Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations

TL;DR

This paper demonstrates that de Broglie relations between energy, momentum, and wave properties hold rigorously in curved spacetimes for Dirac equations, using a canonical form and Whitham's method, clarifying quantum-classical correspondence.

Contribution

It introduces a canonical form of Dirac equations in curved spacetime and applies Whitham's method to rigorously derive generalized de Broglie relations.

Findings

De Broglie relations are valid in curved spacetime for Dirac fields.

The probability current decomposes into convection and spin parts, with spin effects negligible in the Whitham approximation.

The generalized de Broglie relations emerge naturally from Whitham's Lagrangian method.

Abstract

One may ask whether the relations between energy and frequency and between momentum and wave vector, introduced for matter waves by de Broglie, are rigorously valid in the presence of gravity. In this paper, we show this to be true for Dirac equations in a background of gravitational and electromagnetic fields. We first transform any Dirac equation into an equivalent canonical form, sometimes used in particular cases to solve Dirac equations in a curved spacetime. This canonical form is needed to apply the Whitham Lagrangian method. The latter method, unlike the WKB method, places no restriction on the magnitude of Planck's constant to obtain wave packets, and furthermore preserves the symmetries of the Dirac Lagrangian. We show using canonical Dirac fields in a curved spacetime, that the probability current has a Gordon decomposition into a convection current and a spin current, and that the spin current vanishes in the Whitham approximation, which explains the negligible effect of spin on wave packet solutions, independent of the size of Planck's constant. We further discuss the classical-quantum correspondence in a curved spacetime based on both Lagrangian and Hamiltonian formulations of the Whitham equations. We show that the generalized de Broglie relations in a curved spacetime are a direct consequence of Whitham's Lagrangian method, and not just a physical hypothesis as introduced by Einstein and de Broglie, and by many quantum mechanics textbooks.