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Quantum mechanics as electrodynamics of curvilinear waves

The suggested theory is the new quantum mechanics (QM) interpretation.The research proves that QM represents the electrodynamics of the curvilinear closed (non-linear) waves. It is entirely according to the modern interpretation and explains the particularities and the results of the quantum field theory.


1.INTRODUCTION
The attempts to explain the quantum theory and the particle structure were already undertaken in the last century (Lord Kelvin, H.A.Lorentz, Lois de Broglie et.al.) and are continued until today [1].
In the following article we show that the quantum mechanics is mathematically isomorphic to the electrodynamics of the curvilinear wave. The basis and the results of the theory consist in the following: 1) It is accepted as a consequence of Maxwell's theory and the uncertainty relation of Heisenberg that the photon represents a classical relativistic string, which has one wave length. 2) It is accepted as a hypothesis that the reason for the formation of the particles is the movement of a photon along the closed curvilinear trajectory.
3) It is shown that the electron and the positron appears as the consequence of the division of the twirled photon into two completely antisymmetric twirled semi-photons. 4) It is shown that to this corresponds the decomposition of the wave equation of a photon into two Dirac's equations about the electron and the positron. 5) It is shown that the reason of the occurrence of the currents (charges) particles is the movement of the electric vector of a photon on a curvilinear trajectory (that corresponds to the occurrence of the additional members in the linear equations). 6) It is shown that the equations of the particle structure are nonlinear and this nonlinearity is of the same type, as in the equations of the theory of Standard Model. 7) It is shown that the twirled string (photon) has an integer spin (i.e. it is a boson), and the twirled semi-photon has a halfinteger spin (i.e. it is a fermion). 8) It is shown that all parities of QM correspond precisely to the electrodynamics of the curvilinear waves and have formally the same kind, as in Maxwell's theory. 9) It is also shown that all the postulates of QM (for example, the statistical interpretation of the wave function, dualism a waveparticle, a principle of conformity, sense of the Dirac's matrix representation, and so forth), and also all the mathematical particularities of Dirac's theory follow from the electromagnetic form of the theory, as consequences. Thus, the below research proves that the QM represents the electrodynamics of the curvilinear closed (non-linear) waves.
The article contains two parts: A. the quasi-classical theory, and B. the exact theory.
A. QUASI-CLASSICAL THEORY. In the world of the elementary particles the following approximate electron model theory is the same as the hydrogen planetary model theory in the atom theory.

BASIC HYPOTHESES
These hypotheses don't conflict with the modern theory.

Particles production hypotheses
Let's consider the particle-antiparticle production conditions. One γ -quantum cannot turn spontaneously into the electronpositron pair, although it interacts with the electron-positron vacuum. For the pair production, at first, the following mass correlation is necessary: 2 2 c m e p ≥ ε (where p ε is the photon energy, e m is the electron mass and c is the light velocity). At second, the presence of the other particle, having the electromagnetic field, is needed. For example, we have the typical reaction: , which means that, while moving through the nucleus field N the photon takes some transformation, which corresponds to the pair production. Considering the fact that Pauli's matrices describe the vector space rotations and taking in account the optical-mechanical analogy analysis also, we can assume that the above transformation is a field distortion. From this follows the distortion hypothesis: By the fulfilment of the pair production conditions, the distortion of the electromagnetic field of photon can take place; as result photon is able to move along the closed trajectory, making some stable construction named elementary particle.
(The necessity to introduce some other hypothesis, division hypothesis, will be understood from the content of the following chapters).

ELECTROMAGNETIC MODEL OF ELECTRON
According to the modern experiments and ideas, the leptons don't have a structure. That's the reason why it is said that the leptons are point particles. But as it's known, it means only that the electrodynamics is right on any short distance, by modern experiments until 2x10 -18 m. So, this fact not only does contradict to our supposition about electron, «created» from the electromagnetic field, but also verifies it.

STRUCTURE AND PARAMETERS OF THE ELECTRON MODEL
According to the hypotheses the electron must be presented as the torus. It's obvious that the torus radius is equal to π λ 2 p t r = , where p λ is the photon's wavelength.
In our case the photon characteristics are defined by the electron-positron pair production conditions: by the photon energy Let us calculate this current for the wave, moving along the ring. The electrical field vector " E (let it have direction from the centre), which moves along the distortion trajectory, can be written in form: " " E E n = − ⋅ , (3.2) where E E " = is the absolute value and " n is the normal unit-vectors.
Derivative of " E with respect to t can be represented as: Here the first term has the same direction, as " E . The existence of the second term shows that at the wave distortion the supplementary displacement current appears. It is not difficult to show that it has a direction, tangent to the ring: where " τ is the tangential unit-vectors, c p = υ is the photon velocity, t p r r ≡ is the radius of the twirled photon (i.e torus). Then the displacement current of the ring wave can be written: The currents " j n and " j τ are always mutually perpendicular, so that we can write in complex form: where i = −1 . As we see, the appearance of an imaginary unity in the quantum mechanics is tied with the tangent current.
The same results can be obtained by using the general methods of the distortion field investigation (see below chapter 4.2.).

