Wave and particle – from dualism to unity
Abstract
Examples are given when a macroscopic object manifests itself both as a wave and as a tangible object. It is proven that elementary particles are both waves and particles at the same time, and not alternately. This proof is obtained as new solutions to Maxwell's equations. The proof is not comprehensive - only cubic, spherical and disk particles are considered. This publication is a review and addition to already published articles and books.
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1 Khmelnik S.I. Wave and particle – from dualism to unity Examples are given when a macroscopic object manifests itself both as a wave and as a tangible object. It is proven that elementary particles are both waves and particles at the same time, and not alternately. This proof is obtained as new solutions to Maxwell's equations. The proof is not comprehensive - only cubic, spherical and disk particles are considered. This publication is a review and addition to already published articles and books. Content 1. Introduction 2. Cubic WAP 3. Spherical WAP 4. Disc WAP 5. Vacuum, dark matter, dark energy 1. Introduction In quantum physics, there are postulates that relate only to phenomena and objects of the microworld and cannot be applied in the world of macroobjects. And this approach gave brilliant results (we will not list them). We will focus on only one (but the main one for distinguishing quantum physics from classical) postulate, which declares the existence of corpuscular-wave duality, properties of nature, consisting in the fact that material microscopic objects can, under some conditions, exhibit the properties of classical waves, and under others, the properties of classical particles. Just in case, it was announced that this property is also inherent in large objects, only it is invisible to them. But then macroscopic ball lightning was discovered [11, 12, 13, 14], which passes through the glass like an electromagnetic wave, and a stationary and solid wave of water appears [20], on which ships are broken. There are many things in the world that sages never dreamed of am! We have to admit that corpucular-wave dualism is a property of all physical objects. And the philosophical principle of duality will not help in understanding this property, because we are no longer in the magical world of microobjects, where we can hope for the help of postulated spells. Any physical object can manifest itself both as a wave and as a tangible object. And this fact requires explanation (despite the successes of quantum physics). Below we will prove that particles are both waves and particles at the same time, and not alternately. The particle is a "wave-AND-particle", not a "wave-OR-particle". In what follows, we will use the abbreviation WAP for wave-AND-particle. This proof will be obtained as new solutions to Maxwell’s equations [1]. The proof will not be comprehensive: we will only consider the cubic particle, the spherical particle, and the disk particle. This publication is a review and addition to already published articles. The very idea that such a model should exist is not new. Etkin in [7] reviews a brief history of this idea. In 1900, the famous physicist and astronomer J. Jeans argued that “in nature there are waves and only waves: closed waves, which we call matter, and open waves, which we call radiation or light” [8]. E. Schrödinger held the same views until the end of his life, who wrote: “what we now take for particles are actually waves” [9]. And the author of the concept of “wave-particle” dualism, de Broglie, initially also proceeded from the fact that “the waves described by quantum mechanics are the system itself” [10]. 2. Cubic WAP [2]. 2.1. Mathematical model of cubic WAP Consider some volume V with magnetic permeability and dielectric constant. Let, as a result of some influence, an electromagnetic wave with energy arise in this volume. There is no heat loss in volume V and there is no radiation from it. After some time, the wave parameters will take on stationary
