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

Abstract

Abstract: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.

On the Mechanism of Fermion-Boson Transformation

“…Moreover, the problem mentioned at the end of Section 2 is solved due to ansatz (16)-all components of spinors ψ AḂ ,αȦ are present in (17) and (18). Equations (19) and (20) can be also written as the Dirac equations with some projection operators [18]. Components of the ψ BĊ D have still to fulfill the identities (4) and (5).…”

“…It follows that (19) and (20) are the 3 × 3 equations obtained by splitting the DKP equation, cf. Eqns.…”

“…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.…”

Neutrino-assisted Fermion--Boson Transitions

“…In Section 2 we have reviewed the procedure of splitting the Dirac equation into two 3 × 3 equations in the non-interacting [12] as well as interacting case [13]. These equations can be written in covariant form as Dirac equations with some projection operators but do not contain all components of spinors used in the splitting.…”

“…In the next Section we review 3 × 3 subequations of the Dirac and the spin-0 DKP equations in special external fields and we raise the problem of their covariance. Covariance of these equations was established in [13] while the problem of covariance of their solutions was solved, to some extent, in Ref. [9].…”