# Quantum Speed Limit for Relativistic Spin-0 and Spin-1 Bosons on Commutative and Noncommutative Planes

Authors: Kang Wang; Yu-Fei Zhang; Qing Wang; Zheng-Wen Long; Jian Jing
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

Quantum speed limits of relativistic charged spin-0 and spin-1 bosons in the background of a homogeneous magnetic field are studied on both commutative and noncommutative planes. We show that, on the commutative plane, the average speeds of wave packets along the radial direction during the interval in which a quantum state is evolving from an initial state to the orthogonal final one can not exceed the speed of light, regardless of the intensities of the magnetic field. However, due to the noncommutativity, the average speeds of the wave packets on noncommutative plane will exceed the speed of light in vacuum provided the intensity of the magnetic field is strong enough. It is a clear signature of violating Lorentz invariance in the relativistic quantum mechanics region.


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## Body

## 1. Introduction

Duffin-Kemmer-Petiau (DKP) equation is a first-order relativistic wave equation [1–3]. Different from Dirac equation which describes spin-1/2 fermions, DKP equation describes spin-0 and spin-1 bosons. DKP equation takes the form(1)iβμ∂μ-m0ψ=0,where m0 is the rest mass and the matrices βμ satisfy the algebraic relation(2)βμβνβα+βαβνβμ=gμνβα+gανβμ.Here, gμν=gμν=diag(+,-,-,-) is the metric tensor. The algebra (2) has three different representations: a (one-dimensional) trivial representation, a five-dimensional representation describing spin-0 bosons, and a ten-dimensional representation describing spin-1 bosons. As a Dirac-type relativistic quantum mechanical model, DKP equation with various potentials has been studied in the past years [4–8]. The magnetic coupling in DKP equation is also considered [9]. When magnetic field is taken into consideration, one should introduce the magnetic potentials by the minimal coupling (we set ħ=c=1), pi→pi+qAi, where q and Ai are the charge and the magnetic potentials, respectively. Since DKP equation is analogous with Dirac equation, the author of [9] compares it with Dirac equation in detail in that paper.On the other hand, the minimum time of a quantum state evolving from an initial state to the orthogonal final one in Hilbert space is of great importance in the field of quantum computation, quantum control, and quantum metrology. In fact, it has attracted attention for a long time [10]. At present, there are two different descriptions of the minimum time for a quantum system evolving from an initial state to the orthogonal final state. One is given by the expression Tmin=πħ/2ΔE, in which ΔE is the energy variance, defined by ΔE=〈ψ|H2|ψ〉-〈ψ|H|ψ〉2, with H being the Hamiltonian of the system and |ψ〉 being a specific superposition of eigenstates of H [11]. The other is given in [12], which states that the minimum time is given by Tmin=πħ/2(E¯-E0), where E¯ and E0 are the mean energy and the lowest energy of the state which participates in the superposition. Obviously, the results of [11, 12] will be equivalent if the condition ΔE=E¯-E0 is satisfied. This condition can be simply satisfied by superposing two steady states homogeneously. According to the results in [11, 12], it is natural to assume that the minimum time should be given by Tmin=max⁡{πħ/2ΔE,πħ/2(E¯-E0)} [13]. A unified bound which contains both ΔE and E¯ is considered [14].An interesting connection between the minimum time of the quantum state evolving in Hilbert space and the average speed of an electron wave packet travelling in spatial space is constructed in a recent paper [15]. In this paper, the authors study a relativistic electron coupling to a homogeneous magnetic field. They find that the average speed of this electron wave packet moving in radial direction during the interval in which a quantum state is evolving from an initial state to the orthogonal final one in Hilbert space is less than the speed of light in vacuum, regardless of intensities of the magnetic field one applies. It seems that, as expected, Lorentz invariance is not violated in this relativistic quantum mechanical model. However, Lorentz invariance would be violated in the nonrelativistic limit of this model since the average speed of the electron wave packet in radial direction exceeds the speed of light in vacuum provided the intensity of the magnetic field is strong enough.The work of [15] is generalized to noncommutative case recently. Noncommutativity becomes one of the foci of theoretical research due to superstring theories in recent years [16–19]. There are a large number of papers studying quantum field theories in noncommutative space [20–25]. Nonrelativistic noncommutative quantum mechanical models, such as noncommutative harmonic oscillator and noncommutative Landau problem, have been studied extensively [26–32]. The relativistic quantum mechanical models on noncommutative space are investigated since the work of [33]. Some geometrical phases in noncommutative relativistic quantum theory are studied in [34–36] recently. DKP equation in noncommutative space, especially the DKP oscillator in noncommutative space, has also been investigated by some authors [37–42]. Interestingly, it is found that the noncommutativity even has some relationships with Jaynes-Cumming model in quantum optics context [43].In [44], the authors study the noncommutative (both the coordinates and momenta are noncommutative simultaneously) Dirac equation. They find that Lorentz invariance will be violated in noncommutative Dirac equation since the average speed of an electron wave packet exceeds the speed of light in vacuum if the magnetic field is strong enough. In fact, the problem of violating Lorentz invariance has been considered [45–47]. In [46], the authors find that because of noncommutativity, Lorentz invariance will be violated in the noncommutative quantum electrodynamics (QED) since the electromagnetic wave travels in different speeds along different directions at the presence of a background magnetic field. A similar result is also obtained in [47]. This problem is also considered semiclassically from both noncommutative and gravitational points of view [48–50].In this paper, we investigate the problems of whether Lorentz invariance is violated for spin-0 or spin-1 relativistic bosons in two-dimensional spatial space. We study the commutative case firstly and then generalize to the noncommutative case. The organization of this paper is as follows. In the next section, we study spin-0 and spin-1 charged bosons coupling to homogeneous magnetic fields on commutative plane. Then, in Section3, we study the noncommutative case. Some remarks and further discussions will be presented in the last section.


