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: (i) the “classical” part, that is, the velocity ~ w = ~p/m of the centerof-mass (CM), and (ii) the so-called “quantum” part, that is, the velocity ~ V of the motion in the CM frame (namely, the internal “spin motion” or zitterbewegung). By inserting such a complete, composite expression of the velocity into the kinetic energy term of the non-relativistic classical (i.e., newtonian) lagrangian, we straightforwardly get the appearance of the so-called “quantum potential” associated, as it is known, with the Madelung 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 ~v = ~ w + ~ V , it being now necessary to pass to relativistic (classical) physics; and we show that the proper time entering the definition of the four-velocity v for spinning particles has to be the proper time τ of the CM frame. Inserting the correct Lorentz factor into the definition of v leads to completely new kinematical properties for v. The important constraint pμv = m , identically true for scalar particles, but just assumed a priori in all previous spinning particle theories, is herein derived in a self-consistent way.