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| An '''infraparticle''' is an electrically charged particle and its surrounding cloud of [[soft photon]]s—of which there are infinite number, by virtue of the [[infrared divergence]] of [[quantum electrodynamics]].<ref>
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| {{cite arxiv
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| |last=Schroer |first=B.
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| |year=2008
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| |title=A note on infraparticles and unparticles
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| |class=hep-th
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| |eprint=0804.3563
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| }}</ref> 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 particle|virtual]] soft photons become [[real particle]]s. However, only a finite number of these photons are detectable, the remainder falling below the measurement threshold.<ref>
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| {{Cite book
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| |last=Kaku |first=M.
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| |year=1993
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| |title=Quantum Field Theory: A Modern Introduction
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| |publisher=[[Oxford University Press]]
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| |pages=177–184, Appendix A6
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| |isbn=0-19-507652-4
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| }}</ref>
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| 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.<ref name=Buchholz1986>
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| {{Cite journal
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| |last=Buchholz |first=D.
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| |year=1986
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| |title=Gauss' law and the infraparticle problem
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| |journal=[[Physics Letters B]]
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| |volume=174 |pages=331
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| |doi=10.1016/0370-2693(86)91110-X
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| |bibcode = 1986PhLB..174..331B
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| |issue=3 }}</ref>
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| Because of their infraparticle properties, charged particles do not have a sharp [[Dirac delta function|delta function]] density of states like an ordinary particle, but instead the density of states rises like an inverse power at the mass of the particle. This collection of states which are very close in mass to m consist of the particle together with low-energy excitation of the electromagnetic field.
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| ==Noether's theorem for gauge transformations==
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| In [[electrodynamics]] and [[quantum electrodynamics]], in addition to the [[global symmetry|global]] [[unitary group|U(1)]] symmetry related to the [[electric charge]], there are also position dependent [[gauge transformation]]s.<ref>
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| {{Cite journal
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| |last=Weyl |first=H.
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| |year=1929
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| |title=Elektron und Gravitation I
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| |journal=[[Zeitschrift für Physik]]
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| |volume=56 |pages=330–352
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| |doi=10.1007/BF01339504
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| |bibcode = 1929ZPhy...56..330W
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| |issue=5–6 }}</ref> [[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]]).<ref>
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| {{cite journal
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| |last=Noether |first=E.
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| |last2=Tavel |first2=M.A. (transl.)
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| |year=2005 [1971, 1918]
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| |title=Invariant Variation Problems
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| |doi=10.1080/00411457108231446
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| |journal=Transport Theory and Statistical Physics
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| |volume=1
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| |issue=3
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| |pages=235–257
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| |arxiv=physics/0503066
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| |bibcode = 1971TTSP....1..186N }}
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| :Translation of {{Cite journal
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| |last=Noether |first=E.
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| |year=1918
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| |title=Invariante Variationsprobleme
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| |journal=[[Nachrichten von der Königlicher Gesellschaft den Wissenschaft zu Göttingen]], Math-phys. Klasse
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| |volume= |pages=235–257
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| |doi=
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| }}</ref>
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| If this is applied to the global U(1) symmetry, the result
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| :<math>Q=\int d^3x \rho(\vec{x})</math> (over all of space) | |
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| is the conserved charge where ρ is the [[charge density]]. As long as the surface integral
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| :<math>\oint_{S^2} \vec{J}\cdot d\vec{S}</math>
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| at the boundary at spatial infinity is zero, which is satisfied if the [[current density]] '''J''' falls off sufficiently fast, the quantity ''Q''<ref>''Q'' is the integral of the time component of the [[four-current]] '''J''' by definition. See {{Cite book
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| |last=Feynman, R.P.
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| |year=2005
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| |title=[[The Feynman Lectures on Physics]]
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| |volume=2 |pages=
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| |publisher=[[Addison-Wesley]]
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| |edition=2nd
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| |isbn=978-0-8053-9065-0
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| }}</ref>{{Page needed|date=February 2010}} is conserved. This is nothing other than the familiar electric charge.<ref>
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| {{Cite journal
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| |last=Karatas |first=D.L.
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| |last2=Kowalski |first2=K.L.