## 2. Spin-0 and Spin-1 Charged Bosons Coupling to a Homogenous Magnetic Field on Commutative Plane

In this section, we study spin-0 and spin-1 bosons coupling to a homogeneous magnetic field on a commutative plane. We start our studies from the spin-0 bosons.As stated before, the five-dimensional representation of the algebra (2) describes spin-0 boson. The explicit expressions of five-dimensional βμ matrices are(3)β0=Θ0¯0¯T0,βi=0~ρi-ρiT0,in which(4)Θ=0110,ρ1=-100000,ρ2=0-10000,ρ3=00-1000.Here 0¯, 0~, and 0 are 2×3, 2×2 and 3×3 zero matrices, respectively. Choosing symmetric gauge Ai=-1/2Bϵijxj and introducing Larmor frequency ωL=qB/2m0, we write (1) in the form(5)β0E+β1p1-m0ωLx2+β2p2+m0ωLx1-m0ψx1,x2,t=0,where ψ(x1,x2,t) is a five-component wave function ψ(x1,x2,t)=(ψ1,ψ2,ψ3,ψ4,ψ5)T(x1,x2,t).Substituting the explicit expressions ofβμ matrices (3) into the above equation, we get a set of equations(6a)-m0ψ1+Eψ2-p1-m0ωLx2ψ3-p2+m0ωLx1ψ4=0,(6b)Eψ1-m0ψ2=0,(6c)p1-m0ωLx2ψ1-m0ψ3=0,(6d)p2+m0ωLx1ψ1-m0ψ4=0,(6e)ψ5=0.Obviously, the five components (ψ1,ψ2,ψ3,ψ4,ψ5) are not independent to each other.Combining the above equations, we get the dynamical equation of the componentψ1. It is(7)E2-m02ψ1=2m0HLψ1,where HL is the Landau Hamiltonian:(8)HL=12m0p12+p22+12m0ωL2x12+x22+ωLLz,with Lz=xpy-ypx being the angular momentum along z direction. Obviously, the Hamiltonian (8) describes a planar nonrelativistic charged particle interacting with a homogeneous perpendicular magnetic field.Equation (7) is easily solved. The eigenvalues and eigenfunctions, respectively, are [51](9)En,ml=±m02+2m0n+ml+1ωL,(10)ψ1=Fn,mlr,φ=-1n-ml/2n-ml/2!πn+ml/2!n-ml/2!×ααrmlLn-ml/2mlα2r2e-1/2α2r2eimlφ,where n=0,1,2,…,ml=-n,-n+2,…,n-2,n, α=m0ωL=qB/2, and L(n-|ml|/2)|ml| is Laguerre’s polynomials. Thus, the multicomponent wave function ψ can be obtained if one chooses a specific solution of the component ψ1.In order to simplify our calculation further and make a comparison with the spin-1/2 case [15], we choose two steady states whose first components, respectively, are (we only consider the positive-energy sector)(11)ψ1=F0,0r,φ=1παe-1/2α2r2,(12)ψ1=F2,0=-1πα1-α2r2e-1/2α2r2.Then, after some direct calculations, we get these two steady states. They are(13)ϕ0,0=N0,0F0,0E0,0m0F0,0iαm0F1,1αm0F1,10e-iE0,0t,ϕ2,0=N2,0F2,0E2,0m0F2,0iαm02F3,1-F1,-1αm0F3,1+F1,-10e-iE2,0t,where(14)N0,0=m0m02+E0,02+2α2,N2,0=m0m02+E2,02+6α2are two normalization constants.As we have mentioned, there are some controversies on the minimum time of a quantum state evolving from an initial state to the orthogonal one. In order to avoid these problems, we superpose these two steady states (13) homogeneously; that is, (15)Ψr,φ,t=12ϕ0,0r,φe-iE0,0t+ϕ2,0r,φe-iE2,0t.According to [11, 12], the minimum time for a state Ψ(r,φ,t) evolving from the initial state Ψ(r,φ,0) to the final orthogonal state Ψ(r,φ,Tmin) is given by Tmin=πħ/2(E¯-E0,0), or, equivalently, Tmin=πħ/2ΔE, where E¯ and ΔE are the mean energy and energy variance on the state |Ψ(t)〉, respectively. After direct calculation, we get(16)Tmin=πm02+3qB-m02+qB.The average radial displacement of the spinless boson in the interval[0,Tmin] is