Physics Letters A 375 (2011) 2596–2600 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Spinless bosons embedded in a vector Duffin–Kemmer–Petiau oscillator L.B. Castro, A.S. de Castro ∗ UNESP – Campus de Guaratinguetá, Departamento de Física e Química, 12516-410 Guaratinguetá SP, Brazil a r t i c l e i n f o a b s t r a c t Article history: Received 23 November 2010 Received in revised form 12 May 2011 Accepted 31 May 2011 Available online 7 June 2011 Communicated by R. Wu Keywords: DKP equation Nonminimal vector coupling DKP oscillator Some properties of minimal and nonminimal vector interactions in the Duffin–Kemmer–Petiau (DKP) formalism are discussed. The conservation of the total angular momentum for spherically symmetric nonminimal potentials is derived from its commutation properties with each term of the DKP equation and the proper boundary conditions on the spinors are imposed. It is shown that the space component of the nonminimal vector potential plays a crucial role for the confinement of bosons. The exact solutions for the vector DKP oscillator (nonminimal vector coupling with a linear potential which exhibits an equally spaced energy spectrum in the weak-coupling limit) for spin-0 bosons are presented in a closed form and it is shown that the spectrum exhibits an accidental degeneracy. © 2011 Elsevier B.V. All rights reserved. 1. Introduction The first-order Duffin–Kemmer–Petiau (DKP) formalism [1,2] describes spin-0 and spin-1 particles. The DKP equation for a free boson is given by [2] (βμpμ − m)ψ = 0 (with units in which h̄ = c = 1), where the four beta matrices satisfy the alge- bra βμβνβλ + βλβνβμ = gμνβλ + gλνβμ and the metric tensor is gμν = diag(1,−1,−1,−1). The algebra expressed by those matri- ces generates a set of 126 independent matrices whose irreducible representations are a trivial representation, a five-dimensional rep- resentation describing the spin-0 particles and a ten-dimensional representation associated to spin-1 particles. A well-known con- served four-current is given by jμ = ψβμψ/2, where the adjoint spinor ψ is given by ψ = ψ†η0 with ημ = 2βμβμ − gμμ in such a way that (η0βμ)† = η0βμ (the matrices βμ are Hermitian with re- spect to η0). Despite the similarity to the Dirac equation, the DKP equation involves singular matrices, the time component of jμ is not positive definite and the case of massless bosons cannot be obtained by a limiting process. Nevertheless, the matrices βμ plus the unit operator generate a ring consistent with integer-spin alge- bra [3] and j0 may be interpreted as a charge density. The factor 1/2 multiplying ψβμψ , of no importance regarding the conserva- tion law, is in order to hand over a charge density conformable to that one used in the Klein–Gordon theory and its nonrelativistic limit (see e.g. [4]). Then the normalization condition ∫ dτ j0 = ±1 can be expressed as ∫ dτ ψβ0ψ = ±2, where the plus (minus) sign must be used for a positive (negative) charge. * Corresponding author. E-mail addresses: benito@feg.unesp.br (L.B. Castro), castro@pq.cnpq.br (A.S. de Castro). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.05.067 The DKP formalism has been used to analyze relativistic inter- actions of spin-0 and spin-1 hadrons with nuclei. A number of different couplings in the DKP formalism, with scalar and vector couplings in analogy with the Dirac phenomenology for proton- nucleus scattering, has been employed in the phenomenological treatment of the elastic meson-nucleus scattering at medium ener- gies with a better agreement to the experimental data when com- pared to the Klein–Gordon and Proca based formalisms [5–10]. Re- cently, there has been an increasing interest on the so-called DKP oscillator [11–15]. That system is a kind of tensor coupling with a linear potential which leads to the harmonic oscillator problem in the weak-coupling limit. A nonminimal vector potential, added by other kinds of Lorentz structures, has already been used success- fully in a phenomenological context for describing the scattering of mesons by nuclei [5,6,8,10], and a sort of vector DKP oscilla- tor (nonminimal vector coupling with a linear