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arXiv:2503.21404 · Physical Review A

Effects of superradiance on relativistic Foldy-Wouthuysen densities

F. Daem, A. Matzkin

Published May 27, 2025• 2 citations

Abstract

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 wavefunction and the related space-time density. Although both wavefunctions 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 wavepacket 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 wavepacket dynamics indicating that the density based on a Foldy-Wouthuysen transformation of the wavefunction 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.

DOI: 10.1103/d38q-h3qw

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We show, in the framework of a space-time resolved relativistic quantum field theory approach to tunneling, that microcausality precludes superluminal tunneling dynamics. More specifically in this work dealing with Dirac and Klein-Gordon fields, we first prove that microcausality holds for such fields in the presence of a background potential. We then use this result to show that an intervention performed on a localized region of an initial wave packet subsequently scattering on a potential barrier does not result in any effect outside the light cone emanating from that region. We illustrate these results with numerical computations for Dirac fermions and Klein-Gordon bosons.

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