Theory of the Electron as a Composite Particle

Zoran B.TodorovićFaculty of Sciences, University of Prishtina, Serbia&Montenegro

On the basis of derived modified quantum commutation relations for sub-spins have been shown that electron could be a composite particle .Applying quantum commutations has been calculated fractal size of each sub-particle which build the internal structure of the electron.

PACS:Electron, Internal structure, Composite particle.

The beginnings of quantum mechanics dated in 1897 when J. J. Thomson defined the ratio of the electric charge of an electron to its mass. Consequences of such discovery could not have been predicted, and none could have had the faintest idea that the electron would assume one of the major roles in creation of new technological civilization of the twentieth century.

Nevertheless, some questions concerning the electron have still remained open. The Dehmelt indicated the possibility that the electron might have an internal structure. By discovering and studying the unknowns of the electron, physicists solve numerous problems in the area of experimental and theoretical physics. In such problems, the electron is treated as a point-particle with electric charge e, Einstein’s energy mc2, impulse , Bohr΄s magneton

and spin of /2. And this is, according to such opinion, quite sufficient . The other properties of

the electron, such as: possible spatiality and internal structure appear to be useless and may well be designated as academic by character. And that, simply because the electron remained inconceivable in its very essence, because no empirical parameter had been explained theoretically . Even after the turn of the twenty-first century, it remained an unsolved puzzle, a riddle that still inflames imagination of physicists who hope that one- day they will solve it and remove the veil behind which it is so well concealed.

We will see how electron as a point is “broken” apart, and how that points hides an internal structure. This means that we cannot speak of the mathematical concept of point any more; it becomes, because it contains three other points within itself, an entity with certain spatiality.

Due to its internal structure, the electron loses its property of an elementary particle; it becomes equal to the proton, at least concerning the number of sub-particles it is made of, and at the same time, the communication between the electron and the proton in the hydrogen atom becomes symmetrical; three sub-particles in the electron communicate with three quarks in the proton, thus making the hydrogen atom stable. How to explain communication of the electron as a point particle, as considered by the Standard Model, with the proton as a complex particle in an atom? Gel-Mann

and Zweig postulated the theory of possible existence of quarks in the proton, which was experimentally confirmed. However, the electron still keeps its status of a point particle in quantum electrodynamics. This means that, according to mathematical definition of point, it has no spatial dimensions; and such definition of the electron is still in effect; the electron has mass, electric charge, spin and magnetic moment, internal energy, depending on the state of motion, . All of its physical properties are measured through experiments, but on the location where it occupies zero space. In other words, there is something nowhere, which is not only hard to imagine, but is impossible to understand; it is counterintuitive and collides with common ense. The electron as a point particle in the theory of quantum electrodynamics gives much trouble to physicists; divergent integrals show up. Fortunately enough, physicists have found a mathematics-compliant theory applying which the divergent integrals disappear; they obtain results that are experimentally confirmed, which makes everybody happy. But, they are still weighed down by worries, as if having the sword of Damocles hung above their heads, that this is not the right theory to describe natural phenomena related to the electron without spatial dimensions. This gives rise to speculations and considerations of a possible internal structure of the electron, which cancels it as a point particle and gives it some, although miniscule, spatial dimensions.The attidudes and the opinion, about some open and stil actual problems, in the modern physics, by the Russian physicist A.S. Davidov introduced in his book „Quantum Mechanics“ of which some especially related to the electron , even after his book was published, have remained stil enigmatic.The Davidov's words will be partialy quoted in below (pp234/235)as follows:„Novadays, the numerous particles are known such as :electrons, protons, mesons (mesotrons), mesons-pions, K-mesons (kaons) etc , which are called << elementary particles>>, but on the present level of knowledge it is impossible to discuss about <<internal structure>> of the particles.These particles are characterized by dhe definite value of proper mass (mass at the state of rest), so they might be the neutral, or charged with electricity (positive or negative) .The absolute electrical value of all stable particles charged with electricity is equal…one of the most important features of the elementary particles is appeared at the moment of their birth, disappearing and mutual transformation during their mutual affection. In that way the photons will be born at the change of the motion

