Talk:Color charge

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Random stuff that was removed from the page. Kept it here so that it could be incorporated later as actual text. -Seth Mahoney 21:08, 9 Apr 2004 (UTC)

Contents 1 Analogy with electrical charge 2 Description 3 Out-take from the page 3.1 Dialogue concerning 3 world systems :) 3.2 Back to out-take 4 Color vs Colour

[edit] Analogy with electrical charge

electrical charge -> color charge (sometimes color number)
electromagnetic force -> color force
charged particules -> quarks & gluons
exchange of photon -> exchange of gluons (difference: gluons are also 'color' charged)

[edit] Description

-> new quantum state.

color force => force which binds quarks together to form hadrons, the strong nuclear force => sort of residual of the color force which binds baryons together.

Six kinds: 'red, 'green', 'blue' & 'anti-red', 'anti-green' & 'anti-blue'

hadrons contain quarks leptons don't contain quarks, so are always color-neutral

All observed particles are always color neutral.

when interacting, gluons & quarks must change, color resulting must be color neutral example: quark(red) -> quark(blue) + gluons(red, anti-blue). => conservation of color charge

Mathematics of color charge: SU(3)

Only 8 sorts of gluons: ...

I've heard the 8 gluons assertion before from sufficiently trustworthy sources that I believe it, but could someone please explain how the various color pairs result in 8 gluons and not 6? I would expect

• red, anti-blue
• red, anti-green
• blue, anti-red
• blue, anti-green
• green, anti-red
• green, anti-green

What is the function of the remaining 2?

--Peter Farago 03:21, 4 Apr 2005 (UTC)

---

For instance, a red quark combined with an antired antiquark do not form a color-neutral state, like red mixed with cyan to produce white light; only the equal quantum superposition of red-antired, green-antigreen, and blue-antiblue is color-neutral.

What is the difference between "combining" a red quark with an antired and creating an "equal quantum superposition"? red-antired, etc. is precisely the configuration of the muons (quark-antiquark pairs). I believe the red-cyan analogy holds quite well in this case.

--Peter Farago 03:17, 4 Apr 2005 (UTC)

[edit] Out-take from the page

I've removed the following material from the page because is verbose (and wrong in some major parts). It can go back, but only after it is polished into a simple paragraph.

No, it doesn't need to go back. What you wrote is much better :) What I wrote was a late-friday-night after-several-beers ramble; someone had asked me why it was called "charge" and I tried to explain that as simply as possible. linas 05:53, 10 Jun 2005 (UTC)

Thanks. Bambaiah

Do we now agree that its not actually wrong? linas 05:20, 13 Jun 2005 (UTC)

If you agree with my remarks of June 12 in the next subsection, then I think what we have agreed on is the following more careful statement "Charge is quantized in classical non-Abelian pure gauge theories (meaning theories without classical particles). However, there is no observed phenomenon that needs such a classical field theory. Classically observed charges (ie, the electric charge) is not quantized." From a physicist's point of view, the last sentence is crucial, from a mathematician's, the first may be sufficient. Bambaiah 08:17, Jun 13, 2005 (UTC)

Argghh. This is very frustrating; we should let this conversation drop. All I wanted to say was that the theory of Lie groups can provide insight into charge quantization. The point is that the electric charge is quantized, and one does not need to quantize the gauge field to get this result. The appearance of QM is incidental, it is only needed to get something that couples to the gauge field (the particle phase). I suppose SQUID's are another example. I suppose I now need to concoct an experiment to convince you. For what its worth, there are purely classical-mechanical systems that exhibit the so-called Berry phase; the canonical examples are robot arms and satelites in orbit around earth. The math is similar to the math here. While doing that math, the Heisenberg group shows up, and it shows up in a purely classical-mechanical context, the context is of sub-Riemannian geometry. I dunno, it just may be possible to take a classical electric dipole and spin it around on the end of a robot arm in some crazy way, and come up with the proper phase effect, in such a way that only a gauge field can explain the resulting effect. Yes, this seems far-fetched to me, and I suspect that even if we did come up with a classical experiment that required gauge theory, we'd still be arguing charge quantization. Lets stop here. linas 15:13, 13 Jun 2005 (UTC)

