Talk:Yang–Mills theory/Archive 1

Archive 1

Total lack of information

Ok, I'm going to get the ball rolling by pointing out that there is pretty much no information here. Unfortunately, I don't know enough about Yang-Mills theories to contribute, so I think we really need an expert. StewartMH (talk) 20:54, 18 April 2008 (UTC)

Not a *total* lack of information. I thought some info is better than none! --Michael C. Price ^talk 22:15, 18 April 2008 (UTC)

Renaming

This voice should be renamed from "Yang-Mills" to "Yang-Mills theory". Pra1998 (talk) 10:55, 24 November 2008 (UTC)

Yeah, I've made the request to move it.Headbomb {^ταλκ_{κοντριβς} – WP Physics} 11:35, 24 November 2008 (UTC)

And now it's moved.Headbomb {^ταλκ_{κοντριβς} – WP Physics} 17:39, 24 November 2008 (UTC)

Image of Feynman's rules

I have prepared a jpeg file with latex containing Feynman's rules for Yang-Mills theory. I would like to insert this image into this article as it is in need of it. Please, could you help me? Thanks beforehand. Pra1998 (talk) 11:21, 25 November 2008 (UTC)

I'm not very familiar with uploading images, but User:Mike Peel said he could help with things like that.Headbomb {^ταλκ_{κοντριβς} – WP Physics} 21:16, 25 November 2008 (UTC)

Rating

I think that any encyclopedia must have an article about Yang-Mills theory. The reason is that Yang-Mills theories describe strong and electro-weak interactions and when these are discussed one is forced to recall them anyway.

About the quality of the article, I am not fully convinced that the B class is the right one. But it is no more at a starting level and I have substantially put forward a well developed scheme to build upon. The aim is to reach a higher level of quality making this article useful both to students and researchers. Of course, any suggestion about is welcome. Pra1998 (talk) 16:31, 2 December 2008 (UTC)

Mathematical rating

Yang-Mills theory represents a great mathematical challenge and so also wikipedia should consider as such entering into the WikiProject Mathematics. Pra1998 (talk) 16:34, 2 December 2008 (UTC)

Presumed speculative ideas

I take this chance to thank Michael for his intervention. Section about integrable solutions gives no other than an a class of exact classical solutions of Yang-Mills equations and this is always true independently on any theoretical construction one can ever do. --Pra1998 (talk) 10:24, 25 February 2009 (UTC)

Peter Woit

As you may know, Peter Woit is a critic of science. By "critic of science" one means the same as a movie critic that does not produce any original work by his own but is very active in criticizing other work. This section contains no other than a class of exact solutions of classical Yang-Mills equation and this is plain mathematics without further claim. I could have as well cited the Smilga's book that proposed such solutions and the result would be the same.

The right approach here would be eventually to remove any claim about Frasca's work maintaining the exact solutions of Yang-Mills equations that are true independently on Woit point of view.

Addendum: There is currently, in our community, the idea that an ignored idea is a wrong idea. Of course, this is plainly false as history of physics taught us. Rather, fashions make the path and new ideas may find serious difficulties to affirm. What is really important is that there exist a lot of ideas that are published in physics journals everyday. It is this that makes our field really sane.--Pra1998 (talk) 09:21, 26 February 2009 (UTC)

--Pra1998 (talk) 08:15, 26 February 2009 (UTC)

I don't know anything about anything here, but I wonder how you can characterize Peter Woit as some armchair critic of science (see WP:NPA btw). Perhaps the debate here is equivalent to debating whether or not complex exponentials are solutions to second order differential equations, in which case I would agree that the criticism has no ground to stand on. Regardless, the sources for including these solutions seem to be reliable (Smilga's book has generally positive reviews, including a recommendation from Mathematical Reviews, with the negative reviews focusing on the mathematical complexity and lack of efforts made to make it more accessible), independent of each other (i.e. not from two coworkers), and not made by people known to be crackheads. It would take more than simply saying "it's not because it's published that people are paying attention to it" to convince me that this should not be included. If this cannot be resolved, I suggest asking for feedback at WP:PHYS. Headbomb {^ταλκ_{κοντριβς} – WP Physics} 09:56, 26 February 2009 (UTC)