given by(17)Δr=ΨTminrΨTmin-Ψ0rΨ0=2ϕ0,0rϕ2,0.Substituting (13) and (15) into the above equation and after some direct calculations, we arrive at(18)Δr=14+338πα.Thus, the average speed of the wave packet of this charged boson along the radial direction during this interval is given by v¯=Δr/Tmin. The result is(19)v¯=2+3382πqBm02+3qB-m02+qB.The average speed v¯ reaches its maximum when B→∞. It is(20)v¯max=limB→∞v¯≐0.5254<1.It shows that the average radial speed of this spinless boson wave packet is less than the speed of light in vacuum (c=1) no matter the intensities of the magnetic field. Therefore, Lorentz invariance is not violated in this relativistic quantum mechanics model.Now, we study the DKP equation with10-dimensional representation of the algebra (2). As is known, it describes spin-1 bosons. The explicit expressions of β matrices are(21)β0=00ˇ0ˇ0ˇ0ˇT03×3I3×303×30ˇTI3×303×303×30ˇT03×303×303×3,βi=00ˇei0ˇ0ˇT03×303×3-iSi-eiT03×303×303×30ˇT-iSi03×303×3,in which(22)0ˇ=0,0,0,e1=1,0,0,e2=0,1,0,e3=0,0,1,S1=00000-i0i0,S2=00i000-i00,S3=0-i0i00000.Using the explicit expressions ofβ matrices, we write the DKP equation in the component form(23a)p1-m0ωLx2ψ5+p2+m0ωLx1ψ6=m0ψ1,(23b)Eψ5+p2+m0ωLx1ψ10=m0ψ2,(23c)Eψ6-p1-m0ωLx2ψ10=m0ψ3,(23d)Eψ7-p2+m0ωLx1ψ8+p1-m0ωLx2ψ9=m0ψ4,(23e)Eψ2-p1-m0ωLx2ψ1=m0ψ5,(23f)Eψ3-p2+m0ωLx1ψ1=m0ψ6,(23g)Eψ4=m0ψ7,(23h)p2+m0ωLx1ψ4=m0ψ8,(23i)-p1-m0ωLx1ψ4=m0ψ9,(23j)-p2+m0ωLx1ψ2+p1-m0ωLx2ψ3=m0ψ10.According to (23d), (23g), (23h), and (23i), we get the dynamical equation of the component ψ4(24)E2-m02ψ4=2m0HLψ4,where HL is given in (8).Therefore, the solutions ofψ4 in spin-1 case take the same form as (9) and (10); that is,(25)En,ml=±m02+2m0n+ml+1ωL,(26)ψ4=Fn,mlr,φ=-1n-ml/2n-ml/2!πn+ml/2!n-ml/2!×ααrmlLn-ml/2mlα2r2e-1/2α2r2eimlφ.We choose a special solution for componentsψ2 and ψ3, that is, ψ2=0 and ψ3=0. As a result, the components ψ1=ψ5=ψ6=0. Thus, the ten-component eigenfunction can be obtained if we select a specific solution for ψ4 from (26). In order to simplify our calculations and make a comparison with [15], we choose two specific solutions for ψ4. They are(27)ψ4=F0,0=1παe-1/2α2r2,ψ4=F2,0=-1πα1-α2r2e-1/2α2r2.Thus, the explicit expressions for two steady states we want to superpose are(28)ϕ0,0=N0,0000F0,000E0,0m0F0,0αm0F1,1-iαm0F1,10,ϕ2,0=N2,0000F2,000E2,0m0F2,0αm02F3,1+F1,-1-iαm02F3,1-F1,-10,where N0,0, N2,0 are identical to (14).We superpose these two steady states (28) homogeneously in the same manner as (15). The eigenvalues are equal to the ones in spin-0 case; thus the minimum time for the superposition state evolving from the initial state Ψ(r,φ,0) to the final orthogonal state Ψ(r,φ,Tmin) is the same as (16). The radial displacement of the wave packet during the interval [0,Tmin] can be calculated straightforwardly. The result is identical to (17). Therefore, the average velocity along radial direction of this spin-1 boson is the same as in (19). The maximum average speed is achieved when the intensity of the magnetic field tends to infinity. The result is identical to (20). It is less than the speed of light in vacuum. Thus, it shows that Lorentz invariance is not violated.


