Политематический сетевой электронный научный журнал Кубанского государственного аграрного университета, 2012, №75


Научный журнал КубГАУ, №71(07), 2012 года

1

УДК 531.9+539.12.01                           UDC 531.9+539.12.01                                  
КВАНТОВАЯ ЭЛЕКТРОДИНАМИКА                     LORENTZ QUANTUM ELECTRODYNAMICS                      
ЛОРЕНЦА                                                                                            
Трунев Александр Петрович                     Alexander Trunev                                     
к.ф.-м.н., Ph.D.                              Cand.Phys.-Math.Sci., Ph.D.                          
Директор, A&E Trounev IT Consulting, Торонто, Director, A&E Trounev IT Consulting, Toronto,        
Канада                                        Canada                                               
Обсуждается вопрос о расширении электро-      The question of extending the Lorentz electrodynamics
динамики Лоренца до квантовой теории.         to quantum theory is discussed. The system of        
Сформулирована система уравнений квантовой    equations of the Lorentz quantum electrodynamics was 
электродинамики Лоренца                       established                                          
Ключевые слова: КВАНТОВАЯ                     Keywords: QUANTUM ELECTRODYNAMICS                    

ЭЛЕКТРОДИНАМИКА

Lorentz electrodynamics (brief overview)
      In classical electrodynamics, an electron, just like any other charge, has the electric and magnetic field. To describe self-fields of the electron Lorentz [1] using scalar and vector potentials satisfying the equations:

27^ 1 d2

        V* -  д2 ф = -P


(1)

1 д² c² д t1

V² A 
A =

                        u

                        PP c


       Here - c, p, u the speed of light, the charge density and the velocity of the electron, respectively. In the special case of motion with constant velocity vector potential is expressed through the scalar potential in the form [1]



A U Ж
A = — ф c

(2)


      Hence, the problem of determining the fields of the electron is reduced to the problem of congestion charges potential with a given density and velocity of center

http://ej.kubagro.ru/2012/01/pdf/83.pdf

Научный журнал КубГАУ, №71(07), 2012 года

2

of mass. As it know, to solve this problem, Lorenz [1] used transformation of variables (Lorentz transformation), which allows reducing the problem to the Poisson equation:
x - ut
X = I     =T
1в


        p = р¹-в’ ф = ф¹-в

(3)
   Here в = u / c ■

        д2Ф_ д2Ф_ д2ф^₌_, "\/2 + 2 +  2 P

                          дx   dу   dz


(4)



       Thus, the problem of finding the potential of moving charges reduced to the problem of finding the electrostatic potential of fixed charges with a given density. Indeed, the unknown electric and magnetic fields are determined by potential gradients, which is the solution of equation (4), we have


    дф         -1 дф

V" =⁽¹ ⁻e ) T^’ дx           дx

        дФ=(i - в²)-у² дф д z           дz

        д^=(i - e-r:ⁱ дф‘, ду          ду


    Ax = рф, Ay = Az= 0

        д-^- = -р² c дФ dt        дx


http://ej.kubagro.ru/20i2/0i/pdf/83.pdf

Научный журнал КубГАУ, №71(07), 2012 года

3

       Hence we find the electric and magnetic fields by formulas

1 dA

    E =----
    c d t


        -\ф=

    (⁽¹ ⁻ в⁾ф 1

Фу


    H = vxA = в -Фz

        к фу J


( 0 1

к

Ф
J

      The transition from the system (1) to (4) actually is a Lorentz transformation.

This transformation includes the Galileo transformation, x ^ x


ut , as well as



the similarity transformation (3), familiar from relativistic theories.
      However, the charge density remains an unknown quantity, which in Abraham theory [2] is replaced by the surface charge distribution on a rigid sphere, i.e. boundary condition. This approach allows us to solve completely the problem of the electric and magnetic field of an electron moving with constant velocity [12]. Indeed, the initial rigid sphere under the transformations (3) is transformed into an ellipsoid for which the solution of electrostatic problem is well known. Using

this solution, Abraham [2] calculated the energy and electromagnetic field of an electron in the form of

momentum of the

E = ——
8п1<в

(, 1 + в
ln——
к 1 - в

^ в
J



G = p = —e-x  8nRc0²

((ЦЛ ₁п1+в

к2

(5)

1 - в



^ в
J

       Here R - radius of the sphere.