character of the electrons located in the atoms as or the proton in the atomic nuclea. At the collision of the nuclei of the high energy the pions are born. Neutron, as emited electron and antineutrino will be transformed into the proton. On the other side , the protons, located in the structure of the atomic nuclea, by emiting neutrino and positron , could be transformed into neutron. The photons in the nuclea's field could be transformed into the pair of electron-positron etc.The discovery of the possibilities (in accordance with the law suistainable nergy, the law of electricity load and some other laws of the suistainibility) the birth, the annulment (disappearing) and the mutual transformation of the elementary particles have been appearing as one of the most outstanding accomplishments in the perception of the objective (real) characteristics of the surrounding world and the mutual relationship of the different phenomena in nature.In the connestion with the notion of <<the elementary>> and <<the isolattion>of some particles in the relation to the others has become more and more the indefinite one.. In accordance with the present level of knowledge , the mutual interaction among the same type of particles is effected by the other type of particles . In that way , e.g. the charged with electricity or the neutral pion transfer the nuclear mutual affection among the nuclei..To say vice versa , protons and neutrons look as they are surounded by the mezon's cloud, across which ones their mutual affection is realized ; the mezon's cluds appeared as the integral part of the protons and the neutrons, so mostly they determine their features. On the other side, protons-neutrons by themselves determine the numerous pion's features.In this sense the notion (idea) of <<the isolated particles>> in any form becomes senseless .So, the idea about <<free motion of particles>> could be only the rough idealisation of the reality. In relativistic quantum mechanics of the single

particle it is not possible to locate the particle in the space, which linear dimensions are less than , where

means the mass of the particle at the state of rest, but in the opposite case respect Heisenberg's uncertainty

relation to the particles, will be transferred the energy , sufficient to create the pair of particles. That is why, the perception of the single particle can be consider provided the external fields are absent, which leads to the <<location of the particle>> in the space, which linear dimensions are less than Compton's wavelength of the

corresponding particle. For the border relativistic particle- light quantum - the concept of the coordinate of the particle in the ordinary sense is completely missed”.Sokolov stated the following:”Until now all the attempts have been made in order to form electrodynamics with the

final value of the electron radius which equals with the classical radius of electron m (Lorentz's theory,

non-linear Born-Infeld theory, theory Bopp-Podoljsky with the higher derivatives etc.) was not submitted to any convincing results. Even more, all these theories have been gained the principle difficulties during their quantum generalization .On the other side, the theory of the punctual-point electron, if leads to the endless value of the mass, as well in the classical as in the quantum theory (p.304)”.The problems still opened, as Davidov and Sokolov explained a few decades ago, are still present, what is unambiguously possible to find in the work M.V. Tokarev ”:The fundamental problems of the high energy physics are in the origin of the mass, the spin and the electron charge…The new ideas and the theories are connected with the extra dimensions, anisotropy, and fractal space-time, the structures of the quarks, and the theories of the Grand Unification”.As we are going to discuss here about the internal structure of the electron, we are under the obligation to point out some of the circumstances as the facts not contributed to the theory of electron as the point particle. It would be useful to remind the problem known as the Lamb's shift.It presents the energetic displacement of the electron within the hydrogen atom as the consequence of the influence of the electromagnetic field vacuum on the electron.The theoreticians have found the solution for this problem using the principle of the regularization, by defining the lower and the upper border line of the spectrum integration with the method known as <<cut off>>.We consider that electron has interaction with electromagnetic field of vacuum, but always in the association with the virtual electron, where the virtual electron presents the part of the virtual system electron-positron. In that way, the united electron and the virtual electron in the simplest case where their spins are anti-parallel, in the electromagnetic field of vacuum, which we consider as the homogenous one, and with the definite value of the amplitude of that within

the field exposed through the middle value, as << cuted off >>on , present the boson which

according to the rules of quantum mechanics applies Schrődinger`s equation ( -mass of the electron at the state of

rest; c- the light velocity in the vacuum; Bohr ُs magneton).