2 values, determined by the values , , and volume size. These parameters are the electric field strength and the magnetic field strength as a function of Cartesian coordinates and time, i.e. (,,,) and (,,,). Naturally, they satisfy the system of Maxwell equations of the form − − =0, (1) − − =0, (2) − − =0, (3) − + =0, (4) − + =0, (5) − + =0, (6) + + =0, (7) + + =0. (8) Consider the following functions (proposed in [3]) that satisfy this system of equations: (,,,)=cos()sin()sin()sin(), (9) (,,,)=sin()cos()sin()sin(), (10) (,,,)=sin()sin()cos()sin(), (11) (,,,)=ℎsin()cos()cos()cos(), (12) (,,,)=ℎcos()sin()cos()cos(), (13) (,,,)=ℎcos()cos()sin()cos(), (14) where,,,ℎ,ℎ,ℎ- constant amplitudes of functions, - constants. Differentiating (9-14) and substituting the result into (1-8), after reducing the common factors, we obtain:, , , ℎ−ℎ+=0, (15) ℎ−ℎ+=0, (16) ℎ−ℎ+=0, (17) −−ℎ=0, (18) −−ℎ=0, (19) −−ℎ=0, (20) ++=0, (21) ℎ+ℎ+ℎ=0. (22) Let us consider the solution to the resulting system of equations found in [4]. Since the system is symmetric, we accept ==. (23) In this case, the system of equations (15-22) takes the form: ℎ−ℎ+ ⁄=0, (24) ℎ−ℎ+ ⁄=0, (25) ℎ−ℎ+ ⁄=0, (26) −−ℎ ⁄=0, (27) −−ℎ ⁄=0, (28) −−ℎ ⁄=0, (29) ++=0, (30) ℎ+ℎ+ℎ=0. (31) In the system of equations (24-31), equations (30, 31) follow directly from the previous ones. Indeed, adding equations (27-29), we get (31), and adding (24-26), we get (30). The first 6 equations in the system (24-31) with 6 unknowns are independent and from them the amplitudes of the functions ,,,ℎ,ℎ,ℎ can be found. We will look for a solution to system (24-29) at ℎ=0. (32) In this case we find:
3 ℎ=−ℎ, (33) =− , (34) =, (35) =−2, (36) =− . (37) From (34, 37) we find: = . (38) From (34, 38) we find: =− =−ℎ , (39) or ℎ=− . (40) 2.2. Energy WAP Let us write the tensions (9-14) in the form =(,,,) (,,,) (,,,)= cos()sin()sin() sin()cos()sin() sin()sin()cos()sin(), (41) =(,,,) (,,,) (,,,)=ℎ ℎ ℎsin()cos()cos() cos()sin()cos() cos()cos()sin()cos(). (42) Let us denote the time-independent parts of these expressions: =(,,) (,,) (,,)= cos()sin()sin() sin()cos()sin() sin()sin()cos(), (43) =(,,) (,,) (,,)=ℎ ℎ ℎsin()cos()cos() cos()sin()cos() cos()cos()sin(). (44) Let us now find the squared modulus of the total tensions: =++, (45) =++. (46) From (45-46) we find: =++sin(), (49) =++cos(). (50) Let's denote: ||=++, (51) ||=++/ (52) Then we get: =(||sin()), (53) =(||cos()). (54) Let us find and . First of all, we will show that there exists a parallelepiped in which the total energy remains constant in time. Let the segments OA and OB on the oz axis have equal length Z, which meets the condition. ∙ =,. (55) where is integer. Obviously, the condition is satisfied
4 ∫cos() =∫sin()=. (56) Let us consider a volume in which conditions similar to (55, 56) are satisfied along any coordinate, and we will call such a volume a agreed volume. Let's find the value of the agreed volume. From (55) we find the length from the coordinates: 2=2 ⁄,2=2 ⁄ ,2=2 ⁄. (57) Then the total agreed volume =8=8 ⁄, (58) and the minimum agreed volume =8 ⁄ (59) or, taking into account (38), =8 . . (60) y x o B A z Fig. 1. Let us write expressions (43, 44) using the solution obtained above (32, 33, 35, 40): =(,,) (,,) (,,)=11 −2, (61) =(,,) (,,) (,,)= −1 10, (62) where =(,,) (,,) (,,)=cos()sin()sin() sin()cos()sin() sin()sin()cos(), (63) =(,,) (,,) (,,)=sin()cos()cos() cos()sin()cos() cos()cos()sin() (64) From (51, 61, 63) we obtain: ||=++=11 −2=114=cos()sin()sin() sin()cos()sin() sin()sin()()114=
5 ⎩ ⎨ ⎧ cos()sin()sin()+ sin()cos()sin()+ 4sin()sin()() ⎭ ⎬ ⎫ or ||=6(). (65) The last transformation follows from (56). Similarly, from (52, 62, 64, 56) we obtain: ||=++= −1 10= 110=sin()cos()cos() cos()sin()cos() cos()cos()sin() 110= sin()cos()cos()+ cos()sin()cos()+ 0 (66) or ||= 2(). (66) Thus, for the agreed volume from (65, 66) we obtain: ||||= . (67) From (65-67) it follows: =||=||=6(). (68) The energy density is =+. (69) From (53, 54, 69) we get: =||sin()+||cos(). (70) From (68, 70) it follows that =(sin()+cos())=, (71) i.e. in a agreed volume, the energy density in the volume does not depend on time and has a constant value throughout the entire WAP volume. In other words, a standing wave is created in a agreed volume that does not radiate. The value is a constant. Therefore, for a agreed volume, the expression for the energy in the entire volume is =∙. (72) For a minimum volume of WAP, as follows from (68), ==6. (72a) From (72, 72a, 60) we find the energy of the minimum volume of WAP: =6∙8 . =∙ ⁄, (73) where =483...=2.4∙10... (74) Consequently, in a constant agreed volume, the energy of an electromagnetic wave does not depend on time, i.e. remains constant. This means that under the specified conditions, Statement 1. WAP, like a standing electromagnetic wave, can exist in a agreed volume. 2.3. Flows of energy Energy flux densities along coordinates are determined by the formula = =(×)=− − −, (75) where the functions , are determined from (9-14). Obviously, in a consistent volume at the boundaries of the coordinate axes the following conditions are satisfied: sin()=sin()=sin(). (76)