## 3. Spin-0 and Spin-1 Bosons on Noncommutative Plane

As we have shown, the average speeds of the wave packets of spin-0 and spin-1 bosons along the radial direction will not exceed the speed of light in vacuum, regardless of the intensity of the magnetic field. A natural question is does the Lorentz invariance get violated in the noncommutative DKP equation? In the following, we will investigate DKP equation in noncommutative2+1-dimensional phase space.There are two ways to study noncommutative theories in noncommutative space. One is to replace the ordinary product by the Moyal (∗) product [52]:(29)f∗gxI=fxIei/2∂←/∂xIθIJ∂→/∂xJgxI,where f, g are two N-dimensional functions and θIJ, I,J=1,2,…,N is a rank-two antisymmetrical tensor. When the noncommutativity between momenta is taken into account, the above ∗-product should be generalized to the form(30)f∗gxI,pI=fxI,pJexp⁡i2∂←∂xIθIJ∂→∂xJ+∂←∂pIηIJ∂→∂pJgxI,pI,in which ηIJ, I,J=1,2,…,N is also a rank-two antisymmetrical tensor.The other equivalent way is to introduce the noncommutative algebra among variables(xi,pi). For the noncommutative 2+1-dimensional phase space, the noncommutative algebra is given by(31)x^i,x^j=iθϵij,p^i,p^j=iηϵij,i,j=1,2,where θ and η are two real parameters and ϵij is the 2-dimensional antisymmetric tensor. It can be easily verified that the ∗-product (30) can reproduce the commutator (31) directly for 2-dimensional phase space.For the sake of consistency, the commutator among coordinates and momenta should be modified as [53] (in order to avoid the problem of unitarity, we only consider noncommutativity among coordinates and momenta)(32)x^i,p^j=i1+θη4δij.By applying the latter way to study the noncommutative theories, one assumes that the dynamical equations in noncommutative quantum mechanics take the same form as their commutative counterparts. However, variables in dynamical equation are replaced by the corresponding noncommutative ones. Therefore, the noncommutative version of spin-0 DKP equations is nothing but to replace the variables(xi,pi) by (x^i,p^i) which satisfy the algebraic relations (31) and (32) in (6a), (6b), (6c), (6d), and (6e). Choosing the symmetric gauge A^i=-B/2ϵijx^j [54] and combining them, we get(33)E2-m02ψ1=2m0H^LNCψ1,where(34)H^LNC=12m0p^12+p^22+12m0ωL2x^12+x^22+ωLL^z,with L^z=x^p^y-y^p^x being the noncommutative angular momentum along z direction. Hamiltonian (34) describes a planar nonrelativistic charged particle interacting with a homogeneous perpendicular magnetic field on the noncommutative planes (31) and (32).We map noncommutative variables(x^i,p^i) to commutative ones (xi,pi) which satisfy the standard Heisenberg algebra(35)xi,xj=pi,pj=0,xi,pj=iδij.It is straightforward to check that the map from noncommutative variables to commutative ones can be realized by [53](36)x^i=xi-θ2ϵijpj,p^i=pi+η2ϵijxj.In terms of commutative variables(xi,pi), we find that the dynamical equation for component ψ1 takes the same form as (33). The corresponding Hamiltonian (34) in terms of commutative variables becomes(37)HLNC=12Meffp12+p22+12MeffΩ2x12+x22+ΩLz,in which(38)Meff=m0κ2,Ω2=1m0κγ2,where κ=1-qθB/4, γ=qB/2-η/2 are two parameters.The eigenvalues and eigenfunctions of Hamiltonian (37) are(39)En,ml=±m02+2m0n+ml+1Ω,(40)ψ1=Fn,mlr,φ=-1n-ml/2n-ml/2!πn+ml/2!n-ml/2!