A Classical and Spinorial Description of the Relativistic Spinning Particle

Abstract

In a previous work we showed that spin can be envisioned as living in a phase space that is dual to the standard phase space of position and momentum. In this work we demonstrate that the second class constraints inherent in this "Dual Phase Space" picture can be solved by introducing a spinorial parameterization of the spinning degrees of freedom. This allows for a purely first class formulation that generalizes the usual relativistic description of spinless particles and provides several insights into the nature of spin and its relationship with spacetime and locality. In particular, we find that the spin motion acts as a Lorentz contraction on the four-velocity and that, in addition to proper time, spinning particles posses a second gauge invariant observable which we call proper angle. Heuristically, this proper angle represents the amount of Zitterbewegung necessary for a spin transition to occur. Additionally, we show that the spin velocity satisfies a causality constraint, and even more stringently, that it is constant along classical trajectories. This leads to the notion of "half-quantum" states which violate the classical equations of motion, and yet do not experience an exponential suppression in the path integral. Finally we give a full analysis of the Poisson bracket structure of this new parametrization.