Hydrodynamical Reformulation and Quantum Limit of the Barut-Zanghi Theory

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 1 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 for a hydrodynamical reformulation of the BZ theory. Working out a "probabilistic fluid," we are allowed to reinterpret the original classical spinors as quantum wave-functions for the electron. We can pass to "quantize" the BZ theory: by 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 nonlinear, Diraclike equation, which can be regarded as the actual "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 So, a zbw is to be added to the usual drift, translational, or "external,"motion of the CM with velocity p/ε (which is the only one present in the case of scalar particles). In the Barut-Zanghi (BZ) theory [8,9], the classical electron was actually characterized, besides by the usual pair of conjugate variables (x µ , p µ ), by a second pair of conjugate classical spinorial variables (ψ, ψ), representing internal degrees of freedom, which are functions of the (proper) time τ measured in the electron CMF; the CMF being the one -let us remind-in which p = 0 at any instant of time. Barut and Zanghi introduced, namely, a classical Lagrangian which in the free case (i.e., when the external electromagneticwhere λ has the dimension of an action, and ψ and ψ ≡ ψ † γ 0 are ordinary I C 4 -bispinors, the dot meaning derivation with respect to τ . The four Euler-Lagrange equations, with −λ =h = 1, yield the following motion equations:besides the hermitian adjoint of Eq.(2a), holding for ψ. From Eq. (2) one can also see thatis a constant of the motion (and precisely is the energy in the CMF). [8] Since H is the BZ Hamiltonian in the CMF, we can suitably set H = m, where m is the particle rest-mass. The general solution of the equations of motion (3) can be shown to be:with p µ = constant; p 2 = m 2 ; consequently, by inserting Eqs.(5a) and (5b) in the r.h.s. of Eq.(3b), we finally get: Notice that, instead of adopting the variables ψ and ψ, we can work in terms of the "spin variables," i.e., in terms of the set of dynamical variables4 whereis the particle spin tensor; then, we would get the following motion equations:ṗThe last equation expresses the conservation of the total angular momentum J µν , the sum of the orbital angular momentum L µν and of

Electron Mass Predicted From Substructure Stability in Electrodynamical Model

“…Several such zbw models, identifying spin with orbital angular momentum, were interpreted classically [27][28][29]. Subsequent electrodynamical or hydrodynamical models involved fluids with spin [30], current loops of a certain thickness [31], Dirac-like Equations [32,33], moving charged membranes [34], plasmoid fibers [35], or toroidal geometry [35,36]. Wondering whether zbw could be a real phenomenon, Hestenes emphasized the need to investigate the electron substructure, suggested zbw could originate in the electron self-interaction [37], and showed zbw was compatible with the ring models [38].…”