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

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

2011-03-16

arxiv.org/abs/1103.3201v3

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

PDF, 28 pages in referee format. Added significant material on canonical forms of Dirac equations. Simplified Theorem 1 for no

Scientific paper

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 do this by applying Whitham's Lagrangian method to derive covariant equations describing wave packet motion which preserve the symmetries of the Dirac Lagrangian, and in particular, conserve the probability current. We show that generalized de Broglie relations emerge from the Whitham equations after transforming each Dirac equation into an equivalent canonical form via a local similarity transformation of the type first introduced by Pauli, and used by other authors to simplify the Dirac equation in a curved spacetime. We show that COW and Sagnac type terms also appear in the Whitham equations. 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 method applied to each Dirac equation transformed into equivalent canonical form, and not just a physical hypothesis as introduced by Einstein and de Broglie, and by many quantum mechanics textbooks.

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