potential [14,16]) has also been an item of recent investigation. Vector DKP oscilla- tor is the name given to the system with a Lorentz vector coupling which exhibits an equally spaced energy spectrum in the weak- coupling limit. The name distinguishes from that system called DKP oscillator with Lorentz tensor couplings of Refs. [11–15]. The nonminimal vector coupling with square step [17] and smooth step potentials [18] have also appeared in the literature. The one-dimensional vector DKP oscillator was treated in Ref. [16] but we show in this Letter that the three-dimensional case has some very special features such as the question of con- servation of the total angular momentum �J , boundary conditions on the spinor and degeneracy of the spectrum. The conservation of �J is derived from its commutation properties with each term of the DKP equation. The proper boundary condition at the ori- gin follows from the absence of Dirac delta potentials, avoiding in this manner to recourse to plausibility arguments regarding http://dx.doi.org/10.1016/j.physleta.2011.05.067 http://www.ScienceDirect.com/ http://www.elsevier.com/locate/pla mailto:benito@feg.unesp.br mailto:castro@pq.cnpq.br http://dx.doi.org/10.1016/j.physleta.2011.05.067 L.B. Castro, A.S. de Castro / Physics Letters A 375 (2011) 2596–2600 2597 the self-adjointness of the momentum and the finiteness of the kinetic energy, as done by Greiner [19] in the case of the non- relativistic harmonic oscillator. The exact solutions are presented in a closed form and the spectrum presents, beyond the essential degeneracy omnipresent for any central force field, an accidental degeneracy. 2. Vector interactions in the DKP equation With the introduction of interactions, the DKP equation can be written as ( βμpμ − m − V ) ψ = 0 (1) where the more general potential matrix V is written in terms of 25 (100) linearly independent matrices pertinent to the five(ten)- dimensional irreducible representation associated to the scalar (vector) sector. In the presence of interactions jμ satisfies the equation ∂μ jμ + i 2 ψ ( V − η0 V †η0)ψ = 0. (2) Thus, if V is Hermitian with respect to η0 then the four-current will be conserved. The potential matrix V can be written in terms of well-defined Lorentz structures. For the spin-0 sector there are two scalar, two vector and two tensor terms [20], whereas for the spin-1 sector there are two scalar, two vector, a pseudoscalar, two pseudovector and eight tensor terms [21]. The tensor terms have been avoided in applications because they furnish noncausal ef- fects [20,21]. Considering only the vector terms, V is in the form V = βμ A(1) μ + i [ P , βμ ] A(2) μ (3) where P is a projection operator (P 2 = P and P † = P ) in such a way that ψ Pψ behaves as a scalar and ψ[P , βμ]ψ behaves like a vector. A(1) μ and A(2) μ are the four-vector potential functions. No- tice that the vector potential A(1) μ is minimally coupled but not A(2) μ . One very important point to note is that this matrix potential leads to a conserved four-current but the same does not happen if instead of i[P , βμ] one uses either Pβμ or βμ P , as in [5,6,8, 10,12]. As a matter of fact, in Ref. [5] is mentioned that Pβμ and βμ P produce identical results. The DKP equation is invariant under the parity operation, i.e. when �r → −�r, if �A(1) and �A(2) change sign, whereas A(1) 0 and A(2) 0 remain the same. This is because the parity operator is P = exp(iδP )P0η 0, where δP is a constant phase and P0 changes �r into −�r. Because this unitary operator anticommutes with �β and [P , �β], they change sign under a parity transformation, whereas β0 and [P , β0], which commute with η0, remain the same. Since δP = 0 or δP = π , the spinor components have definite parities. The charge-conjugation operation changes the sign of the minimal interaction potential, i.e. changes the sign of A(1) μ . This can be ac- complished by the transformation ψ → ψc = Cψ = C Kψ , where K denotes the complex conjugation and C is a unitary matrix such that Cβμ = −βμC . The matrix that satisfies this relation is C = exp(iδC )η0η1. The phase factor exp(iδC ) is equal to ±1, thus E → −E . Note