CHARGE APPEARANCE AND DIVISION HYPOTHESIS
It's not difficult to calculate now the charge density of the twirled photon: The full charge of the particle (i.e. of the twirled photon) can be defined by integrating along all the torus volume It's easy to understand these results: because the ring current changes its direction every half-period, the full charge is equal to zero.
Therefore, the above model may represent only the non-charged particle. It's clear that a charge particle, must contain only one half-period of wave.
For the solution of this problem we suggest the following division hypothesis: at the moment when the photon begins to roll up, the spontaneous photon division in two half-periods is possible (see fig.2). We can suppose, that the cause of a photon's division is the mutual repulsion of oppositely directed currents.
It's also clear that the division process corresponds to the process of the particle-antiparticle pair production. It's proved out by the fact, that the daughter's twirled semi-photons B and C are completely anti-symmetric and can't be transformed one to each other by any transformation of co-ordinates (if it is not to be accompanied by the changing of field direction).
It is interesting that both the twirled semi-photon and the twirled photon radii must be the same. This fact follows from the momentum conservation low. Really, the twirled photon momentum is equal to: In accordance with the momentum conservation low: . From this it also follows that the volumes and areas of the twirled photon and the semi-photon will be the same: 2) the difference between positive and negative charges: this difference follows from the difference of the field and tangent current direction of the twirled semi-photons after pair production (by condition that the Pauly -principle is true).
3) Zitterbewegung. The results obtained by E.Schroedinger in his well-known articles about the relativistic electron [3] are the most important confirmation for the electron structure model. He showed, that electron has a special inner motion "Zitterbewegung", which has , and velocity of light υ = c .
The attempts to explain this motion had not given results (see e.g. [4,5]). But if the electron is the twirled semi-photon, then we receive the simple explanation of Schroedinger's analysis. 4) the difference between bosons and fermions is well explained: the bosons contain the even number and the fermions contain the odd number of the twirled semi-photons. 5) Infinite problems: in the above theory the charge and mass infinite problems don't exist.

CHARGE AND FIELD MASS OF TWIRLED SEMI-PHOTON
We can now calculate the semi-photon charge: , we obtain: To calculate the mass we must calculate first the energy density of the electromagnetic field:   As long as in our calculations we didn't take into account the physical vacuum polarization, it would be right to consider the s r as the "bare" electron radious. Then we can suppose that o r is the real electron radius.
Although the constant α q has the indeterminate value ζ , its calculation should be understood as a theory success. The formula (3.25) shows that in our theory the electric charge is defined only by the universe constants; that means that there are no free charges less than this one. At the same time, our theory doesn't limit the mass value of twirled semi-photon; this fact is also according to the experimental data.

STABILITY OF THE ELECTRON MODEL
This problem is very important for the existence of our model. In the classical electron theory the forces, which can keep the parts of charge together, had not been found. In our theory such forces exist. The magnetic component of Lorentz's force, which acts in opposite direction to the Coulomb's electric force, appears in the twirled semiphoton.
In fact, as result of the current " j τ and magnetic field " H interaction, the magnetic force density appears: It is difficult to say something more about the electron model field distribution here (note only that the twirled semi-photon fields are like the tokamak fields). In this area there is a number of achievements both old authors and new (see e.g. [6,7]).
Here we shall be satisfied by the fact that the electron stability is the consequence of the exact theory (see below).

SPIN OF MODEL
Using the data of the torus model we can calculate the spin of the twirled semi-photon: (3.28)

MAGNETIC MOMENT OF MODEL
Magnetic moment accordingly with definition is: where I is electron ring current and S I is the current ring square.
In our case we have: (3.31) Using these formulae, we find: If we put e q = , the value (3.32) is equal to half of the experimental value of the magnetic momentum of the electron. Taking into account Thomas's precision we obtain the experimental value of the electron magnetic momentum.
B. EXACT THEORY. It is entirely according to the modern theory.

