On the probability interpretation of wave functions in the Dirac theory
aa r X i v : . [ qu a n t - ph ] J a n On the probability interpretation of wave functions in the Dirac theory
Yu.M. Poluektov ∗ National Science Center “Kharkov Institute of Physics and Technology”,Akhiezer Institute for Theoretical Physics, 61108 Kharkov, Ukraine
It is suggested symmetric relative to particles and antiparticles formulation of the Dirac theory,in which the states with negative energy are excluded. The fields of particles and antiparticles areassociated with the wave functions, for which there is valid the Born interpretation of being theprobability amplitudes. In doing so, one eliminates different kinds of “paradoxes” in the theory,which existence is caused by an incorrect account of the states with negative energy.
Key words : Dirac equation, electron, positron, particle, antiparticle, wave function, probabilityamplitude, charge conjugation
PACS numbers: 03.65.Pm Relativistic wave equations; 14.60.Cd Electrons (including positrons);
I. INTRODUCTION
The relativistic equation for an electron was obtained by Dirac in his classical works [1]. A presentation of theDirac theory is available in a large number of textbooks and monographs, some of which became classical as well[2–15]. Like the nonrelativistic Schr¨odinger equation describing the spatial and temporal evolution of the complexfunction, the Dirac equation describes the complex function incorporating the four components. The solutions of thenonrelativistic Schr¨odinger equation for a free particle with positive energy constitute a full set of states by which anarbitrary solution can be decomposed. In the Dirac theory, along with the solutions with positive energy, there arethe solutions with the opposite sign of energy. In contrast to the nonrelativistic theory, the solutions of the Diracequation corresponding to the states with positive energy do not constitute a full set of states, and in order to obtainthe general solution one have to take into account also the solutions with negative energy. In this connection, after itbecame clear that it is impossible to eliminate negative energies from the theory, the problem has arisen regarding thephysical interpretation of such solutions [1]. With this aim Dirac assumed that in the nature almost all states withnegative energy are occupied, and the unoccupied states (holes) behave themselves as the positively charged particles.Since the electrons occupying the states with negative energy are charged, they should have created the electric fieldwith an infinite density of energy. To overcome this difficulty Dirac made a rather exotic assumption that the electricfield is produced only by the electrons situated over “the electron sea” or by the holes. Initially Dirac identified thepositive states with the protons, but from the symmetry consideration it followed that the mass of electrons andholes should be equal. The result of awareness of this circumstance was the hypothesis that the holes correspond tothe new particles unknown to the science at that time and with the mass equal the electron’s mass but the oppositecharge. The Dirac theory describes both the positively and negatively charged particles in a completely equal way,whereas the hypotheses about “the electron sea” breaks the symmetry between them. The Dirac interpretation whichin itself does not follow from the mathematical formalism was rather coldly accepted by a lot of leading physicists.Only the discovery of the positron changed the attitude towards it. However, the experimental confirmation of somepredictions of a theory by no means makes a theory in itself logically perfect.Although eventually it became clear that the Dirac theory does not need additional constructions for its interpreta-tion, the treatment at the level “particle-hole” is reproduced in many books on the relativistic quantum mechanics andfield theory [6]. At present, there exists quite consistent interpretation of the Dirac theory not attracting additional,internally unnatural to it, qualitative considerations [2], although in our opinion this right point of view is far fromalways being consistently advanced. In the interpretation of the theory one deals with a central question: “What isthe physical meaning of the complex multicomponent field, which is described by the Dirac equation?”. This questionis note merely a speculative one, but it has an important practical meaning, since depending on the answer to it thereshould be established the rules for the calculation of the observable quantities. While according to the Born inter-pretation the complex Schr¨odinger field is the probability amplitude and its phase-invariant combinations correspondto the probability density and the probability flow density, the general solution of the Dirac equation contains thecontributions of the states with both positive and negative energies and does not allow such an interpretation. Sincesuch general solution of the Dirac equation does not have the meaning of the probability amplitude, it cannot be used ∗ Electronic address: [email protected] for the calculation of the average values of the observable quantities and other probability characteristics. The usualprobability interpretation is allowed only for the solutions with positive energies, through which one can also expressthe general solution of the Dirac equation. But this general solution is not a new allowable state in accordance withthe superposition principle since it includes the operation of the charge conjugation. The neglecting of this fact leadsto the appearance of various “paradoxes”, such as the Klein paradox [2, 6], and also to such phenomena in the theoryas the “jittering” of electrons [6, 13].In this work we propose a variant of the Dirac theory, in which the equations for the particles with the oppositecharges and equal masses are considered simultaneously and symmetrically and where the independent variablesdescribing the physically realizable states are the solutions for the particles with the opposite signs of charges andpositive energies. The solutions of the equations to which correspond the negative energies are not independent butthey are expressed through the solutions with positive energies by means of the operation of the charge conjugation.In this way the problem of negative energies in the theory is resolved. The complex wave functions describing thestates of particles and antiparticles allow the usual quantum-mechanical interpretation and have the meaning of theprobability amplitudes. The equations for a particle and an antiparticle prove to be independent so that they shouldbe considered separately.
