About kinematics and hydrodynamics of spinning particles: some simple considerations

In the first part (Sections 1 and 2) of this paper - starting from the Pauli current, in the ordinary tensorial language - we obtain the decomposition of the non-relativistic field velocity into two orthogonal parts: the classical part, that is the velocity w p/m of the center-of-mass (CM), and the so-called quantum part, that is, the velocity V of the motion in the CM frame (namely, the integral spin motion or Zitterbewegung). By inserting such a complete, composite expression of the velocity into the kinetic energy term of the non-relativistic classical (Newtonian) Lagrangian, we straightforwardly get the appearance of the so-called quantum potential associated, as it is know, with the Madelueng fluid. This result carries further evidence that the quantum behaviour of micro-systems can be a direct consequence of the fundamental existence of spin. In the second part (Sections 3 and 4), we fix our attention on the total velocity vector v vector w + vector V, being now necessary to pass to relativistic (classical) physics; and we show that the proper time entering the definition of the four-velocity vμ for spinning particles has to be the proper time τ of the CM frame. Inserting the correct Lorentz factor into the definition of vμ leads to completely new kinematical properties for v2. The important constraint pμ vμ identically true for scalar particles, but just assumed a priori in all previous spinning particle theories, is herein derived in a self-consistent way. (author). 24 refs