ELECTRODYNEMICS' FORM OF DIRAC'S EQUATION
The twirled semi-photons, as fermions, must be defined by the spinors, which have other transformation properties than the vectors of Maxwell's electromagnetic field. On the other hand the theory of elementary particles, describing the non-linear wave, must be nonlinear.
Therefore, the Dirac equations must be the limit of the non-linear equations and some modification of Maxwell's equations. Let us prove it.
Consider the linear photon, moving along y -axis. In a more general case it has the two possible polarisations and contains the field vectors . Such photon can form the ring only on the plain. The bispinor form of Dirac's equations can be written as one equation [8]: where " $ , $ α β -are Dirac's matrices by [8]: are the operators of energy and momentum, ψ is the wave function named bispinor, which has four component. Put the following semi-photon bispinor: Taking into account that ( ) ψ ψ = y , from (4.1) we obtain: Eqs. (4.3) are the Maxwell equations with the imaginary currents. It is interesting that together with the electrical current the magnetic current also exists here.
This current is equal to zero by Maxwell's theory, but its existence by Dirac doesn't contradict to the quantum theory. (According to the full theory the magnetic current appearance is related to the origin photon circular polarization).
So, our first suppositions are right. Now we must show that all the quantum mechanics can have the electrodynamics' form.
Note also that the formal matrix form of the Maxwell's equations, known for a long time, acquires in our theory a physical sense.

ELECTRODYNAMICS' AND QUANTUM FORM OF ELECTRON-POSITRON PAIR PRODUCTION THEORY
Using Eq. (4.2), we can write the equation of the electromagnetic wave moved along the y -axis in form: The Eq. (4.4) can also be written in the following form: In fact, taking into account that , we see that Eqs.(4.4) and (4.5) are equivalent. Factorising Eq. (4.5) and multiplying it from left on Hermithianconjugate function ψ + we get: The Eq. (4.6) may be disintegrated on two equations: It is not difficult to show (using Eq. (4.2)) that the Eqs. In accordance with our assumption, the reason for current appearance is the electromagnetic wave motion along a curvilinear trajectory. We showed the current appearance in the vector form (see chapter 3.2.). The same results can be obtained, using the general methods of the distortion field investigation [9]. The question is about the tangent space introduction at every point of the curvilinear space, in which the orthogonal axis system moves. This corresponds to the fact, that the wave motion along a circular trajectory is accompanied by the motion of the rectangular basis, built on vectors ( is the Pointing vector.
For the generalisation of Dirac,s equation in Riemann's geometry [9] it is necessary to replace the usual derivative with the covariant derivative: are the summing indexes), where ν Γ is the analogue of Christoffel's symbols. When a spinor moves along the line, all 0 = Γ ν , and we have a usual derivative. If a spinor moves along the curvilinear trajectory, then not all the ν Γ are equal to zero and a supplementary term appears. Typically, the last one is not the derivative, but is equal to the product of the spinor itself with some coefficient ν Γ .
Thus we can assume that the supplementary term a longitudinal field is, i.e. it is a current.
As the increment in spinor ν Γ is a 4-vector and has here the energy-momentum dimension [9] it is logical (taking into account the above results) to identify ν Γ with 4-vector of energy-momentum of the own electron field: (4.9) Then Eqs. (4.7) and (4.8) in the curvilinear space will be have a view: According to the energy conservation law we can write: (4.12) Substituting Eq.(4.12) in Eqs.(4.10) and (4.11) we will arrive at the usual kind of Dirac's equation with the mass: (4.14) Therefore, our supposition is correct: the electron is the twirled semiphoton.
Note that the alternative set of the electromagnetic field vectors corresponds to the alternative set of Dirac's matrices.
Note also: because is true by any transformation, there is Dirac's electron equation property that the bispinor has only four components.

ELECTRODYNAMICS' SENSE OF BILINEAR FORMS AND STATISTICAL INTERPRETATION OF THE QUANTUM MECHANICS
It is known that there are 16 Dirac's matrices of 4x4 dimensions.
(We use the set of matrices by [8], which we will name α -set).
Enumerate main Dirac's matrices: where 1) is a scalar, 2) is a 4-vector, 3) is a pseudoscalar. Using (where ( + ) is the Hermithian conjugation sign) it is easy to obtain the electrodynamics' expression for the bilinear forms of these matrices: is the probability density, and is probability-flux density. Using the above results we can obtain: is Pointing's vector. Taking in account the above results from (4.15-18) we get the equation: which is the energy (mass) conservation law of the electron electromagnetic field. Therefore, the Eq. It is not difficult to understand the method of the quantum values calculations as the operators' eigenvalues: if the elementary particles are the twirled electromagnetic waves, their interaction leads to the waves interference. In such case the calculation of probability density is the calculation of the energy maximums of this interference. As it is well known, in classical physics such calculation is the eigenvalue problem.

