# Synthesis of Relativistic Wave Equations: The Noninteracting Case

Authors: Andrzej Okniński
Journal: Advances in Mathematical Physics
 (2015)

---

## Abstract

We study internal structure of the Duffin-Kemmer-Petiau equations for spin 0 and spin 1 mesons. We show that in the noninteracting case full covariant solutions of thes
=
0 and s
=
1 DKP equations are generalized solutions of the Dirac equation.


---

## Body

## 1. Introduction

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
2
s
+
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 thes
=
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 Section2 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.


## 2. Relativistic Wave Equations

In what follows we use conventions and definitions introduced in [22]. The Dirac equation, describing spin 1
/
2 elementary particles, is(1)
γ
μ
p
μ
Ψ
=
m
Ψ
,where γ
μ are 4
×
4 matrices fulfilling [26, 27](2)
γ
μ
γ
ν
+
γ
ν
γ
μ
=
2
g
μ
ν
I
4
×
4
,where g
μ
ν
=
diag
⁡
1
,
-
1
,
-
1
,
-
1 and I
4
×
4 is 4
×
4 unit matrix. In the spinor representation of the Dirac matrices we have Ψ
=
ξ
A
,
η
B
˙
T [28], where T denotes transposition of a matrix.The DKP equations for spin0 and spin 1 mesons are written as(3)
β
μ
p
μ
Ψ
=
m
Ψ
,where β
μ are 5
×
5 and 10
×
10 matrices, respectively, obeying the following commutation relations [1–3]:(4)
β
λ
β
μ
β
ν
+
β
ν
β
μ
β
λ
=
g
λ
μ
β
ν
+
g
ν
μ
β
λ
.In the case ofs
=
0 representation equation (3) can be written as(5)
p
μ
ψ
=
m
ψ
μ
,
p
ν
ψ
ν
=
m
ψ
,with Ψ in (3) defined as(6)
Ψ
=
ψ
μ
,
ψ
T
=
ψ
0
,
ψ
1
,
ψ
2
,
ψ
3
,
ψ
T
.In the case ofs
=
1 (3) reduces to(7)
p
μ
ψ
ν
-
p
ν
ψ
μ
=
m
ψ
μ
ν
,
p
μ
ψ
μ
ν
=
m
ψ
ν
,with Ψ in (3) equalling(8)
Ψ
=
ψ
μ
ν
,
ψ
λ
T
=
ψ
01
,
ψ
02
,
ψ
03
,
ψ
23
,
ψ
31
,
ψ
12
,
ψ
0
,
ψ
1
,
ψ
2
,
ψ
3
T
,where ψ
λ are real and ψ
μ
ν are purely imaginary (alternatively, -
∂
μ
ψ
ν
+
∂
ν
ψ
μ
=
m
ψ
μ
ν and ∂
μ
ψ
μ
ν
=
m
ψ
ν, where ψ
λ and ψ
μ
ν are real). The s
=
1 condition, p
ν
ψ
ν
=
0, follows from the second equation of (7) due to antisymmetry of tensor ψ
μ
ν. Equations for spin 1 bosons (7) were first written by Proca [29].


