Classical dynamics of particles with non-abelian gauge charges

TL;DR

This paper develops the classical equations of motion for particles with non-abelian gauge charges and spin on curved manifolds, highlighting their relation to conservation laws and the need for Grassmann variables.

Contribution

It provides a systematic derivation of classical dynamics for particles with non-abelian charges and spin, including extensions with spin interactions and the role of Grassmann variables.

Findings

Equations of motion match conservation law conditions

Classical equations cannot be derived from an action without Grassmann variables

Systematic derivation of constants of motion from symmetries

Abstract

The classical dynamics of particles with (non-)abelian charges and spin moving on curved manifolds is established in the Poisson-Hamilton framework. Equations of motion are derived for the minimal quadratic Hamiltonian and some extensions involving spin-dependent interactions. It is shown that these equations of motion coincide with the consistency conditions for current and energy-momentum conservation. The classical equations cannot be derived from an action principle without extending the model. One way to overcome this problem is the introduction of anticommuting Grassmann co-ordinates. A systematic derivation of constants of motion based on symmetries of the background fields is presented.