6 The function is present in the definition of one of the functions specified in condition (75). Therefore, from (75, 76) it follows that energy flows directed perpendicular to the faces are equal to zero, i.e. this volume does not exchange energy with the environment.sin Statement 2. WAP can exist within an agreed volume. In addition, for such a volume, Statement 1 is satisfied. Thus, WAP can exist in such a volume. First of all, let us consider the cubic form proposed in [4]. Consider, for example, the energy flux density along the axis . From (75) we find: =− (77) Combining this formula with formulas (9, 10, 12, 13, 23), we find: =(sin()cos()cos()ℎsin()cos()sin() −cos()sin()cos()ℎcos()sin()sin())sin(2) Taking into account (33, 35, 40), from (77) we obtain: =sin()cos()cos()3/sin()cos()sin() −cos()sin()cos()3/cos()sin()sin()sin(2) or =3 sin()cos()cos()sin()cos()sin() +cos()sin()cos()cos()sin()sin()sin(2) or = sin(2)sin()cos() +cos()sin()sin(2) (78) or = sin(2)sin(2) (78) or = sin(4+4), (79) We have obtained an equation for the energy flux density along the axis . This flux varies with time. It is equal to zero on the faces of the cube in the case when on the faces of the cube, i.e. when = (see Fig. 1) conditions of the form sin(2)=0 are met. These conditions are met to the agreed extent - see (55). Let us consider the energy flux density along the axis . From (75) we find: =− (80) Combining this formula with formulas (10, 11, 13, 23), we find: =12−sin()sin()cos()ℎcos()sin()cos()sin(2) Taking into account (35, 32, 36, 33, 40), from (80) we obtain: =12−2sin()sin()cos()3/cos()sin()cos()sin(2) or =− sin(2)sin()sin(2)sin(2) (81) Since on the faces of the cube sin(2)=0, then on the faces of the cube =0. Consider the energy flux density along the axis . From (75) we find: =− (82) Combining this formula with formulas (9, 11, 12, 23), we find: =sin()sin()cos()ℎsin()cos()cos()sin(2) Taking into account (36, 33, 40), from (82) we obtain:
7 =2sin()sin()cos()3 sin()cos()cos()sin(2) or = sin()sin(2)cos()sin(2) (83) From equations (78, 81, 83) it follows that flows of electromagnetic energy circulate in the cube along all axes. species (78, 84, 85). Consider the vector sum = + + . (84) Obviously, many vectors circulate in the cube and at each point of the cube there is a certain vector that has a module - the density of the total vector of the electromagnetic energy flow. From (78, 81, 83, 75, 63, 64) it follows that = =− − −sin(2). (85) From (78, 81, 83, 85) it follows that =о sin(2), (86) where о =−+−+−. (87) Thus, inside the cube there are lines formed by vectors . Obviously, such a line represents some kind of “spatial entangled spiral” (hereinafter simply a spiral). Such spirals are closed. Through every point where о ≠0 there is a single spiral, and through every point where о ≠0 there are many spirals. At each point of this spiral, the magnitude of the flow fluctuates in time, as sin(2). The amplitude of these fluctuations changes at a given point and depends on the location of this point in the cube. You can consider the development of this spiral. Let us denote the coordinate of a point on this scan as . Then we get a sinusoid with an amplitude that is a function of this coordinate: (,)=о()∙sin(2), (88) where ,о is a more convenient notation for functions ,о , respectively. Let's expand the function о() into a trigonometric series: о()=оо+∑ (оsin()) (89) Accordingly, function (88) will take the form: =ооsin(2)+∑ (оsin()sin(2)) . (90) Each term of this sum can be represented as: оsin()sin(2)=оsin()cos 2−=+, (91) Where =оsin−+2, (92) =оsin+−2. (93) Each of these two features a traveling wave. Consequently, the function under consideration (90) represents the sum of many traveling waves of electromagnetic energy flow. So, many running waves of energy flow circulate along each spiral. These waves have a common frequency, but differ in direction of movement, phase and amplitude. The total amplitude of the flow of these waves is equal to спираль= ∑о (94)
8 2.4. Weight In the existing theory, electromagnetic mass is the mass of an electromagnetic wave that is created by a moving particle [5]. In our case, it is the wave that creates the WAP particle, and in this wave there are no particles that form it. But at the same time, we cannot use this approach to determine the mass. We will use the well-known Umov formula, which connects the energy density and energy flow with the speed of energy movement: = . (95) It is also known that the pulse density = , (96) and the mass == . (97) Hence, = . (98) In this case, for a wave with known intensities, one can find the energy density , electromagnetic energy flux density and mass density according to (98). It is shown above that in the cube there are trajectories along which flows of electromagnetic energy propagate. At the same time, many such flows pass through each point of the WAP