Possibility of Small Electron States

TL;DR

This paper investigates claims about the minimum size of electron states described by the Dirac equation, analyzing counterexamples that challenge previous size constraints and examining their implications for electron properties.

Contribution

It clarifies how certain counterexamples bypass established size constraints for electron states, questioning previous assumptions about minimum electron size.

Findings

Counterexamples violate the small momentum spread assumption.

Energy and charge flow analysis addresses FTL motion prohibition.

Previous minimum size proofs do not hold for these counterexamples.

Abstract

Some authors have claimed that there exists a minimum size (on the order of the Compton radius) for electron states composed entirely of positive-frequency solutions to the free Dirac equation. Other authors have put forward counterexamples to such claims. This article asks how the counterexamples of A. J. Bracken and G. F. Melloy [J. Phys. A. 32, 6127 (1999)] bypass two arguments against their possibility. The first is an old argument that, because of the prohibition on faster-than-light motion, the electron must be larger than a certain minimum size if it is to have the correct angular momentum and magnetic moment. This challenge can be addressed by analyzing the flow of energy and charge for the counterexample states. The second argument is an explicit proof (presented in C.-P. Chuu et al., [Solid State Commun. 150, 533 (2010)]) that there is a minimum size for purely positive-frequency electron states. This proof hinges on the assumption of a small spread in momentum space, which is violated by the counterexamples that have been put forward.