II. THE CHARGE-SYMMETRIC FORM OF THE DIRAC EQUATIONS
The Dirac equation describes the evolution of the four-component complex function ψ ( x ) ≡ ψ σ ( x , t ), where theindex takes the values σ = 1 , , ,
4. In the matrix notation the Dirac equation for the function ψ ( x ) and the conjugatefunction ψ ( x ) = ψ + ( x ) γ has the form ~ cγ µ ∂ψ ( x ) ∂x µ + mc ψ ( x ) = 0 , ~ c ∂ψ ( x ) ∂x µ γ µ − mc ψ ( x ) = 0 , (1)where x ≡ x µ ≡ ( x , x ) = ( x , ix ) = ( x , ict ), m – the mass of a particle, c – the speed of light, ~ – the Planck constant, γ µ – the hermitian 4 × ψ + ( x ) denotes the hermitian conjugation. A summation iseverywhere implied over the repeated indices. We will mainly follow the notations of the book [2]. The scalar productof the two 4-vectors a = ( a , a ) = ( a , ia ) and b = ( b , b ) = ( b , ib ) is written in the form ab = ab + a b = ab − a b .The account for the interaction with the electromagnetic field is performed by means of the famous replacement ∂∂x µ → ∂∂x µ − ie ~ c A µ ( x ) , (2)where A µ ( x ) is the 4-vector potential, and the charge e = ∓| e | , or e = 0 for electrically neutral particles. Furtherwe deal with the charged particles. Usually when writing down the Dirac equation one chooses the electron chargein (2). Thus from the beginning a certain asymmetry is introduced to the theory. However, since the oppositelycharged particles enter into the theory completely symmetrically, then for the charge-symmetrical consideration it isconvenient to introduce the two fields with the equal masses but opposite charges – the “negative” field ψ which isconsidered to be the field of a particle and the “positive” field η which is referred to as the field of an antiparticle.Traditionally, a particle with negative charge e = −| e | (electron) is believed to be a “particle” and a particle withpositive charge e = | e | (positron) – an “antiparticle”. We do not discuss here the asymmetry between particles andantiparticles observed in the nature which is, quite probably, not related to the breaking of the charge symmetry inthe fundamental equations. Note also, that the state of a particle can be additionally characterized by the “inner”(lepton) quantum number Λ, which is assumed to be positive for a particle whereas an antiparticle is characterized bythe lepton quantum number of the opposite sign. The existence of the “inner” quantum number allows to distinguishthe fields of particles and antiparticles also in the absence of the electromagnetic field and in the case of neutralparticles. Thus, the Dirac equations for the fields of a charged particle and antiparticle in the electromagnetic field A µ = ( A , iA ) take the form a ) ~ cγ µ (cid:18) ∂ψ Λ ∂x µ − i e ~ c A µ ψ Λ (cid:19) + mc ψ Λ = 0 , b ) ~ cγ µ (cid:18) ∂η − Λ ∂x µ + i e ~ c A µ η − Λ (cid:19) + mc η − Λ = 0 , (3)Let us also give the equations for the conjugate functions ψ Λ ≡ ψ +Λ γ and η − Λ ≡ η + − Λ γ : a ) ~ c (cid:18) ∂ψ Λ ∂x µ + i e ~ c A µ ψ Λ (cid:19) γ µ − mc ψ Λ = 0 , b ) ~ c (cid:18) ∂η − Λ ∂x µ − i e ~ c A µ η − Λ (cid:19) γ µ − mc η − Λ = 0 . (4)In the following the index of the inner quantum number Λ will be everywhere omitted for brevity. The functions ψ and η are not independent. Their relationship can be established