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The Relativistic Quantum Hydrodynamic Representation of Klein-Gordon Equation

The quantum hydrodynamic-like equations for two real variables (i.e., the phase and the amplitude of the wave function) of the relativistic Klein-Gordon equation are derived in the present paper. The paper also shows that in classical limit the hydrodynamic Klein-Gordon equations lead to the Madelung pseudo-potential as well as to the quantum pseudo potential for a charged particle given by Janossy and by Bialynicki et al. The origin of the non-local interactions of quantum mechanics in the hydrodynamic model is discussed both for free and charged particles. The generalization to the stochastic case is preliminarily depicted.