## 3. From the DKP Equations to Generalized Solutions of the Dirac Equation

### 3.1. Spin 0

Equations (5) can be written within spinor formalism as(9)
p
A
B
˙
ψ
=
m
ψ
A
B
˙
,
p
A
B
˙
ψ
A
B
˙
=
2
m
ψ
.Splitting the last equation of (9), p
A
B
˙
ψ
A
B
˙
=
p
1
1
˙
ψ
1
1
˙
+
p
1
2
˙
ψ
1
2
˙
+
p
2
1
˙
ψ
2
1
˙
+
p
2
2
˙
ψ
2
2
˙
=
2
m
ψ; we obtain two sets of equations involving components ψ
1
1
˙
,
ψ
1
2
˙
,
ψ and ψ
2
1
˙
,
ψ
2
2
˙
,
ψ, respectively:(10)
p
1
1
˙
ψ
=
m
ψ
1
1
˙
,
p
1
2
˙
ψ
=
m
ψ
1
2
˙
,
p
1
1
˙
ψ
1
1
˙
+
p
1
2
˙
ψ
1
2
˙
=
m
ψ
,
(11)
p
2
1
˙
ψ
=
m
ψ
2
1
˙
,
p
2
2
˙
ψ
=
m
ψ
2
2
˙
,
p
2
1
˙
ψ
2
1
˙
+
p
2
2
˙
ψ
2
2
˙
=
m
ψ
,each of which describes particle with mass m (we check this substituting, e.g., ψ
1
1
˙ and ψ
1
2
˙ or ψ
2
1
˙ and ψ
2
2
˙, into the third equations). The splitting preserving p
μ
p
μ
ψ
=
m
2
ψ is possible due to spinor identities, p
1
1
˙
p
1
1
˙
+
p
2
1
˙
p
2
1
˙
=
p
μ
p
μ and p
1
2
˙
p
1
2
˙
+
p
2
2
˙
p
2
2
˙
=
p
μ
p
μ (cf. [22]). Thus (10) and (11) are equivalent to DKP equations (9). We described similar equations in [21]. From each of (10) and (11) an identity follows: (12a)
p
1
2
˙
ψ
1
1
˙
=
p
1
1
˙
ψ
1
2
˙
,
(12b)
p
2
2
˙
ψ
2
1
˙
=
p
2
1
˙
ψ
2
2
˙
.Equation (10) and the identity (12a), as well as (11) and the identity (12b), can be written in form of the Dirac equations: (13)
0
0
p
1
1
˙
p
2
1
˙
0
0
p
1
2
˙
p
2
2
˙
p
1
1
˙
p
1
2
˙
0
0
p
2
1
˙
p
2
2
˙
0
0
ψ
1
1
˙
ψ
1
2
˙
ψ
0
=
m
ψ
1
1
˙
ψ
1
2
˙
ψ
0
,
(14)
0
0
p
1
1
˙
p
2
1
˙
0
0
p
1
2
˙
p
2
2
˙
p
1
1
˙
p
1
2
˙
0
0
p
2
1
˙
p
2
2
˙
0
0
ψ
2
1
˙
ψ
2
2
˙
0
ψ
=
m
ψ
2
1
˙
ψ
2
2
˙
0
ψ
,respectively, with one component equalling zero. Since in (13) and (14) there is the same differential operator we can write these equations as a single Dirac equation. Substituting explicit formulae for the spinors p
A
B
˙ and p
A
B
˙ (see [22]), we have(15)
p
0
γ
0
-
p
1
γ
1
-
p
2
γ
2
-
p
3
γ
3
A
=
m
A
,where A
=
ψ
1
1
˙
ψ
2
1
˙
ψ
1
2
˙
ψ
2
2
˙
ψ
0
0
ψ is a generalized (matrix) wavefunction and γ
μ matrices read(16)
γ
0
=
0
0
1
0
0
0
0
1
1
0
0
0
0
1
0
0
,
γ
1
=
0
0
0
1
0
0
1
0
0
-
1
0
0
-
1
0
0
0
,
γ
2
=
0
0
0
-
i
0
0
i
0
0
i
0
0
-
i
0
0
0
,
γ
3
=
0
0
1
0
0
0
0
-
1
-
1
0
0
0
0
1
0
0
.Let us note that this is the modified spinor representation with γ
i
→
-
γ
i
i
=
1,2
,
3 and Ψ
=
ξ
A
,
η
B
˙
T
→
Ψ
=
η
B
˙
,
ξ
A
T with respect to [28]. In what follows we will use a shorthand A
=
ψ
A
B
˙
,
ψ
I
2
×
2
T, where I
2
×
2 is the 2
×
2 unit matrix.


