the photon will coincide.

The ratio of the magnetic moment of the photon under consideration to the Bohr magneton is equal to . Note that in [20], based on the astrophysical data, there is a restriction of the magnetic dipole moment of the photon, which must not exceed . In the previous section we found that the Lorentz factor  increases in proportion to the nuclear charge number  in the hydrogen-like atom or in proportion to the square root of the photon energy: . Then from (44) it follows that the low-energy photons have an increased magnetic moment. In addition, the magnetic moment of the photon must increase in transitions with large quantum numbers , corresponding to the electron orbits, which are distant from the nucleus.

In [9] we studied the structure of various neutrinos and the ways of their formation, and it was shown that the muon and electron neutrinos contain the fluxes of the same particles (electron neutrinos of the praon level of matter) and differ in the energy spectrum and in the method of ordering (helicity) of the angular momenta of these particles as well as their fluxes in space. Due to this, electron and muon neutrinos can partially transform into each other. Neutrinos emerge in the weak interaction processes and differ from photons by their internal structure. Photons are most often generated during interaction of the praons’ fluxes of the vacuum field with the accelerated lepton matter; and neutrinos are more typical for the processes inside the hadrons matter.

Despite the different structures of the photon and neutrino, we assume that neutrinos of the atomic level of matter, like photons, consist of praons and have a magnetic moment. Therefore we will present the estimates of the dipole magnetic moment of neutrinos: not more than , according to [21]; not more than , according to [22]; not more than , according to [23]. As we can see, the dipole magnetic moment (44) for the photon in the hydrogen atom does not exceed the values, which is expected for the neutrino.

 

9. The photon mass

In the special theory of relativity there is a well-known formula, connecting the relativistic energy , momentum  and invariant mass  (the rest mass) of the particle:

 

.                                                      (45)

 

In [24] the restriction was given for the photon’s rest mass: eV/s2 in energy units, and in [25] it is assumed that eV/s2. As a rule, it is believed that the rest mass of the photon is zero, , and then the photon energy depends only on its momentum: . The latter ratio allows us to find the photon’s momentum using the energy or angular frequency of the photon. In this case the photon must move at the speed of light .

Let us consider the case of the photon mass in the substantial model described above. From (38) we can determine the relativistic energy of the particles inside the photon by multiplying the number of praons by the average rest mass of one particle , by the Lorentz factor and by the square of the speed of light:

 

 .                                          (46)

 

In (46)  denotes the angular frequency of the photon,  is the photon energy, and the product  in its meaning is the averaged speed of rotation of the electron in the atom during the photon emission. Consequently, the relativistic energy of praons is  times greater than the photon energy. By the order of magnitude, the difference between the energies  and  is about tens of thousands and more.

The invariant mass of the photon, understood as the invariant mass of the praons that make up the photon, in view of (46) is equal to:

 

 .                                               (47)

 

Substituting into (46) and (47) the data for the photon, emitted by the hydrogen atom during the electron’s transition from the second to the first level in the Lyman series: s-1, , , we obtain the following estimates: J or 170 keV, kg or eV/c2 in energy units. It turns out that the rest mass of the photon’s particles  is not equal to zero, though it is quite low.

In (40) it was found that , while  and . Therefore, , that is the total rest mass of the particles of the emitted photon increases in the atomic transitions with large quantum numbers  and reaches the maximum for the hydrogen atom with the nuclear charge number .

Instead of (45), for the photon we can write:

 

.                                                     (48)

 

Since the rest energy of the photon’s particles  is very low, in (48) the relativistic energy of the photon’s particles  is close to the product of the momentum by the speed of light: . This is due to the fact that the Lorentz factor  of the particles inside the photon is very large.

From the stated above we can see why in (48) the rest mass of the photon’s particles  is not equal to zero, while in (45) the photon mass  is equated to zero. This follows from the difference between the energies  and  – if the energy  is associated only with the rotation energy of the praons inside the photon and with the energy of their fields, then the energy  also takes into account the energy of praons’ motion at the velocity , almost reaching the speed of light. This additional energy is not transferred to the praons from the electrons in the photon emission from the atom, but they had this energy at the time of interaction of the praons’ fluxes with the electron. Not taking into account the initial energy of praons in the energy  leads to the loss of their rest mass and to zeroing of the mass  in (45) for the photon. In addition, the difference arises between the momentum of praons  in (48) and the generally accepted photon momentum  in (45).

Despite the fact that the rest mass of the photon particles , calculated by us, significantly exceeds the estimates of the photon mass in [24] and in [25], the mass  cannot be directly found in experiments. This is due to the fact that during interaction of the photon with the matter, the photon’s angular momentum of the order of  is transferred to the matter, as well as the corresponding energy and momentum. However, the main part of the photon energy, involved in the relativistic motion of praons, is carried away with them at the moment of the photon decay and its scattering into separate praons.