cube. Let us denote the total power density of such flows as . Then, using (98), we find the density of the electromagnetic mass, which is generated at this point by the very existence of the electromagnetic wave in the WAP. The sum of these masses is the electromagnetic mass of WAP. Consequently, WAP can be considered both as a standing wave and as a volume having a certain mass. 2.5. Conclusion We have established two conditions that must be satisfied by the region in which WAP can exist within a closed and continuous boundary. WAP, like a standing electromagnetic wave, can exist in a consistent volume We have established that WAP forms a closed area and has a certain shape and volume. The results obtained can be applied to any arbitrarily small units of length. The shape of the WAP region is such that multiple WAPs can be adjacent to each other without gaps. Consequently, WAP groups can occupy any volume. Thus, WAP of any size and areas of WAP of any size can exist. WAP does not have its own speed and its mechanical energy is determined by its mass and the speed that it received when interacting with other masses (including other WAP). The internal pressure on the WAP border is equal to the energy density at the border, although WAP does not have any envelope. It can be assumed that WAP behaves like an absolutely elastic body and transmits the received impulse without changing its magnitude. Then the WAP region also behaves as a conductor of the impulse. Obviously, WAP can form elementary particles and larger structures. But we can assume that the vacuum is also woven from WAP. 3. Spherical WAP[6]. 3.1. Maxwell's equations in spherical coordinates In [1], a solution to Maxwell's equations in spherical coordinates was found. The known solution for a spherical electromagnetic wave does not satisfy the law of conservation of energy (it is conserved only on average), the electric and magnetic intensities of the same name (in coordinates) are in phase, only one of Maxwell’s system of equations is satisfied, the solution is not a wave one, there is no energy flow with a real value. The proposed solution is free from these shortcomings. Maxwell's system of equations, being a system of partial differential equations, has many solutions. The applicability of a solution to physics is determined by a single criterion: it must satisfy the law of conservation of energy (LEC). The existing solution does NOT satisfy this law. So, let's consider the system of Maxwell's equations for vacuum, which has the form rot()+ =0, (1)
9 rot()− =0, (2) div()=0, (3) div()=0. (4) where is the electric field strength, is the magnetic field strength, is the absolute magnetic permeability, is the absolute dielectric constant. Next, spherical coordinates are considered - see Fig. 1. Maxwell's equations in spherical coordinates in the absence of charges and currents have the form given in table. 1. Fig. 1. Table 1. 1 3 1 tg ( ) + − sin ( ) + = 0 2 sin ( ) − − + = 0 3 + − + = 0 4 + + tg ( ) + + sin ( ) = 0 5 ( ) + − ( ) − = 0 6 ( ) − − − = 0 7 + − − = 0 8 + + ( ) + + ( ) = 0 In the solutions found, the tensions are determined by formulas of the following form: = Khm(,∝)sin(∝++), (5) = Khm(,∝)cos(∝++ ) , (6) = Khm(,∝)sin(∝++ ) , (7) = Khm(,∝)cos(∝++), (8) = Khm(,∝)sin(∝++), (9) = Khm(,∝)cos(∝+ + ) , (10) where Khm is some function, ∝,,,,ℎ are constants. We will consider a special case when
16 We will not consider the existing explanations for this phenomenon. But this experiment is amazing: a bright spherical radiant cold area, the appearance of which is inexplicable. A pulse with a steep leading edge can be expanded in a Fourier series, where a sinusoidal function with a high frequency will prevail. Thus, it can be assumed that the capacitor is connected to a high-voltage and high-frequency generator. No elements are connected in series with the capacitor, i.e. it is located absolutely symmetrically relative to the generator terminals. In this case, the energy flow into the capacitor comes from two sides. Two equal and oppositely directed energy flows meet exactly at the center of the capacitor. Above we considered a neutrino that was formed when two identical waves met. The object we observe in this experiment can be called a "giant neutrino." Fig. 4. It is proposed to consider neutrinos as DWAP. It fully corresponds to the above description of neutrinos. Above we defined for it (more precisely, for the cylinder-disk, which makes up half of the neutrino) energy according to (13), mass = с – see (9), angular momentum = с according to (17). 