Field Theory of the Electron, Spin and Zitterbewegung

In previous papers, we have investigated the classical theory of Barut and Zanghi (BZ) for the electron spin [which interpreted the Zitterbewegung (zbw) motion as an internal motion along helical paths], and its "quantum" version, by using the language of Clifford algebras. And, in so doing, we ended with a new non-linear Dirac-like equation (NDE). We want to readdress in this Review the whole subject, and extend it, by translating it however into the ordinary tensorial language, within the frame of the first quantization formalism. In particular, we re-derive here the NDE for the electron field, and show it to be associated with a new conserved probability current (which allows us to work out a quantum probabilistic interpretation of our NDE). Actually, the Dirac equation is obtained from the former NDE just by averaging over a zbw cycle. We then derive an equation of motion for the 4-velocity field which will allow us to regard the electron as an extended-type object with a classically intelligible internal structure. We carefully study the solutions of the NDE; with special attention to those implying (at the classical limit) light-like helical motions, which appear to be the most adequate solutions for the electron description from a kinematical and physical point of view, and do cope with the electromagnetic properties of the electron. At last we propose a natural generalization of our approach, for the case in which an external electromagnetic potential A^\mu is present; it happens to be based on a new system of five first-order differential field equations.