We assume that the velocities  of the fluxes of praons in the vacuum field are of the order of the speed of light, . At the same time, the photons are moving at the velocity , and we should have . Some difference between  and  is explained by the fact that the praons in the photon do not only move along the axis , which is perpendicular to the plane of the electron disk at the moment of the photon emission, but they also rotate around this axis by some spirals. Rotation of the praons depends on the photon frequency and energy, which should influence the velocity of the photons  and lead to some initial velocity dispersion of the photons of different frequencies.

 

10. Conclusion

In Section 2, we show that the frequency of the photon, emitted from the atom, is equal to the rotation frequency of the electron cloud’s center around the nucleus, averaged with respect to the time of the photon emission during quantum transition of the electron from a certain state to a state with lower energy.

In Section 3 we present the expressions for the electromagnetic field strength in the wave zone away from the charge, rotating around a certain center, which can be used to estimate the electromagnetic energy flux. From these, it follows that most of the energy is emitted from the charge in the rotation axis direction. In other directions the energy flux decreases quite rapidly in magnitude and has an oscillating character, without producing noticeable emission.

Taking as a basis the electromagnetic fields in the near zone and the wave zone from the hydrogen-like atom that undergoes quantum transition, we estimate the action of these fields on the charged particles (praons) of the vacuum field. The electron in the substantial model of the electron [9] is considered as a disk, and the above-mentioned fields cause rotation of the praon fluxes around an axis, which is perpendicular to the electron disk. Based on the pattern of the field in Section 4, in Section 5 we present the corresponding photon structure. The positively charged praons are concentrated near the photon’s axis, and the negatively charged praons form the helical part of the photon. Based on this structure, with the help of the idea of strong gravitation at praon level we solve the problem of stability of the positively and negatively charged praons inside the photon, explaining the long-term stability of the photon by the huge value of the strong gravitational constant . Rotation of the negative praons inside the photon leads to the fact that photons can easily interact with electrons and other charged particles, exchanging the energy with each other.

The photon is emitted along the axis of the electron disk, but some part of the energy in the form of electromagnetic emission leaves the excited atom in other directions. This emission is in phase with the oscillations inside the photon. The latter can explain the results of the Young’s interference experiment with low light intensity, when interference between single photons is observed. In this case, each photon passes through a particular slit and the coherent emission from the atom, associated with it, passes through another slit, which as a result gives the interference pattern.

In Sections 6-8, we estimate the values of the fields inside the photon, the Lorentz factor for the praons and the energy fluxes, we calculate the longitudinal magnetic field and the magnetic dipole moment of the photon. Here we use Lorentz transformations in order to turn to the reference frame, moving synchronously with the photon.

We finish development of the substantial model of the photon by considering the question of the invariant mass of the photon, which in the special theory of relativity is assumed to be equal to zero. In contrast, in the substantial model we calculate the rest mass of the praons that make up the photon, which obviously cannot be zero. Using this mass and the total momentum of the praons, we determine the relativistic energy of these praons, which tens of thousands times exceeds the energy of the photon in the classical theory. The difference between these energies is explained by the fact that the generally accepted photon energy does not include the relativistic energy of the praons, moving almost at the speed of light. As a result, in our model the photon is described by a standard relativistic formula, which relates its energy, momentum and the nonzero rest mass.

According to the ordinary interpretation, the photon is considered as an elementary particle, which is a quantum of electromagnetic emission, and its uniqueness is enhanced by the absence of the rest mass. In quantum electrodynamics, the photon is also a gauge boson, while the carriers of electromagnetic interaction are considered virtual photons. However, in [14] we have shown that the electromagnetic forces can occur under the action exerted on the charged bodies by multiple fluxes of praons that exist in the vacuum field. In this article we consider the processes that take place in quantum electron transitions in the atom, which allow forming photons from the fluxes of praons. Thus, the concept of praons allows us not only to understand the photon structure and to find its mass, but also to give a general explanation of the main electromagnetic phenomena.

If the photon has non-zero mass, then how could it change our understanding of the effect of light deflection by massive bodies under the action of gravitation? According to [15], gravitation is explained in Le Sage’s model as the result of the action exerted on the bodies by the fluxes of gravitons in the vacuum field. As a whole the vacuum field consists of two components, the field of gravitons and the field of charged praons, generating gravitational and electromagnetic forces, respectively. The photon is an object composed of praons, tightly bound by strong gravitation and electromagnetic forces. The fluxes of praons near massive bodies deflect from their initial direction under the action of the graviton field. In this case the photon is not just a flux of praons, it carries additional energy and angular momentum, acquired at the time of emission. Moving at relativistic velocity, the photon must be influenced by the graviton field, just like other particles having the same velocity are influenced by this field. For example, in the covariant theory of gravitation [26], the full angle of deflection is given by the formula: , where is the body mass,  and  denote the velocity and the impact parameter of the relativistic particle at infinity. If we assume for the photon , we will arrive at the formula , which would be valid for the photon in the general theory of relativity as well. In this formula, there is no dependence on the mass of the relativistic particle or on the photon mass. Therefore, emerging of the photon’s rest mass will change the full angle of deflection  by a very small quantity, arising from the difference between the photon velocity and the speed of light.

