Field Theory of the Electron Spin and Zitterbewegung

In a recent paper, 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, just by using thelanguage of Clifford algebras. And, in so doing, we have ended with a new non-linear Dirac--like equation (NDE). We want to readdress here the whole subject, and extend it, by translating it however into the ordinary tensorial language, within the frame of a 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, we propose this equation in substitution for the Dirac equation, which is obtained from the former by averaging over a zbw cycle. We then derive a new 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, since they 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 {\em five} first--order differential field equations.