### 3.2. Spin 1

We will now describe in two steps splitting of the DKP equations. To achieve first level of splitting we write DKP equations (7) in spinor notation as [30, 31] (17)
p
A
B
˙
ζ
C
B
˙
+
p
C
B
˙
ζ
A
B
˙
=
2
m
η
A
C
,
p
B
˙
A
ζ
A
D
˙
+
p
D
˙
A
ζ
A
B
˙
=
2
m
χ
B
˙
D
˙
,
p
A
C
˙
χ
B
˙
C
˙
+
p
B
˙
C
η
A
C
=
-
2
m
ζ
A
B
˙
.Equations (17) are now splitted to yield two separate equations for spinors χ
B
˙
D
˙ and ζ
A
B
˙ and η
A
C and ζ
A
B
˙:(18)
p
A
B
˙
ζ
C
B
˙
=
m
η
A
C
,
η
A
C
=
η
C
A
,
p
B
˙
C
η
A
C
=
-
m
ζ
A
B
˙
,
(19)
p
B
˙
A
ζ
A
D
˙
=
m
χ
B
˙
D
˙
,
χ
B
˙
D
˙
=
χ
D
˙
B
˙
,
p
A
D
˙
χ
B
˙
D
˙
=
-
m
ζ
A
B
˙
,respectively. The splitting was achieved due to appropriate spinor identities; see equation (
11
) in [22]. Indeed, solutions of (18) and (19) obey the DKP equations (17). This derivation was described in [22] and is included here for the sake of completeness.Equations (18) and (19), first written by Dirac [32, 33], are known to describe spin 1 bosons where spinors η
C
A and χ
D
˙
B
˙ correspond to self-dual or anti-self-dual antisymmetric tensors ψ
μ
ν, respectively [34]. Each of the above equations is covariant except from space reflection but (18) and (19) considered together are fully covariant. These equations written in tensor form, β
μ
p
μ
Ψ
=
m
Ψ, Ψ
=
ψ
01
,
ψ
02
,
ψ
03
,
ψ
0
,
ψ
1
,
ψ
2
,
ψ
3
T, where ψ
μ
ν are self-dual or anti-self-dual antisymmetric tensors, with 7
×
7 matrices β
μ fulfilling the DKP algebra (4), are the Hagen-Hurley equations [35–38]. Explicit formulae for these β
μ matrices are given in [38].We will now split thes
=
1 Dirac equations (18). Substituting expressions for p
A
B
˙ and p
B
˙
C (cf. [22]), we obtain a system of eight equations: (20a)
-
p
1
+
i
p
2
ζ
1
1
˙
-
p
0
-
p
3
ζ
1
2
˙
=
m
η
11
,
p
0
+
p
3
ζ
1
1
˙
+
p
1
-
i
p
2
ζ
1
2
˙
=
m
η
21
,
-
p
1
-
i
p
2
η
11
-
p
0
-
p
3
η
12
=
-
m
ζ
1
1
˙
,
p
0
+
p
3
η
11
+
p
1
+
i
p
2
η
12
=
-
m
ζ
1
2
˙
,
(20b)
-
p
1
+
i
p
2
ζ
2
1
˙
-
p
0
-
p
3
ζ
2
2
˙
=
m
η
12
,
p
0
+
p
3
ζ
2
1
˙
+
p
1
-
i
p
2
ζ
2
2
˙
=
m
η
22
,
-
p
1
-
i
p
2
η
21
-
p
0
-
p
3
η
22
=
-
m
ζ
2
1
˙
,
p
0
+
p
3
η
21
+
p
1
+
i
p
2
η
22
=
-
m
ζ
2
2
˙
,where all equations are arranged into two subsets (20a) and (20b) and we have not assumed yet that η
12
=
η
21.Demanding nowη
12
=
η
21
≡
η we achieve splitting of (20a) and (20b) obtaining two Dirac-like equations: (21a)
p
0
γ
~
0
-
p
1
γ
~
1
-
p
2
γ
~
2
-
p
3
γ
~
3
Ψ
=
m
Ψ
,with Ψ
=
ζ
1
1
˙
,
ζ
1
2
˙
,
η
11
,
η
T and(21b)
p
0
γ
~
0
-
p
1
γ
~
1
-
p
2
γ
~
2
-
p
3
γ
~
3
Ψ
~
=
m
Ψ
~
,with Ψ
~
=
ζ
2
1
˙
,
ζ
2
2
˙
,
η
,
η
22
T, where γ
~
μ matrices are expressed by γ
μ matrices (16):(22)
γ
~
0
=
i
γ
2
,
γ
~
1
=
-
γ
3
,
γ
~
2
=
i
γ
0
,
γ
~
3
=
γ
1
.The Dirac-like equations (21a) and (21b) are highly nonstandard because of several reasons: they contain higher-order spinors η
A
B and ζ
A
B
˙ and one common component (η
12
=
η
21
≡
η) and their solutions, Ψ and Ψ
~, are not fully covariant since, considered separately, they do not involve all components of the spinors η
A
B and ζ
A
B
˙. On the other hand, these equations are fully covariant when considered as a whole.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
~
μ matrices unitarily to get γ
μ defined in (16):(23)
γ
μ
=
V
γ
~
μ
V
†
,
V
=
1
0
0
0
0
1
0
0
0
0
0
1
0
0
-
1
0
.Now, we can write (21a) and (21b) as a single Dirac equation and with the same representation of γ
μ matrices as in (15):(24)
p
0
γ
0
-
p
1
γ
1
-
p
2
γ
2
-
p
3
γ
3
B
=
m
B
,where B
=
V
ζ
1
1
˙
ζ
2
1
˙
ζ
1
2
˙
ζ
2
2
˙
η
11
η
η
η
22
=
ζ
1
1
˙
ζ
2
1
˙
ζ
1
2
˙
ζ
2
2
˙
-
η
1
1
-
η
2
1
-
η
1
2
-
η
2
2 is a generalized matrix wavefunction. In what follows we will write B
=
ζ
A
B
˙
,
-
η
C
D
T.