4.6. More about neutrinos Above, we examined a neutrino, which was formed when two identical waves met, in which all characteristics coincided, except for the direction of flight and direction of rotation. This is, of course, an unlikely case. Now consider the general case when the parameters ,, differ and denote them for the first and second waves as ,, and ,,. In this case, from (12, 13) we obtain: W ()= , (21) ()= с , (22) W ()= , (23) ()= с . (24) After the waves meet, the newly formed particle flies towards a more massive wave (with the same speed с), and both halves of it (rotating, as before, in opposite directions with the same speeds) acquire a new parameter value and a new value of the angular velocity of rotation that is common to both halves . We assume that the newly formed neutron flies towards the first wave. In this case, the kinetic power of the second half W =0. According to the law of conservation of energy, similarly to (3.1), we find: W ()+ ()+ W ()+ ()= W ()+ ()+ () , (25) where
17 W ()= , (26) ()= с , (27) ()= с . (28) Formula (25) is an equation with one unknown . In this case, the total energy of the pair remains constant. From here and from (9) it follows that the total mass of the pair also remains constant, i.e. the appearance of a neutron does not change the ratio of mass and energy. 5. Vacuum, dark matter, dark energy [21]. 5.1. Introduction The structure of the vacuum is studied by quantum field theory, which never gets tired make it look very complicated and, indeed, does not offer anything to describe the structure of the vacuum that is consistent with the ideas of classical physics. Below we propose such a structure, which follows only from the solution of Maxwell’s equations - no additional assumptions are made. This structure can be the structure of vacuum, dark matter, dark energy, any region of space... Here we will not establish the scope of application of this structure. On the contrary, the author would like to hear a discussion of this idea, which was outlined back in 2020 [22] (Chapter 5). But the public is sternly silent. It was proven above that there can be a cubic WAP, which is a cubic volume of vacuum, and in which a standing volumetric wave pulsates. It is important to note that this volume does NOT have any boundaries - physical or formed by the heterogeneity of the environment. WAP does NOT radiate through the faces of the cube, but on each face there is an electrical intensity, the vector of which is directed perpendicular to this face. The amount of energy, frequency, and tension at the cube's faces are functions of the size of the cube only. Apparently, there is a smallest volume of a cube, determined by the minimum energy quantum. Many of these WAP can fill the space entirely, without gaps. And it is precisely this structure that is described below. This structure occurs in nature [20] – in Fig. 1 and Fig. 2 show so-called square waves on the sea. Fig. 1. Fig. 2.
18 In [21, 22], Chapter 5, it is shown that there are several variants of square WAP. In Fig. Figure 1 shows one of the options - magnetic strength emerging from the faces of the cube are shown. It is important to note that there is no tension in this case, although it is shown in Fig. 3. On faces with a negative value, the stress coordinates are directed in the negative direction. The energy flow does not leave the face perpendicular to the x axis, but circulates along this face, because flux densities and on this face are not equal to zero. For example, =−. Here ≠0. Consequently, on this face, as well as in the entire volume, there is energy with a density that does not change over time. Consequently, on this face and, in general, on all faces, there is a constant pressure equal to the energy density. х у Нх Нх L=pi/alfa Нy Нy Нz Нz Fig. 3. 5.2. Vacuum structure Let us now consider the set of WAP. The cubic shape of WAP suggests that many WAP form a continuous volume - see fig. 4. Various combinations of WAP are possible. There may be a space filled with WAP, creating only magnetic strengths on the edges or only electrical strengths on the edges. There may be a space filled with only symmetrical WAPs or only asymmetrical WAPs. In the latter case, a direction should arise in space in which there is no tension in any direction. Such a vacuum must somehow exhibit anisotropic properties. It can be assumed that nature uses all options and there are heterogeneous spaces. Thus, each WAP remains autonomous, but together they form a continuous volume of vacuum. It can be assumed that all WAP have the same volume and then there is a single vacuum frequency. It can also be assumed that there are different regions of space with different (but common for a given region) volume of WAP. Then these areas should have different vacuum frequencies. Fig. 4.