LORENTZ'S FORCE
The expression of Lorentz's force by the energy-momentum tensor of electromagnetic field τ µ ν is well known [2]: This tensor is symmetrical and has the following components:  Using (4.18) it can be written:  As we see the Lorenz's force direction does not coincide with the y -axis, but is perpendicular to it.

THE ELECTRODYNAMICS' FORM OF ELECTRON THEORY LANGRANGIAN
As it is known [2,9,10], the Lagrangian of Maxwell's theory on the free field is the first invariant of the theory 1 Transfer the each term of (4.28) in electrodynamics' form we obtain for the twirled semi-photon Lagrangian: where ω e mc = 2 2 ! ;(note that we must differ the complex conjugate field vectors " E * and " E , " H * and " H ).
The Eq.  we take: which is the energy-momentum conservation law equation with current. According to (4.29) for the own electron electromagnetic wave the kind of Maxwell's equation Lagrangian exists, which differs from (4.26): (4.33)

ELECTROMAGNETIC FORM OF THE FREE ELECTRON EQUATION SOLUTION
In accordance with above results the electromagnetic form of the solution of the Dirac's free electron equation must be the twirled electromagnetic wave. From the electron model point of view for the y -direction photon two solutions must exist: 1) for the wave, rotating around the OZ -axis      At third, it is not difficult to calculate the correlation between components of the field vectors, putting ε + = mc 2 and φ π = 2 And also for ε − = −mc 2 :  Father, from solution we can see that the magnetic field amplitude is two times less, than the electric field amplitude.

NON-LINEAR EQUATIONS OF TWIRLED SEMI-PHOTON
The stability of twirled photon is possible only by the photons parts self-action. Obviously, the equation that describes the twirled semi-photon structure must be non-linear and like the equation, which describes photon-photon interaction [9,10].
We could introduce the self-field interaction to the twirled semi-photon equation like the external field is introduced to the quantum [9,10] and classical [11] field equations (putting the photon mass equal to zero). But this equation may be obtained more rigorously. For the electron the energy conservation equation is true: where e m is the electron mass and ε and " p are not operators in this case, but are the physical value of energy and momentum. The linear conservation law can be written then as: The relation (5.2) is rigorously, like as (5.1), since the second degree of (5.2) give the (5.1). According to relationship (5.2) for the twirled semi-photon we can find: 2) in Dirac's electron equation we obtain non-linear integral-differential equation: which is, as we propose, the common form of the twirled semi-photon equation.
In the electromagnetic form we have: where ∆τ is the semi-photon volume, which according to our model is equal approximate: It is not difficult to see that Eq. (5.12) is a non-linear equation of the same type, which was investigated by Heisenberg e.al. [9,12] and which played for a while the role of the unitary field theory equation. Moreover, Eq. (5.12) is obtained logically rigorously, contrary to the last one. Self-action constant s r [9] appears in Eq. (5.12) automatically and expresses the radius of the «bare» electron.

THE EQUATION OF TWIRLED SEMI-PHOTON MATTER MOTION
As it is known [8,10], the motion equations can be found from the next operator equation: where $ L is the physical value operator, whose variation we want to find, and $ H is the Hamilton operator of Dirac's equation. The Hamilton operator of the electron equation is equal: where P " is the full momentum of electron.
Let us s P " be the full momentum of the twirled semi-photon. For -velocity of the semiphoton (electron) matter, we obtain: Let us analyse the physical means of Eq. (5.18), considering the motion equation of the ideal liquid in form of Lamb-Gromeka's equation [13]. In case the external forces are absent, this equation is: L r m f a " " = ρ , (5.23) As a consequence of this analysis we can say that the twirled photons is like the vortex rings in ideal liquid.

LAGRANGIAN OF NON-LINEAR TWIRLED SEMI-PHOTON EQUATION
The Lagrangian of the non-linear equation is not difficult to obtain from Lagrangian of the linear Dirac's equation: using the method by which we find the first degree non-linear equation.
Using (5.10) and (5.11) we represent (5.25) in explicit quantum form: Normalising ψ -function we obtain: Here accordingly (4.29) the first summand may be replaced through ( ) As we see, the Lagrangian of the non-linear twirled semi-photon equation contains only the invariant of Maxwell's theory. Accordingly to our conception the Eq. (5.31) is Lagrangian of the semi-photon itself interaction. In quantum mechanics [9,10] is known the Lagrangian of photon-photon interaction: ]...,  [14], which is the Lagrangian of the relativistic superconductivity theory and gives the solution to the problem of the appearance of the elementary particles mass by the mechanism of the vacuum symmetry spontaneous breakdown. Note that in our theory, the breakdown of symmetry also takes place when a mass of particles appears (it corresponds to the Cooper's pair decay on electron and «hole» in the superconductivity theory).