also that jμ → − jμ , as should be expected for a charge current. Meanwhile C anticommutes with [P , βμ] and the charge-conjugation operation entails no change on A(2) μ . The invariance of the nonminimal vector potential under charge conju- gation means that it does not couple to the charge of the boson. In other words, A(2) μ does not distinguish particles from antiparticles. Hence, whether one considers spin-0 or spin-1 bosons, this sort of interaction cannot exhibit Klein’s paradox. For massive spinless bosons the projection operator is given by [20] P = 1 3 ( βμβμ − 1 ) . (4) Defining Pμ = Pβμ and μ P = βμ P , one can obtain the follow re- lations [22] βμ = Pμ +μ P , Pμβν = P gμν,( Pμ ) P = P (μ P ) = 0, ( Pμ )( Pν ) = (μ P )(ν P ) = 0. (5) Applying P and Pν to the DKP equation and using the relations (5), we have i ( Dμ − A(2) μ )( Pμψ ) = m(Pψ) (6) and i ( Dμ + A(2) μ ) (Pψ) = m(Pμψ), (7) respectively. Here, Dμ = ∂μ + i A(1) μ . Combining these results we obtain[ DμDμ + m2 + ( ∂μ A(2) μ ) − ( A(2) ) μ ( A(2) )μ] (Pψ) = 0. (8) On the other hand, by using (5) jμ can be written as jμ = − 1 m Im [ (Pψ)† Dμ(Pψ) ] . (9) One sees that A(2) μ does not intervene explicitly in the current and, in the absence of the nonminimal potential, (8) reduces to the Klein–Gordon equation in the presence of a minimally coupled po- tential and that all elements of the column matrix Pψ are scalar fields of mass m. It is instructive to note that the form of the two distinct vector couplings in the generalized Klein–Gordon equation has become obvious because the interaction operates under the umbrella of the DKP theory. Otherwise, only the minimal vector coupling could be obtained by applying the minimal substitution ∂μ → ∂μ + i A(1) μ to the free Klein–Gordon equation. 3. The nonminimal vector interaction In this stage, we concentrate our efforts in the nonminimal vector potential A(2) μ = Aμ and use the representation for the βμ matrices given by [11,23] β0 = ( θ 0 0T 0 ) , �β = ( 0̃ �ρ − �ρT 0 ) (10) where θ = ( 0 1 1 0 ) , ρ1 = (−1 0 0 0 0 0 ) , ρ2 = ( 0 −1 0 0 0 0 ) , ρ3 = ( 0 0 −1 0 0 0 ) . (11) 0, 0̃ and 0 are 2 × 3, 2 × 2 and 3 × 3 zero matrices, respectively, while the superscript T designates matrix transposition. The five- component spinor can be written as ψ T = (ψ1, . . . ,ψ5). With this representation the projection operator is P = diag(1,0,0,0,0). In this case P picks out the first component of the DKP spinor. If the terms in the potential Aμ are time-independent one can write ψ(�r, t) = φ(�r)exp(−iEt), where E is the energy of the boson, in such a way that the time-independent DKP equation becomes[ β0 E + iβ i∂i − ( m + i [ P , βμ ] Aμ )] φ = 0. (12) 2598 L.B. Castro, A.S. de Castro / Physics Letters A 375 (2011) 2596–2600 In this case jμ = φβμφ/2 does not depend on time, so that the spinor φ describes a stationary state. In the time-independent case (7) becomes φ2 = 1 m (E + i A0)φ1, (13) �ζ = ( �∇ − �A)φ1 (14) where �ζ = m i (φ3, φ4, φ5) T (15) and (8) furnishes(−∇2 + �∇ · �A + �A2)φ1 = k2φ1 (16) where k2 = E2 − m2 + A2 0. (17) Meanwhile, j0 = E m |φ1|2, �j = 1 m Im ( φ∗ 1 �∇φ1 ) . (18) If we consider spherically symmetric potentials Aμ(�r) = ( A0(r), Ar(r)r̂ ) , (19) then the DKP equation permits the factorization φ1(�r) = uκ (r) r Ylml (θ,ϕ) (20) where Ylml is the usual spherical harmonic, with l = 0,1,2, . . . , ml = −l,−l + 1, . . . , l, ∫ dΩ Y ∗ lml Yl′ml′ = δll′δmlml′ and κ stands for all quantum numbers which may be necessary to characterize φ1. For r �= 0 the radial function u obeys the radial equation d2u dr2 + [ k2 − 2 Ar r − dAr dr − l(l + 1) r2 − A2 r ] u = 0 (21) and because ∇2(1/r) = −4πδ(�r), unless the potentials contain a delta function at the origin, one must impose the homogeneous Dirichlet condition u(0) = 0 [24]. Furthermore, from the normal- ization condition ∫ dτ j0 = ±1 one sees that u must be normalized according to E m ∞∫ 0 dr |u|2 = ±1. (22) Therefore, for motion in a central field, the solution of the three- dimensional DKP equation can be found by solving a Schrödinger- like equation. The other components are obtained through of (13) and (14). Note that the DKP spinor