The spinning electron: Hidrodynamical formulation, and quantum limit, of the Barut-Zanghi theory

TL;DR

This paper reformulates the Barut-Zanghi classical theory of spinning electrons using hydrodynamics, leading to a quantum-like non-linear Dirac equation and insights into electron spin, wave-functions, and fundamental symmetries.

Contribution

It introduces a hydrodynamical reformulation of the BZ theory, deriving a non-linear Dirac-like equation and analyzing the quantum limit of classical spinor dynamics.

Findings

Reformulation of BZ theory as a probabilistic fluid

Derivation of a non-linear Dirac-like equation as quantum limit

Insight into the gauge, parity, and charge conjugation symmetries

Abstract

One of the most satisfactory pictures for spinning particles is the Barut-Zanghi (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 the remarkable correlation between the gauge transformation U(1) and a general rotation on the spin plan, we clarify and comment on the two-valuedness nature of the fermionic wave-function, and on the parity and charge conjugation transformations.