On the Supersymmetry of the Klein–Gordon Oscillator

Georg Junker

doi:10.3390/sym13050835

Abstract

The three-dimensional Klein–Gordon oscillator exhibits an algebraic structure known from supersymmetric quantum mechanics. The supersymmetry is unbroken with a vanishing Witten index, and it is utilized to derive the spectral properties of the Klein–Gordon oscillator, which is closely related to that of the nonrelativistic harmonic oscillator in three dimensions. Supersymmetry also enables us to derive a closed-form expression for the energy-dependent Green’s function.

Highlights

To the best of our knowledge, this is the first quantum mechanical system with an unbroken N = 2 SUSY but vanishing Witten index, implying that the spectrum of H is fully symmetric with respect to the origin, as we see
As was recently shown [26], SUSY in a relativistic Hamiltonian implies the existence of a Foldy–Wouthuysen transformation, which brings that Hamiltonian into a block-diagonal form
We showed that the Klein–Gordon oscillator (KGO) exhibits a SUSY structure, closely following the general approach of [26]

Summary

Introduction

1960s [4,5,6], but attracted considerable attention only with the seminal work by Moshinsky and Szczepaniak [7] (see Quesne and Moshinsky [8]) Inspired by this so-called Dirac oscillator, the Klein–Gordon oscillator (KGO) was studied by various authors [9,10,11]. We set up the stage with a brief discussion on the KGO Hamiltonian in three space dimensions and show that this Hamiltonian exhibits a SUSY structure by mapping it onto a quantum mechanical SUSY system. This is utilized to derive explicit results of the system.

Supersymmetry

Spectral Properties

Green’s Function

+ 41 , 5. Summary and Outlook