Comment on "Nondispersive analytical solutions to the Dirac equation"

TL;DR

This paper critiques a previous claim that certain Dirac equation solutions describe electron beams, emphasizing the importance of accounting for pair creation in time-dependent fields for physical validity.

Contribution

It highlights the limitations of existing solutions in time-dependent electromagnetic fields and clarifies that they do not accurately represent electron trajectories without considering pair processes.

Findings

Solutions do not directly describe electron trajectories.

Pair creation and annihilation are essential in time-dependent fields.

Mathematical solutions have pathological properties like zitterbewegung.

Abstract

In our Comment we question the validity of the claim made by the authors of \cite{cc} that their solutions of the Dirac equation in an external {\em time-dependent} electromagnetic field describe beams of electrons. In every time-dependent field, no matter how weak, which has {\em infinite} time duration, there is a continuous electron-positron pair creation and annihilation. Without the proper accounting for these processes, the mathematical solutions of the Dirac equation are not directly applicable to realistic physical situations. In particular, the time evolution of the average values $\langle x\rangle$ and $\langle y\rangle$ does not describe the electron trajectory but the motion of some combination of the electron and positron charge distributions with pathological properties (zitterbewegung).