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This article was nominated for deletion on 10 December 2009 (UTC). The result of the discussion was no consensus.

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Article reduced

Considering concerns on the AFD and on my talk page, I've removed the OR portions of the article and kept the first paragraph only. It will be almost like re-building from scratch. --JForget 20:27, 31 December 2009 (UTC)

What happened to this article???

This used to be the single most informative Misplaced Pages article! I had no idea what an "infraparticle" was exactly until I read this article, which explained the idea so clearly. It was nominated for deletion, and now it has been completely decimated. I will first restore the relevant text to the talk page for safekeeping.

Old (excellent) Article content

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

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

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

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

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 ${\displaystyle \delta \psi ({\vec {x}})=iq\alpha ({\vec {x}})\psi ({\vec {x}})}$ 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.

Using integration by parts,

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

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

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

is a c-number and a conserved quantity. Using the result that states with different charges exist in different superselection sectors, 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, 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 Lorentz transformations). 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 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.

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, there might exist a similar situation in other quantum field theories besides QED. The name "infraparticle" still applies in those cases.

Explanation

This article explained two things which are both related: one of which is the Noether procedure for gauge symmetries. You expect that a gauge symmetry will produce an infinite number of conservation laws, one for each point in space, because there are an infinite number of symmetries, one for each point in space. Everybody knows that this doesn't work, because the Noether procedure gives a charge which is a perfect divergence.

But what this article shows is that this procedure actually does work, because the gauge transformation at infinity gives rise to a conserved current for each gauge transformation at infinity. This then allows you to define different superselection sectors, which are defined by the electric and magnetic fields at infinity. This is encyclopedic content of the highest quality, written by one of the best physics contributors ever (User:Phys, who left the site very early on). It is important not to delete what you don't understand--- just leave it alone. If it really is bad content, someone who does understand it will delete it. This used to be, hands down, the best article in the encyclopdia.Likebox (talk) 06:32, 18 February 2010 (UTC)

AfD

Everything in this article can be easily verified from first principles by any competent physicist. It is not OR, since the concept of infraparticle is at least 40 years old. I request that if you don't understand it, don't nominate it for deletion. This material is hard to find, occurs only sporadically in the literature, and is the chief source of value of the encyclopedia for working physicists. Without articles such as this, Misplaced Pages would not be useful.Likebox (talk) 06:58, 18 February 2010 (UTC)

I agree that the maths content should not be deleted, unless it is demonstrably false. Instead stick it all in a maths section or something. We should retain links to concepts such as infrared divergence. Even so, the text needs drastic improvement, and I couldn't tell what the concept was about. For example:

1. By "charged", do we mean "electrically charged"
2. "delta function of states at infinity"?? What states? Photon states? Or the particle's states?
3. Are all "charged" particles infraparticles?
4. Same effect with gravitons?

--Michael C. Price 08:49, 18 February 2010 (UTC)

1. yes
2. they're mixed up--- the infraparticle is an amalgam of particle+field
3. yes (electrically charged)
4. yes (almost certainly--- although the analysis is more difficult and I haven't thought how to do it)

The issue here is that an article was butchered while no-one was looking, and it was only by an accident that I noticed the change. The reason is that the original author left Misplaced Pages, possibly because he/she was annoyed at the too-low level of science discussions.Likebox (talk) 22:35, 18 February 2010 (UTC)

I noticed now you put the word "charged" in quotes, so I suppose you are asking if objects in other field theories which can be thought of as charged are infraparticles. Some surprises--- scalars interacting with scalars in 4d are not infraparticles (I can't verify this, I read it somewhere) so Nucleons interacting with massless pions are ok. The infrared divergences in gauge theory really give rise to new conservation laws at infinity, and this is the best way to express the effect.