×α′α′rmlLn-ml/2mlα′2r2e-1/2α′2r2eimlφ,where n=0,1,2,…,ml=-n,-n+2,…,n-2,n and (41)α′=MeffΩ.Other components can be calculated directly from the commutative version of (6a), (6b), (6c), (6d), and (6e).In order to make a comparison with commutative version, we choose two solutions ofψ1 as(42)ψ1=F0,0=1πα′e-1/2α′2r2,ψ1=F2,0=-1πα′1-α′2r2e-1/2α′2r2.Then, the corresponding two steady states are(43)ϕ0,0=N0,0F0,0E0,0m0F0,0iα′κm0F1,1α′κm0F1,10,ϕ2,0=N2,0F2,0E2,0m0F2,0iα′κm02F3,1-F1,-1α′κm02F3,1+F1,-10,where N0,0 and N2,0 are two normalization constants. They are(44)N0,0=m0m02+E0,02+2α′2κ2,N2,0=m0m02+E2,02+6α′2κ2,respectively.We superpose two steady states (43) homogeneously in the same manner as (15). According to [11, 12], the minimum time for the superposition state evolving from the initial state Ψ(r,φ,0) to the final orthogonal one Φ(r,φ,Tmin) is given by Tmin=πħ/2(E¯-E0,0)=πħ/2ΔE. Substituting the eigenvalues (39) for states ϕ0,0 and ϕ2,0 into the expression of the minimum time, we get(45)Tmin=πm02+6m0Ω-m02+2m0Ω.Accordingly, the displacement along the radial direction during the period of time Tmin is given by the expression Δr=|〈Ψ(Tmin)|r|Ψ(Tmin)〉-〈Ψ(0)|r|Ψ(0)〉|=2|〈ϕ0,0|r|ϕ2,0〉|. Substituting this superposition state into the expressions of Δr and taking the limit of B→∞, we get(46)Δr=ϕ0,0rϕ2,0=2+338α′.Therefore, the average speed along the radial direction of wave packet during the interval [0,Tmin] in the limit B→∞ is given by(47)v¯=ΔrTmin≐0.5254κ.Compared with the commutative case, we find that due to spatial noncommutativity, there is an extra factor|κ|=|1-qθB/4|. It is this factor which enables average radial speed of the wave packet to exceed the speed of light in vacuum provided the intensity of the magnetic field is strong enough. It is a clear evidence of violating Lorentz invariance in this noncommutative relativistic quantum mechanical model.Now, we study the noncommutative spin-1 DKP equation. In noncommutative plane, the dynamical equations take the same form as (23a), (23b), (23c), (23d), (23e), (23f), (23g), (23h), (23i), and (23j) except that variables (xi,pi) are replaced by noncommutative ones (x^i,p^i).Mapping noncommutative variables(x^i,p^i) to commutative ones (xi,pi), we get(48a)κp1-γx2ψ5+κp2+γx1ψ6=m0ψ1,(48b)Eψ5+κp2+γx1ψ10=m0ψ2,(48c)Eψ6-κp1-γx2ψ10=m0ψ3,(48d)Eψ7-κp2+γx1ψ8+κp1-γx2ψ9=m0ψ4,(48e)Eψ2-κp1-γx2ψ1=m0ψ5,(48f)Eψ3-κp2+γx1ψ1=m0ψ6,(48g)Eψ4=m0ψ7,(48h)κp2+γx1ψ4=m0ψ8,(48i)-κp1-γx2ψ4=m0ψ9,(48j)-κp2+γx1ψ2+κp1-γx2ψ3=m0ψ10.The dynamical equation for componentψ4 can be obtained from above equations. It is(49)E2-m02ψ4=2m0HLNCψ4,where HLNC is given in (37). The solutions to (49) can be obtained in the standard way. The eigenvalues are identical to (39) and the corresponding solutions of component ψ4 are identical with the solutions of ψ1 in noncommutative spin-0 case (40).Similar with the commutative version, we setψ2=ψ3=0. Then the other components can be determined by (48a), (48b), (48c), (48d), (48e), (48f), (48g), (48h), (48i), and (48j). For the sake of comparing with the corresponding commutative case, we choose two specific solutions of ψ4:(50)ψ4=F0,0=1πα′e-1/2α′2r2,ψ4=F2,0=-1πα′1-α′2r2e-1/2α′2r2.Then, the two steady states we prepared are(51)ϕ0,0=N0,0000F0,000E0,0m0F0,0α′κm0F1,1-iα′κm0F1,10,ϕ2,0=N2,0000F2,000E2,0m0F2,0α′κm02F3,1+F1,-1-iα′κm02F3,1-F1,-10.Superposing these two steady states homogeneously, we get the superposition state which takes the same form as (15). Then the minimum time for this state evolving from the initial state Ψ(r,φ,0) to the orthogonal final one Ψ(r,φ,Tmin) is identical to (45). The displacement along the radial direction can also be calculated directly. It is equivalent to (46) in the limit of B→∞. Thus, the average speed of the wave packet during interval [0,Tmin] is given by (47). Compared with the commutative counterpart, there is an extra factor |κ|=|1-qθB/4| which enables the average speed along the radial direction to exceed light in vacuum. Therefore, it means that Lorentz invariance is also violated in this relativistic spin-1 boson due to noncommutativity.


