An infraparticle is an electrically charged particle together with its surrounding cloud of soft photons—of which there are an infinite number, by virtue of the infrared divergence of quantum electrodynamics.
Whenever electric charges accelerate they emit Bremsstrahlung radiation, whereby an infinite number of the virtual soft photons become real particles.
However, only a finite number of these photons are detectable, the remainder falling below the measurement threshold.
[2] The form of the electric field at infinity, which is determined by the velocity of a point charge, defines superselection sectors for the particle's Hilbert space.
[3] Because of their infraparticle properties, charged particles do not have a sharp delta function density of states like an ordinary particle, but instead the density of states rises like an inverse power at the mass
[4] Noether's theorem states that for every infinitesimal symmetry transformation that is local (local in the sense that the transformed value of a field at a given point only depends on the field configuration in an arbitrarily small neighborhood of that point), there is a corresponding conserved charge called the Noether charge, which is the space integral of a Noether density (assuming the integral converges and there is a Noether current satisfying the continuity equation).
As long as the surface integral at the boundary at spatial infinity is zero, which is satisfied if the current density J falls off sufficiently fast, the quantity Q[6][page needed] is conserved.
[3] Using integration by parts, This assumes that the state in question approaches the vacuum asymptotically at spatial infinity.
The conclusion is that both electrons lie in different superselection sectors no matter how tiny the velocity is.
[3] At first sight, this might appear to be in contradiction with Wigner's classification, which implies that the whole one-particle Hilbert space lies in a single superselection sector, but it is not because m is really the greatest lower bound of a continuous mass spectrum and eigenstates of m only exist in a rigged Hilbert space.
This is related to the expanding electromagnetic waves spreading outwards at the speed of light (the soft photons).
More generally, there might exist a similar situation in other quantum field theories besides QED.