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| |year=1990
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| |title=Noether's theorem for Local Gauge Transformations
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| |url=http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?198908224
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| |journal=[[American Journal of Physics]]
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| |volume=58 |issue=2 |pages=123–131
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| |doi=10.1119/1.16219
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| |bibcode = 1990AmJPh..58..123K }}</ref><ref>
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| {{Cite journal
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| |last=Buchholz |first=D.
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| |last2=Doplicher |first2=S.
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| |last3=Longo
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| |year=1986
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| |first3=R
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| |title=On Noether's Theorem in Quantum Field Theory
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| |journal=[[Annals of Physics]]
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| |volume=170 |issue=1 |pages=1–17
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| |doi=10.1016/0003-4916(86)90086-2
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| |bibcode = 1986AnPhy.170....1B }}</ref>
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| But what if there is a position-dependent (but not time-dependent) infinitesimal [[Gauge theory|gauge transformation]] <math>\delta \psi(\vec{x})=iq\alpha(\vec{x})\psi(\vec{x})</math> where α is some function of position? <!--Unencyclopedic tone, rephrase this-->
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| The Noether charge is now
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| :<math>\int d^3x \left[\alpha(\vec{x})\rho(\vec{x})+\epsilon_0 \vec{E}(\vec{x})\cdot \nabla\alpha(\vec{x})\right]</math>
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| where <math>\vec{E}</math> is the [[electric field]].<ref name=Buchholz1986/>
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| Using [[integration by parts]],
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| :<math>\oint_{S^2} \alpha \vec{E}\cdot d\vec{S} + \int d^3x \alpha\left[\rho-\epsilon_0 \nabla\cdot \vec{E}\right].</math>
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| 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 [[Gauss law]]. 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 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.<!--Rephrase--> But for other functions, we also get conserved charges (which are not so well known).<ref name=Buchholz1986/>
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| This conclusion holds both in classical electrodynamics as well as in quantum electrodynamics. If α is taken as the [[spherical harmonics]], 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 [[Coleman–Mandula theorem]] as there is no [[mass gap]].<ref>
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| {{Cite journal
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| |last=Coleman |first=S.
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| |last2=Mandula |first2=J.
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| |year=1967
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| |title=All Possible Symmetries of the S Matrix
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| |journal=[[Physical Review]]
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| |volume=159 |pages=1251–1256
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| |doi=10.1103/PhysRev.159.1251
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| |bibcode = 1967PhRv..159.1251C
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| |issue=5 }}</ref> In particular, for each direction (a fixed ''θ'' and ''φ''), the quantity
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| :<math>\lim_{r\rightarrow \infty}\epsilon_0 r^2 E_r(r,\theta,\phi)</math>
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| is a [[c-number]] and a conserved quantity. Using the result that states with different charges exist in different [[superselection sector]]s,<ref>
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| {{Cite web
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| |last=Giulini |first=D.
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| |year=2007
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| |title=Superselection Rules
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| |url=http://philsci-archive.pitt.edu/archive/00003585/01/SSR-QMC-NetVersion.pdf
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| |work=[http://philsci-archive.pitt.edu/ PhilSci Archive]
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| |accessdate=2010-02-21
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| }}</ref> the conclusion that states with the same electric charge but different values for the directional charges lie in different superselection sectors.<ref name=Buchholz1986/>
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| Even though this result is expressed in terms of a particular spherical coordinates with a given [[Origin (mathematics)|origin]], translations changing the origin do not affect spatial infinity.
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| ==Implication for particle behavior==
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| 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 [[Lorentz transformation]]s). The conclusion is that both electrons lie in different superselection sectors no matter how tiny the velocity is.<ref name=Buchholz1986/> 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]]. The electron, and other particles like it is called an infraparticle.<ref>
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| {{Cite journal
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| |last=Buchholz |first=D.
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| |year=1982
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| |title=The Physical State Space of Quantum Electrodynamics
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| |journal=[[Communications in Mathematical Physics]]
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| |volume=85|pages=49
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| |doi=10.1007/BF02029133
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| |bibcode = 1982CMaPh..85...49B }}</ref>
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| The existence of the directional charges is related to [[soft photon]]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).
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| More generally, there might exist a similar situation in other [[quantum field theory|quantum field theories]] besides QED. The name "infraparticle" still applies in those cases.
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| ==References==
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| {{Reflist}}
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| [[Category:Electrodynamics]]
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| [[Category:Quantum field theory]]
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