Kinematics and Hydrodynamics of Spinning Particles

In the first part (Secs. I and II) of this paper, starting from the Pauli current, we obtain the decomposition of the nonrelativistic field velocity into two orthogonal parts: (i) the “classical” part, that is, the velocity w=p/m in the center of mass (c.m.), and (ii) the “quantum” part, that is, the velocity V of the motion of the c.m. 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 nonrelativistic 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 provides further evidence of the possibility that the quantum behavior of microsystems is a direct consequence of the fundamental existence of spin. In the second part (Secs. III and IV), we fix our attention on the total velocity v=w+V, now necessarily considering relativistic (classical) physics. 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 c.m. frame. Inserting the correct Lorentz factor into the definition of vμ leads to completely different kinematical properties for v2. The important constraint pμvμ=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.