Parabose-Parafermi Supersymmetry

TL;DR

This paper explores the algebraic structure and symmetries of parabose-parafermi supersymmetry with p=2, introduces a new framework called supersymmetric paraquantum mechanics, and explicitly solves the parabose-parafermi oscillator.

Contribution

It provides a detailed analysis of the algebraic structure, symmetries, and spectrum of p=2 parabose-parafermi supersymmetry, establishing a new quantum mechanical framework.

Findings

The parabose-parafermi oscillator has two supersymmetries.

The symmetry algebra includes supersymmetric quantum mechanics with a central charge.

Explicit energy eigenvalues and eigenvectors are obtained for the oscillator.

Abstract

The ($p=2$) parabose-parafermi supersymmetry is studied in general terms. It is shown that the algebraic structure of the ($p=2$) parastatistical dynamical variables allows for (symmetry) transformations which mix the parabose and parafermi coordinate variables. The example of a simple parabose-parafermi oscillator is discussed and its symmetries investigated. It turns out that this oscillator possesses two parabose- -parafermi supersymmetries. The combined set of generators of the symmetries forms the algebra of supersymmetric quantum mechanics supplemented with an additional central charge. In this sense there is no relation between the parabose-parafermi supersymmetry and the parasupersymmetric quantum mechanics. A precise definition of a quantum system involving this type of parabose- parafermi supersymmetry is offered, thus introducing ($p=2$) Supersymmetric Paraquantum Mechanics. The spectrum degeneracy structure of general ($p=2$) supersymmetric paraquantum mechanics is analyzed in detail. The energy eigenvalues and eigenvectors for the parabose-parafermi oscillator are then obtained explicitly. The latter confirms the validity of the results obtained for general supersymmetric paraquantum mechanics.