Fractional Supersymmetry and Quantum Mechanics

TL;DR

This paper introduces fractional supersymmetric quantum mechanics, where Hamiltonians are powers of conserved charges, realized through parafermionic variables, and explores their classical transformations and invariant actions.

Contribution

It develops a new framework of fractional supersymmetry in quantum mechanics using parafermionic variables and constructs invariant actions under these transformations.

Findings

Hamiltonians as Fth powers of conserved charges

Realization via parafermionic variables with =0

Invariant actions under fractional supersymmetry transformations

Abstract

We present a set of quantum-mechanical Hamiltonians which can be written as the $F^{\,\rm th}$ power of a conserved charge: $H=Q^F$ with $[H,Q]=0$ and $F=2,3,...\, .$ This new construction, which we call {\it fractional}\/ supersymmetric quantum mechanics, is realized in terms of \pg\ variables satisfying $\t^F=0$. Furthermore, in a pseudo-classical context, we describe {\it fractional}\/ supersymmetry transformations as the $F^{\,\rm th}$ roots of time translations, and provide an action invariant under such transformations.