Hmm, it is well known that the geodesics on the Heisenberg group are identical to the motions of a classically charged particle in a uniform magnetic field (the equations of motion are identical)... linas 15:48, 13 Jun 2005 (UTC)

[edit] Dialogue concerning 3 world systems :)

It wrongly claims that charge is quantized in a classical theory. See Landau and Lifschitz for the true classical theory of electrodynamics: there is no charge quantization here. The fallacy is to take a classical field theory of coupled Dirac and Maxwell fields and to call it classical electrodynamics. A classical charged particle is not a Dirac field!

Hmmm. Although Landau and Lifschitz were brilliant and then a bit more so, they hadn't yet heard of fiber bundles and Yang-Mills. I may be mistaken, and may be recalling/explaining this incorrectly, but if you have a fibre bundle where the fiber is a Lie group (a principal G-bundle), then the natural coupling of the field is via the root system of the Lie algebra. In this sense, one gets charge quantization; the roots are what they are, they cannot be made something else. The color charges of quarks sit on the corners of that su(3) triangle. The size of that triangle is determined by the properties of the Lie algebra, and not by any quantization that gets performed. Now there is also a coupling constant (e.g. the square root of the fine structure constant for electrodynamics), but even second quantization does not "quantize" that.

The problem with electromagentism is that its much harder to "see" that the charge is "1", because it is multiplied by the value "e" (which is just "some scalar"). Multiply "1" by anything and the "1" just sort-of vanishes :) Once you go to a non-abelian theory, the distinction becomes clear: you can't scale the three color charges independently: the shape of the triangle they're on is fixed. The size of the triangle is fixed as well. Now one can clearly distinguish the color coupling constant "g" from the color charges, which are (-1/2 - i sqrt(3)/2, etc). The color charges got quantized due to the theory of Lie algebras, and not due to a second quantization of the quark or gluon fields. The color charge is quantized even in a classical treatment of classical (non-abelian) fields on a fibre bundle.

Once one sees the non-abelian version, one can go back to the case of classical electrodynamics, and see that the charge was quantized there all along; its just that it was hidden; one didn't see the charge because it was too trivial to see.

Or are you just saying that you can't call a fiber bundle a "classical"? I don't know what else to call it; its sure not quantum. Its also unrelated to any dirac particles. The gluons have color charges too (the octet); these are quantized, even if one has a classical theory with no fermions in it. linas 05:53, 10 Jun 2005 (UTC)

It is classical, but it is not the classical theory of anything. The Dirac field is a classical field, but classical particles have nothing to do with spinors. L&L write the field theory which is truly the classical theory and agrees with experiments. Of course, Maxwell knew it before them. :) Bambaiah 13:33, Jun 10, 2005 (UTC)

Not sure what you mean by "its not the classical theory of anything". It is universally accepted that the SU(3) fiber bundle is the "classical" field theory behind QCD. Its universally accepted (by mainstream physics; by most professors) that the U(1) line bundle encompasses classical electrodynamics; certainly Maxwell's equations are a natural by-product of the formulation. What I found curious was that the line bundle explains not only Maxwell's equations, but it also explains the Bohm-Aharonov effect. Thus, in this certain sense, Bohm-Aharonov comes out as a by-product of geometry, and not a by-product of quantization. Thus, one can say that the line bundle is a "broader, more powerful" view of classical electrodynamics than maxwell's equations are. So, given the insight that Lie algebras provide for the notion of "charge" in non-abelian theories, I don't think its crazy to reverse that insight and apply it to the notion of "charge" in the Lie group U(1). This is not fringe theory here; as far as I know, most theoretical field-theory/string/supergravity types would know this stuff well; its not exceptionally deep, either; I picked it up as a grad student, by osmosis.

