Charge density and electric charge in quantum electrodynamics

TL;DR

This paper examines the mathematical and physical aspects of charge densities and charged states in quantum electrodynamics, emphasizing the importance of time smearing and gauge modifications for physical state construction.

Contribution

It provides a detailed analysis of charge integral convergence, the role of time smearing, and a gauge modification to construct physical charged states in QED.

Findings

Time smearing is crucial for removing vacuum polarization effects.

Gauge modification of Dirac exponential factors removes infrared divergences.

Physical charged states can be constructed as limits of local state vectors.

Abstract

The convergence of integrals over charge densities is discussed in relation with the problem of electric charge and (non-local) charged states in Quantum Electrodynamics (QED). Delicate, but physically relevant, mathematical points like the domain dependence of local charges as quadratic forms and the time smearing needed for strong convergence of integrals of charge densities are analyzed. The results are applied to QED and the choice of time smearing is shown to be crucial for the removal of vacuum polarization effects responible for the time dependence of the charge (Swieca phenomenon). The possibility of constructing physical charged states in the Feynman-Gupta-Bleuler gauge as limits of local states vectors is discussed, compatibly with the vanishing of the Gauss charge on local states. A modification by a gauge term of the Dirac exponential factor which yields the physical Coulomb fields from the Feynman-Gupta-Bleuler fields is shown to remove the infrared divergence of scalar products of local and physical charged states, allowing for a construction of physical charged fields with well defined correlation functions with local fields.