19 Any facet of WAP may end up on the border of an empty region of space. Then tension will arise on the border of this area - the tension that is present on the specified border of WAP. This tension is the given tension that forms a standing wave. Thus, the tension on the edge of some WAP generates a standing wave in the empty space and thereby creates a new WAP. In this way, WAP multiplies, filling the entire vacuum. It can be assumed that the Universe arose from one WAP. 5.3. Casimir effect Let us consider the right side surface of the vacuum fragment in Fig. 3. Assume that this surface is the boundary of the WAP region. On the open surfaces of WAP in their center the vectors of tensions entering and leaving these surfaces are shown. The thick line going around the ends of these vectors conventionally depicts a wave of tension on open surfaces. These tensions vary sinusoidally in time. Thus, there is a standing wave of tensions on the surface of the WAP domain border. But, most importantly, there is constant pressure on exposed WAP surfaces. If some body is adjacent to these surfaces, then it must experience this pressure. Thus, the body, located in a vacuum filled with WAP, experiences vacuum pressure from all sides. Each WAP area also puts pressure on the neighboring area. Consequently, WAP seeks to fill internal voids. One can argue, following Torricelli, that “a vacuum does not tolerate a emptiness.” In more detail, the book proves that what has been said is nothing more than a proposed explanation of the Casimir effect - two parallel mirror surfaces located at short distances in a vacuum attract each other. In the existing vacuum model, the cause of the Casimir effect is considered to be “energy fluctuationsphysical vacuumdue to constant birth and disappearance in itvirtual particles…. This occurs due to the fact that only standing waves can exist in the space between the plates, the amplitude of which on the plates is zero. As a result, the pressure of virtual photons from the inside on the two surfaces turns out to be less than the pressure on them from the outside, where the birth of photons is not limited in any way.“In addition, when explaining this effect, the existence of negative energy is recognized. These references are provided to highlight the apparent contradiction between the proposed and existing theories (PT and ET). In PT it is proved that there is a volumetric standing wave with certain intensities at the nodes, and in ST it is stated that the amplitude of the intensities at the nodes (on the plates) is equal to zero (it can be proven that the law of conservation of energy is not satisfied in this case). In PT it is proved that real particles fill the vacuum, and in ET the existence of virtual particles is assumed, the birth of which is not limited by anything, and the disappearance of which is inexplicable. In PT it is proven that there is a constant vacuum pressure on bodies, and in ET it is assumed that such pressure is created by waves of virtual particles that constantly appear and disappear. The ET proves the existence of negative energy, while the PT maintains respect for the law of conservation of energy. The reader is invited to choose what he likes best. References 1.Khmelnik S.I. New solutions to Maxwell's equations. Version 25, pp. 1–471, "MiC" - Mathematics in Computer Corp., https://doi.org/10.5281/zenodo.10658891 2. Khmelnik S.I. Quantum mechanics: a particle is a volumetric standing wave (second part). Papers of independent authors, ISSN 2225-6717, no. 51, https://doi.org/10.5281/ZENODO.4072758 3. Khmelnik S.I. Variational Principle of Extremum in Electromechanical and Electrodynamic Systems. Publisher by “MiC”, printed in USA, Lulu Inc. ISBN 9780557082315, 2014, Fourth Edition, pp. 1-347, https://doi.org/10.5281/zenodo.3926034 4. Khmelnik S.I. Electromagnetic Keeper of Energy and Information. Canadian Journal of Pure and Applied Sciences, 13, Okt 2019 (No. 3), 4853–4859, https://doi.org/10.5281/zenodo.3518396 5. R. Feynman, R. Layton, M. Sands. Feynman lectures on physics. T. 6. Electrodynamics. Moscow, ed. "World", 1966. 6. Khmelnik S.I. Particle – spherical standing wave, Papers of independent authors, ISSN 22256717, no.62, preprint, https://doi.org/10.5281/ZENODO.10827358
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