SLIGHTLY GENERALIZED MAXWELL CLASSICAL ELECTRODYNAMICS CAN BE APPLIED TO INNERATOMIC PHENOMENA V.M.Simulik and I.Yu.Krivsky Institute of Electron Physics, Ukrainian National Academy of Sciences, 21 Universitetska Str., 88000 Uzhgorod, Ukraine e-mail: [email protected] 2 Abstract 0 0 In order to extend the limits of classical theory application in the microworld some 2 weakgeneralization ofMaxwellelectrodynamicsissuggested. Itisshownthatweaklygen- n eralized classical Maxwell electrodynamics can describe the intraatomic phenomena with a J thesamesuccess as relativistic quantummechanics can do. Group-theoretical groundsfor 1 the description of fermionic states by bosonic system are presented briefly. The advan- 2 tages of generalized electrodynamics in intraatomic region in comparison with standard Maxwell electrodynamics are demonstrated on testing example of hydrogen atom. We are 1 v able to obtain some results which are impossible in the framework of standard Maxwell 0 electrodynamics. The Sommerfeld - Dirac formula for the fine structure of the hydrogen 6 1 atom spectrum is obtained on the basis of such Maxwell equations without appealing to 1 the Dirac equation. The Bohr postulates and the Lamb shift are proved to be the con- 0 sequences of the equations under consideration. The relationship of the new model with 2 0 theDirac theory is investigated. Possible directions of unification of such electrodynamics / with gravity are mentioned. h t - p e 1 Introduction h : v There is no doubt that the Maxwell classical electrodynamics of macroworld (without any gen- i X eralization) is sufficient for the description of electrodynamical phenomena in macro region. On r a the other hand it is well known that for micro phenomena (inneratomic region) the classical Maxwell electrodynamics (as well as the classical mechanics) cannot work and must be replaced by quantum theory. Trying to extend the limits of classical electrodynamics application to the intraatomic region we came to the conclusion that it is possible by means of generalization of standard Maxwell classical electrodynamics in the direction of the extencion of its symmetry. We also use the relationships between the Dirac and Maxwell equations for these purposes. Furthermore, the relationships between relativistic quantum mechanics and classical micro- scopical electrodynamics of media are investigated. Such relationships are considered here not only from the mathematical point of view - they are used for construction of fundamentals of a non-quantum-mechanical model of microworld. Ournon-quantum-mechanical modelofmicroworldisamodelofatomonthebasisofslightly generalized Maxwell’s equations, i. e. in the framework of moderately extended classical microscopical electrodynamics of media. This model is free from probability interpretation and can explain many intraatomic phenomena by means of classical physics. Despite the fact that we construct the classical model, for the purposes of such construction we use essentially the analogy with the Dirac equation and the results which were achieved on the basis of this equation. Note also that electrodynamics is considered here in the terms of field strengths without appealing to the vector potentials as the primary (input) variables of the theory. 