Relation between Classical and Pseudo-classical Spinning Particle

TL;DR

This paper explores the relationship between classical and pseudo-classical descriptions of relativistic particle spin, establishing a natural connection via constraints and transformations, and deriving the pseudoclassical Lagrangian from the classical one.

Contribution

It provides a detailed link between Poincaré group variables and Grassmann variables, including a natural derivation of the pseudoclassical Lagrangian from the classical framework.

Findings

Established a natural relation between Poincaré variables and Grassmann variables.

Derived the pseudoclassical Lagrangian from the classical one.

Connected the Hopf fibration to the transformation of spin momentum.

Abstract

The spin degrees of freedom for the relativistic particle are described by either Poincar\'{e} 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 Poincar\'{e} 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 Poincar\'{e} group. And with the relation just mentioned, pseudoclassical lagrangian is derived naturally from the classical one.