Pseudosupersymmetric quantum mechanics: General case, orthosupersymmetries, reducibility, and bosonization

TL;DR

This paper explores pseudosupersymmetric quantum mechanics, providing explicit solutions, establishing connections with orthosupersymmetries, and presenting new matrix realizations that lead to bosonized operators with nonlinear spectra.

Contribution

It introduces new matrix realizations of PsSSQM using generalized deformed oscillator algebras and analyzes their reducibility and spectral properties.

Findings

Any orthosupersymmetric system has pseudosupersymmetry.

Matrix realizations of PsSSQM are fully reducible.

Irreducible components yield bosonized operators with nonlinear spectra.

Abstract

Pseudosupersymmetric quantum mechanics (PsSSQM), based upon the use of pseudofermions, was introduced in the context of a new Kemmer equation describing charged vector mesons interacting with an external constant magnetic field. Here we construct the complete explicit solution for its realization in terms of two superpotentials, both equal or unequal. We prove that any orthosupersymmetric quantum mechanical system has a pseudosupersymmetry and give conditions under which a pseudosupersymmetric one may be described by orthosupersymmetries of order two. We propose two new matrix realizations of PsSSQM in terms of the generators of a generalized deformed oscillator algebra (GDOA) and relate them to the cases of equal or unequal superpotentials, respectively. We demonstrate that these matrix realizations are fully reducible and that their irreducible components provide two distinct sets of bosonized operators realizing PsSSQM and corresponding to nonlinear spectra. We relate such results to some previous ones obtained for a GDOA connected with a $C_3$-extended oscillator algebra (where $C_3 = {\rm Z}_3$) in the case of linear spectra.