Open Access
2011 Classical-Quantum Correspondence and Wave Packet Solutions of the Dirac Equation In a Curved Space-Time
Mayeul Arminjon, Frank Reifler
J. Geom. Symmetry Phys. 24: 77-88 (2011). DOI: 10.7546/jgsp-24-2011-77-88


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 space-time 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 four-velocity field which obeys exactly the classical equation of motion. The complete de Broglie relations are then derived as exact equations.


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Mayeul Arminjon. Frank Reifler. "Classical-Quantum Correspondence and Wave Packet Solutions of the Dirac Equation In a Curved Space-Time." J. Geom. Symmetry Phys. 24 77 - 88, 2011.


Published: 2011
First available in Project Euclid: 25 May 2017

zbMATH: 1253.81048
MathSciNet: MR2932787
Digital Object Identifier: 10.7546/jgsp-24-2011-77-88

Rights: Copyright © 2011 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences

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