Quantum delocalization of the electric charge

TL;DR

This paper investigates the quantum behavior of electric charge localization, revealing that quantum corrections prevent well-defined charge support, contrasting classical solutions where charge can be localized.

Contribution

It demonstrates that quantum effects cause charge delocalization, making it impossible to define a localized electric charge support in quantum electrodynamics.

Findings

Quantum corrections decay no faster than Coulomb fields at spacelike infinity.

Charge density moments do not vanish outside bounded regions.

Classical localized charge solutions do not have quantum analogs.

Abstract

The classical Maxwell-Dirac and Maxwell-Klein-Gordon theories admit solutions of the field equations where the corresponding electric current vanishes in the causal complement of some bounded region of Minkowski space. This poses the interesting question of whether states with a similarly well localized charge density also exist in quantum electrodynamics. For a large family of charged states, the dominant quantum corrections at spacelike infinity to the expectation values of local observables are computed. It turns out that certain moments of the charge density decrease no faster than the Coulomb field in spacelike directions. In contrast to the classical theory, it is therefore impossible to define the electric charge support of these states in a meaningful way.