Relation between Classical and Pseudo-classical Spinning Particle

Abstract

The spin degrees of freedom for the relativistic particle are described by either Poincaré group variables (classically) or Grassmann variables (pseudo-classically). The relationship between those two descriptions are given. In doing that, appropriate constraints are constructed to put into the lagrangian. Especially a natural relation of Poincaré group variables and Grassmann variables is obtained. Hopf fibration relating the spin momentum to the group is just the right transformation of the spin momentum under Poincaré group. And with the relation just mentioned, pseudoclassical lagrangian is derived naturally from the classical one.