by means of the unitary matrix of the chargeconjugation, which satisfies the conditions [2]: C + C = CC + = 1 , Cγ µ C + = − e γ µ , C = − e C. (5)The symbol ∼ denotes the transposition. This matrix can be chosen in the form C = γ γ . Then from equations (3)and (4) there follow the relations which connect the solutions of the Dirac equations with the opposite charges: a ) ψ = C ∗ e η, b ) η = C ∗ e ψ. (6)Let us decompose the solution of the Dirac equation into the Fourier integral: ψ ( x ) = Z ∞−∞ c ( ω ) ψ ( x , ω ) e − iωt dω = ψ + ( x ) + ψ − ( x ) , (7)where the positive-frequency and negative-frequency functions are defined by the formulas: ψ ± ( x ) = Z ∞ c ( ± ω ) ψ ( x , ± ω ) e ∓ iωt dω. (8)It is assumed that there exists the integral 12 π Z ∞−∞ ψ ( x ) e iωt dt < ∞ . (9)Note that the integration in (8) is performed over only positive frequencies. In a similar way one can represent thesolution of the equation with the opposite sign of the charge: η ( x ) = Z ∞−∞ b ( ω ) η ( x , ω ) e − iωt dω = η + ( x ) + η − ( x ) , (10)where η ± ( x ) = Z ∞ b ( ± ω ) η ( x , ± ω ) e ∓ iωt dω. (11)Then from equations (6) there follow the relations expressing the negative-frequency functions through the positive-frequency functions: ψ − ( x ) = C ∗ e η + ( x ) , ψ − ( x ) = e η + ( x ) C,η − ( x ) = C ∗ e ψ + ( x ) , η − ( x ) = e ψ + ( x ) C. (12)Thus, the general solutions of the Dirac equation with the charge of an arbitrary sign can be expressed through onlythe positive-frequency particular solutions ψ + ( x ) and η + ( x ) of the Dirac equations with the opposite signs of thecharge, which are those to be considered as the wave functions of a particle and an antiparticle allowing the Bornprobability interpretation: ψ ( x ) = ψ + ( x ) + C ∗ e η + ( x ) , η ( x ) = η + ( x ) + C ∗ e ψ + ( x ) . (13)Since the positive-frequency particular solutions are interpreted as the wave functions of a particle and an antiparticlehaving the meaning of the probability amplitudes, they should be normalized by the conditions: Z | ψ + ( x ) | d x = 1 , Z | η + ( x ) | d x = 1 . (14)The general solutions of the Dirac equations (13) are expressed through both the wave function of a particle and thewave function of an antiparticle, but they are not the linear superposition of these functions since contain the antilineartransformation of the complex conjugation, and so they do not have the meaning of the probability amplitudes. Thus,the functions (13) containing the contribution of the states with negative energies are not the physically realizablestates having the meaning of the probability amplitudes and, therefore, they cannot be used in calculating of thetransition probabilities and average values of the operators of the observables. It is precisely the improper use incalculations of such functions including the states with negative energies that leads to the appearance in the theoryof various “paradoxes”. III. THE DIRAC EQUATIONS FOR THE PROBABILITY AMPLITUDES