The solution of the equation gives the wave function as the key used to calculate all physical electron׳s characteristics. Because of its important role in the theory , we write down the expression for psi function..It is

expressed in the form of .Already the density of probability by itself

gives us very important parameter , and it is the volume which is encompassed

by the smeared electron in the space of the Compton׳s wavelength.It makes out .It is also

possible to determine it from the relation .The distribution of the mass and

the electric charge of electron in this field is according to the Gauss׳s law, and the size of the volume depends of the

state of the electron in the relation to the external fields, so it is changed according to the law

.Already in that relation we noted the forbiden state as evidently , because it is in collision with theory of relativity , from here we can draw the conclusion of the impossibility to obtain the zero volume for ; it means that the electron cannot be the point particle, because the point particle according to the mathematical definition of the point presents the entity out of the spatial dimension. Another important link used for description of the electron is in the application of Maxwell׳s relation for density of the electromagnetic energy on the electron.Namely, Maxwell׳

srelation for density of the electromagnetic energy is As we know E and B are

the transverzal components of the electric and magnetic field of the photon.If we include in this expression the<< cut off>> values of the magnetic field components and the equivalent value of the magnetic induction (

; is the frequency of the oscillator , formed by electron-virtual electron system)

and so, we consequently get the density of substance building up electron. It makes out .In

the arbital state, while electron is moved under the influence of the external field , it is possible to demonstrate that

the density of the electron substance is changed consequently to the law: .If this density is

multiplied by the volume of the smeared electron in that condition we obtain:

, which is well -known Einstein׳s of changing the electron׳s

proper energy during its movement . If the electron were the point particle result of the left side would be zero, which is automatically in the collision with the theory of relativity for the electron energy..According to this , it is completely clear that the electron cannot be the point particle.In this case the questions are broadened:if it is not the point particle how big are its spatial dimensions, and if it could have the spatial dimensions , consequently it could be one complex object.Our purpose is to introduce through the work the possibility that electron may be a complex object , it means the object with internal structure. Until now we suppose for necessity of symmetry, as an example within hydrogen atom, in the interaction between electron and proton having three quarks and a repercussion on the stability of the atom, this offers the reason for existing inner structure of the electron.Before we pass to the commutative relations , from which we will get the compounded fractal parts of the electron , we bear in ouar mind necessity to comment the facts in the quantum mechanics accepted without explanation- automatically , and which one the observed interest has not be devoted. Namely, it is concerned by the known relation for the operator of impulse and energy. To remind they are written the following well-known relations : :

for impulse and . for energy. If the bozons are concerned the problem is understandable, the bozon׳s spin is the wholenumber value derived from Plank׳s constant. But, if the fermions are concerned, concrete for electron, why wholenumber value is still written of Plank׳s constant, since we know that its spin is equal to the half value of Plank׳s constant? The answer may be single and it explains:in the wholenumber value of Plank׳s constant the spin of electron is included which we observe according to the Schrödinger׳s or Dirac׳s equation or the commutators, and the spin of the virtual electron, which we have not taken in the mind. The obtained theoretical results were in accordance with the experiments, so it was satisfied.In the following, it will be prooved that by taking into consideration the virtual electron we will get spontaneously to the internal structure of electron.. Why the virtual electron must be taken into the consideration? The answer is in the fact that the electron may not exist in the real world as an isolated particle, so it is always in the link with its virtual electron; virtual electron is the part of the virtual pair electron-positron, and electron permanently has been followed by ithat pair as its shadow .

