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In electrodynamics and quantum electrodynamics, in addition to the global U(1) symmetry related to the electric charge, we also have position dependent gauge transformations. Noether's theorem states for every infinitesimal symmetry transformation which is local in the sense that the transformed value of a field at a given point only depends upon the field configuration in an arbitrarily small neighborhood of the point in question, there corresponds a conserved charge called the Noether charge which is the space integral of a Nother density (assuming that the integral converges) and there is a Noether current satisfying the continuity equation.

If we apply this to the global U(1) symmetry, we get

${\displaystyle Q=\int d^{3}x\rho ({\vec {x}})}$ (over all of space)

as the conserved charge where ρ is the charge density. As long as the surface integral

${\displaystyle \oint _{S^{2}}{\vec {J}}\cdot d{\vec {S}}}$

at the boundary at spatial infinity is zero, which is satisfied if the current density J falls of sufficiently fast, the quantity Q is conserved. This is nothing other than the electric charge that we all know and love.

But what if we have a position dependent (but not time dependent) infinitesimal gauge transformation ${\displaystyle \delta \psi ({\vec {x}})=iq\alpha ({\vec {x}})\psi ({\vec {x}})}$ where α is some function of position?

The Noether charge is now

${\displaystyle \int d^{3}x\left}$

where ${\displaystyle {\vec {E}}}$ is the electric field.

If we integrate by parts, we get

${\displaystyle \oint _{S^{2}}\alpha {\vec {E}}\cdot d{\vec {S}}+\int d^{3}x\alpha \left}$

We assume that the state in question approaches the vacuum asymptotically at spatial infinity. The first integral is the surface integral at spatial infinity and the second integral is zero by Gauss law. Let's also assume that α(r,θ,φ) approaches α(θ,φ) as r approaches infinity (in polar coordinates). Then, the Noether charge only depends upon the value of α at spatial infinity but not upon the value of α at finite values. This is consistent with the idea that symmetry transformations not effecting the boundaries are gauge symmetries wheareas those which do are global symmetries. If α(θ,φ)=1 all over the S, we get the electric charge. But for other functions, we also get conserved charges (which are not so well known).

This conclusion holds both in classical electrodynamics as well as in quantum electrodynamics. If we let α be the spherical harmonics, we get conserved scalar charges (the electric charge) as well as conserved vector charges and conserved tensor charges. This is not a violation of the Coleman-Mandula theorem as there is no mass gap. In particular, for each direction (a fixed θ and φ), the quantity

${\displaystyle \lim _{r\rightarrow \infty }\epsilon _{0}r^{2}E_{r}(r,\theta ,\phi )}$

is a c-number and a conserved quantity. Using the result that states with different charges exist in different superselection sectors, we conclude that states with the same electric charge but different values for the directional charges lie in different superselection sectors.

Even though this result is expressed in terms of a particular spherical coordinates, in particular with a given origin, it's easy to see that translations changing the origin do not affect spatial infinity.

The directional charges are different for an electron which has always been at rest and an electron which has always been moving at a certain nonzero velocity. (because of the Lorentz transformations). The startling conclusion is that both electrons lie in different superselection sectors no matter how tiny the velocity is. 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 isn't because m is really the greatest lower bound of a continuous mass spectrum and eigenstates of m only exist in a rigged Hilbert space. The electron, and other particles like it is called an infraparticle.

The existence of the directional charges is related to soft photons. The directional charge at ${\displaystyle t=-\infty }$ and ${\displaystyle t=\infty }$ are the same if we take the limit as r goes to infinity first and only then take the limit as t approaches infinity. If we interchange the limits, the directional charges change. This is related to the expanding electromagnetic waves spreading outwards at the speed of light (the soft photons).

More generally, we might have a similar situation in other quantum field theories besides QED. The name "infraparticle" still applies in those cases.

• Electrodynamics
• Quantum field theory