Now let us substitute the functions (13) into the equations (3) and (4). As a result we obtain the equationscontaining only the functions with positive frequencies: ~ cγ µ ∂∂x µ (cid:16) ψ + + C ∗ e η + (cid:17) − ieA µ γ µ (cid:16) ψ + + C ∗ e η + (cid:17) + mc (cid:16) ψ + + C ∗ e η + (cid:17) = 0 , (15) ~ c ∂∂x µ (cid:0) ψ + + e η + C (cid:1) γ µ + ieA µ (cid:0) ψ + + e η + C (cid:1) γ µ − mc (cid:0) ψ + + e η + C (cid:1) = 0 , (16) ~ cγ µ ∂∂x µ (cid:16) η + + C ∗ e ψ + (cid:17) + ieA µ γ µ (cid:16) η + + C ∗ e ψ + (cid:17) + mc (cid:16) η + + C ∗ e ψ + (cid:17) = 0 , (17) ~ c ∂∂x µ (cid:16) η + + e ψ + C (cid:17) γ µ − ieA µ (cid:16) η + + e ψ + C (cid:17) γ µ − mc (cid:16) η + + e ψ + C (cid:17) = 0 . (18)It is convenient to introduce the notations Q ( x ) ≡ ~ cγ µ ∂ψ + ∂x µ − ieA µ γ µ ψ + + mc ψ + , Π( x ) ≡ ~ cγ µ ∂η + ∂x µ + ieA µ γ µ η + + mc η + . (19)Then in these notations the equations (15) – (18) take the form Q ( x ) + C ∗ e Π( x ) = 0 , Q ( x ) + e Π( x ) C = 0 , Π( x ) + C ∗ e Q ( x ) = 0 , Π( x ) + e Q ( x ) C = 0 , (20)where Q ( x ) ≡ Q + ( x ) γ , Π( x ) ≡ Π + ( x ) γ . Each of the three equations (20) are the consequence of the fourth equation,so that the relations (20) give the various forms of presentation of the single equation. First we mainly consider thecase of the stationary electromagnetic field, setting A µ ( x ) = A µ ( x ). Using the decompositions (8) and (11) for thefunctions ψ + ( x ) and η + ( x ), we obtain Q ( x ) = Z ∞ c ( ω ) Q ( x , ω ) e − iωt dω, Π( x ) = Z ∞ b ( ω )Π( x , ω ) e − iωt dω, (21)where Q ( x , ω ) = h ~ c (cid:16) γ ∇ − ωc γ (cid:17) + mc − ieA µ ( x ) γ µ i ψ ( x , ω ) , Π( x , ω ) = h ~ c (cid:16) γ ∇ − ωc γ (cid:17) + mc + ieA µ ( x ) γ µ i η ( x , ω ) . (22)In these notations the equations (20) are equivalent to the equation Z ∞ h c ( ω ) Q ( x , ω ) e − iωt + b ∗ ( ω ) C ∗ e Π( x , ω ) e iωt i dω = 0 . (23)Multiplying (23) first by e iω ′ t , where ω ′ >
0, and integrating over time, and then multiplying by e − iω ′ t and alsoperforming integration over time, we obtain that there must be Q ( x , ω ) = 0 and e Π( x , ω ) = 0. Thus, we arrive atthe two independent equations for the wave functions of a particle and an antiparticle, which with account of thenotations (22) has the form h ~ c γ (cid:16) ∇ − i e ~ c A ( x ) (cid:17) − ( ~ ω − eA ( x )) γ + mc i ψ ( x , ω ) = 0 , (24) h ~ c γ (cid:16) ∇ + i e ~ c A ( x ) (cid:17) − ( ~ ω + eA ( x )) γ + mc i η ( x , ω ) = 0 . (25)Here the functions ψ ( x , ω ) and η ( x , ω ), according to (8) and (11), depend only on positive frequency. From equations(24), (25) there follow the conditions of orthonormality for the electron and positron wave functions: Z d x ψ + ( x , ω ) ψ ( x , ω ′ ) = δ ( ω − ω ′ ) , Z d x η + ( x , ω ) η ( x , ω ′ ) = δ ( ω − ω ′ ) . (26)The conditions of orthogonality of the electron and positron wave functions are also satisfied: Z d x ψ + ( x , ω ′ ) γ C ∗ η ∗ ( x , ω ) = 0 , Z d x e η ( x , ω ) Cγ ψ ( x , ω ′ ) = 0 . (27)For the wave functions depending on time, the equations for particles and antiparticles have the form of the Diracequations which differ only by the sign of the charge: ~ cγ µ (cid:18) ∂ψ + ( x ) ∂x µ − i e ~ c A µ ( x ) ψ + ( x ) (cid:19) + mc ψ + ( x ) = 0 , ~ cγ µ (cid:18) ∂η + ( x ) ∂x µ + i e ~ c A µ ( x ) η + ( x ) (cid:19) + mc η + ( x ) = 0 , (28)Thus, in the stationary fields the equations for the wave functions of particles and antiparticles with positive energiesare unconnected and, therefore, the states of particles and antiparticles should be considered independently. IV. LAGRANGIAN FORMALISM
