Dynamical Relativistic Systems and the Generalized Gauge Fields of Manifestly Covariant Theories

TL;DR

This paper reviews relativistic charged particle dynamics under electromagnetic fields, critiques standard assumptions, and proposes a more complete off-shell electromagnetic theory based on gauge invariance and Stueckelberg mechanics.

Contribution

It introduces a generalized off-shell electromagnetic framework derived from Stueckelberg mechanics, expanding the scope of relativistic particle-field interactions.

Findings

Standard Lorentz force derived from covariant mechanics

Identification of limitations in traditional assumptions

Proposal of a more complete off-shell electromagnetic theory

Abstract

The problem of the classical non-relativistic electromagnetically kicked oscillator can be cast into the form of an iterative map on phase space. The original work of Zaslovskii {\it et al} showed that the resulting evolution contains a stochastic flow in phase space to unbounded energy. Subsequent studies have formulated the problem in terms of a relativistically charged particle in interaction with the electromagnetic field. We review the standard derivation of the covariant Lorentz force, and review the structure of the relativistic equations used to study this problem. We show that the Lorentz force equation can be derived as well from the manifestly covariant mechanics of Stueckelberg in the presence of a standard Maxwell field. We show how this agreement is achieved, and criticize some of the fundamental assumptions underlying these derivations. We argue that a more complete theory, involving ``off-shell'' electromagnetic fields should be utilized. We then discuss the formulation of the off-shell electromagnetism implied by the full gauge invariance of the Stueckelberg mechanics (based on its quantized form), and show that a more general class of physical phenomena can occur.