## 3.1. Spin 0

Equations (5) can be written within spinor formalism as(9)
p
A
B
˙
ψ
=
m
ψ
A
B
˙
,
p
A
B
˙
ψ
A
B
˙
=
2
m
ψ
.Splitting the last equation of (9), p
A
B
˙
ψ
A
B
˙
=
p
1
1
˙
ψ
1
1
˙
+
p
1
2
˙
ψ
1
2
˙
+
p
2
1
˙
ψ
2
1
˙
+
p
2
2
˙
ψ
2
2
˙
=
2
m
ψ; we obtain two sets of equations involving components ψ
1
1
˙
,
ψ
1
2
˙
,
ψ and ψ
2
1
˙
,
ψ
2
2
˙
,
ψ, respectively:(10)
p
1
1
˙
ψ
=
m
ψ
1
1
˙
,
p
1
2
˙
ψ
=
m
ψ
1
2
˙
,
p
1
1
˙
ψ
1
1
˙
+
p
1
2
˙
ψ
1
2
˙
=
m
ψ
,
(11)
p
2
1
˙
ψ
=
m
ψ
2
1
˙
,
p
2
2
˙
ψ
=
m
ψ
2
2
˙
,
p
2
1
˙
ψ
2
1
˙
+
p
2
2
˙
ψ
2
2
˙
=
m
ψ
,each of which describes particle with mass m (we check this substituting, e.g., ψ
1
1
˙ and ψ
1
2
˙ or ψ
2
1
˙ and ψ
2
2
˙, into the third equations). The splitting preserving p
μ
p
μ
ψ
=
m
2
ψ is possible due to spinor identities, p
1
1
˙
p
1
1
˙
+
p
2
1
˙
p
2
1
˙
=
p
μ
p
μ and p
1
2
˙
p
1
2
˙
+
p
2
2
˙
p
2
2
˙
=
p
μ
p
μ (cf. [22]). Thus (10) and (11) are equivalent to DKP equations (9). We described similar equations in [21]. From each of (10) and (11) an identity follows: (12a)
p
1
2
˙
ψ
1
1
˙
=
p
1
1
˙
ψ
1
2
˙
,
(12b)
p
2
2
˙
ψ
2
1
˙
=
p
2
1
˙
ψ
2
2
˙
.Equation (10) and the identity (12a), as well as (11) and the identity (12b), can be written in form of the Dirac equations: (13)
0
0
p
1
1
˙
p
2
1
˙
0
0
p
1
2
˙
p
2
2
˙
p
1
1
˙
p
1
2
˙
0
0
p
2
1
˙
p
2
2
˙
0
0
ψ
1
1
˙
ψ
1
2
˙
ψ
0
=
m
ψ
1
1
˙
ψ
1
2
˙
ψ
0
,
(14)
0
0
p
1
1
˙
p
2
1
˙
0
0
p
1
2
˙
p
2
2
˙
p
1
1
˙
p
1
2
˙
0
0
p
2
1
˙
p
2
2
˙
0
0
ψ
2
1
˙
ψ
2
2
˙
0
ψ
=
m
ψ
2
1
˙
ψ
2
2
˙
0
ψ
,respectively, with one component equalling zero. Since in (13) and (14) there is the same differential operator we can write these equations as a single Dirac equation. Substituting explicit formulae for the spinors p
A
B
˙ and p
A
B
˙ (see [22]), we have(15)
p
0
γ
0
-
p
1
γ
1
-
p
2
γ
2
-
p
3
γ
3
A
=
m
A
,where A
=
ψ
1
1
˙
ψ
2
1
˙
ψ
1
2
˙
ψ
2
2
˙
ψ
0
0
ψ is a generalized (matrix) wavefunction and γ
μ matrices read(16)
γ
0
=
0
0
1
0
0
0
0
1
1
0
0
0
0
1
0
0
,
γ
1
=
0
0
0
1
0
0
1
0
0
-
1
0
0
-
1
0
0
0
,
γ
2
=
0
0
0
-
i
0
0
i
0
0
i
0
0
-
i
0
0
0
,
γ
3
=
0
0
1
0
0
0
0
-
1
-
1
0
0
0
0
1
0
0
.Let us note that this is the modified spinor representation with γ
i
→
-
γ
i
i
=
1,2
,
3 and Ψ
=
ξ
A
,
η
B
˙
T
→
Ψ
=
η
B
˙
,
ξ
A
T with respect to [28]. In what follows we will use a shorthand A
=
ψ
A
B
˙
,
ψ
I
2
×
2
T, where I
2
×
2 is the 2
×
2 unit matrix.