       The limiting value of the total energy of the electron is at в ^ ⁰

http://ej.kubagro.ru/2012/01/pdf/83.pdf

Научный журнал КубГАУ, №71(07), 2012 года

4

E0 =E(0)=

e²

8nR

      Combining the first and second equation (5) we find the dispersion relation, which we write in the form f (u / c) = cG / E - see Figure 1.

Using the momentum expression (6), we can determine the longitudinal and

transverse mass, according to the formulas

mₗ

1 dG=f² -,„1±£ ’ | c dp 6nRcв ^ 1 - в² 1 - в J

mₜ

e²

6nRc² в

f .
 - 2в + (1 + в ²)ln

1+в I
1 - в J

(6)

1G c P

<

       The longitudinal mass characterizes the inertia of the body when the velocity changes in magnitude, and the transverse mass characterizes the inertia only when the velocity direction changes, for example, an electron moving in a uniform

http://ej.kubagro.ru/2012/01/pdf/83.pdf

Научный журнал КубГАУ, №71(07), 2012 года

5

magnetic field. Limit values for longitudinal and transverse mass (6) coincide
 в —— 0 ti< й
at , in this case we have

mₗ = mₜ

6nRc²

(7)

       Assuming that the mass limit (7) is equal to the rest mass, we can define the classical electron radius

Re

2
6nm ₀ c

(8)

      Kaufman [3] performed a number of experiments in which he measured deflection of beta-electrons in the electric and magnetic field. He showed that the experimental dependence of the transverse mass on the velocity corresponds well to the Abraham theory - see Figure 2. Kaufman believed that the electron mass is entirely of electromagnetic energy, as follows from expressions (5) - (6).
      Lorentz theory of electrons [1], as well as Abraham theory [2], based on the idea of the existence of a luminiferous ether - a continuous medium in which

electromagnetic waves propagate. The negative result of Michelson-Morley experiment to detect the influence of the earth's motion through the ether on the speed of light, forced to reconsider the basis of Lorentz theory of electrons [1[. As a result, Lorentz formulated a general form of the electrodynamics equations transformation, which, together with the transformations (3) include the conversion time by the formula

t'

t — fix / c
1—вг

(9)

http://ej.kubagro.ru/2012/01/pdf/83.pdf

Научный журнал КубГАУ, №71(07), 2012 года

6

      Lorenz suggested that all material bodies are experiencing reduction in the size along the direction of movement by the formula (3), which in this case can be written as

l = l' у[1-в²


(10)

      The new model is the electron conducting sphere in the coordinate system, where it rests, and in the moving frame the electron is an ellipsoid of rotation, compressed into the direction of motion according to the equation (10). Lorenz

found that, in this case, the longitudinal and transverse electron mass is converted asas follows

    m, = mo(1 - в²)m, = mo(1 - Л-¹'²


(11)

      Lorenz has shown that these equations are applicable not only to electrons and atoms, but also to any material bodies.
      Einstein [4-5] developed Lorenz ideas, basing them on the principle of relativity and the constancy of the speed of light. Once published the first paper on relativity theory [4], Kaufmann repeated his experiments to compare the theory by Abraham [2], Bucherer [6] and Lorentz-Einstein [1, 4]. Although the results of Kaufman [3] more consistent with the Abraham theory [2], nevertheless Lorenz [1] and Einstein [5] praised these experiments, never doubting their authenticity. In subsequent years, was put a lot of experiments [7-9] that confirmed the theory of Lorentz-Einstein. However, the Abraham theory was rejected as a bankrupt after the publication of data [9], obtained in the electrostatic analyzer, which contained only three points - see Figure 2.

http://ej.kubagro.ru/2o12/o1/pdf/83.pdf

Научный журнал КубГАУ, №71(07), 2012 года

7

       Figure 2: Dependence of the transverse electron mass on velocity according to the Abraham theory [2] and Lorentz-Einstein theory [1, 4], and according to experimental data [3, 7-9].

