# Duffin–Kemmer–Petiau and Dirac Equations—A Supersymmetric Connection

## Abstract

**:**

## 1. Introduction

## 2. Relativistic Wave Equations

^{μv}= diag (1,−1,−1,−1) and we shall always sum over repeated indices. For example, . Four-momentum operators are defined as where natural units have been used: c = 1, . The interaction will be introduced via minimal coupling,

^{μ}and a charge q. In what follows we shall work with external fields of special configuration, so-called crossed and longitudinal fields, non-standard but Lorentz covariant, see [10]. We shall also need elements of spinor calculus. Four-vectors and spinors are related by the formula :

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

#### 2.1. The Dirac Equation

^{j}are the Pauli matrices and σ

^{0 }is again the 2 × 2 unit matrix. The wave function is a bispinor, i.e., consists of 2 two-component spinors ξ, η: where T denotes transposition of a matrix. Sometimes it is more convenient to use the standard representation:

#### 2.2. Subsolutions of the Dirac Equation and Supersymmetry

^{5}= −iγ

^{0}γ

^{1}γ

^{2}γ

^{3}anticommutes with γ

^{μ}p

_{μ}:

^{5}= diag (−1,−1,1, 1) and thus , and separate equations for ξ, η follow:

^{μ}matrices can be also separated in form of second-order Equations:

_{4}= diag (1,1,1,0), P

_{3}= diag (1,1,0,1) and spinor representation of the γ

^{μ}matrices Equation (5). Equations analogous to (15,16) appear also in the Duffin–Kemmer–Petiau theory of massive bosons [9].

#### 2.3. The Duffin–Kemmer–Petiau Equations

^{μ}, respectively, which fulfill the following commutation relations [26,27,28,29]:

^{μ}matrices Equation (17) is equivalent to the following set of equations:

^{μ }Equation (17) reduces to:

^{λ}are real and Ψ

^{μν}are purely imaginary (in alternative formulation we have , , where Ψ

^{λ}, Ψ

^{μν}are real). Because of antisymmetry of Ψ

^{μν}we have p

_{ν}Ψ

^{ν }= 0what implies spin 1 condition. The set of Equation (21) was first written by Proca [30,31] and in a different context by Lanczos, see [32] and references therein. More on the history of the formalism of Duffin, Kemmer and Petiau can be found in [33].

## 3. Splitting the Dirac Equation in Longitudinal External Fields

_{4}is the projection operator, P

_{4}= diag (1,1,1,0) in the spinor representation of the Dirac matrices and . There are also other projection operators which lead to analogous three component equations, P

_{1}= diag (0,1,1,1), P

_{2}= diag (1,0,1,1), P

_{3}= diag (1,1,0,1). Acting from the left on Equation (37) with P

_{4}and (1−P

_{4})we obtain two Equations:

^{μ}matrices, Equation (38) is equivalent to Equation (33) while Equation (39) is equivalent to the identity Equation (31), respectively. The operator P

_{4}can be written as where γ

^{5}= iγ

^{0}γ

^{1}γ2γ

^{3}(similar formulae can be given for other projection operators P

_{1, }P

_{2, }P

_{3}, see [13] where another convention for γ

^{μ }matrices was however used). It thus follows that Equation (37) is given representation independent form and is Lorentz covariant (in [9] subsolutions of form Equation (37) were obtained for the free Dirac equation).

## 4. Separation of Variables in Subequations

## 5. Splitting the Spin 0 Duffin–Kemmer–Petiau Equations in Crossed Fields

## 6. A Supersymmetric Link between Dirac and DKP Theories

^{μ}= p

^{μ}− qA

^{μ}, A

^{μ }obeying condition of longitudinality Equation (23).

^{μ}, , cf. Equations (65) and (66), and π

^{μ}= p

^{μ}− qA

^{μ}, A

^{μ }obeying condition Equation (57)—fulfilled by crossed fields.

^{μ}→ π

^{μ}= p

^{μ}− qA

^{μ}, lead to subsolutions of the Dirac Equations (63) and (64) in the case of longitudinal fields Equation (23), while for crossed fields Equation (57) yield DKP subsolutions Equations (67) and (68).

## 7. Discussion

^{μ}π

_{μ}:

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Okniński, A.
Duffin–Kemmer–Petiau and Dirac Equations—A Supersymmetric Connection. *Symmetry* **2012**, *4*, 427-440.
https://doi.org/10.3390/sym4030427

**AMA Style**

Okniński A.
Duffin–Kemmer–Petiau and Dirac Equations—A Supersymmetric Connection. *Symmetry*. 2012; 4(3):427-440.
https://doi.org/10.3390/sym4030427

**Chicago/Turabian Style**

Okniński, Andrzej.
2012. "Duffin–Kemmer–Petiau and Dirac Equations—A Supersymmetric Connection" *Symmetry* 4, no. 3: 427-440.
https://doi.org/10.3390/sym4030427