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. That is, it is a dressed particle rather than a bare particle. 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. 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. This is unlike the usual Fock space description, where the Hilbert space includes particle states with different velocities. 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 m {\displaystyle m} of the particle. These states which are very close in mass to m {\displaystyle m} consist of the particle together with low-energy excitations of the electromagnetic field.
Noether's theorem for gauge transformations In electrodynamics and quantum electrodynamics, in addition to the global U(1) symmetry related to the electric charge, there are also position dependent gauge transformations. 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). If this is applied to the global U(1) symmetry, the result
Q = ∫ d 3 x ρ ( x → ) {\displaystyle Q=\int d^{3}x\rho ({\vec {x}})} (over all of space) is the conserved charge where ρ is the charge density. As long as the surface integral
∮ S 2 J → ⋅ d S → {\displaystyle \oint _{S^{2}}{\vec {J}}\cdot d{\vec {S}}}
at the boundary at spatial infinity is zero, which is satisfied if the current density J falls off sufficiently fast, the quantity Q is conserved. This is nothing other than the familiar electric charge. But what if there is a position-dependent (but not time-dependent) infinitesimal gauge transformation δ ψ ( x → ) = i q α ( x → ) ψ ( x → ) {\displaystyle \delta \psi ({\vec {x}})=iq\alpha ({\vec {x}})\psi ({\vec {x}})} where α is some function of position? The Noether charge is now
∫ d 3 x [ α ( x → ) ρ ( x → ) + ϵ 0 E → ( x → ) ⋅ ∇ α ( x → ) ] {\displaystyle \int d^{3}x\left[\alpha ({\vec {x}})\rho ({\vec {x}})+\epsilon _{0}{\vec {E}}({\vec {x}})\cdot \nabla \alpha ({\vec {x}})\right]}
where E → {\displaystyle {\vec {E}}} is the electric field. Using integration by parts,
ϵ 0 ∮ S 2 α E → ⋅ d S → + ∫ d 3 x α [ ρ − ϵ 0 ∇ ⋅ E → ] . {\displaystyle \epsilon _{0}\oint _{S^{2}}\alpha {\vec {E}}\cdot d{\vec {S}}+\int d^{3}x\alpha \left[\rho -\epsilon _{0}\nabla \cdot {\vec {E}}\right].}
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