## 3.2. Spin 1

We will now describe in two steps splitting of the DKP equations. To achieve first level of splitting we write DKP equations (7) in spinor notation as [30, 31] (17)
p
A
B
˙
ζ
C
B
˙
+
p
C
B
˙
ζ
A
B
˙
=
2
m
η
A
C
,
p
B
˙
A
ζ
A
D
˙
+
p
D
˙
A
ζ
A
B
˙
=
2
m
χ
B
˙
D
˙
,
p
A
C
˙
χ
B
˙
C
˙
+
p
B
˙
C
η
A
C
=
-
2
m
ζ
A
B
˙
.Equations (17) are now splitted to yield two separate equations for spinors χ
B
˙
D
˙ and ζ
A
B
˙ and η
A
C and ζ
A
B
˙:(18)
p
A
B
˙
ζ
C
B
˙
=
m
η
A
C
,
η
A
C
=
η
C
A
,
p
B
˙
C
η
A
C
=
-
m
ζ
A
B
˙
,
(19)
p
B
˙
A
ζ
A
D
˙
=
m
χ
B
˙
D
˙
,
χ
B
˙
D
˙
=
χ
D
˙
B
˙
,
p
A
D
˙
χ
B
˙
D
˙
=
-
m
ζ
A
B
˙
,respectively. The splitting was achieved due to appropriate spinor identities; see equation (
11
) in [22]. Indeed, solutions of (18) and (19) obey the DKP equations (17). This derivation was described in [22] and is included here for the sake of completeness.Equations (18) and (19), first written by Dirac [32, 33], are known to describe spin 1 bosons where spinors η
C
A and χ
D
˙
B
˙ correspond to self-dual or anti-self-dual antisymmetric tensors ψ
μ
ν, respectively [34]. Each of the above equations is covariant except from space reflection but (18) and (19) considered together are fully covariant. These equations written in tensor form, β
μ
p
μ
Ψ
=
m
Ψ, Ψ
=
ψ
01
,
ψ
02
,
ψ
03
,
ψ
0
,
ψ
1
,
ψ
2
,
ψ
3
T, where ψ
μ
ν are self-dual or anti-self-dual antisymmetric tensors, with 7
×
7 matrices β
μ fulfilling the DKP algebra (4), are the Hagen-Hurley equations [35–38]. Explicit formulae for these β
μ matrices are given in [38].We will now split thes
=
1 Dirac equations (18). Substituting expressions for p
A
B
˙ and p
B
˙
C (cf. [22]), we obtain a system of eight equations: (20a)
-
p
1
+
i
p
2
ζ
1
1
˙
-
p
0
-
p
3
ζ
1
2
˙
=
m
η
11
,
p
0
+
p
3
ζ
1
1
˙
+
p
1
-
i
p
2
ζ
1
2
˙
=
m
η
21
,
-
p
1
-
i
p
2
η
11
-
p
0
-
p
3
η
12
=
-
m
ζ
1
1
˙
,
p
0
+
p
3
η
11
+
p
1
+
i
p
2
η
12
=
-
m
ζ
1
2
˙
,
(20b)
-
p
1
+
i
p
2
ζ
2
1
˙
-
p
0
-
p
3
ζ
2
2
˙
=
m
η
12
,
p
0
+
p
3
ζ
2
1
˙
+
p
1
-
i
p
2
ζ
2
2
˙
=
m
η
22
,
-
p
1
-
i
p
2
η
21
-
p
0
-
p
3
η
22
=
-
m
ζ
2
1
˙
,
p
0
+
p
3
η
21
+
p
1
+
i
p
2
η
22
=
-
m
ζ
2
2
˙
,where all equations are arranged into two subsets (20a) and (20b) and we have not assumed yet that η
12
=
η
21.Demanding nowη
12
=
η
21
≡
η we achieve splitting of (20a) and (20b) obtaining two Dirac-like equations: (21a)
p
0
γ
~
0
-
p
1
γ
~
1
-
p
2
γ
~
2
-
p
3
γ
~
3
Ψ
=
m
Ψ
,with Ψ
=
ζ
1
1
˙
,
ζ
1
2
˙
,
η
11
,
η
T and(21b)