1 The first step in our consideration is the unitary relationship (and wide range analogy) between the Dirac equation and slightly generalized Maxwell equations [1]. Our second step is the symmetry principle. On the basis of this principle we introduced in [2] the most symmetrical form of generalized Maxwell equations which now can describe both bosons and fermions because they have (see [2]) both spin 1 and spin 1/2 symmetries. On the other hand, namely these equations are unitarily connected with the Dirac equation. So, we have one more important argument to suggest these equations in order to describe intraatomic phenomena, i. e. to be the equations of specific intraatomic classical electrodynamics. In our third step we refer to Sallhofer, who suggested in [3] the possibility of introduction of interaction with external field as the interaction with specific media (a new way of introduction of the interaction into the field equations). Nevertheless, our model of atom (and of electron) [1] is essentially different from the Sallhofer’s one. On the basis of these three main ideas we are able to postulate the slightly generalized Maxwell equations as the equations for intraatomic classical electrodynamics which may work in atomic, nuclear and particle physics on the same level of success as the Dirac equation can do. Below we illustrate it considering hydrogen atom within the classical model. The interest to the problem of relationship between the Dirac and Maxwell equations dates back to the time of creation of quantum mechanics [4]. But the authors of these papers dur- ing long time considered only the most simple example of free and massless Dirac equation. The interest to this relationship has grown in recent years due to the results [3], where the investigations of the case m = 0 and the interaction potential Φ = 0 were started. Another 0 6 6 approach was developed in [5], where the quadratic relations between the fermionic and bosonic amplitudes were found and used. In our above mentioned papers [1, 2], in publications [6] and herein we consider the linear relations between the fermionic and bosonic amplitudes. In [6] we have found the relationship between the symmetry properties of the Dirac and Maxwell equa- tions, the complete set of 8 transformations linking these equations, the relationship between the conservation laws for the electromagnetic and spinor fields, the relationship between the Lagrangians for these fields. Here we summarize our previous results and give some new details of the intraatomic electrodynamics and its application to the hydrogen atom. The possibilities of unification with gravitation are briefly discussed. 2 New classical electrodynamical hydrogen atom model Consider the slightly generalized Maxwell equations in a medium with specific form of sources: curl−→H ∂ ǫ−→E = −→j , curl−→E +∂ µ−→H = −→j , − 0 e 0 mag (1) divǫ−→E = ρ , divµ−→H = ρ e mag, where −→E and −→H are the electromagnetic field strengths, ǫ and µ are the electric and magnetic permeabilities of the medium being the same as in the electrodynamical hydrogen atom model of H. Sallhofer [3]: Φ(x)+m Φ(x) m −→ 0 −→ 0 ǫ(x) = 1 , µ(x) = 1 − (2) −→ −→ − ω − ω where Φ(x) = Ze2/r (we use the units: h¯ = c = 1, transition to standard system is fulfilled −→ − by the substitution ω h¯ω, m m c2). The current and charge densities in equations 0 0 −→ −→ (1) have the form 2 −→je = gradE0, −→j mag = −gradH0, (3) ρ = ǫµ∂ E0 +−→Egradǫ, ρ = ǫµ∂ H0 +−→Hgradµ, e 0 mag 0 − − where E0,H0 is the pair of functions (two real scalar fields) generating the densities of gradient- like sources. One can easily see that equations (1) are not ordinary electrodynamical equations known from the Maxwell theory. These equations have the additional terms which can be considered as the magnetic current and charge densities - in one possible interpretation, or equations (1) can be considered as the equations for compound system of electromagnetic −(E→, −H→) and scalar E0,H0 fields in another possible interpretation. The reasons of our slight generalization of the classical Maxwell electrodynamics are the following. 