More generally, I think there's a broader understanding by mainstream physics that quantum mechanics isn't the only way to get things to happen at integer values. For example, the skyrme model provides a quantized baryon number topologically, as a soliton, without having to invoke quantum mechanics in any way. There are similar effects in solid-state physics; topological dislocations come to mind. linas 00:41, 11 Jun 2005 (UTC)

Aharonov-Bohm does not happen with classical particles. Classical field theory is not classical mechanics. The correct, experimentally verifiable, classical theory of EM is not the classical field theory of Gauge field + Dirac field. It is the theory of Gauge field + classical point particle. Classical gauge theory in the sense that you are talking of is a perfectly nice construct but cannot be used analyze any classical experiment whatsoever. It has many mathematical uses; and is useful as soon as you analyze quantum physics because of stuff like Aharonov-Bohm. So here is the actual mathematical and physical structure of these things

1. A gauge + dirac classical field theory, whose physical reality we are discussing and is apparent in step 3.
2. A classical gauge + classical particle theory, which accords with physical reality (experiments with classical particles and fields)
3. A gauge + dirac quantum field theory, which accords with physical reality (experiments with quantum particles and fields). It is reached in two steps from theory 2: first by quantizing the particle into a Dirac field to get the quantum mechanics (this is the closest physical theory one has to theory 1) and next (by second quantizing) to get to the QFT. Or, in one small step from theory 1 using, for example, canonical quantization.

Bottom line: theory 1 accords with an intermediate physical reality where experiments involve a quantum particle but a classical field: this is exactly the setting of the Aharonov Bohm effect with its classical magnetic field (or the stationary energy levels of the Hydrogen atom with its classical coulomb field). Irresistible footnote: Study of transitions between energy levels of H, the particle number changes (because a photon is created or destroyed). So, strictly speaking, stationary levels are QM but transitions are QFT. Of course one usually computes transition amplitudes by sticking an operator between states in QM. When you think about it, you find this is just 1st order perturbation theory in QFT, which works because the in and out states are those you got from QM.

As you say, not deep, but sometimes confusing. I had to clear this up as a grad student myself (you might recall a wonderful RMP article on classical gauge theory, which at the same time was so deeply confusing). The thing to hold on to is the clear distinction between a classical and quantum particle.

It's been good to have this discussion because it clarifies the confusion in the article on quantization. Bambaiah 05:25, Jun 11, 2005 (UTC)

Well, I'm slightly irked, because you seem to be putting words in my mouth, words I never said, and then demonstrating that they're wrong. I never said Aharonov-Bohm happened with classical particles. I said Aharonov-Bohm is an effect explainable by a theory of the electromagnetic field that does not require the electromagnetic field to be quantized. I never said classical mechanics is about Dirac particles or fields. I'm not sure what you mean when you say "Classical gauge theory... cannot be used to analyze experiments". Classical gauge theory, that is, the U(1) line bundle, is more or less synonymous with Maxwell's equations. I agree that 19th century experiments with wires and batteries won't demonstrate the existance of a gauge field. You can neither prove nor disprove the correctness of the line-bundle formulation with 19th century experiments. However, 20th century experiments can do this, and the AB effect is one such experiment. The point I was trying to make was that not only do the experimentally verifiable Maxwell equations come from gauge theory, but in fact the gauge theory is able to provide additional results that the Maxwell equations alone are not able to provide, viz. the Bohm-Aharonov effect.

I agree with your conceptual split 1,2,3. I guess my hand-waving obscured the differences between 1&2. The argument was that theory 1 or 2, with gauge group SU(3), shows that charge is quantized. Also, theory 1, with gauge group U(1), is experimentially verified by Bohm-Aharonov. (Actually, don't need Dirac, I think a plain scalar field is enough, right?). Also, if we take away the particle field in theory 1, it is equivalent to theory 2. Therefore, we may conclude that a) gauge theories describe physical reality, and that b) gauge theories quantize charge, and that c) charge quantization does not require QFT. More precisely, charge quantization does not require even quantum mechanics, and that instead, charge quantization is a by-product of the theory of Lie groups, and not a byproduct of quantum mechanics. I don't want to quibble about the physical reality of particle fields and whether or not a particle field should be called a first-quantized particle or not. I think we are now on the same page.

