Equivalent Forms of Dirac Equations in Curved Spacetimes 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 spacetime. This canonical form is needed to apply Whitham's Lagrangian method. The latter method, unlike the WentzelKramersBrillouin 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 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 classicalquantum 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.
 Publication:

Brazilian Journal of Physics
 Pub Date:
 April 2013
 DOI:
 10.1007/s1353801201110
 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
 EPrint:
 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