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

TL;DR

This paper derives the Dirac equation in curved spacetime from classical Hamiltonian mechanics, connecting wave mechanics with classical trajectories and providing exact de Broglie relations.

Contribution

It presents a derivation of the Dirac equation in curved spacetime from classical Hamiltonian and dispersion relations, incorporating geometrical optics approximation.

Findings

Derived the classical Hamiltonian in curved spacetime with electromagnetic fields.

Connected wave equations with classical equations of motion via geometrical optics.

Established exact de Broglie relations in a curved spacetime context.

Abstract

The idea of wave mechanics leads naturally to assume the well-known relation $E=\hbar \omega $ 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.