1. The standard Maxwell electrodynamics cannot work in intraatomic region and its equa- tions are not mathematically equivalent to any of quantum mechanical equations for electron (Schrodinger equation, Dirac equation, etc...) 2. The existence of direct relationship between the equations (1) and the Dirac equation for the massive particle in external electromagnetic field in the stationary case can be applied. Namely these equations were shown in papers [1] to be unitary equivalent with such Dirac equation (see also Sec. 3 below). 3. Equations (1) can be derived from the principle of maximally possible symmetry - these equations have both spin 1 and spin 1/2 Poincar´e symmetries and in the limit of vanishing of the interaction with medium, where ǫ = µ = 1, they represent [2] the maximally symmetrical form of the Maxwell equations. This fact means first of all that from the group-theoretical point of view of Wigner, Bargmann - Wigner (and of modern field theory in general) Eqs. (1) can describe both bosons and fermions (for more details see Sec. 4. below). As a consequence of this fact one can use these equations particularly for the description of the electron. On the other hand, this fact means that intraatomic classical electrodynamics of electron needs further (relatively to that having been done by Maxwell) symmetrization of Weber - Faraday equations of classical electromagnetic theory which leads to the maximally symmetrical form (1). Below we demonstrate the possibilities of the equations (1) in the description of testing example of hydrogen atom. Contrary to[1], herethe equations (1)aresolved directly by means ofseparationof variables method. It is useful to rewrite these equations in the mathematically equivalent form where the sources are maximally simple: curl−→H ǫ∂ −→E = −→j , curl−→E +µ∂ −→H = −→j , − 0 e 0 mag (4) div−→E =ρ∼e, div−→H =ρm∼ag, where −→je = gradE0, −→j mag = −gradH0, ρ∼e= −µ∂0E0, ρm∼ag= −ǫ∂0H0. (5) Consider thestationary solutions of equations (4). Assuming the harmonictime dependence for the functions E0,H0 E0(t, x) = E0(x)cosωt+E0(x)sinωt, →− A −→ B −→ (6) H0(t, x) = H0(x)cosωt+H0(x)sinωt, −→ A −→ B −→ we are looking for the solutions of equations (4) in the form 3 −→E(t, x) = −→E (x)cosωt+−→E (x)sinωt, −→ A −→ B −→ (7) −→H(t,−→x) = −→HA(−→x)cosωt+−→HB(−→x)sinωt. For the 16 time-independent amplitudes we obtain the following two nonlinked subsystems curl−H→ ωǫ−E→ = gradE0, curl−E→ ωµ−H→ = gradH0, A − B A B − A − B (8) div−E→ = ωµE0, div−→H = ωǫH0, B A A − B curl−H→+ωǫ−E→ = gradE0, curl−E→+ωµ−H→ = gradH0, B A B A B − A (9) div−E→ = ωµE0, div−→H = ωǫH0. A − B B A Below we consider only the first subsystem (8). It is quite enough because the subsystems (8) and (9) are connected with transformations E H, H E, ǫE µH, µH ǫE, −→ −→ − −→ −→ − (10) ǫ µ, µ ǫ, −→ −→ which are the generalizations of duality transformation of free electromagnetic field. Due to this fact the solutions of subsystem (9) can be easily obtained from the solutions of subsystem (8). Furthermore, it is useful to separate equations (8) into the following subsystems: ωǫE3 ∂ H2 +∂ H1 +∂ E0 = 0, B − 1 A 2 A 3 A ωǫH0 +∂ H1 +∂ H2 +∂ H3 = 0, B 1 A 2 A 3 A (11) ωµE0 +∂ E1 +∂ E2 +∂ E3 = 0, − A 1 B 2 B 3 B ωµH3 ∂ E2 +∂ E1 ∂ H0 = 0, A − 1 B 2 B − 3 B and ωǫE1 ∂ H3 +∂ H2 +∂ E0 = 0, B − 2 A 