Sorry, I was mistaken about your position, and therefore the first lines of my previous posting: no offence was meant. Now we agree on the split 1,2,3; and I agree with you that if we leave out the particle, then 1 and 2 are the same: for U(1) it is nothing but Maxwell theory. Now, pure Maxwell theory is has no charge in it, so no disagreement yet. Next we look for a classical non-Abelian gauge theory of the kind 2 (even without a classical particle in the picture) which one can do experiments with. And we can't find one: not a single example of a pure glue classical non-Abelian theory in nature. It seems that you need to go to theory 3 before you encounter a non-Abelian gauge theory. I don't know whether there is anything deep in this, but it is so remarkable that there must be something to learn from here. Anyway, there is no classical experiment that one can discover charge quantization with. This is what I meant when I made the somewhat provocative statement earlier that these gauge theories are classical all right, but they are not classical theories of anything. Bambaiah 10:16, Jun 12, 2005 (UTC)

Don't know what RMP is; nor have I reviewed the WP article on quantization that carefully. I did find very recently R. Bott On some recent Interactions Between Mathematics and Physics Canad. Math. Bull. Vol 28 (2), 1985. which I haven't read (yet?) but does seem to provide a readable review of a whole slew of "semi-classical" topics of this sort. Thanks for the Yukawa potential diagram! linas 15:49, 11 Jun 2005 (UTC)

I'd first learnt about classical gauge theories from The geometrical setting of gauge theories of the Yang-Mills type by M. Daniel and C. M. Viallet in Reviews of Modern Physics, vol 52, 175, 1980. I haven't read Bott's review. One book which is kind of a favourite of mine is Solitons and Instantons, by R. Rajaraman (Elsevier, 1987) [ISBN 0444870474]. Bambaiah

[edit] Back to out-take

Furthermore, much of this material belongs to a section on mathematical aspects of classical field theories, not in an article on "color charge". So why not write this article? Bambaiah 16:36, Jun 9, 2005 (UTC)

I have no desire to write such an article. My plan for editing WP articles was to stay away from physics articles. Somehow, I've gotten drawn in. linas 05:53, 10 Jun 2005 (UTC)

Beginning of out-take

"== Mathematics of color charge =="

Mathematically, color charge is described by the representation theory of the Lie group SU(3). A color-neutral state, such as any hadron, corresponds to the one-dimensional representation of SU(3); the quarks and antiquarks correspond to the two three-dimensional representations, and the gluons (typically for the force-carrying particles of a gauge theory) are in the adjoint representation, which in this case is eight-dimensional.

This is a completely different application of the group SU(3) from Murray Gell-Mann, Yuval Ne'eman and George Zweig's use of it to categorize the hadrons. They thought of an approximate SU(3) symmetry which, in its three-dimensional representation, acts on the flavors of the up, down, and strange quarks, which were the constituents of all the hadrons known at the time.

More precisely, the color charge corresponds to the eigenvectors of the generators of a semi-simple representation of a Lie algebra. They are essentially points on a weight lattice or root system of a Lie algebra. The reason for this is that the canonical Yang-Mills formulation of a particle (such as a quark) interacting with a vector field (such as a QCD color field) is by means of the principal bundle of the associated Lie group. The connection on such a bundle is Lie-algebra valued. The Hamiltonian mechanics of a classical (not quantum!) particle coupled to the principal bundle is such that the momentum can be written as

pq = (d + A)q

where d is the exterior derivative on the base space of the bundle (or the covariant derivative, if the base space is curved) and A is the connection on the principal G-bundle. Here q is the charge (or color charge); it is a vector in the vector space of the representation of the Lie algebra. It multiplies the Lie-algebra-valued A so that p becomes a vector in the representation space of the Lie algebra as well. It should be immediately recognized that p is the covariant derivative on the fiber bundle. Strict mathematicians may call p itself the connection, whereas physicists usually reserve the term connection for the so-called vector field A. When the underlying manifold (the base space) is physical space-time, then p can be understood to be the momentum of a particle with a given color charge, being acted on by the gauge field A.

