Field Theory of the Spinning Electron: II — The New Non-Linear Field Equations

One of the most satisfactory picture of spinning particles is the Barut-Zanghi (BZ) classical theory for the relativistic electron, that relates the electron spin to the so-called zitterbewegung(zbw). The BZ motion equations constituted the starting point for two recent works about spin and electron structure, co-authored by us, which adopted the Clifford algebra language. Here, employing on the contrary the tensorial language, more common in the (first quantization) field theories, we “quantize” the BZ theory and derive for the electron field a non-linear Dirac equation (NDE), of which the ordinary Dirac equation represents a particular case. We then find out the general solution of the NDE. Our NDE does imply a new probability current J μ , that is shown to be a conserved quantity, endowed (in the center-of-mass frame) with the zbw frequency ω = 2m, where m is the electron mass. Because of the conservation of Jμ , we are able to adopt the ordinary probabilistic interpretation for the fields entering the NDE. At last we propose a natural generalization of our approach, for the case in which an external electromagnetic potential A μ is present; it happens to be based on a new system of five first-order differential field equations.