Relativistic Schrodinger equation and probability currents for spinless particles

In this work, we face the problem of quantizing the relativistic Hamiltonian of a spinless particle (electrons, photons, etc.). In tempered distribution state spaces, we find the natural way to define the relativistic Hamiltonian operator and its associated relativistic Schrodinger equation. Then we deduce the equivalent relativistic continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine the possible probability currents and flux velocity fields associated with the particle dynamics. We provide the relativistic invariant expression for both relativistic Schrodinger equation and probability flux continuity equations.