3 A 1 A ωǫE2 ∂ H1 +∂ H3 +∂ E0 = 0, B − 3 A 1 A 2 A (12) ωµH1 ∂ E3 +∂ E2 ∂ H0 = 0, A − 2 B 3 B − 1 B ωµH2 ∂ E1 +∂ E3 ∂ H0 = 0. A − 3 B 1 B − 2 B Assuming the spherical symmetry case, when Φ(x) = Φ(r), r x , we are making the −→ −→ ≡ | | transition into the spherical coordinate system and looking for the solutions in the spherical coordinates in the form (E,H)( r ) = R (r)f (θ,φ), (13) −→ (E,H) (E,H) where E E0,→−E , H H0,−→H . We choose for the subsystem (11) the d’Alembert Ansatz ≡ ≡ in the form(cid:16) (cid:17) (cid:16) (cid:17) − − E−0=C− R Pm4e im4φ, A E4 H4 lH4 − − − E−Bk=C−Ek REkPlmEkke−imkφ, (14) − − H−0=C− R Pm4e im4φ, k = 1,2,3. B H4 E4 lE4 − H−Ak=C−Hk RHkPlmH−kke−im−kφ, 4 We use the following representation for ∂ ,∂ ,∂ operators in spherical coordinates 1 2 3 ∂ CRPme imφ = e∓imφC cosφ R Pm+1 R Pm+1 +e i(m 1)φC m PmR, ∂1CRPlme∓imφ = e∓2ilm+φ1C sinφ(cid:16)R,l+1Plm−+11 −R,−lPlm++11(cid:17) e∓i(m−1)φCsiimnθPlmRr, (15) 2 l ∓ 2l+1 ,l+1 l−1 − ,−l l+1 ∓ ∓ − sinθ l r ∂ CRPme imφ = e∓imφC(cid:16)R (l+m)Pm +R (cid:17)(l m+1)Pm . 3 l ∓ 2l+1 ,l+1 l−1 ,−l − l+1 (cid:16) (cid:17) Substitutions (14) and (15) together with the assumptions R = R , l = l , R = R , l = l , Eα E Eα E Hα H Hα H m− =m− =m− 1 =m− 1 = m, 1 2 3 4 − − C− = i C− , C− = i C− , C− = i C− , C− = i C− , H1 H2 E2 − E1 H4 − E3 H3 − E4 − C−I =C−I (lI +m+1), C−I = CI C−I, H2 E4 H E3 − E4 ≡ (16) C−I =C−I (lI m), lI = lI 1 lI, E1 E3 E − H E − ≡ − C−II= C−II (lII m), C−II= CII C−II, H2 − E4 H − E3 − E4 ≡ C−II= C−II (lII +m+1), lII = lII +1 lII E1 − E3 E H E ≡ into the subsystem (11) guarantee the separation of variables in these equations and lead to the pair of equations for two radial functions R ,R (for the subsystem (12) the procedure is E H similar): ǫωRI RI = 0, µωRI +RI = 0, (17) E − H,−l H E,l+2 d a ǫωRII RII = 0, µωRII +RII = 0; R + R. (18) E − H,l+1 H E,−l+1 ,a ≡ dr r! For the case Φ = ze2/r the equations (17), (18) coincide exactly with the radial equations − for the hydrogen atom of the Dirac theory and, therefore, the procedure of their solution is the same as in well-known monographs on relativistic quantum mechanics. It leads to the well-known Sommerfeld - Dirac formula for the fine structure of the hydrogen spectrum. We note only that here the discrete picture of energetic spectrum in the domain 0 < ω < m c2 is 0 guaranteed by the demand for the solutions of the radial equations (17), (18) to decrease on infinity (when r ). From the equations (17), (18) and this condition the Sommerfeld - → ∞ Dirac formula m c2 ω = ωhyd = 0 (19) nj h¯ 1+ α2 2 (nr+√k2 α2) r − follows, wherethenotationsoftheDiractheory(see, e. g., [7])areused: n = n k,k = j+1/2, r − α = e2/h¯c. Let us note once more that the result (19) is obtained here not from the Dirac equation, but from the Maxwell equations (1) with sources (3) in the medium (2). Substituting (16)into(14)onecaneasyobtaintheangularpartofthehydrogensolutionsfor the −(→E,−H→,E0,H0) field and calculate according to (3) the corresponding currents and charges. Let us write down the explicit form for the set of electromagnetic field strengths −(→E,−→H), which 5 are the hydrogen solutions of equations (1), and also for the currents and charges generating these field strengths (the complete set of solutions is represented in [1]: ( l+m 1)Pm cosmφ (l+m+1)Pmsinmφ − − l+1 l −E→I = RI (l m+1)Pm sinmφ , −H→I = RI (l+m+1)Pmcosmφ , E (cid:12)(cid:12)(cid:12)(cid:12) −−Plm++11cos(ml+1+1)φ (cid:12)(cid:12)(cid:12)(cid:12) H (cid:12)(cid:12)(cid:12)(cid:12) −Plm+1sin(ml +1)φ (cid:12)(cid:12)(cid:12)(cid:12) (20) −→jI =(cid:12)(cid:12)gradRI