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The quantum hydrodynamic like equations as a function of two real sets ofvariables, the 4x4 action matrix and the 4 dimensional wave function modulusvector of the Dirac equation, are derived in the present work. The paper showsthat in the low velocity limit the equations lead to the hydrodynamicrepresentation of the Pauli equation for charged particle with spin given byJanossy and by Bialynicki.The Lorentz invariance of the relativistic quantumpotential that generates the non-local behavior of the quantum mechanics isdiscussed.

The quantum hydrodynamic formulation of Dirac equation and its generalized stochastic and non-linear analogs