On the occasion of the “ Anniversary of the Discovery of the Electron” , a Russian physicist A.A. Bogus says:”The electron, as much as 100 years ago, remains a natural phenomenon not completely resolved. By the time being, strictly speaking, no adequate theoretical explanations have been found for parameters of most important physical properties of the electron, such as; mass, electric charge, spin; they remain to be empirical characteristics taken over from experiments”. As we are going to discuss here about possible electron's structure, which was experimentally measured by Purdue University physicists and others , as well we are under the obligation to point out some of the circumstances, as the facts not contributed to the theory of electron as the point particle. It would be useful to remind the problem known as the Lamb's shift. It presents the energetic displacement of the electron within the hydrogen atom as the consequence of the influence of the electromagnetic field vacuum on the electron. The theoreticians have found the solution for this problem using the principle of the regularization, by defining the lower and the upper border line of the spectrum integration with the method known as <<cut off>>.After all, our aim in this work is to verify by quantum-mechanical theory the experimental data according to measurements by Purdue University physicists et.al , and then apply that theory on the possible internal structure within “bare” electron as na central core of the complex electron..According to experimental data they stated following: “All charges are surrounded by clouds of virtual photons, which spend part of their existence dissociated into fermion-antifermion pairs. Electron may not be a simple negative point charge, as previously thought; surrounding the electron’s

core is a fuzzy “cloud” of virtual particles, which wink in and out of existence in pairs. One particle in the pair is positively charged, the other negatively charged. The cloud consists of polarized pairs which essentially cancel each other out so that they do not “add” any net electric charge to the electron, Koltick say, the cloud plays a key role in how we perceive the electromagnetic force from the electron. The virtual fermions with charges opposite to the bare charge (electron) will be, on average, closer to the bare charge than those virtual particle of like sign”.On the basis of such experimentally founded picture of the electron, our obligation is to form quantum-mechanical theory, which would be to describe that experimentally facts.

Theory of the electron as a complex particle

So, we consider that, one bare electron and one virtual fermion from the virtual pair make together one pair. Such pair (bare electron-virtual electron) is in the interaction with electromagnetic field of the vacuum. . The spins of the united electron and the virtual electron in the simplest case are anti-parallel. We suppose that the electromagnetic field of vacuum, which feel both bare electron and associated virtual electron, is the homogenous within the field exposed

through the middle value, as << cuted off >>on . The pair: bare electron-virtual electron presents

the boson which according to the rules of quantum mechanics applies Schrődinger`s equation ( -mass of the

electron at the state of rest; c- the light velocity in the vacuum; Bohr ُs magneton).We think that for the pair: bare electron-virtual electron, charged by 2e , we may describe their state at the rest, in the cilindrical coordinate eystem, by the following Schrődinger's equation

(1)

(where is effecive mass equal to ,because we consider bare electron at rest and

therefore mass of the virtual electron tends to

; )

which solution is the wave function in the following form:

(2)

as a key parameter for the calculation all physical electron׳s characteristics.Already the density of probability by itself

gives us very important parameter , and it is the volume which is encompassed

by the smeared electron in the space of the Compton׳s wavelength.It makes out .It is also

possible to determine it from the relation .The distribution of the mass and

the electric charge of electron in this field is according to the Gauss׳s law , and the size of the volume depends of

the state of the electron in the relation to the external fields, so it is changed according to the law

.Already in that relation we noted the forbiden state as evidently , because it is in collision with theory of relativity , from here we can draw the conclusion of the impossibility to obtain the zero volume for ; it means that the electron cannot be the point particle, because the point particle according to the mathematical definition of the point presents the entity out of the spatial dimension. Another important link used for description of the electron is in the application of Maxwell׳s relation for density of the electromagnetic energy on the electron.Namely, Maxwell׳

srelation for density of the electromagnetic energy is As we know E and B are

the transverzal components of the electric and magnetic field of the photon.If we include in this expression the<< cut off>> values of the magnetic field components H and the equivalent value of the magnetic induction (

; is the frequency of the oscillator , formed by electron-virtual electron system)

and so, we consequently get the density of substance building up electron. It makes out .In

the arbital state, while electron is moved under the influence of the external field , it is possible to demonstrate that

the density of the electron substance is changed consequently to the law: .If this density is

multiplied by the volume of the smeared electron in that condition we obtain:

, which is well -known Einstein׳s of changing the electron׳s

proper energy during its movement . If the electron were the point particle result of the left side would be zero, which is automatically in the collision with the theory of relativity for the electron energy.. According to this , it is completely clear that the electron may not be a point particle.In this case the questions are broadened:if it is not the point particle how big are its spatial dimensions, and if it could have the spatial dimensions , consequently it could be one complex object.Our aim is to calculate, by using function (2), physical characteristic of the bare electron.First of all, we find rotor of the magnetic field pair.