Let us formulate the developed approach for the description of particles and antiparticles in terms of the probabilityamplitudes on the basis of the Lagrangian formalism, which will allow to obtain the energy-momentum tensor andthe conservation laws. The Dirac equations (28) can be obtained if the Lagrangian function density is represented asa sum of the Lagrangians of a particle Λ ψ ( e ) and an antiparticle Λ η ( − e ): Λ = Λ ψ ( e ) + Λ η ( − e ), whereΛ ψ ( e ) = − c ~ ψ + γ µ ∂ψ + ∂x µ − ∂ψ + ∂x µ γ µ ψ + ! + ieA µ ( x ) ψ + γ µ ψ + − mc ψ + ψ + , (29)Λ η ( − e ) = − c ~ (cid:18) η + γ µ ∂η + ∂x µ − ∂η + ∂x µ γ µ η + (cid:19) − ieA µ ( x ) η + γ µ η + − mc η + η + . (30)As the independent dynamical variables we consider the functions ψ + , ψ + and η + , η + . Since, as was shown, in thestationary field particles and antiparticles are described independently, then it is sufficient to consider the case ofparticles and the similar relations for antiparticles are obtained if there are performed the replacements e → − e and ψ + → η + . From the Euler-Lagrange equations, with account of the form of the Lagrangians (29), (30), there followthe equations (28), which were obtained above directly from the Dirac equation. From the condition of the invarianceof the particle Lagrangian (29) relative to the phase transformations: ψ + ( x ) → ψ ′ + ( x ) = ψ + ( x ) e iα , ψ + ( x ) → ψ ′ + ( x ) = ψ + ( x ) e − iα , (31)where α is a real parameter, there follows the continuity equation for the particle’s probability density ∂j ψµ ∂x µ = 0 , (32)where the 4-vector of the probability flow density has the form j ψµ = icψ + γ µ ψ + . (33)From (32) there follows the law of conservation of the total probability for a particle R d x j ψ ( x ) = const. Theserelations are similar to those taking place in the nonrelativistic quantum theory [16].The Lagrangian function densities (29), (30) depend on the wave functions of a particle and an antiparticle and onthe electromagnetic field which is considered as an external field, but they do not depend explicitly on x . Consequentlythe form of the Lagrangians should not change under the translation of the whole system, including an external field,by an arbitrary 4-vector a . From these considerations we find the equation for the energy-momentum tensor ∂T ψµν ∂x ν = − ∂ Λ ψ ∂A ν ( x ) ∂A ν ( x ) ∂x µ , (34)where the energy-momentum tensor pertaining to a particle is defined by a known relation T ψµν = ∂ Λ ψ ∂ ∂ψ + ∂x ν ∂ψ + ∂x µ + ∂ψ + ∂x µ ∂ Λ ψ ∂ ∂ψ + ∂x ν − Λ ψ δ µν . (35)It is customary to introduce the 4-vector of the energy-momentum P ψµ = ic Z T ψµ d x , (36)where P ψµ ≡ (cid:18) P ψ , ic W ψ (cid:19) . Thus the total momentum P ψ and the total energy W ψ are defined by the formulas P ψi = ic Z T ψi d x , W ψ = Z T ψ d x . (37)With account of the form of the Lagrangian (29) and also the fact that this Lagrangian vanishes for the functions ψ + , ψ + satisfying the Dirac equation, we find the energy-momentum tensor expressed through the wave functions ofa particle with positive energy: T ψµν = − c ~ ψ + γ ν ∂ψ + ∂x µ + c ~ ∂ψ + ∂x µ γ ν ψ + . (38)Taking into account the law of conservation of probability (32), the 