The problem of the classical non-relativistic electromagnetically kickedoscillator can be cast into the form of an iterative map on phase space. Theoriginal work of Zaslovskii {\it et al} showed that the resulting evolutioncontains a stochastic flow in phase space to unbounded energy. Subsequentstudies have formulated the problem in terms of a relativistically chargedparticle in interaction with the electromagnetic field. We review the standardderivation of the covariant Lorentz force, and review the structure of therelativistic equations used to study this problem. We show that the Lorentz force equation can be derived as well from themanifestly covariant mechanics of Stueckelberg in the presence of a standardMaxwell field. We show how this agreement is achieved, and criticize some ofthe fundamental assumptions underlying these derivations. We argue that a morecomplete theory, involving ``off-shell'' electromagnetic fields should beutilized. We then discuss the formulation of the off-shell electromagnetismimplied by the full gauge invariance of the Stueckelberg mechanics (based onits quantized form), and show that a more general class of physical phenomenacan occur.
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