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| In ] and ], in addition to the ] ] symmetry related to the ], there are also position dependent ]s. ] 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 ], which is the space integral of a Noether density (assuming the integral converges and there is a ] satisfying the ]). | In ] and ], in addition to the ] ] symmetry related to the ], there are also position dependent ]s. ] 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 ], which is the space integral of a Noether density (assuming the integral converges and there is a ] satisfying the ]). | ||
| If this is applied to the global U(1) symmetry, the result | |||
| :<math>Q=\int d^3x \rho(\vec{x})</math> (over all of space) | |||
| is the conserved charge where ρ is the ]. As long as the surface integral | |||
| :<math>\oint_{S^2} \vec{J}\cdot d\vec{S}</math> | |||
| at the boundary at spatial infinity is zero, which is satisfied if the ] '''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 <math>\delta \psi(\vec{x})=iq\alpha(\vec{x})\psi(\vec{x})</math> where α is some function of position? | |||
| The Noether charge is now | |||
| :<math>\int d^3x \left</math> | |||
| where <math>\vec{E}</math> is the ]. | |||
| Using ], | |||
| :<math>\oint_{S^2} \alpha \vec{E}\cdot d\vec{S} + \int d^3x \alpha\left</math> | |||
| This assumes 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 the ]. Also assume that α(''r'',θ,φ) approaches α(θ,φ) as ''r'' approaches infinity (in ]). 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 affecting the boundaries are gauge symmetries whereas those that do are global symmetries. If α(θ,φ)=1 all over the ''S''<sup>2</sup>, 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 α is taken as the ], conserved scalar charges (the electric charge) are seen as well as conserved vector charges and conserved tensor charges. This is not a violation of the ] as there is no ]. In particular, for each direction (a fixed θ and φ), the quantity | |||
| :<math>\lim_{r\rightarrow \infty}\epsilon_0 r^2 E_r(r,\theta,\phi)</math> | |||
| is a ] and a conserved quantity. Using the result that states with different charges exist in different ]s, the conclusion 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 ], it is easy to see that translations changing the origin do not affect spatial infinity. | |||
| The directional charges are different for an electron that has always been at rest and an electron that has always been moving at a certain nonzero velocity (because of the ]s). The 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 ], which implies that the whole one-particle ] 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 ]. The electron, and other particles like it is called an '''infraparticle'''. | |||
| The existence of the directional charges is related to ]s. The directional charge at <math>t=-\infty</math> and <math>t=\infty</math> 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, there might exist a similar situation in other ] besides QED. The name "infraparticle" still applies in those cases. | |||
| ] | ] | ||
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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).
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