The Spinning Electron: Hydrodynamical Reformulation, and Quantum Limit, of the Barut–zanghi Theory

One of the most satisfactory pictures for spinning particles is the BarutZanghi (BZ) classical theory for the relativistic extended-like electron, that relates spin to zitterbewegung (zbw). The BZ motion equations constituted the starting point for recent works about spin and electron structure, co-authored by us, which adopted the Clifford algebra language. This language results to be actually suited and fruitful for a hydrodynamical re-formulation of the BZ theory. Working out, in such a way, a “probabilistic fluid”, we are allowed to re-interpret the original classical spinors as quantum wave-functions for the electron. Thus, we can pass to “quantize” the BZ theory employing this time the tensorial language, more popular in first-quantization. “Quantizing” the BZ theory, however, does not lead to the Dirac equation, but rather to a non-linear, Dirac–like equation, which can be regarded as the actual “quantum limit” of the BZ classical theory. Moreover, an original variational approach to the the BZ probabilistic fluid shows that it is a typical “Weyssenhoff fluid”, while the Hamilton-Jacobi equation (linking together mass, spin and zbw frequency) appears to be nothing but a special case of de Broglie’s famous energy-frequency relation. Finally, after having discussed (†) Work partially supported by INFN, MURST, CNR, and by CNPq.