The term charge comes from analogy to the electromagnetic field. When formulated mathematically, the electromagnetic field can be understood to be a line bundle or a principle G-bundle with G-fibre U(1). The strength of the electromagnetic field (literally, the strength of the electric field and the magnetic field) is given by the curvature F = dA of the connection A. In Minkowski spacetime, one of the four components of A is called the electric potential and the other three are called the magnetic vector potential. The alternating product nature of the exterior derivative gives rise to the traditional divergence and curl expressions seen in, for example, Maxwell's equations. (To be more precise, the Faraday tensor is F whereas the Hodge dual *F is the Maxwell tensor. It is the Maxwell tensor that is used to form the Maxwell equations, along with the Hodge dual of the electric current.) More generally, the curvature of a principle bundle is given by F = dA + [A,A] but the second term vanishes when the Lie algebra is Abelian. In the non-Abelian case, F is known as the Yang-Mills field. The traditional coupling of an electrically charged particle is given through its momentum p=d+A as above, the momentum being important as it gives the particle's motion through space, and, in the fourth component, through time. The momentum can essentially be understood to be the covariant derivative on the electromagnetic line bundle.

Although the line-bundle formulation of electrodynamics may at first seem to be a needlessly mathematically convoluted and glorified treatment of an otherwise rather simple subject, the correctness of the formulation is verified in a most peculiar and unexpected way: holonomy curves on the line bundle express themselves as the Aharonov-Bohm effect. Without the principle bundle formulation, the Aharonov-Bohm effect appears to be quite spooky and mysterious, an odd action-at-a-distance effect not inherently derivable from the Maxwell equations. It is the strength of such results that affirms the correctness of the principal bundle formulation.

The root system for U(1) is trivial: it is just the scalar 1. Thus, the fiber-bundle interpretation immediately gives some insight to the problem of the quantization of the electric charge: the charge is not just any value; it is a particular value, and that value is 1. One need not quantize either the electromagnetic field or the particle field to come to this conclusion: electric charge, as well as color charge, is essentially a classical outcome, not a quantum outcome, of coupling a particle to a gauge field by means of a principal fiber bundle.

In quantum chromodynamics, the relevant Lie group is SU(3) and the Lie algebra is the smallest irreducible representation su(3). Actually, su(3) has a pair of conjugate representations; these look identical except that they are complex conjugates. The root system can best be understood to be two pairs of three vectors each, oriented 120 degrees apart, forming a Star of David. The vectors are of length 1/2; one triangle belongs to one representation, the other to its conjugate. Quarks transform as one representation, and anti-quarks transform as the conjugate representation. That is, the color charge of a quark is thus associated with one of the three vectors, and the charge of the anti-quark is associated with one of the vectors of the conjugate representation.

Generalizations to supersymmetry and to curved base spaces follow the same pattern laid out here: one defines a principle bundle over some group, and couples fields by means of the covariant derivative on the bundle. For good measure, one recognizes that four-dimensional space-time can be represented by a pair of spinors, each spinor transforming under su(2) or its complex conjugate. Spinors naturally have an anti-symmetric algebra, by means of the Pauli exclusion principle; they essentially transform as a sub-algebra of a Clifford algebra. One fundamental problem of supersymmetry is that there are so many different Lie groups, couplings, and representations one can choose from, leading to a bewildering number of fields, charges, and particles; it is difficult to guess which of the many choices might be the one that represents physical reality.