Higgsed gauge fields don't show up at infinity, and neither do confined fields. So really, gravity is the only option. That and certain weird theories which are believed to be conformal in the infrared, like QCD with 16 quark flavors, or certain supersymmetric models.Likebox (talk) 23:33, 18 February 2010 (UTC)

(deindent) The reason this article is hard to read is because it assumes that the reader is intuitively familiar with a more advanced concept from quantum field theory: the density of states of interacting quantum fields. This is a delta function when you have a particle with definite mass M, it is a superposition of delta-functions (the scattering amplitude gets a cut) for two interacting particles of masses M and m, and the cut starts at m+M. and in this case, the cut coincides with the pole, because the photon is massless. This can do nothing unusual, or it can smear out the delta-function in the density of states to a power-divergence at M. If the smearing happens, you have an infraparticle (an unparticle is also a particle with a powerlaw density of states, which is why the author of the current reference provided by Headbomb tries to identify infraparticle and unparticle physics--- I am not sure if this identification is valid).Likebox (talk) 00:37, 19 February 2010 (UTC)

Source

The original source for this seems to be this article: D. Buchholz, Phys. Lett. B 174 (1986) 331. There, the discussion of Gauss's law is extended for the infraparticle case, and this parallels the discussion here. I have not had time to review this source in any detail (it was just linked from the one reference found here), and there might be an earlier one.Likebox (talk) 08:23, 18 February 2010 (UTC)

Proposed expansion

Infraparticles are charged particles permanently surrounded by an electric field. They have strange quantum mechanical properties, because the classical field can be thought of as an infinite cloud of soft photons. Charged particles radiate an infinite number of soft photons whenver they change directions, these are the cause of the infrared divergences of quantum electrodynamics. 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

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

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

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

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

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 ${\displaystyle \delta \psi ({\vec {x}})=iq\alpha ({\vec {x}})\psi ({\vec {x}})}$ 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.

Using integration by parts,

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

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

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

is a c-number and a conserved quantity. Using the result that states with different charges exist in different superselection sectors, 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, it is easy to see that translations changing the origin do not affect spatial infinity.

Implication for particle behavior

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 transformations). 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 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.

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, there might exist a similar situation in other quantum field theories besides QED. The name "infraparticle" still applies in those cases.

References

1. Bert Schroer (2008). "A note on infraparticles and unparticles". arXiv:0804.3563 .
2. D. Buchholz, Phys. Lett. B 174 (1986) 331
3. D. Buchholz, Phys. Lett. B 174 (1986) 331

I think this is fine, now that the original source is present. It is important to emphasize that there are dozens if not hundreds of other sources for infraparticle behavior, starting in the 1960s. I don't know all this literature.Likebox (talk) 03:03, 19 February 2010 (UTC)

If there are dozens if not hundreds of sources for this, it should be easy to find a few that supports this material. Currently, most of the text is not supported by any references, and thus is removable per WP:Original Synthesis. I'm not objected to an expansion of the article, but it has to be sourced and verifiable. Basing most of this article on a preprint is also worrysome. Headbomb {_{κοντριβς} – WP Physics} 17:59, 19 February 2010 (UTC)

I found a few, and I will give them here as soon as I review them (google for Buchholz). But I am curious--- what part of this discussion are you objecting to? I don't see anything the least bit objectionable. If you delete this, you are destroying the best article on Misplaced Pages, and its best reason for existence.Likebox (talk) 18:06, 19 February 2010 (UTC)

Everything, as this is unsourced, and this doesn't not match anything I come across when search google. From WP:V:

This policy requires that a reliable source in the form of an inline citation be supplied for any material that is challenged or likely to be challenged, and for all quotations, or the material may be removed. This is strictly applied to all material in the mainspace—articles, lists, and sections of articles—without exception

The material is not only likely to be challenged, it is challenged. Either source it, or live with the stub version. Headbomb {_{κοντριβς} – WP Physics} 18:40, 19 February 2010 (UTC)

(deindent) What are you challenging exactly, so I can know what you want sourced? What statements don't you agree with?Likebox (talk) 18:55, 19 February 2010 (UTC)

These statements. Headbomb {_{κοντριβς} – WP Physics} 19:15, 19 February 2010 (UTC)

That's a joke! This is a long article, and I really have no idea what you want sourced. Most of the discussion is of old Noether's theorem for electric charge, which appears in all textbooks. The specific application of Noether's theorem for infraparticles appear in Buchholz. Putting them side by side does not constitute any synthesis.

Perhaps you are objecting to the mention of "rigged hilbert space?" I can source that too. Perhaps you are objecting to the "power law tail of the density of states?" I can source that as well. Please explain in detail what your objections are, so that they can be met.Likebox (talk) 19:36, 19 February 2010 (UTC)

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