(3)

.It is important to see, if we compute the effective value of the electrical current of the bare electron

(3a)

the Ampere’s law is satisfied. If we use the vector potential of the pair:bare electron-virtual electron , which rotor is equal to::

(4)

we may calculate the quantum-mechanical average value of :

(5)

Apparently, the electron's spin is equal to

(6)

and quantum of magnetic flux is:

(7)

Comment:no matter of the electron state in motion, its spin and quantum magnetic flux are constants.In the simplest and generall case, when the electron is at the arbitrary state of motion, we may calculate magnetic moment as a kinematical quantity by the following formula:

(8)

Theory of the internal structure of the electron

Before we introduce expressions for the commutative relations of the sub-particles, as the internal electron structure of the electron, we should remind what they are the own values for the spin projections on the coordinate axes using Pauli׳s matricies for the operator of the electron׳s spin

where the Pauli׳s matricies are equal:

The commutative relations are:

corresponding to the anticommutative relations

Let׳s find the solution in details for the first commutative relation in the system (3):

As it is evident we deliberately separated the Plank׳s constant on the two halves , in order to get the definite formula for commutative relations , which one because of the unknown reasons have not be realized to the end..Now, there is opportunity to write the definite formulae for commutative relations, by simple decoding of the formula (5), which, before the final generalization, is written in the more complete following: form:

where and are the projections of the spin electron on the suitable axes , because as we very well know

are the operators of the electron׳s spin projections on the axes׳s coordinates.

Now, it is possible to see the definite look of the commutative formula, so we are going to generalize on the system (3), which complete form is the following:

It could get an impression that the relations (7) are formally correct .However, the mechanism of making generation of the third operator is possible to achieve trough the spin of the electron including the spin of virtual electron

. In that way the system (7) in the definite form would be:

(7a)

The suplement of the commutative relations (3) with (7a) we had to do , because of necessity for making of the commutative relations for the sub-particles in the electron. It will be possible to see very soon that the commutative relations introduced in the form (7a) could be applied completely on the electron sub-particles.

Commutative relations of the electron sub-particles

Therefore, as we said we will apply the system of the commutative relations (7a) on the electron sub-particles, taking into consideration the suitable sub-spins of the electron and the virtual electron expressed in the absolute values, never mind of their mutual direction of the possible projections. With this observation we write the detailed commutative relations of sub-particles.Suppose that electron has three sub-particles, which are described by the sub-spins, through

the fractal values , where are the unknowns which should be determined. In

this case we will have the sub-Pauli׳s matrices

(8)

.So, the commutative relations (7a) by the fractals would be:

(8a)

Let׳s write in details the commutative relation by which two sub-particles, arbitrary choosen, with the fractals

generate the third sub-particle with fractal :

(9)

Let׳s describe the relation (9). The commutator fractal 1. (one) and the fractal 2 (two) is equal to the sum of these fractals in the electron and as the sum of the suitable fractals in the virtual electron , gives the third fractal. The similar matter for the rest of the commutators , so it is possible to present the general formula for all three cases:

As the result of the commutation two final commutators will give:

(9b)

As first we have to underline the minus symbol is taken when the commutative relations in the normal order are expressed by the Pauli׳s matrices. For example when each sub-particle is followed by Pauli׳s matrices (9a) ; in the other circumstances the symbol minus was not taken into the consideration . In the formula (7) spins each of them were the half of the Plank׳s constant, and in that way the electron is defined as the full-whole;and so the spin of the electron and the spin of the virtual electron are collected together, they had resulted as the operator of the electron spin, described by the third arbitrary taken Pauli׳s matrix. However, the internal structure of the electron and the virtual electron is composed of the fractals .We see how the operator of the fractal according to the third Pauli׳s matrix is created :on::two fractals arbitrary taken in the electron followed by the sub-spins plus two equivalent fractals of the virtual electron will “give birth“-generate the third fractal with its own spin.