4-vector of the energy-momentum can be repre-sented in the form P ψµ = − i ~ Z ψ ++ ∂ψ + ∂x µ d x , (39)so that the total momentum P ψ and energy W ψ = − icP ψ of a particle are given by the relations P ψ = − i ~ Z ψ ++ ∇ ψ + d x , W ψ = i ~ Z ψ ++ ∂ψ + ∂t d x . (40)These formulas are similar to those for the calculation of the average momentum and the average energy in thenonrelativistic quantum mechanics. The peculiarity of the field relations obtained here consists in that the fieldstherein have the meaning of the complex probability amplitudes for particles and antiparticles, and the temporaldependence of the fields is determined by the Fourier decompositions (8), (11) over only positive frequencies. V. FREE PARTICLES AND ANTIPARTICLES
Let us apply the proposed interpretation of the Dirac theory to the description of the free particles and antiparticles.In the absence of an external field the equations for a particle and an antiparticle are the same: (cid:2) ~ c γ ∇ − Eγ + mc (cid:3) ψ + (cid:0) x , E (cid:1) = 0 , (cid:2) ~ c γ ∇ − Eγ + mc (cid:3) η + (cid:0) x , E (cid:1) = 0 , (41)where the notation E ≡ ~ ω is introduced for positive energy. We look for the solutions of these equations in the formof plane waves ψ + (cid:0) x , E (cid:1) = 1 √ V ψ ( k ) e i kx , η + (cid:0) x , E (cid:1) = 1 √ V η ( k ) e i kx . (42)In this case: (cid:2) i ~ c k γ − Eγ + mc (cid:3) ψ ( k ) = 0 , (cid:2) i ~ c k γ − Eγ + mc (cid:3) η ( k ) = 0 . (43)The bispinors can be written in the form of the columns of the spinors ψ ( k ) = " ϕ ( k ) χ ( k ) , η ( k ) = " ζ ( k ) υ ( k ) . (44)In the Dirac-Pauli representation γ ≡ " − i σ i σ , γ ≡ " − (45)the equation for a particle takes the form " − E + mc ~ c σ k − ~ c σ k E + mc ϕ ( k ) χ ( k ) = 0 . (46)From here it follows the expression for positive energy of a particle: E = q(cid:0) ~ ck (cid:1) + (cid:0) mc (cid:1) . (47)Since there are considered the functions (8), (11) for which the Fourier decomposition is carried out over only positivefrequencies, then the root with the negative sign for energy should not be taken into account. Similar relations arevalid for an antiparticle. Thus, the solutions of the equations (43) can be written in the form ψ ( k ) = ϕ ( k ) ~ c σ k E + mc ϕ ( k ) , η ( k ) = ζ ( k ) ~ c σ k E + mc ζ ( k ) . (48)For the fulfilment of the normalization for the bispinors ψ + ( k ) ψ ( k ) = 1 and η + ( k ) η ( k ) = 1 the following normalizationof the spinors is necessary: ϕ + ( k ) ϕ ( k ) = ζ + ( k ) ζ ( k ) = 12 (cid:18) mc E (cid:19) . (49)Thus, the general solution of the Dirac equation for a free particle with the momentum ~ k can be represented in oneof the two forms: ψ (cid:0) x , t (cid:1) = 1 √ V h ψ ( k ) e i ( kx − ωt ) + C ∗ e η ( k ) e − i ( kx − ωt ) i ,η (cid:0) x , t (cid:1) = 1 √ V h η ( k ) e i ( kx − ωt ) + C ∗ e ψ ( k ) e − i ( kx − ωt ) i , (50)where ω = E (cid:14) ~ = q(cid:0) ~ ck (cid:1) + (cid:0) mc (cid:1) . ~ is the positive frequency. However, as was outlined, these functions do nothave the meaning of the probability amplitudes and cannot be used for the calculation of the probability characteristics.As in the nonrelativistic theory, the wave functions (42) having the meaning of the probability amplitudes describethe