On an intrinsic approach of the guiding-center anholonomy and gyro-gauge-arbitrariness
Résumé
The guiding-center anholonomy and gyro-gauge arbitrariness are considered in the light of a recently proposed gauge-independent gyro-angle. The difficulties are shown to completely disappear in this coordinate system, directly given by the physical state, but to have regular counterparts, related to intrinsic properties of the system. The basic differential operators do not behave just as partial derivatives. Covariant gradients with non-zero commutators show up, as well as a connection arbitrariness, because the coordinate system is constrained. Within the traditional coordinate system, the difficulties are found to come from the requirement for the coordinates to fit with the basic derivative operators, which also explains the non-global existence of the standard gyro-angle coordinate. To make covariant gradients commute, a connection is introduced for the pitch-angle as well, induced by its physical definition, but non-zero commutators are not avoided in phase-space. This provides an interesting framework to identify existence conditions for a global splitting of the momentum into scalar coordinates for both the pitch-angle and the gyro-angle, with conclusions extending previous results on the existence of a gauge-dependent gyro-angle. Last, a coordinate system both gauge-independent and unconstrained is obtained by generalizing the method of intrinsic coordinates non-adapted to the basic derivative operators of the theory. The need for covariant derivatives is then removed and the scheme is simplified by eliminating both the gauge- or connection-arbitrariness and the non-zero commutators or anholonomy.
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