is an eigenstate of the par- ity operator. This happens because η0 = diag(1,1,−1,−1,−1) and the parity of �ζ is opposite to that one of φ1 and φ2. Furthermore, the spin operator Sk = iεklmβlβm [2] satisfies the commutation re- lations[ Sk, β 0] = [ Sk, [ P , β0]] = 0,[ Sk, β l] = iεklmβm, [ Sk, [ P , βl]] = iεklm [ P , βm] (23) so that the total angular momentum �J = �L + �S satisfies[�J , βμpμ ] = [�J , βμ A(1) μ ] = [�J , [P , βμ ] A(2) μ ] = �0 (24) in such a way that the DKP spinor is also an eigenstate of �J 2 and J3. Accordingly, the DKP spinor can be classified by the parity, by the total angular momentum, and its third component, quan- tum numbers. As a matter of fact, �S = ( 0̃ 0 0T �s ) (25) where sk are the 3 × 3 spin-1 matrices (sk)lm = −iεklm . As a result, �S does not act on the two upper components of the DKP spinor. This means that the orbital angular momentum quantum numbers of φ1 and φ2 are good quantum numbers. With the orbital angular momentum quantum number l referring to the two upper com- ponents of the DKP spinor, as before, �ζ in (14) can be written as [25] �ζ = �Yl,l−1,ml √ l 2l + 1 ( d dr + l + 1 r − A(2) r ) u(r) r − �Yl,l+1,ml √ l + 1 2l + 1 ( d dr − l r − A(2) r ) u(r) r . (26) In this last expression, �Y Jlm J (θ,ϕ) are the so-called vector spher- ical harmonics. They result from the coupling of the three- dimensional unit vectors in spherical notation to the eigenstates of orbital angular momentum, form a complete orthonormal set and satisfy �J 2 �Y Jlm J = J ( J + 1)�Y Jlm J , �L2 �Y Jlm J = l(l + 1)�Y Jlm J , J3 �Y Jlm J = m J �Y Jlm J (27) and �Yl,l±1,ml transforms under parity as �Yl,l±1,ml (θ − π,ϕ + π) = (−1)l+1 �Yl,l±1,ml (θ,ϕ). (28) One sees that if the two upper components of the DKP spinor are eigenfunctions of �L2 with an orbital angular momentum quantum number l, the three lower components will be a linear superposi- tion of two types of eigenfunctions of �L2. One of those with orbital angular momentum quantum number l + 1 and the other with l − 1. The fact that the upper and lower components of the DKP spinor have different orbital angular momentum quantum num- bers is related to the fact that �L is not a conserved quantity in the DKP theory. Nevertheless, the orbital angular momentum quantum number of the first component of the DKP spinor equals the to- tal angular momentum quantum number of the DKP spinor, as it should be since φ1 describes a spinless particle. It follows that the parity of the DKP spinor is given by (−1)l . 4. The vector DKP oscillator Let us consider a nonminimal vector linear potential in the form A(2) 0 = m2λ0r, A(2) r = m2λrr (29) where λ0 and λr are dimensionless quantities. Our problem is to solve (21) for u and to determine the allowed energies. One finds that u obeys the second-order differential equation d2u dr2 + [ K 2 − λ2r2 − l(l + 1) r2 ] u = 0 (30) where K = √ E2 − m2 − 3m2λr, λ = m2 √ λ2 r − λ2 0. (31) With u(0) = 0 and ∫ ∞ 0 dr |u|2 < ∞, the solution for (30) with K and λ real is precisely the well-known solution of the Schrödinger equation for the three-dimensional harmonic oscillator (see, e.g. L.B. Castro, A.S. de Castro / Physics Letters A 375 (2011) 2596–2600 2599 [19]). For λ = i|λ| (and K = |K | or K = i|K |), the case of an in- verted harmonic oscillator, the energy spectrum will consist of a continuum corresponding to unbound states. We shall limit our- selves to study the case of bound-state solutions. The asymptotic behavior of (30) and the conditions u(0) = 0 and ∫ ∞ 0 dr |u|2 < ∞ dictate that the solution close to the origin valid for all values of l can be written as being proportional to rl+1, and proportional to e−λr2/2 as r → ∞. It is convenient to introduce the following new variable and parameters: z = λr2, a = 1 2 ( l + 3 2 − K 2 2λ ) , b = l + 3 2 (32) so that the solution for all r can be expressed as u(r) = rl+1 × e−λr2/2 w(r), where w is a regular solution of the confluent hyper- geometric equation (Kummer’s equation) [26] z d2 w dz2 + (b − z) dw dz − aw = 0. (33) The general solution of (33) is given by [26] w = A M(a,b, z) + B z1−b M(a − b + 1,2 − b, z) (34) with arbitrary constants A and B . The confluent hypergeometric function (Kummer’s function) M(a,b, z), or 1 F1(a,b, z), is M(a,b, z) = �(b) �(a) ∞∑ n=0 �(a + n) �(b + n) zn n! , (35) and �(z) is the gamma function. The second term in (34) has a singular point at z = 0, so we set B = 0. In order to furnish normalizable φ1, the confluent hypergeometric function must be a polynomial. This is because M(a,b, λr2) goes as eλr2 as r goes to infinity unless the series breaks off. This demands that a = −n, where n is a nonnegative integer. We put N = 2n + l, whence the requirement a = −n implies into |E| = m √ 1 + 3λr + (2N + 3) √ λ2 r − λ2 0, N = 0,1,2, . . . . (36) Note that M(−n,b, λr2) is proportional to the generalized Laguerre polynomial L(b−1) n (λr2), a polynomial of degree n. Therefore, by using the normalization condition | ∫ dτ j0| = |E|/m ∫ dτ |φ1|2 = 1, with |E| �= 0, and that the generalized Laguerre polynomial is stan- dardized as [26] ∞∫ 0 dξ ξαe−ξ [ L(α) n (ξ) ]2 = �(α + n + 1) n! (37) one determines A in (34) and obtains φ1 = √√√√2mλl+3/2( N−l 2 )! |E|�( N+l+3 2 ) rle−λr2/2L(l+1/2) N−l 2 ( λr2)Ylml (θ,ϕ). (38) Note that l can take the values 0,2, . . . , N when N is an even number, and 1,3, . . . , N when N is an odd number. All the en- ergy levels with the exception of that one for N = 0 are degen- erate. The degeneracy of the level of energy for a given principal quantum number N is given by (N + 1)(N + 2)/2. Notice that the condition λ ∈ R requires that |λr | > |λ0|, meaning that the space component of the potential must be stronger than its time com- ponent. There is an infinite set of discrete energies (symmetrical about E = 0 as it should be since Aμ does not distinguish parti- cles from antiparticles) irrespective to the sign of λ0. In general, |E| is higher for λr > 0 than for λr < 0. It increases with the principal quantum number and it is a monotonically decreasing function of λ0. For λr < 0 and λ0 = 0 the spectrum acquiesces |E| = m for N = 0. In order to insure the reality of the spectrum, the coupling constants λ0 and λr satisfy the additional constraint√ λ2 r − λ2 0 > −(1 + 3λr)/(2N + 3) in such a way that there can be no bound states for λr < 0 with small principal quantum numbers and |λr | enough small. This means that for λr < 0 and |λr | enough small a number of solutions with the smallest principal quantum numbers does not exist. For |λr | |λ0| we have a very high den- sity of very delocalized states (because λ 0). For |λr | � |λ0| one has that |E| m √ 1 + 3λr + (2N + 3)|λr | (39) so that |E| > m for λr > 0. Concerning λr < 0, as far as λr de- creases, the spectrum moves towards E = 0, except for λ0 = 0 which maintains |E| � m. On the other hand, in the weak-coupling limit, |λr | � 1 and |λ0| � 1, |E| m for small quantum numbers, and (36) becomes |E| m [ 1 + 3 2 λr + ( N + 3 2 )√ λ2 r − λ2 0 ] . (40) Because of this equally spaced energy spectrum, it can be said that the linear potential given by (29) describes a genuine vector DKP oscillator. It is obvious that, despite the effective harmonic oscilla- tor potential appearing in (30) and the spectrum given by (40), in a nonrelativistic scheme would appear the sum of the two inter- vening potentials in the Schrödinger equation and no bound-state solutions would be possible for λr < 0 and |λr | > |λ0|. Therefore, the weak-coupling limit does not correspond to the nonrelativistic limit and so we can say that the nonminimal vector linear poten- tial given by (29) is an intrinsically relativistic potential in the DKP theory. 5. Conclusions We showed that minimal and nonminimal vector interactions behave differently under the charge-conjugation transformation. In particular, nonminimal vector interactions have no counterparts in the Klein–Gordon theory. The conserved charge current plus the charge conjugation operation are enough to infer about the ab- sence of Klein’s paradox under nonminimal vector interactions, or its possible presence under minimal vector interactions. Although Klein’s paradox cannot be treated as unworthy of regard in the DKP theory with minimally coupled vector interactions, it never makes its appearance in the case of nonminimal vector