{'fraction_non_alphanumeric': 0.039423484527342095, 'fraction_numerical': 0.019923696481559984, 'mean_word_length': 4.3564131668558455, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Supersymmetric content of the Dirac and Duffin-Kemmer-Petiau equations

We study subsolutions of the Dirac and Duffin-Kemmer-Petiau equations described in our earlier papers. It is shown that subsolutions of the Duffin-Kemmer-Petiau equations and those of the Dirac equation obey the same Dirac equation with some built-in projection operator. This covariant equation can be referred to as supersymmetric since it has bosonic as well as fermionic degrees of freedom.


Introduction
Recently, subsolutions of the Duffin-Kemmer-Petiau (DKP) equations were found and it was shown that the subsolutions fulfill the appropriately projected Dirac equation [13,14]. On the other hand, massive subsolutions of the Dirac equation were also found and studied [15]. In the present paper we demonstrate that subsolutions of the DKP equations and those of the Dirac equation obey the same Dirac equation with some built-in projection operator. This equation was shown to be covariant in our earlier paper [15]. We shall refer to this equation as supersymmetric since it has bosonic (spin 0 and 1) as well as fermionic (spin 1 2 ) degrees of freedom. Some of the results described below were derived earlier but are included for the sake of completeness. The paper is organized as follows. In Section 2 the Dirac as well as the DKP equations are described shortly. The DKP equations for s = 0 are written as a set of two 3 × 3 equations in Section 3 (the case s = 1 leads to analogous equations [13]) and it is shown that their solutions fulfill the Dirac equation. In Section 4 subsolutions of the Dirac equation are described. Then in Section 5 it is demonstrated that all these subsolutions obey the Dirac equation with built-in projection operator, referred henceforth as the supersymmetric equation. Finally, the supersymmetric equation is written in representation independent form (its covariance was verified in [15]).

Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by Paul Dirac in 1928 providing a description of elementary spin-1 2 particles, such as electrons and quarks, consistent with both the principles of quantum mechanics and the theory of special relativity [1]. The Dirac equation is [2,3,4]: where m is the rest mass of the elementary particle. The γ's are 4 × 4 anticommuting Dirac matrices: γ µ γ ν + γ ν γ µ = 2g µν I where I is a unit matrix.

Duffin-Kemmer-Petiau equations
The DKP equations for spin 0 and 1 are written as: with 5 × 5 and 10 × 10 matrices β µ , respectively, which fulfill the following commutation relations [5,6]: In the case of 5×5 (spin-0) representation of β µ matrices Eq.(2) is equivalent to the following set of equations: if we define Ψ in (2) as: where T denotes transposition of a matrix. Let us note that Eq.(4) can be obtained by factorizing second-order derivatives in the Klein-Gordon equation p µ p µ ψ = m 2 ψ.
In the case of 10 × 10 (spin-1) representation of matrices β µ Eq.(2) reduces to: with the following definition of Ψ in (2): where ψ λ are real and ψ µν are purely imaginary (in alternative formulation we Because of antisymmetry of ψ µν we have p ν ψ ν = 0 what implies spin 1 condition. The set of equations (6) was first written by Proca [8] and in a different context by Lanczos [9]. More on the rich history of the formalism of Duffin, Kemmer and Petiau can be found in [10].
Indeed, demanding in (1) that Ψ = CΨ where C is the charge conjugation operator, CΨ = iγ 2 Ψ * , we obtain in the spinor representation ξ = −iσ 2 η * , η = iσ 2 ξ * and the Dirac equation (1) reduces to two separate Majorana equations for two-component spinors: It follows from the condition Ψ = CΨ that Majorana particle has zero charge built-in condition. The problem whether neutrinos are described by the Dirac equation or the Majorana equations is still open [19,20,18].
Let us note that the Dirac equations (1) in the spinor representation of the γ µ matrices can be also separated in form of second-order equations: Such equations were used by Feynman and Gell-Mann to describe weak decays in terms of two-component spinors [22].

Discussion
We have shown that subsolutions of the Dirac equation as well as of the DKP equations for spin 0 (similar subsolutions arise in the DKP theory for spin 1 [13]) obey the Dirac equation with built-in projection operator (30). Therefore, this covariant equation has bosonic as well as fermionic degrees of freedom and may provide a background for supersymmetric formalism. Let us note here that interaction can be incorporated into (30) via minimal action, p µ → π µ = p µ − eA µ , but in the interacting case Eq.(30) is non-equivalent neither to the Dirac or the DKP equations [15].