PHSYCAL SENSE OF THE PARTICULARITIES OF DIRAC'S EQUATION
The Dirac equation has a lot of particularities. In modern interpretation these particularities are considered as entirely mathematical features that don't have a physical meaning. For example, according to [15]  The mathematical properties of Dirac's matrices are well known: they are anticommutative and hermitian; they build a group of 16 matrices; the bilinear forms of these matrices have definite transformation properties, which correspond to the vector properties of the (e.g.) electrodynamics; repeatedly it was pointed that in the classical physics these matrices describe the rotations, etc. [16].
We have shown above that the Dirac matrices describe the electromagnetic vectors moving along the curvilinear trajectories (not vectors in the curvilinear space!). From this suggestion follows the interpretation of all matrix properties.
Below we will consider how the Dirac matrices are joined with the electromagnetic Dirac's equation forms. We will show that all the electron Dirac's equation particularities have the exact electrodynamics meaning. We will also answer to the question, why the "choice of Dirac's matrices" exists, but the results "don't depend on the special choice". Here we will also analyse the physical sense of the different form of Dirac's equation in regard to various Dirac's equation matrix representation.

PYSICAL SENSE OF THE DIFFERENT FORMS OF DIRAC'S EQUATION
As it is known, there are two Dirac's equation forms: which correspond to the two signs of the relativistic energy expression: Also, as it is known that, for each sign of equation (6.3) there are two hermitian-conjugate Dirac's equations.
Here we consider the physical meaning of all these equations. Let us consider at first two hermitian-conjugate equations, corresponding to the minus sign in (6. Using (4.2) from (6.4) and (6.5) we obtain: As we see, the equation (6.8) and (6.9) are the Maxwell equations with complex currents that we name the twirled semi-photon equations. As we see the equations (6.8) and (6.9) are differed by the current directions. We could foresee this result before the calculations, since the functions + ψ and ψ are differed by the argument signs: Let us compare now the equations corresponding to the both plus and minus signs of (6.3). For the plus sign of (2.2) we have following two equations: Obviously, the electromagnetic form of equation (6.11) will have the opposite signs of the currents compared to (6.12). Comparing (6.12) and (6.8) we see that the equation (6.12) can be considered as Maxwell's equation in the left co-ordinate system. So as not to use the left co-ordinate system, together with the wave function of the electron ele ψ we can consider the wave function of positron in the form, which correspond to the right co-ordinate system: Then, contrary to the system (6.12) we get the system (6.9). The transformation of the function ele ψ to the function pos ψ is named the charge conjugation operation.
Note that the electron and positron wave functions can be considered as the retarded and advanced waves. So the above result links also with the theory of advanced waves of Wheeler and Feynman [17].

PHYSICAL SENSE OF THE MATRIX CHOICE
Now we will consider the bispinor form of Dirac's equation from the point of view that it describes the twirled object.
As we saw above this matrix sequence ) , , (  The question arises how to describe the photons having initially x and z -directions? It is not difficult to see [15] that the matrices' sequence is not determined by the some special requirements. In fact, this matrices' sequence can be changed without breaking some quantum electrodynamics results. So we can write three groups of matrices, each of which corresponds to the one and only one direction, introducing the axes' indexes, which indicate the photon direction: Let us choose now the wave function forms, which give the correct Maxwell equations. We will take as initial the form for the ydirection, which we already used, and from them, by means of the indexes' transposition around the circle (see fig.4) we will get other forms for x and y -directions. Since in this case the Poynting vector has the minus sign, we can suppose that the transposition takes place counterclockwise.
Let us check the Poynting vector values: (6.14) As we see, we took the correct result: by the counterclockwise indexes' transposition the wave functions describe the photons, which are moved in negative directions of the corresponding co-ordinate axes.
We may hope that by the clockwise indexes' transposition, the wave functions will describe the photons, which are moved in positive directions of co-ordinate axes. Prove this:  In the same way for all the other Dirac's equation forms the analogue results can be obtained.
Obviously, it is possible via canonical transformations to choose the Dirac matrices so that the initial photon could have not only x , y or z directions, but any other direction.

THE ELECTRODYNAMICS SENSE OF CANONICAL TRANSFORMATIONS
The choice of the matrixes α , made by us, is not unique. As it is known there is the canonical transformation of a kind  It is not difficult to check, that all these relations have a certain sense in the electromagnetic theory. Using (6.6) and (6.26) we have for ' ψ -functions the following relationships: