  The quantum hydrodynamic like equations as a function of two real sets of
variables, the 4x4 action matrix and the 4 dimensional wave function modulus
vector of the Dirac equation, are derived in the present work. The paper shows
that in the low velocity limit the equations lead to the hydrodynamic
representation of the Pauli equation for charged particle with spin given by
Janossy and by Bialynicki.The Lorentz invariance of the relativistic quantum
potential that generates the non-local behavior of the quantum mechanics is
discussed.