Pm+1cos(m+1)φ, (cid:12)(cid:12)−j−I→ = grad(cid:12)(cid:12)RIPm+1sin(m+1)φ, (cid:12)(cid:12) e (cid:12) H l (cid:12) mag − (cid:12) e l+1 (cid:12) ρI = εRI Pm+1cos(m+1)φ, ρI = µRI Pm+1sin(m+1)φ, e − E ,l+2 l mag − H , l l+1 (cid:16) (cid:17) (cid:16) (cid:17)− (l+m)Pm cosmφ ( l+m)Pmsinmφ l 1 − l −E→II = RII ( l m)P−m sinmφ , −H−→II = RII ( l+m)Pmcosmφ E (cid:12)(cid:12)(cid:12)(cid:12) −Plm−+11cos(lm−1+1)φ (cid:12)(cid:12)(cid:12)(cid:12) H (cid:12)(cid:12)(cid:12)(cid:12) −−Plm+1sin(lm+1)φ (cid:12)(cid:12)(cid:12)(cid:12) (21) −j→II = grad(cid:12)(cid:12)RIIP−m+1cos(m+1)φ,(cid:12)(cid:12) −j−I→I = grad(cid:12)(cid:12)RIIPm+1sin(m+1)φ(cid:12)(cid:12), e (cid:12) H l (cid:12) mag − (cid:12) E l 1 (cid:12) ρII = εRII Pm+1cos(m+1)φ, ρII = µRII − Pm+1sin(m+1)φ. e − E , l+1 l mag − H ,l+1 l−1 (cid:16) (cid:17)− (cid:16) (cid:17) In one of the possible interpretations the states of the hydrogen atom are described by these field strength functions −→E,→−H generated by the corresponding currents and charge densities. It is evident from (1) that currents and charges in (20), (21) are generated by scalar fields (E0,H0). Corresponding to (20), (21) (E0,H0) solutions of equations (1) are the following: EI0 = RI Pm+1cos(m+1)φ, HI0 = RI Pm+1sin(m+1)φ, H l E l+1 (22) EII0 = RIIPm+1cos(m+1)φ, HII0 = RIIPm+1sin(m+1)φ. H l E l 1 − As in quantum theory, the numbers n = 0,1,2,...; j = k 1 = l 1 (k = 1,2,...,n) − 2 ∓ 2 and m = l, l + 1,...,l mark both the terms (19) and the corresponding exponentially de- − − creasing field functions −→E,−→H (and E0,H0) in (20)-(22), i. e. they mark the different discrete states of the classical electrodynamical field (and the densities of the currents and charges) which by definitions describes the corresponding states of hydrogen atom in the model under consideration. Note that the radial equations (17), (18) cannot be obtained if one neglects the sources in equations (1), or one (electric or magnetic) of these sources. Moreover, in this case there is no solution effectively concentrated in atomic region. Now we can show on the basis of this model that the assertions known as Bohr’s postulates are the consequences of equations (1) and of their classical interpretation, i. e. these assertions can be derived from the model, there is no necessity to postulate them from beyond the frame- work of classical physics as it was in Bohr’s theory. To derive the first Bohr’s postulate one can calculate the generalized Pointing vector for the hydrogen solutions (20)-(22), i. e. for the compound system of stationary electromagnetic and scalar fields −(→E,−H→,E0,H0) −→P = d3x(−→E −→H −→EE0 −→HH0). (23) gen × − − Z The straightforward calculations show that not only vector (23) is identically equal to zero but the Pointing vector itself and the term with scalar fields (E0,H0) are also identically equal to zero: −→P = d3x(−→E −→H) 0, d3x(−→EE0 +−→HH0) 0. (24) × ≡ ≡ Z Z 6 This means that in stationary states hydrogen atom does not emit any Pointing radiation neither due to the electromagnetic −(→E,→−H) field, nor to the scalar (E0,H0) field. That is the mathematical proof of the first Bohr postulate. The similar calculations of the energy for the same system (in formulae (23)-(25) the func- tions −(→E,−H→,E0,H0) are taken in appropriate physical dimension which is given by the formula (49) below) P0 = 12 d3xE†E = 21 d3x(−→E2 +−→H2 +E02 +H02) = ωnhjyd (25) Z Z give a constant W , depending on n,l (or n,j) and independent of m. In our model this nl constant is to be identified with the parameter ω in equations (1) which in the