Supersymmetry of Relativistic Hamiltonians for Arbitrary Spin

TL;DR

This paper reveals a supersymmetric structure in relativistic Hamiltonians for particles of arbitrary spin, enabling an exact transformation to separate positive and negative energy states, with applications to various spin cases.

Contribution

It demonstrates the supersymmetry of relativistic Hamiltonians for arbitrary spin and derives an exact Foldy-Wouthuysen transformation for these systems.

Findings

Supersymmetry exists in relativistic Hamiltonians for arbitrary spin.

An exact Foldy-Wouthuysen transformation can be constructed.

Application to particles in magnetic fields across different spins.

Abstract

Hamiltonians describing the relativistic quantum dynamics of a particle with an arbitrary spin are shown to exhibit a supersymmetric structure when the even and odd elements of the Hamiltonian commute. For such supersymmetric Hamiltonians an exact Foldy-Wouthuysen transformation exits which brings it into a block-diagonal form separating the positive and negative energy subspaces. Here the supercharges transform between energy eigenstates of positive and negative energy. The relativistic dynamics of a charged particle in a magnetic field is considered for the case of a scalar (spin-zero) boson obeying the Klein-Gordan equation, a Dirac (spin one-half) fermion and a vector (spin-one) boson characterised by the Proca equation.