interactions be- cause they do not couple to the charge. Nonminimal vector interac- tions have the very same effects on both particles and antiparticles and so in the case of a pure nonminimal vector coupling, both par- ticle and particle energy levels are members of the spectrum, and the particle and antiparticle spectra are symmetrical about E = 0. If the interaction potential is attractive (repulsive) for bosons it will also be attractive (repulsive) for antibosons. However, there is no crossing of levels because possible states in the strong field regime with E = 0 are in fact unnormalizable. These facts imply that there is no channel for spontaneous boson–antiboson creation and for that reason the single-particle interpretation of the DKP equation is ensured. The charge conjugation operation allows us to migrate from the spectrum of particles to the spectrum of antipar- ticles and vice versa just by changing the sign of E . This change induces no change in the nodal structure of the components of the DKP spinor and so the nodal structure of the four-current is pre- served. We showed that nonminimal vector couplings have been used improperly in the phenomenological description of elastic meson- nucleus scatterings potential by observing that the four-current 2600 L.B. Castro, A.S. de Castro / Physics Letters A 375 (2011) 2596–2600 is not conserved when one uses either the matrix Pβμ or βμ P , even though the bilinear forms constructed from those matrices behave as true Lorentz vectors. The space component of the non- minimal vector potential cannot be absorbed into the spinor and we showed that the space component of the nonminimal vector potential could be irrelevant for the formation of bound states for potentials vanishing at infinity but its presence is an essential in- gredient for confinement. The complete solution of the DKP equation with spherically symmetric nonminimal vector potentials was found by recurring to vector spherical harmonics due to the expression appearing in (14) with �∇ in spherical coordinates acting on a function of r multiplied by Ylml (θ,ϕ). A similar procedure resulting in a set of coupled dif- ferential equations for the components of the spinor has already appeared in the literature [23]. Here, instead of a set of coupled first-order equations, the DKP equation was mapped into a Sturm– Liouville problem for the first component of the spinor and the remaining components were expressed in terms of the first one in a simple way. In this process, the conserved four-current was also expressed in terms of the first component of the DKP spinor in such a way that the searching for the solutions of the DKP equa- tion becomes more clear and transparent. The conservation of the total angular momentum was derived from its commutation prop- erties with each term of the DKP equation. The solution for a nonminimal linear potential was found by solving a Schrödinger-like problem for the nonrelativistic harmonic oscillator for the first component of the spinor. The behavior of the solutions for this sort of DKP oscillator was discussed in de- tail. Instead of imposing boundary conditions at the origin by re- curring to plausibility arguments regarding the self-adjointness of the momentum and the finiteness of the kinetic energy, as done by Greiner [19] in the case of the nonrelativistic harmonic os- cillator, the proper boundary conditions were imposed in a sim- ple way by observing the absence of Dirac delta potentials. The exact solutions were presented in a closed form and the spec- trum presents, beyond the essential degeneracy omnipresent for any central force field, an accidental degeneracy. That model rein- forced the absence of Klein’s paradox for nonminimal vector inter- actions. Acknowledgements We acknowledge financial support from CAPES and CNPq. References [1] G. Petiau, Acad. R. Belg., A. Sci. Mém. Collect. 16 (1936) 1; N. Kemmer, Proc. R. Soc. A 166 (1938) 127; R.J. Duffin, Phys. Rev. 54 (1938) 1114. [2] N. Kemmer, Proc. R. Soc. A 173 (1939) 91. [3] R.A. Krajcik, M.M. Nieto, Phys. Rev. D 10 (1974) 4049; R.A. 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