       At the time, it suggested, rather, a loss of interest in the problem, rather than the desire to reconcile the Lorentz-Einstein theory with experiment. Indeed, at the time of publication [9] in 1940 already existed Dirac relativistic quantum theory of the electron [10], and the spectrum of the hydrogen atom was explained on the basis of the nonrelativistic Schrodinger equation [11]. The special theory of relativity (STR) was accepted by leading theoretical physicists without further discussion, as the basis for the construction of elementary particle physics.
       The basis of Dirac's relativistic quantum theory [10] is STR, derived by Einstein [4], and based on the analysis of Maxwell-Hertz equations and Lorentz electrodynamics [1], in which the electron is described by equations (1). But the data in Figure 2, painstakingly collected by a whole generation of experimentalists http://ej.kubagro.ru/2012/01/pdf/83.pdf

Научный журнал КубГАУ, №71(07), 2012 года

8

to confirm Lorentz-Einstein theory, obtained by analyzing the trajectories of electrons, rather than spinors in the Dirac theory [10]. If, however, perform data analysis [3, 7-9] on the basis of the Dirac theory, we obtain the unexpected result that the scatter of the data in Fig. 2 may be associated with the excitation frequency of natural oscillations of electrons in a magnetic field [12] - Figure 3.
      In this sense, data [3, 7-9] and others can be regarded as fit to a known result, by adjusting the setting on the playback of that particular frequency of quantum oscillations. Consequently, the Dirac theory is at odds with the original hypothesis of Lorentz, so it does not allow testing this hypothesis experimentally. For example, it is impossible to distinguish the Abraham theory [2] from the Lorentz theory [1], which is equally consistent with the theory of Dirac, which, in turn, describes well the entire set of known experimental data - see Fig. 3.
      Another apparent paradox is that, apart quantum mechanics and

electrodynamics are the linear theory, which holds the principle of superposition,

whereas the quantum electrodynamics (QED), which unites the Dirac theory and

the electrodynamics, is a nonlinear theory / 13 /.

http://ej.kubagro.ru/2012/01/pdf/83.pdf

Научный журнал КубГАУ, №71(07), 2012 года

9

      Figure 3: Dependence of the electron momentum on the velocity in a homogeneous magnetic field calculated for two quantum numbers [12], and according to experimental data [3, 7].

      Thus, historically Lorentz electrodynamics provided the basis for the withdrawal of the Lorentz transformation, based on which, in turn, there was the Dirac relativistic quantum theory [10]. Consequently, these theories must be linked, as describe one particle - an electron.

Mills theory and its generalization
      For paradoxes described above was found an unexpected solution in the Mills theory [14]. Mills main hypothesis [14] is that the electric charge density in the right-hand side of equations (1) is described by the wave equation of the form
V7 2       1 d ²       _ ₙ
v p - V ъёp = 0                                  ⁽¹²⁾
      Here - v speed charge-density waves.
      The hypothesis (12) together with the assumption of the charge distribution in the atom on the surface of the sphere of fixed radius, determined from Bohr’s theory, allows one to calculate the energy levels of multielectron atoms and to determine the anomalous magnetic moment of the electron [14]. In this sense, the Mills theory is an alternative to quantum electrodynamics (QED), as it allows the calculation of all relativistic quantum effects with the same accuracy as QED, but it is a linear theory.
      The basic equation (12) of Mills theory, which he used to solve the problems of atomic physics, chemistry, and physics of elementary particles, at first, not directly related to quantum mechanics. Moreover, Mills asserts that quantum mechanics is a profoundly erroneous theory that contains internal logical http://ej.kubagro.ru/2012/01/pdf/83.pdf

Научный журнал КубГАУ, №71(07), 2012 года

10

contradictions. In reality, however, it is easy to show using the results of our paper [12] that the equation (2) can be obtained from the relativistic quantum Dirac equation - see [15]. Indeed, as shown in [12], the charge density and potential of an electron subject to both of the Dirac equation and the Lorenz equations (1) are related by:
p=[- hes⁽фE -c(Ap))+1^ф - A²)⁻m¥~ ф              (13)
^ H C               H C            H J


~                                         ~~
     It is indicated E,p is the energy and momentum of electron, ф, A - the scalar and vector potential of the external electromagnetic field. Hence we find that in the absence of external fields, the charge density depends linearly on the potential of the electron:

p=

                              22 m₀c — ф

(14)

      Substituting this expression into the first equation (1), we finally obtain


1 d²
⁻ T^Tp p = c d t

2 2
m 0 c —p

v² p

(15)


       Mills used his basic equation (12) only to find the singular solutions in which the density distribution along the radial coordinate is described by the Dirac delta function. On such solutions eq. (15) takes the Mills form (12)
                        V7 2        1 d ²        _
                        v p - wp =⁰                                                (16)

      We give a simple derivation of equation (13), based on a special recording of the Dirac equation in the form of second-order equation / 13 /

http://ej.kubagro.ru/2012/01/pdf/83.pdf