stationary states of −(→E,−H→,E0,H0) field appears to be equal to the Sommerfeld - Dirac value ωhyd (19). nj By abandoning the h¯ = c = 1 system and putting arbitrary ”A” in equations (1) instead of h¯ we obtain final ωhyd with ”A” instead of h¯. Then the numerical value of h¯can be obtained by nj comparison of ωhyd containing ”A” with the experiment. These facts complete the proof of the nj second Bohr postulate. This result means that in this model the Bohr postulates are no longer postulates, but the direct consequences of the classical electrodynamical equation (1). Moreover, this means that together with Dirac or Schrodinger equations we have now the new equation which can be used for finding the solutions of atomic spectroscopy problems. In contradiction to the well-known equations of quantum mechanics our equation is the classical one. Being aware that few interpretations of quantum mechanics (e.g.: Copenhagen, statistical, Feynman’s, Everett’s, transactional, see e. g. [8]) exist, we are far from thinking that here the interpretation can be the only one. But the main point is that now the classical interpretation (without probabilities) is possible. Today we prefer the following interpretation of hydrogen atom in the approach, when one considers only the motion of electron in the external field of the nucleon. In our model the interacting field of the nucleon and electron is represented by the medium with permeabilities ǫ,µ given by formulae (2). The atomic electron is interpreted as the stationary electromagnetic- scalarwave−(→E,−H→,E0,H0)inmedium(2),i.e. asthestationaryelectromagneticwaveinteracting with massless scalar fields (E0,H0), or with complex massless scalar field 0 = E0 iH0 with E − spin s = 0. In other words, the electron can be interpreted as an object having the structure consisting of a photon and a massless meson with zero spin connected, probably, with leptonic charge. The role of the massless scalar field is the following: it generates the densities of electric and magnetic currents and charges (ρ,−→j ), which are the secondary objects in such model. The mass is the secondary parameter too. There is no electron as an input charged massive corpuscle in this model! The mass and the charge of electron appear only outside such atom according to the law of electromagnetic induction and its gravitational analogy. That is why no difficulties of Rutherford - Bohr’s model (about different models of atom see, e. g., [9]) of atom are present here! The Bohr postulates are shown to be the consequences of the model. This interpretation is based on the hypothesis of bosonic nature of matter (on the speculation of the bosonic structure of fermions) according to which all the fermions can be constructed from different bosons (something like new SUSY theory). Of course, before the experiment intended to observe the structure of electron and before the registration of massless spinless meson it is only the hypothesis but based on the mathematics presented here. We note that such massless spinless boson has many similar features with the Higgs boson and the transition 7 here from intraatomic (with high symmetry properties) to macroelectrodynamics (with loss of many symmetries) looks similarly to the symmetry breakdown mechanism. The successors of magnetic monopole can try to develop here the monopole interpretation (see [10] for the review and some new ideas about monopole) - we note that there are few interesting possibilities of interpretation but we want to mark first of all the mathematical facts which are more important than different ways of interpretation. 