p
0
γ
~
0
-
p
1
γ
~
1
-
p
2
γ
~
2
-
p
3
γ
~
3
Ψ
~
=
m
Ψ
~
,with Ψ
~
=
ζ
2
1
˙
,
ζ
2
2
˙
,
η
,
η
22
T, where γ
~
μ matrices are expressed by γ
μ matrices (16):(22)
γ
~
0
=
i
γ
2
,
γ
~
1
=
-
γ
3
,
γ
~
2
=
i
γ
0
,
γ
~
3
=
γ
1
.The Dirac-like equations (21a) and (21b) are highly nonstandard because of several reasons: they contain higher-order spinors η
A
B and ζ
A
B
˙ and one common component (η
12
=
η
21
≡
η) and their solutions, Ψ and Ψ
~, are not fully covariant since, considered separately, they do not involve all components of the spinors η
A
B and ζ
A
B
˙. On the other hand, these equations are fully covariant when considered as a whole.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
~
μ matrices unitarily to get γ
μ defined in (16):(23)
γ
μ
=
V
γ
~
μ
V
†
,
V
=
1
0
0
0
0
1
0
0
0
0
0
1
0
0
-
1
0
.Now, we can write (21a) and (21b) as a single Dirac equation and with the same representation of γ
μ matrices as in (15):(24)
p
0
γ
0
-
p
1
γ
1
-
p
2
γ
2
-
p
3
γ
3
B
=
m
B
,where B
=
V
ζ
1
1
˙
ζ
2
1
˙
ζ
1
2
˙
ζ
2
2
˙
η
11
η
η
η
22
=
ζ
1
1
˙
ζ
2
1
˙
ζ
1
2
˙
ζ
2
2
˙
-
η
1
1
-
η
2
1
-
η
1
2
-
η
2
2 is a generalized matrix wavefunction. In what follows we will write B
=
ζ
A
B
˙
,
-
η
C
D
T.


## 4. Discussion

We have demonstrated that, in the noninteracting case, full covariant solutions of thes
=
0 and s
=
1 DKP equations are generalized solutions of the same Dirac equation; see (1), (15), and (24). More exactly, if we choose the modified spinor representation of the Dirac matrices defined in (16) then the following functions Ψ
=
η
B
˙
,
ξ
A
T, A
=
ψ
A
B
˙
,
ψ
I
2
×
2
T, and B
=
ζ
A
B
˙
,
-
η
C
D
T with η
C
D
=
η
D
C are solutions of the same Dirac equation and correspond to s
=
1
/
2, s
=
0, and s
=
1 cases, respectively. We note that A and B are generalized (matrix) solutions. It follows that these solutions of the Dirac equation provide synthesis of relativistic equations for s
=
1
/
2 and s
=
0,1.Similar generalized solutions exist also in the interacting case. Indeed, although, in thes
=
0 case, (3) and (15) are not equivalent in general fields, they are equivalent in crossed fields [39]. Similarly, the Dirac equations (18) or (19) are not equivalent to the s
=
1 DKP equations (17) in general fields. It remains to determine if the s
=
1 Dirac equations are equivalent to the s
=
1 DKP equations in some special fields.



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*Source: 528484-2015-11-05.xml*