Hydrodynamical Reformulation and Quantum Limit of the Barut-Zanghi Theory

The Barut-Zanghi (BZ) classical theory for the relativistic extended electron relates spin to zitterbewegung (zbw). The BZ equations are the starting point for recent work about electron-spin using Clifford algebra. This approach is suited to a hydrodynamical reformulation of the BZ theory. Working with a “probabilistic fluid”, we reinterpret the original classical spinors as wave-functions for the electron. We can “quantize” the BZ theory by employing the tensorial language. “Quantizing” the BZ theory, however, does not lead to the Dirac equation, but rather to a nonlinear, Dirac—like equation, which can be regarded as the “quantum limit” of the BZ classical theory. Moreover, a new variational approach to the BZ probabilistic fluid shows that it is a typical “Weyssenhoff fluid”, while the Hamilton-Jacobi equation (linking mass, spin and zbw frequency together) appears to be nothing but a special case of the de Broglie energy—frequency relation. Finally, after having discussed the remarkable relation existing between the gauge transformation U(1) and a general rotation on the spin plane, we clarify the two-valuedness nature of the fermionic wave-function, and the parity and charge conjugation transformations.