The fractals׳ determination

If we apply any of the commutative relations from the system (9a) we will get the equation of the same form, which links the three unknowns:

(10)

which should be determined.The fractals, mean the parts of a single unit, so it is clear that their total sum must be equal with 1 (one):

The expression (10) we can find in the quadrat of the expression (11):

(12)

If we insert the (10) into (12), we will obtain:

(13)

Now, it is absolutally clear, that the fractals have to satisfied the conditions of the three equations in the system

(14)

The only solutions of the system (14) we find if we suppose that two of three unknowns are equal in its mutual relation. Take, for example, on the arbitrary way that the first and the second fractals are equal in the mutual relation, then system (14) becomes:

(15)

If we change from the third relation the value of the first fractal into the first one , gives us the value of the third fractal:

(16)

and the first and the second fractal, which are mutualy equal, gives:

(17)

So, according to the commutative relations we got, that the fractals of the electron could be:

(18)

Equation of motion

According to measurements by Purdue University physicists, electron may not be a simple negative point charge, but surrounding the electron`s core is a fuzzy “cloud” of virtual particles, which are in pairs.One particle in the pair is positively charged, the other negatively charged.Now it is clear why we write the impulse operator for the electron in the form

(19)

where and are appropriate spin projections of the electron and virtual electronTaking into account a hypotesis that both electron and virtual electron are made of sub-particles, we can write the impulse operators for the appropriate sub-pairs:two of them made of one sub-electron and one sub-virtual electron, and the third is made of one sub-positron and one sub-virtual positron.For such pairs we may write the following formulas for appropriate impulse operators;

Using these operators () applies Schrődinger`s equations, because one pair made of electron and one virtual electron from the „cloud“ behaves like a boson.This is reason why for sub-pairs we may write appropriate Schrődinger`s equations of motion at the state of rest.These equations read:

(21)

where is effective mass of the pairs, made of both sub-particle and sub-virtual particle, is equal to:

(22)

because we introduce the condition at the state of rest.

Especially, for each sub-pair, which is in the interaction with the electromagnetic field of the physical vacuum expressed by vecor potential ,we my write appropriate Schrődinger`s equations:

in the cilindrical coordinate system.

(24)

while energy of each pair can be computed by formulas:

(25) To determine the distribution of electron charge and mass we will use the formulae for probability current density

(27)

Considering the fact that the wave function is a real function, equation (27) may be reduced to the following form:

(28)

If we muliplay equation (28) by the sub-electron rest mass, we will obtain the mass density distribution:

(29)

The sub-spins are computed if we use the appropriate mass current densities in the following formula:

(30)

By multiplying the probability current density by appropriate electric charge of the sub-particle

(31)

we will obtain sub-electrical currents density.To calculate Dirac's magnetic moments we will use the formulas from electrodynamics:

Differential element of the effective value of the electrical sub-currents are

(33)

By integration (33) ones obtain effective values of the electrical sub-currents:

(34)

Rotor of magnetic field, directed along z-axism of the sub-pair :sub-bare electron and sub-virtual electron, is equal to:

(35)

gives the electrical current density.Integral of the differential element of magnetic field is

(36)

and rotor of the vector potential of the sub-pair is equal to the magnetic field induction:

(37)

From formula (37) we find :

(38)

Spin of the electron will be, as a result of the averaged quantum-mechanical quantity:

:

(38)

If we remind formula () we see that should to multiplay by fractal , so one obtains:

()

