Supersymmetric Content of the Dirac and Duffin-Kemmer-Petiau Equations

Abstract

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.

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

“…Equations analogous to (15,16) appear also in the Duffin-Kemmer-Petiau theory of massive bosons [9]. Let us note finally that as shown in [24] the square of the Dirac operator is indeed supersymmetric, and this can be used for a convenient description of fluctuations around a self-dual monopole.…”

“…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 ( [9]). For m = 0 we can define new quantities:…”

“…The operator P 4 can be written as P 4 = 1 4 (3+γ 5 − γ 0 γ 3 + iγ 1 γ 2 ) where γ 5 = iγ 0 γ 1 γ 2 γ 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).…”

“…Our results derived lately fit into this broader picture. We have demonstrated that certain subsolutions of the free Duffin-Kemmer-Petiau (DKP) and the Dirac equations obey the same Dirac equation with some built-in projection operators [9]. 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.…”

On the Mechanism of Fermion-Boson Transformation

“…Our approach is A. Okniński ( ) PolitechnikaŚwiȩtokrzyska, Al. 1000-lecia PP 7, 25-314 Kielce, Poland e-mail: fizao@tu.kielce.pl based on the 3 × 3 equations, reviewed in the next Section, which are subequations of both the Dirac and DKP equations [18] (see also [19] for the interacting case) and thus provide a link between these equations. We interpret the 3 × 3 equations in Section 3, showing that they can be transformed nonlocally into the form which can be obtained from the Dirac equation by application of the unitary Melosh transformation.…”