A New and Unifying Approach to Spin Dynamics and Beam Polarization in Storage Rings

TL;DR

This paper develops a unifying mathematical framework for spin dynamics and beam polarization in storage rings using principal bundle theory, connecting various models and invariant structures.

Contribution

It introduces four major theorems linking invariant fields, normal forms, and invariant sets, unifying spin-vector and spin-tensor dynamics in a comprehensive approach.

Findings

Unified description of spin-1/2 and spin-1 particle dynamics

New theorems relating invariant fields and invariant sets

Connections between different dynamical systems in beam polarization

Abstract

With this paper we extend our studies [1] on polarized beams by distilling tools from the theory of principal bundles. Four major theorems are presented, one which ties invariant fields with the notion of normal form, one which allows one to compare different invariant fields, and two that relate the existence of invariant fields to the existence of certain invariant sets and relations between them. We then apply the theory to the dynamics of spin-1/2 and spin-1 particles and their density matrices describing statistically the particle-spin content of bunches. Our approach thus unifies the spin-vector dynamics from the T-BMT equation with the spin-tensor dynamics and other dynamics. This unifying aspect of our approach relates the examples elegantly and uncovers relations between the various underlying dynamical systems in a transparent way.