3 The unitary relationship between the relativistic quan- tum mechanics and classical electrodynamics in medium Let us consider the connection between the stationary Maxwell equations curl−→H ωǫ−→E = gradE0, curl−→E ωµ−→H = gradH0, − − − (26) div−→E = ωµE0, div−→H = ωǫH0, − which follow from the system (8) after ommitting indices A,B, and the stationary Dirac equa- tion following from the ordinary Dirac equation iγµ∂ m +γ0Φ Ψ = 0, Ψ (Ψα), (27) µ 0 − ≡ (cid:16) (cid:17) with m = 0 and the interaction potential Φ = 0. Assuming the ordinary time dependence 6 6 Ψ(x) = Ψ(x)e iωt = ∂ Ψ(x) = iωΨ(x), (28) −→ − 0 ⇒ − for the stationary states and using the standard Pauli - Dirac representation for the γ matrices, one obtains the following system of equations for the components Ψα of the spinor Ψ: iωǫΨ1 +(∂ i∂ )Ψ4 +∂ Ψ3 = 0, 1 2 3 − − iωǫΨ2 +(∂ +i∂ )Ψ3 ∂ Ψ4 = 0, − 1 2 − 3 (29) iωµΨ3 +(∂ i∂ )Ψ2 +∂ Ψ1 = 0, 1 2 3 − − iωµΨ4 +(∂ +i∂ )Ψ1 ∂ Ψ2 = 0, 1 2 3 − − where ǫ and µ are the same as in (2). After substitution in Eqs. (29) instead of Ψ the following column Ψ = column H0 +iE3, E2 +iE1,E0 +iH3, H2 +iH1 . (30) − − − (cid:12) (cid:12) one obtains Eqs. (26). A com(cid:12)plete set of 8 such transformations can be obta(cid:12)ined with the help (cid:12) (cid:12) of the Pauli - Gursey symmetry operators [11] similarly to [6]. It is useful to represent the right-hand side of (30) in terms of components of the following complex function −→ E = column E1 iH1,E2 iH2,E3 iH3,E0 iH0 , (31) E ≡ (cid:12) 0 (cid:12) − − − − (cid:12) E (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) where −→ = −→E i−→H(cid:12)(cid:12) is t(cid:12)(cid:12)he well-kno(cid:12)wn form for the electromagnetic field us(cid:12)ed by Majorana E − as far back as near 1930 (see, e.g., [4]), and 0 = E0 iH0 is a complex scalar field. In these E − terms the connection between the spinor and electromagnetic (together with the scalar) fields has the form = WΨ, Ψ = W , (32) † E E 8 where the unitary operator W is the following: 0 iC 0 C − − 0 C 0 iC 1 W = (cid:12)(cid:12)(cid:12) iC −0+ C 0+ (cid:12)(cid:12)(cid:12); C∓ ≡ 2(C ∓1), CΨ ≡ Ψ∗, CE ≡ E∗. (33) (cid:12) − − (cid:12) (cid:12) iC 0 C 0 (cid:12) (cid:12) + + (cid:12) (cid:12) (cid:12) The unitarit(cid:12)y of the operator (33) (cid:12)can be verified easily by taking into account that the (cid:12) (cid:12) equations (AC)† = CA†, aC = Ca∗, (aC)∗ = Ca (34) hold for an arbitrary matrix A and a complex number a. We note that in the real algebra (i. e. the algebra over the field of real numbers) and in the Hilbert space of quantum mechanical amplitudes this operator has all properties of unitarity: WW 1 = W 1W = 1, W 1 = W , − − − † plus linearity. The operator (33) transforms the stationary Dirac equation (ω Φ)γ0 +iγk∂ m Ψ(x) = 0 (35) k 0 −→ − − h i from the standard representation (the Pauli - Dirac representation) into the bosonic represen- tation (ω Φ)γ0 +iγk∂ m (x) = 0. (36) k 0 −→ − − E Here the γµ matrices havhe the following unusual exiplicit form e ee e 1 0 0 0 0 0 i 0 0 1 0 0 0 0 0 1 γ0 = (cid:12) (cid:12)C, γ1 = (cid:12) − (cid:12), (cid:12) 0 0 1 0 (cid:12) (cid:12) i 0 0 0 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 0 0 0 1 (cid:12) (cid:12) 0 1 0 0 (cid:12) e (cid:12) − (cid:12) e (cid:12) (cid:12) (37) (cid:12) 0 0 0 1(cid:12) (cid:12) i 0 0 0 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)− (cid:12) 0 0 i 0 0 i 0 0 γ2 = (cid:12) (cid:12), γ3 = (cid:12) − (cid:12) (cid:12) 0 i 0 0 (cid:12) (cid:12) 0 0 i 0 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 