Truncated Harmonic Osillator and Parasupersymmetric Quantum Mechanics

TL;DR

This paper explores the structure of parasupersymmetric quantum mechanics involving truncated harmonic oscillators, revealing that despite algebraic differences, its core consequences mirror those of standard parasupersymmetry, and extends the framework to conformal cases.

Contribution

It introduces a detailed analysis of parasupersymmetry for arbitrary order with truncated harmonic oscillators and reformulates the theory in terms of supercharges, including a conformal extension.

Findings

Parasupersymmetry algebra differs but yields identical consequences.

Quantum mechanics can be expressed with p supercharges, though Hamiltonian form is complex.

Conformal parasupersymmetry forms a closed algebra with supercharges and conformal generators.

Abstract

We discuss in detail the parasupersymmetric quantum mechanics of arbitrary order where the parasupersymmetry is between the normal bosons and those corresponding to the truncated harmonic oscillator. We show that even though the parasusy algebra is different from that of the usual parasusy quantum mechanics, still the consequences of the two are identical. We further show that the parasupersymmetric quantum mechanics of arbitrary order p can also be rewritten in terms of p supercharges (i.e. all of which obey $Q_i^{2} = 0$). However, the Hamiltonian cannot be expressed in a simple form in terms of the p supercharges except in a special case. A model of conformal parasupersymmetry is also discussed and it is shown that in this case, the p supercharges, the p conformal supercharges along with Hamiltonian H, conformal generator K and dilatation generator D form a closed algebra.