end of out-take

[edit] Color vs Colour

The three references in the article all use "color" rather than "colour" in their texts. Based on that evidence and Wikipedia:Naming_conventions_(use_English) I see no justification for the edits made by User:Xerxes314 on 2006-03-02, which were justified with "britishisms reign supreme" and the false claim on User_talk:Xerxes314 that "international English is the standard for science articles." - ChrisKennedy 05:22, 22 April 2006 (UTC)

I apologize for my humourously intended justification. However, the point remains: the standard agreed-upon for science articles (at least for QCD) is International English. The "Use English" convention is fully met by the use of International English, so your criticism does not apply. If you really think it necessary, we can have another discussion of what the convention should be. It really doesn't matter as long as we pick one and stick with it. -- Xerxes 17:32, 24 April 2006 (UTC)

Okay, if it is a "standard" that is fine, but please provide a source for that rather than asserting it and not providing a way for interested parties to verify. I agree that we need to pick one standard and stick with it, because right now there is inconsistent usage, even on this page. - ChrisKennedy 20:16, 24 April 2006 (UTC)

Just passing by and noticed this. 1) I am aware of no Wikipedia standard for science articles (is that what you meant?) that dictates any particular dialect. I see nothing about it here WP:MOS. 2) There is no standard in the science community. 3) There is no such (one) thing as International English. --Cultural Freedom talk 2006-June-26 06:54 (UTC)

There's no official standard other than "use whatever is used throughout most of the article" and "be consistent". Since "International English" (you must not have read the article you linked to because it clearly states: Sometimes International English is used to refer to a general standard that is based on English as spoken in the British Isles and most Commonwealth countries (as opposed to American English).) is used throughout most of these articles, it is therefore the standard to which we must be consistent. -- Xerxes 14:28, 26 June 2006 (UTC)

Agree there's no official standard. About "International English": did you read the article? "[O]thers use the term to refer to a standard based on U.S. English." As I said, the notion of International English, qua (one) standard, does not exist. Following MoS guidelines -- which, as far as I see, make no mention of International English -- on spelling is the best way to avoid conflicts, in my view.

Either way, I may have misunderstood what you meant by "the standard agreed-upon for science articles (at least for QCD) is International English." What did you mean? Where and how was it agreed upon? And what did you mean by International English? Thanks. --Cultural Freedom talk 2006-06-26 14:51 (UTC)

Honestly, I really really really don't care which spelling gets used. Back when the articles were a mishmash of different spellings because non-Americans kept throwing extra u's into it, I thought it would be easiest to just standardize on International/Commonwealth/British/whatever spelling. I thought Americans would be able to rise above the fray, swallow their spelling-patriotism and accept a standard when non-Americans would not. I guess I'm wrong. Can we just pick a spelling a stick to it? Consistency is the only thing I care about. -- Xerxes 16:29, 26 June 2006 (UTC)

The easiest thing is to stick with MoS guidelines: "use first nonstub version" to determine the spelling (in case about articles not about England, or the US, etc.) The first nonstub version used American spelling. Sticking to the guidelines eliminates the fray in the first place! :) (By the way, I've seen little evidence of spelling patriotism among Americans.)

Still curious what you meant by "the standard agreed-upon for science articles (at least for QCD) is International English." Was that about some non-WP standard? --Cultural Freedom talk 2006-06-26 16:34 (UTC)

No, just from the last time we had this argument. In my experience, the International versions have a longer half-life than American versions. -- Xerxes 22:16, 26 June 2006 (UTC)

Another thing just crossed my mind: A first non-stub rule would not ensure consistency, since it would call for different spellings of the same term across different articles. Surely we want color charge, color confinement, color force, and all the uses of "color" in QCD, etc. to have the same spelling. -- Xerxes 22:19, 26 June 2006 (UTC)

What does it matter? Why does it even need to be consistant? Who cares? People who are intelligent enough to read these articles must be intelligent enough to realise that there are different, correct, ways of spelling the same word. Create redirects for alternative spellings then chill. It's trivial. Theresa Knott | Taste the Korn 22:23, 26 June 2006 (UTC)