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| == Noether's theorem for gauge transformations == | == Noether's theorem for gauge transformations == | ||
| {{Refimprovesect|date=February 2010}} | {{Refimprovesect|date=February 2010}} | ||
| 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.{{Citation needed|February 2010}} ] 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 ]).{{Citation needed|February 2010}} | ||
| If this is applied to the global U(1) symmetry, the result | If this is applied to the global U(1) symmetry, the result | ||
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| :<math>\oint_{S^2} \vec{J}\cdot d\vec{S}</math> | :<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. | 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.{{Citation needed|February 2010}} | ||
| 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? | 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? | ||
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| :<math>\int d^3x \left</math> | :<math>\int d^3x \left</math> | ||
| where <math>\vec{E}</math> is the ]. | where <math>\vec{E}</math> is the ].{{Citation needed|February 2010}} | ||
| Using ], | Using ], | ||
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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 ]. 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).<ref name=Buchholz1986/> | 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).<ref name=Buchholz1986/> | ||
| 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 | This conclusion holds both in classical electrodynamics as well as in quantum electrodynamics.{{Citation needed|February 2010}} 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 ].{{Citation needed|February 2010}} 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> | :<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. | is a ] and a conserved quantity.{{Citation needed|February 2010}} 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.{{Clarify|February 2010}}{{Citation needed|February 2010}} | ||
| Even though this result is expressed in terms of a particular spherical coordinates, in particular with a given ], |
Even though this result is expressed in terms of a particular spherical coordinates, in particular with a given ], translations changing the origin do not affect spatial infinity.{{Citation needed|February 2010}} | ||
| ==References== | ==References== | ||
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The term infraparticle refers to an electrically charged particle and its surrounding cloud of soft photons - of which there are 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. However only a finite number of these photons are detectable, the remainder being 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 are accompanied by a soft tail of density of states which consist of all the low energy excitation of the electromagnetic field.
Noether's theorem for gauge transformations
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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).
If this is applied to the global U(1) symmetry, the result
- (over all of space)
(over all of space)is the conserved charge where ρ is the charge density. 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 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 where α is some function of position?
where α is some function of position?
The Noether charge is now
where is the electric field.
is the Using integration by parts,
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, 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 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. In particular, for each direction (a fixed θ and φ), the quantity
is a c-number and a conserved quantity. Using the result that states with different charges exist in different superselection sectors, 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 origin, translations changing the origin do not affect spatial infinity.
References
- Bert Schroer (2008). "A note on infraparticles and unparticles". arXiv:0804.3563 .
- ^ D. Buchholz (1986). "Gauss' law and the infraparticle problem". Physics Letters B. 174: 331. doi:10.1016/0370-2693(86)91110-X.