delocalized particle and antiparticle with the definite momentum and positive energy.In the general case the wave functions of particles and antiparticles decomposed over the plane waves have the form ψ + (cid:0) x , t (cid:1) = 1 √ V X k,r c r ( k ) ψ ( k , r ) e i ( kx − ωt ) , η + (cid:0) x , t (cid:1) = 1 √ V X k,r b r ( k ) η ( k , r ) e i ( kx − ωt ) (51)Here the index r = ± z .For the bispinors in (51) the orthonormality conditions are fulfilled ψ + ( k , r ) ψ ( k , r ′ ) = δ rr ′ , η + ( k , r ) η ( k , r ′ ) = δ rr ′ . (52)For the coefficients of the decomposition (51) the normalization conditions are also fulfilled X k,r (cid:12)(cid:12) c r ( k ) (cid:12)(cid:12) = X k,r (cid:12)(cid:12) b r ( k ) (cid:12)(cid:12) = 1 . (53)The completeness conditions for the system of the wave functions of a particle and an antiparticle with positiveenergies can be written in one of the equivalent forms: X r = ± h ψ ( k , r ) ψ ( k , r ) + C ∗ e η ( − k , r ) e η ( − k , r ) C i = 1 , X r = ± h η ( k , r ) η ( k , r ) + C ∗ e ψ ( − k , r ) e ψ ( − k , r ) C i = 1 . (54)According to (40) the energy and momentum of an electron and a positron are determined by the formulas W = X k,r E ( k ) (cid:2) c ∗ r ( k ) c r ( k ) + b ∗ r ( k ) b r ( k ) (cid:3) , (55) P = X k,r ~ k (cid:2) c ∗ r ( k ) c r ( k ) + b ∗ r ( k ) b r ( k ) (cid:3) , (56)where E ( k ) = q(cid:0) ~ ck (cid:1) + (cid:0) mc (cid:1) . Naturally, in the proposed interpretation the contribution of the states withnegative energy into the total energy is absent. One should use the formulas (51) – (56) in proceeding towards thequantum-field description of electrons and positrons.The Dirac equations describing electrons and positrons in stationary fields have the same form as in the standardapproach [2]. The difference consists in that there is no necessity to take into account the states with negative energies E < − mc which do not exist. Therefore, naturally, there is absent the tunneling probability into such states andthere are absent the “paradoxes” conditioned by taking into account of such states (different variants of “the Kleinparadox” [2, 6]). VI. THE ELECTRON AND POSITRON STATES INTHE NONSTATIONARY ELECTROMAGNETIC FIELD
It was shown above that the states of electrons and positrons with positive energies in the stationary electromagneticfield are independent and they are described by the same unconnected among themselves equations which differ onlyby the sign of the charge. Let us consider the equations in the presence of a nonstationary field. The vector potentialcan be represented as a sum of a stationary and a nonstationary in time terms: A µ ( x ) = A µ ( x ) + A µ ( x ). In this casethe equation (20) takes the form Q ( x ) + C ∗ e Π( x ) = ieA µ ( x ) γ µ (cid:16) ψ + ( x ) + C ∗ e η + ( x ) (cid:17) , (57)where Q ( x ) and Π( x ) are defined by the formulas (19). The equation (57) also can be decomposed into the equationfor particles when η + ( x ) = 0 and the equation for antiparticles when ψ + ( x ) = 0: ~ cγ µ ∂ψ + ∂x µ − ieA µ ( x ) γ µ ψ + + mc ψ + = ieA µ ( x ) γ µ ψ + ( x ) , (58) ~ cγ µ ∂η + ∂x µ + ieA µ ( x ) γ µ η + + mc η + = − ieA µ ( x ) γ µ η + ( x ) . (59)In the alternating external field the energy does not conserve. VII. CONCLUSION