The substantial model provides its solution of the wave-particle duality of the photon. As an estimate of the number of praons in the photon we used, emitted by the hydrogen atom, we will calculate the relation: praons. This entire set of praons is tightly bound by the electromagnetic forces and strong gravitation at the level of praons, ensuring the integrity and long-term stability of the photon as a particle. On the other hand, due to its origin, the photon represents a long and rotating in space periodic structure of small cross section. Due to this structure, the photon exhibits wave properties, in particular by means of its field strengths, periodically varying in space and time.

From the stated above it also follows that due to the small difference of the photon velocity  from the speed of light, the recorded photon velocity would not the same for observers in the reference frames, moving at different speeds. In particular, we assume the existence of the reference frame , in which the longitudinal velocity of the photon as a whole is equal to zero and there is only its proper rotation.

It is convenient to assume that the speed of light is the limiting value for the motion of photons and particles. In transition to the lower levels of matter (to nucleons, praons, etc.) the Lorentz factor increases in the particles that make up the photons of the respective matter level, while their velocities must not exceed the speed of light. Thus the theory of relativity is applied in the theory of infinite nesting of particles. On the other hand, if the proton is accelerated by the electromagnetic forces, actually by the directed and concentrated fluxes of charged, relativistically moving praons, then the proton velocity cannot exceed the velocity of these praons. If any photon and the praons, forming it, are moving at a velocity less than the speed of light, then the proton velocity, accelerated by such praons, will not be able to reach the speed of light. The relativistic praons, that in the aggregate form the field of charged particle of the vacuum field, acquired their energy in the electromagnetic fields near the protons and other charged particles. Only the electromagnetic field is the basic source of relativistic charged particles at all levels of matter, and then it turns out that all these particles are moving more slowly than the speed of light.

According to the theory of infinite nesting of matter and calculations with the use of similarity coefficients, the invariant mass of a positively charged praon must be  kg and the radius must be  m with the proton radius equal to  m, according to [18]. Apparently, in modern experiments individual particles of such low masses and sizes cannot be detected directly. Actually, the search for preons, as the particles that make up quarks and leptons, and for partons, as the constituent particles of nucleons, in experiments on scattering of particles reached only the size of  m. This is much larger than the expected size of praons. As a result, partons were identified as quarks and gluons, and preons as a type of new particles were not detected and remained to be the subject of theoretical research [27].

In our opinion, the maximum possible photon energy , as well as the invariant mass of its particles , depend significantly on the way of the photon formation. If in (39) we equate the doubled density  of the electromagnetic energy inside the photon to the energy density of the charged particles of the vacuum field , and take from [14] the value J/m3, we can estimate the maximum amplitude of the electric field strength inside the photon:

 

 V/m.                                              (49)

 

In (49) it is assumed that the charged particles of the vacuum field (praons) are part of the photons at the time of their formation, and the energy density of the photons  cannot exceed the energy density . The latter follows from the fact that  is associated with rotation of the praons inside the photon, where the rotation speed does not exceed the speed of light, and  is the averaged energy density of the praons in their motion at the velocity of the order of the speed of light. The electric field strength (49) can be compared only with the field strength at the surface of the proton, where V/m. Therefore, formation of high-energy photons can be associated with the relativistic protons and interactions with them, as is the case with the cosmic rays.

In [15] we estimated the temperature of the graviton field, which is part of the vacuum field, for the case when gravitons are the particles similar to photons: K. Let us assume that the field of the charged particles of the vacuum field has the same temperature as the graviton field, and both fields are in temperature equilibrium. Assuming in a first approximation, that the Wien's displacement law holds true for the wavelength  , that the majority of photons have at the given temperature  , with regard to the Wien's displacement constant  image1908  m·K, we find:

 

,                  J.

 

In terms of energy units of particle physics the average photon energy equals GeV and belongs to the range of gamma-quanta. Assuming that in (37-38) the radius  can be replaced by the proton radius  and the speed of light can be used as the limiting speed of praons’ rotation in the photon: , then for the maximum energy of such a photon we find:  GeV.

However, measurements show that the energy of photons can be much higher and can reach 80 TeV [28]. Apparently, the photons with such energies are formed not under condition of temperature equilibrium between the matter and electromagnetic field, but under strongly non-equilibrium conditions. This requires interaction of a relativistic particle with a great number of particles at the same time. The examples are the synchrotron emission of a charged particle in the magnetic field of a sufficiently large and magnetized object, and the inverse Compton effect.

 

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