Abstract
We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell system, proving the existence of solutions in which the Dirac wave function is of the form \begin{document}$φ(x,ω)e^{-iω t}$\end{document} , with \begin{document}$ω∈(-m,ω_ *)$\end{document} for some \begin{document}$ω_ *>-m$\end{document} . The solutions satisfy \begin{document}$φ(\,·\,,ω)∈ H^ 1(\mathbb{R}^3,\mathbb{C}^4)$\end{document} , and are small amplitude in the sense that \begin{document}${\left\| {φ(\,·\,,ω)} \right\|}^2_{L^ 2} = O(\sqrt{m+ω})$\end{document} and \begin{document}${\left\| {φ(\,·\,,ω)} \right\|}_{L^∞} = O(m+ω)$\end{document} . The method of proof is an implicit function theorem argument based on the identification of the nonrelativistic limit as the ground state of the Choquard equation. This identification is in some ways unexpected on account of the repulsive nature of the electrostatic interaction between electrons, and arises as a manifestation of certain peculiarities (Klein paradox) which result from attempts to interpret the Dirac equation as a single particle quantum mechanical wave equation.
Highlights
Introduction and resultsThe Dirac equation, which appeared in [Dir28] just two years after the Schrodinger equation, is the correct Lorentzinvariant equation to describe particles with nonzero spin when relativistic effects cannot be ignored
The perturbative treatment of the Dirac–Maxwell system in the framework of second quantization allows computation of quantities such as the energy levels and scattering cross-sections, which have been compared successfully with experiment, this quantum formalism does not provide the type of tangible description of particles and dynamical processes familiar from classical physics
This has resulted in an enduring interest in the classical Dirac–Maxwell system, both in the physics and mathematics literature
Summary
The Dirac equation, which appeared in [Dir28] just two years after the Schrodinger equation, is the correct Lorentzinvariant equation to describe particles with nonzero spin when relativistic effects cannot be ignored. The physical significance of solitary waves requires existence and stability, and it is to be hoped that the type of detailed information about the solutions which is a consequence of the existence proof in this article, but does not seem to be so accessible from the original variational constructions, will be helpful in future stability analysis (see Remark 6 below) In this context we mention some recent stability results for the nonlinear Dirac equation.
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