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In the present paper we study subsolutions of the Dirac and Duffin–Kemmer–Petiau equations in the interacting case. It is shown that the Dirac equation in longitudinal external fields can be split into two covariant subequations (Dirac equations with built-in projection operators). Moreover, it is demonstrated that the Duffin–Kemmer–Petiau equations in crossed fields can be split into two 3 × 3 subequations. We show that all the subequations can be obtained via minimal coupling from the same 3 × 3 subequations which are thus a supersymmetric link between fermionic and bosonic degrees of freedom.

Recently, several supersymmetric systems, concerned mainly with anyons in 2 + 1 dimensions [

The paper is organized as follows. In

In what follows tensor indices are denoted with Greek letters: ^{μv}

with a four-potential ^{μ}

where ^{0} is the 2 × 2 unit matrix. Spinor with lowered indices

For details of the spinor calculus reader should consult [

The Dirac equation is a relativistic quantum mechanical wave equation formulated by Paul Dirac in 1928 providing a description of elementary spin

where

where ^{j}^{0 }is again the 2 × 2 unit matrix. The wave function is a bispinor,

In the ^{5} = −iγ^{0}γ^{1}γ^{2}γ^{3}^{μ}p_{μ}

In the spinor representation of the Dirac matrices [^{5}

Equations (8) and (9) are known as the Weyl equations and are used to describe massless left-handed and right-handed neutrinos. However, since the experimentally established phenomenon of neutrino oscillations requires non-zero neutrino masses, theory of massive neutrinos, which can be based on the Dirac equation, is necessary [

Although the Majorana equations can be introduced without any reference to the Dirac theory, they are subsolutions of the Dirac Equation [

It follows from the condition

Let us note that the Dirac Equation (4) in the spinor representation of the ^{μ}

Such equations, valid also in the interacting case, were used by Feynman and Gell-Mann to describe weak decays in terms of two-component spinors [

More exotic subsolutions of the Dirac equation, related to supersymmetry, are also possible. In the massless case Simulik and Krivsky demonstrated that the following substitution,

when introduced into the Dirac Equation (4), converts it for

with _{4}_{3}^{μ}

Let us note finally that as shown in [

The DKP equations for spin 0 and 1 are written as:

with 5 × 5 and 10 × 10 matrices ^{μ}

In the case of 5 × 5 (spin 0) representation of ^{μ}

if we define Ψ in Equation (17) as:

Let us note that Equation (19) can be obtained by factorizing second-order derivatives in the Klein–Gordon equation

In the case of 10 × 10 (spin 1) representation of matrices ^{μ }

with Ψ in Equation (17) defined as

Where ^{λ}^{μν}^{λ}^{μν} ^{μν}_{ν}Ψ^{ν }

The interaction is introduced into the Dirac Equation (4) via minimal coupling Equation (1). We consider a special class of four-potentials obeying the condition:

where

This is the case of longitudinal potentials for which several exact solutions of the Dirac equation were found [

The Dirac Equation (4) can be written in spinor notation as [

where

In this Section we shall investigate a possibility of finding subsolutions of the Dirac equation in longitudinal external field, analogous to subsolutions found for the free Dirac equation in ([

where we have:

In spinor notation

The Dirac Equation (25) can be now written with help of Equations (26) and (27) as (we are now using components

It follows from Equations (26) and (27)

Taking into account the identities Equations (31) and (32) we can decouple Equation (30) and write it as a system of the following two Equations:

System of Equations (33) and (34) is equivalent to the Dirac Equation (25) if the definitions Equations (28) and (29) are invoked.

Due to the identities, Equations (31–34) can be cast into form:

Let us consider Equation (35). It can be written as:

where _{4} is the projection operator, _{4} = diag (1,1,1,0) in the spinor representation of the Dirac matrices and _{1}= diag (0,1,1,1), _{2}= diag (1,0,1,1), _{3}= diag (1,1,0,1). Acting from the left on Equation (37) with _{4} and (1−_{4}

In the spinor representation of ^{μ}_{4} can be written as ^{5} = iγ^{0}γ^{1}γ2γ^{3}_{1, }_{2, }_{3}, see [^{μ }

Let us note finally that Equation (36) can be alternatively written as

where

It is possible to separate variables in Equations (33) and (34) following procedures described in [

Taking into account definition of

where

To achieve separation of variables we put:

We now substitute Equation (43) into Equation (42) to get:

where

Combining now Equation (46a) with the first of Equation (33) and rescaling,

with effective mass

On the other hand, combining Equation (46b) with the second of Equation (33) we get equations:

which can be written as the Pauli Equation:

The same procedure applied to Equation (34) yields the equation for

Carrying out separation of variables we get 2D Dirac Equation:

with effective mass

which is written as the Pauli Equation

where the following definitions were used:

We introduce interaction into DKP Equation (19) via minimal coupling Equation (1). We consider four-potentials obeying the condition:

The condition Equation (57) means that

with

Equation (19) in the interacting case can be written within spinor formalism (

Indeed, it follows from Equation (59) that

Let us note now that for fields obeying Equation (57), the following spinor identities hold:

Due to identities Equation (60) we can split the last of Equation (59) and write Equation (59) as a set of two equations:

each of which describes particle with mass

Substituting first two equations into the third one in Equation (61), we get the Klein–Gordon equation

We have shown that subsolutions of the Dirac equation as well as of the DKP equations for spin 0 obey analogous pairs of 3 × 3 Equations (33–62), respectively.

More exactly, Equations (33) and (34) can be written as:

with

and ^{μ} = p^{μ} − qA^{μ}^{μ }

On the other hand, Equations (61) and (62) can be written in analogous form:

with the same matrices ^{μ}^{μ} = p^{μ} − qA^{μ}^{μ }

It thus follows that the 3 × 3 free equations described in [

provide a link between solutions of the Dirac and DKP equations. Namely, Equations (69) and (70) in the interacting case, ^{μ}^{μ} = p^{μ} − qA^{μ}

We have shown that the Dirac equation in longitudinal external fields is equivalent to a pair of 3 × 3 subequations (33) and (34) which can be further written as Dirac equations with built-in projection operators, Equations (37) and (40). Furthermore, we have demonstrated that the Duffin–Kemmer–Petiau equations for spin 0 in crossed fields can be split into two 3 × 3 subequations (61) and (62) (subequations of the DKP equations for spin 1 were discussed in [36]). It was also shown that all the subequations can be obtained via minimal coupling from the same 3 × 3 subequations (69) and (70), which are thus a supersymmetric link between fermionic and bosonic degrees of freedom. It can be expected that for a combination of crossed and longitudinal potentials these subequations should describe interaction of fermionic and bosonic degrees of freedom. We shall investigate this problem in our future work.

Finally, we shall address problem of Lorentz covariance of the subequations. Let us have a closer look at a single subequation of spin 0 DKP equation, say Equation (67). Although both equations, Equation (67) and (68), are covariant as a whole, this subequation alone is not Lorentz covariant. Moreover, it cannot be written as manifestly covariant Dirac equation, ^{μ}π_{μ}

where symbols

It turns out that Equation (71), with

We shall now discuss problem of Lorentz covariance of subequations of the Dirac equation, Equations (63) and (64). Let first note that Equations (69) and (70), as well as Equations (63) and (64), can be written in covariant form as the Dirac equation with one zero component as Equations (15,16,37,40), respectively. However, solutions of Equations (63) and (64) do not involve the whole spinor