## 4. Conclusions and Remarks

In this paper, we investigate the problem of whether Lorentz invariance is violated in noncommutative relativistic quantum mechanics regime. It is known that there are two frames in studying the noncommutative theories [55, 56]. One is the C-frame, in which one works in the commutative space by mapping the noncommutative variables into commutative ones. The other frame is the NC-frame, in which one works in the noncommutative phase spaces (31) and (32) directly. Since we have to calculate the wave functions explicitly, it is more convenient to work in the C-frame in our studies. It is an interesting task to study this problem in NC-frame directly.In fact, the violation of Lorentz invariance due to noncommutativity has been noticed for more than 10 years in [45–47]. Nevertheless, there is a difference between [46, 47] and ours. In [46, 47], the authors investigate the propagation of electromagnetic wave in noncommutative space. The electromagnetic wave, from the quantum point of view, is photons, which are massless.We study the charged massive spin-0 and spin-1 relativistic bosons in the presence of homogeneous magnetic fields in2-dimensional space. Both commutative and noncommutative cases are studied. According to the theory of special relativity, the speed of a massive particle can not exceed the speed of light in vacuum. In our studies, we find that the average radial speeds of wave packets during the interval [0,Tmin] in commutative plane are less than the speed of light in vacuum, no matter how strong the intensity of the magnetic field is. However, when noncommutativity is taken into account, we find that the average radial speed takes the form v¯≐0.5254|κ|=0.5254|1-qθB/4|, which will exceed the speed of light in vacuum if the intensity of the magnetic field is strong enough. It conflicts with the special relativity directly. Therefore, it indicates that Lorentz invariance will be violated in noncommutative space in quantum mechanics regime.It is known that there are redundant degrees of freedom in DKP equation. For the spin-0 case, the physical degree of freedom is easy to obtain. However, it is not straightforward to get the degrees of freedom for spin-1 case [57]. The solutions (28) and (51) are based on ψ2=ψ3=0, which are special solutions to DKP equation. It may be worthwhile to study whether the same conclusions will still be held for the general solutions of the DKP equation.



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