On an intrinsic approach of the guiding-center anholonomy and gyro-gauge-arbitrariness

TL;DR

This paper explores an intrinsic, gauge-independent approach to guiding-center theory, clarifying the concepts of gyro-gauge freedom and anholonomy, and proposing new insights into gyro-angle transport and existence conditions.

Contribution

It introduces a global, gauge-independent gyro-angle coordinate and analyzes multiple covariant derivatives, offering new perspectives on gyro-gauge freedom and anholonomy in guiding-center theory.

Findings

Multiple covariant derivatives exist for gyro-angle transport

A new derivation of the global existence condition for unit vectors perpendicular to magnetic field

Clarification of the physical meaning of gyro-gauge freedom and anholonomy

Abstract

In guiding center theory, the standard gyro-angle coordinate is associated with gyro-gauge dependence, the global existence problem for unit vectors perpendicular to the magnetic field, and the notion of anholonomy, which is the failure of the gyro-angle to return to its original value after being transported around a loop in configuration space. We analyse these three intriguing topics through the lens of a recently proposed, global, gauge-independent gyro-angle. This coordinate is constrained, and therefore necessitates the use of a covariant derivative. It also highlights the intrinsic meaning and physical content of gyro-gauge freedom and anholonomy. There are, in fact, many possible covariant derivatives compatible with the intrinsic gyro-angle, and each possibility corresponds to a different notion of gyro-angle transport. This observation sheds new light on Littlejohn's notion of gyro-angle transport and suggests a new derivation of the recently-discovered global existence condition for unit vectors perpendicular to the magnetic field. We also discuss the relationship between Cartesian position-momentum coordinates and the intrinsic gyro-angle.