Synthesis of Relativistic Wave Equations : The Noninteracting Case

The Duffin-Kemmer-Petiau (DKP) equations [1–3], describing spin 0 and spin 1 mesons, are becoming increasingly useful due to their applications to problems in particle and nuclear physics [4–13]. It is well known that the DKP equations contain redundant components since only 2(2s + 1) components are needed to describe free spin s particles with nonzero rest masses [14, 15] while s = 0 and s = 1 DKP equations contain 5 and 10 components, respectively. The presence of redundant components in DKP equations leads for some interactions to such nonphysical effects as superluminal velocities [16, 17] (see also [18–20] for s = 3/2, 2 cases). On the other hand, physically acceptable equations for arbitrary spin can be obtained by removing redundant components with use of additional covariant condition [14, 15]. It seems that presence of redundant components means that the DKP equations have internal structure. The motivation of this work is our recent discovery that solutions of subequations of the s = 0 and s = 1 DKP equations fulfill the Dirac equation [21, 22]. On the other hand, these solutions do not contain all spinor components and are thus noncovariant solutions of covariant equations. We studied this problem in [23, 24]. In the present work, we show that, in the free case, full covariant solutions of the s = 0 and s = 1 DKP equations are generalized solutions of the Dirac equation. This finding may provide a basis for a synthesis of covariant particle equations, alternative to the classical Foldy programme [25]. The paper is organized as follows. In Section 2 the Dirac equation and the Duffin-Kemmer-Petiau equations for s = 0 and s = 1 are described. It is shown in Section 3 that in the noninteracting case solutions of these DKP equations are generalized matrix solutions of the Dirac equation. We discuss our findings in the last section.

The motivation of this work is our recent discovery that solutions of subequations of the  = 0 and  = 1 DKP equations fulfill the Dirac equation [21,22].On the other hand, these solutions do not contain all spinor components and are thus noncovariant solutions of covariant equations.We studied this problem in [23,24].In the present work, we show that, in the free case, full covariant solutions of the  = 0 and  = 1 DKP equations are generalized solutions of the Dirac equation.This finding may provide a basis for a synthesis of covariant particle equations, alternative to the classical Foldy programme [25].
The paper is organized as follows.In Section 2 the Dirac equation and the Duffin-Kemmer-Petiau equations for  = 0 and  = 1 are described.It is shown in Section 3 that in the noninteracting case solutions of these DKP equations are generalized matrix solutions of the Dirac equation.We discuss our findings in the last section.

Advances in Mathematical Physics
In the case of  = 0 representation equation ( 3) can be written as with Ψ in (3) defined as In the case of  = 1 (3) reduces to with Ψ in (3) equalling where   are real and  ] are purely imaginary (alternatively, where   and  ] are real).The  = 1 condition,  ]  ] = 0, follows from the second equation of ( 7) due to antisymmetry of tensor  ] .Equations for spin 1 bosons (7) were first written by Proca [29].
We will now split the  = 1 Dirac equations (18).Substituting expressions for  Ḃ  and   Ḃ (cf.[22]), we obtain a system of eight equations: where all equations are arranged into two subsets (20a) and (20b) and we have not assumed yet that  12 =  21 .
Since in (21a) and (21b) there is the same differential operator we can write these equations as a single Dirac equation.We note, however, that the Dirac matrices in (16) and (22) are different.We thus first transform γ matrices unitarily to get   defined in (16):

( 23 )
Now, we can write (21a) and (21b) as a single Dirac equation and with the same representation of   matrices as in(15):