Supersymmetry and the physical-phase-space formulation of spinning particles

Abstract

We present an action for the relativistic spin- 1 2 particle, with non-linearly realised Lorentz invariance, in terms of physical phase-space variables only. It is the relativistic counterpart of a standard form of the non-relativistic spin- 1 2 particle action. On quantisation of the non-relativistic particle, galilean supersymmetry is crucial to the recovery of the Levy-Leblond equation as the square root of the Schrödinger equation. The Dirac equation is not obtained in a similar way from quantisation of the relativistic particle because supersymmetry implies positive energy. Instead, the pair of Schrödinger equations for positive and negative energy yield the Foldy-Wouthuysen transform of the Dirac equation.