Extended phase space for a spinning particle

TL;DR

This paper introduces an extended phase space framework for relativistic spinning particles, applicable to various space-times, revealing that such particles may have either two-dimensional trajectories or noncommuting space-time coordinates.

Contribution

It develops a generalized phase space formulation for spinning particles that applies to different homogeneous space-times and Poisson actions, offering new insights into their geometric properties.

Findings

In Minkowski space, spinning particles exhibit either two-dimensional trajectories or noncommuting coordinates.

The extended phase space approach is compatible with de Sitter and other homogeneous space-times.

The formulation provides an alternative perspective on the geometry of relativistic particles with spin.

Abstract

Extended phase space of an elementary (relativistic) system is introduced in the spirit of the Souriau's definition of the `space of motions' for such system. Our formulation is generally applicable to any homogeneous space-time (e.g. de Sitter) and also to Poisson actions. Calculations concerning the Minkowski case for non-zero spin particles show an intriguing alternative: we should either accept two-dimensional trajectories or (Poisson) noncommuting space-time coordinates.