Equivalent Forms of Dirac Equations in Curved Space-times and Generalized de Broglie Relations
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 space-time. This canonical form is needed to apply Whitham's Lagrangian method. The latter method, unlike the Wentzel-Kramers-Brillouin 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 by using canonical Dirac fields in a curved space-time 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 space-time based on both Lagrangian and Hamiltonian formulations of the Whitham equations. We show that the generalized de Broglie relations in a curved space-time 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.
- Publication:
-
Brazilian Journal of Physics
- Pub Date:
- April 2013
- DOI:
- 10.1007/s13538-012-0111-0
- arXiv:
- arXiv:1103.3201
- Bibcode:
- 2013BrJPh..43...64A
- Keywords:
-
- Dirac equation;
- General relativity;
- de Broglie relations;
- Whitham's Lagragian method;
- Mashhoon term;
- Sagnac term;
- COW term;
- Gordon decomposition;
- General Relativity and Quantum Cosmology;
- Mathematical Physics;
- Quantum Physics
- E-Print:
- PDF, 32 pages in referee format. Added significant material on canonical forms of Dirac equations. Simplified Theorem 1 for normal Dirac equations. Added section on Gordon decomposition of the probability current. Encapsulated main results in the statement of Theorem 2