Recent interest in the studies of structured states obtained in relativistic electron beams has highlighted the use of two alternative descriptions, each based on a different wave function and the related space-time density. Although both wave functions obey the Dirac equation (one directly and the other through a Foldy-Wouthuysen transformation), they lead to different dynamics and properties, such as the presence or absence of spin-orbit interactions. In this work we investigate wave-packet dynamics for the Klein-Gordon equation, which displays the same ambiguity regarding the choice of different densities, in a setting involving Klein tunneling across a series of supercritical potential barriers. Relying on the superradiant character of this setting, we obtain solutions to the wave-packet dynamics indicating that the density based on a Foldy-Wouthuysen transformation of the wave function can be locally amplified outside the light cone. In principle, the exponential increase of the charge due to the field inhomogeneities can lead to an arbitrarily large amplification over macroscopic distances. These results question the interpretation of the Foldy-Wouthuysen density as a fundamentally correct probability or charge density.

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• Electron beams & optics
• Quantum foundations
• Relativistic wave equations
• Non-Hermitian systems