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Classical-quantum correspondence and wave packet solutions of the Dirac equation in a curved spacetime

The idea of wave mechanics leads naturally to assume the well-known relation $E=\hbar ω$ in the specific form $H=\hbar W$, where $H$ is the classical Hamiltonian of a particle and $W$ is the dispersion relation of the sought-for wave equation. We derive the expression of $H$ in a curved spacetime with an electromagnetic field. Then we derive the Dirac equation from factorizing the polynomial dispersion equation corresponding with $H$. Conversely, summarizing a recent work, we implement the geometrical optics approximation into a canonical form of the Dirac Lagrangian. Euler-Lagrange equations are thus obtained for the amplitude and phase of the wave function. From them, one is led to define a 4-velocity field which obeys exactly the classical equation of motion. The complete de Broglie relations are then derived exact equations.