1 0 0 0 (cid:12) (cid:12) 0 0 0 i (cid:12) e (cid:12) − (cid:12) e (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) in which γ0 matrix explicitly c(cid:12)ontains operato(cid:12)rC of com(cid:12) plex conjugatio(cid:12)n. We call the represen- (cid:12) (cid:12) (cid:12) (cid:12) tation (37) the bosonic representation of the γ matrices. In this representation the imaginary unit i is reepresented by the 4 4 matrix operator: × 0 1 0 0 − 1 0 0 0 i = (cid:12) (cid:12). (38) (cid:12) 0 0 0 i (cid:12) (cid:12) (cid:12) (cid:12) − (cid:12) e (cid:12) 0 0 i 0 (cid:12) (cid:12) − (cid:12) (cid:12) (cid:12) Duetotheunitarityof theoperator((cid:12)33)theγµ matrice(cid:12)sstill obeytheClifford-Diracalgebra (cid:12) (cid:12) γµγν +γνγeµ = 2gµν (39) and have the same Hermitian properties as the Pauli - Dirac γµ matrices: e e e e γ0 = γ0, γk = γk. (40) † † − e e 9e e Thus, the formulae (37) give indeed an exotic representation of γµ matrices. In the vector-scalar form the equation (36) is as follows icurl−→+[(ω Φ)C m ]−→ = grad 0, div−→ = [(ω Φ)C +m ] 0. (41) 0 0 − E − − E − E E − E Fulfilling the transition to the common real field strengths according to the formula = E E iH and separating the real and imaginary parts we obtain equations (26) which are math- − ematically equivalent to the equations (1) in stationary case. We emphasize that the only difference between the equation (36) in the case of description of fermions and in the case of bosons is the possibility of choosing γµ matrices: for the case of fermions these matrices may be chosen in arbitrary form (in each of representations of Pauli - Dirac, Majorana, Weyl, ...), in the case of the description of bosons the representation of γµ matrices and their explicit form must be fixed in the form (37). In the case of bosonic interpretation of Eq. (35) one must fixes the explicit form of γµ matrices and of Ψ (30). The mathematical facts considered here prove the one-to-one correspondence between the solutions of the stationary Dirac and the stationary Maxwell equations with 4-currents of gradient-like type. Hence, one can, using (30), write down the hydrogen solutions of the Maxwell equations (1) (or (4)) starting from the well-known hydrogen solutions of the Dirac equation (27), i. e. without special procedure of finding the solutions of the Maxwell equations, see [1]. 4 Some group-theoretical grounds of the model Consider briefly the case of absence of interaction of the compound field −(→E,−H→,E0,H0) with media, i. e. the case ǫ = µ = 1, and the symmetry properties of the corresponding equations. In this case equations (1) for the system of electromagnetic and scalar fields −(→E,−H→,E0,H0) have the form: ∂ −→E = curl−→H gradE0, ∂ −→H = curl−→E gradH0, 0 − 0 − − (42) div−→E = ∂ E0, div−→H = ∂ H0. 0 0 − − The Eqs. (42) are nothing more than the weakly generalized Maxwell equations (ǫ = µ = 1) with gradient-like electric and magnetic sources je = ∂ E0, jmag = ∂ H0, i. e. µ − µ µ − µ −→j e = gradE0, −→j mag = gradH0, ρe = ∂0E0, ρmag = ∂0H0. (43) − − − − In terms of complex 4-component object = E iH from formula (31) (and in terms of E − following complex tensor 0 1 2 3 E E E 1 0 i 3 i 2 E = (Eµν) (cid:12) −E E − E (cid:12)) (44) ≡ (cid:12)(cid:12) 2 i 3 0 i 1 (cid:12)(cid:12) (cid:12) −E − E E (cid:12) (cid:12) 3 i 2 i 1 0 (cid:12) (cid:12) −E E − E (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Eqs. (42) can be rewritten in the mani(cid:12)festly covariant forms (cid:12) ∂ ∂ +iε ∂ρ σ = 0, ∂ µ = 0 (45) µ ν ν µ µνρσ µ E − E E E (vector form) and ∂ Eµν = ∂µ 0 (46) ν E 10