So, the sub-spins in explicit form are:

The quantum of magnetic flux will be:

(40)

Taking into account that sub-positron rotates in oposite to the sub-electrons, one obtains:

(41)

So, total magnetic flux on the level of the electron will be equal to the sum of the sub-fluxes:

(42)

Sub-magnetic moments apply the folloving formulas:

In accordance with (18) above, we obtain the following results representing the fractal values of the electron charge e and the pertaining masses:

(43)

From the standpoint of the energy levels, we may say that internal structure of the electron is composed of two energy

levels, namely: a higher energy level occupied by two identical sub-particles of equal masses and electric

charges, and a lower energy level occupied by the sub-positron. The difference between the two energy

levels is:

which is a value of internal rest energy of the electron.

We clearly see that the total energy of all sub-particles in the electron is equal to Einstein’s rest energy of the electron;

this is automatically obtained if each term of the first relation in the formula above is multiplied by :

and the relativistic relation for the squared energy is accounted for, as well:

which is automatically obtained by multiplying each term of the second relation in the formula above by ,

which unmistakably speaks of a unique, yet possible, solution concerning the internal redistribution of energy of the sub-particles forming internal structure of the electron.Iside of each pair of the electron-virtual electron there are the appropriate fractal parts of the Plank's constant:

(46)

and inside one electron and one virtual electron are:

(47)

The reader may easily notice that the sub-particle appears with a positive electric charge

and a negative mass, possesses a negative energy level

.

Conclusion

As we see there is not any similarity in the calculations of the electron properties based on the proposed mathematical model of the electron as a complex particle, and that ones given in today physics , and which have thought an electron as a simple point charge. The obtained results (6), (7), and (8) are in agreement with the experimental data, so in conclusion of this work, they may be considered as the proof:

a) That the electron may not be a simple negative point charge (bare electron), and we do not speak of the mathematical concept of the electron as a point particle any more, as it was defined by Standard Model

b) Electron is a spatially particle, because the results (6), (7) and (8) are obtained by the space integration.

c) Electron in reality is not one isolated-bare particle only; bare charge is always followed by virtual electron, which is the part of the virtual pair fermion- antifermion from the “cloud”.

d) Electron may not be an elementary particle, but it is complex particle; it is consisted of the bare electron and one virtual electron, where virtual electron is the part of the virtual pair fermion- antifermion.

e) We agree with the Koltick's words:”The “cloud”, consisted of virtual particles in the pairs (fermion-antifermion), surrounding the electron's core (bare electron) may be to play a key role in how we perceive the electromagnetic force from the electron” .

What conclusion may be drawn from this mathematical model of the electron implying its internal structure? First, the internal structure of the electron may be composed of three sub-particles; two are identical by mass and negative electric charge, while the third sub-particle is positively charged and has a negative mass (since the mass is always related to the energy, the concept of a negative mass should be construed conditionally; in fact, the concept of a negative mass should always be identified with the concept of a negative energy level that makes a physical sense, (but

it could be mean as an atraction link between all sub-particles), which reminds us of Dirac’s positron. Energetically speaking, the electron internal structure is made of two identical sub-particles

identical by mass and by electric charge and having the energy level . Therefore, this level is degenerated

twice. The third sub-particle - the sub-positron - occupies the energy level . How is it possible that the

electron remains stable when its internal structure is composed of the sub-particles, where one of them having opposite electric charge from the other two; Won’t one of the two identical sub-electrons annihilate with the sub-positron, thus making the electron unstable, because the annihilation process would destroy its structure!The answer is obvious: the sub-electron and the sub-positron cannot annihilate one another (the annihilation process means the integration of two particles of identical masses, but of the opposite and the identical electric charges, which leads to the transformation of their energies; they disappear, and, instead of them, the quantas of electromagnetic radiation - light are emitted in the surrounding space) due to the fact that they do not have equal masses and identical electric charges by the absolute value; the nature bans such processes as they would result in appearance of the remainders in form of electric charge and mass.

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