Headbomb, sorry for the improper comment and thank you for pointing me this out. I apologize if my sentence implied an offense. I think you hit the point and this was the argument I was making. This is just a class of exact solutions for classical Yang-Mills equations and I think they should be there as also other ones that should be inserted. Of course, there is no harm if this implies removing Frasca ref. and pointing just to Smilga's one.--Pra1998 (talk) 10:14, 26 February 2009 (UTC)

If, by "critic of science", you mean well-respected contributor of science, then yes. The characterisation of Peter Woit as a mere "critic of science" is akin to to calling Stanley Kubrick a "film critic." --Logoskakou (talk) 15:49, 26 February 2009 (UTC)

Not to insult Peterwoit here, but he only made 5 (contested) edits in two days. That's hardly a Stanly Kubrick worthy comparison.Headbomb {^ταλκ_{κοντριβς} – WP Physics} 16:42, 26 February 2009 (UTC)

Ah, you're referring to this, not his wikipedia edits. My bad.Headbomb {^ταλκ_{κοντριβς} – WP Physics} 16:48, 26 February 2009 (UTC)

On a general note, I'm beginning to wonder if I'm not smelling some WP:MEAT here. A newly registered editor removes material, then an editor inactive for one year replies and heralds the first one as being "really super". Nothing to warrant ignoring WP:AGF at this point, but there's some red flags being raised. Headbomb {^ταλκ_{κοντριβς} – WP Physics} 16:42, 26 February 2009 (UTC)

Please see Peter Woit's blog (at [1]), I don't think you need to be quite so suspicious (at least not in this case). In any case, Peter Woit is not the issue here, what he's pointing out is quite valid.--Innerproduct (talk) 18:22, 26 February 2009 (UTC)

Marco Frasca

The last part of this article has nothing to do with conventional main-stream understanding of Yang-Mills theory. It is purely the work of Marco Frasca, a physicist who appears to have no institutional affiliation (his papers carry his home address) who I suspect is "Pra1998". There is no reference arguing against these ideas since they are completely ignored by the main-stream. This sort of thing should have no place here. Frasca is free to argue for his ideas on his blog, but he shouldn't be doing it by inserting them into Wikipedia entries. —Preceding unsigned comment added by Peterwoit (talk • contribs) 01:38, 26 February 2009 (UTC)

What about if we hived off the material into its own article? It has been published in a RS, after all. --Michael C. Price ^talk 01:41, 26 February 2009 (UTC)

I don't know much about Wikipedia standards. The bottom line is that the content in question is unconventional speculation due to Frasca, speculation that I don't think anyone else is much interested in or convinced by. Lots of such ideas are published in journals, and then mostly ignored. Personally I don't think they belong on Wikipedia at all, but they certainly don't belong in an entry like this on one of the core ideas of modern physics.Peterwoit (talk) 01:54, 26 February 2009 (UTC)

I know you think they certainly don't belong in an entry like this on one of the core ideas of modern physics -- that's why I'm asking you if it should be hived off into a separate article.--Michael C. Price ^talk 02:15, 26 February 2009 (UTC)

No. If nothing else, the information that is posted by Pra1998 would fall under "original research", though I think it is generous to call it research. As per Wikipedia's policy on original research, it should neither be in this article nor hived in to its own article. --Logoskakou (talk) 15:49, 26 February 2009 (UTC)

I read the section in some details, and while I don't understand one thing about it, it does feels like WP:OR. Especially with sentences like "the infrared theory has been recently formulated" and "the results appear to be in agreement with computations with lattice field theory". I don't know how recent 2006 is in QFT, but this may be too immature to include in WP. Headbomb {^ταλκ_{κοντριβς} – WP Physics} 17:17, 26 February 2009 (UTC)

The material may or may not be original research, but it's not an appropriate topic for the main article on Yang-Mills theory. The material itself is rather specialized. Moreover, the relation between the material and the main concerns of Yang-Mills theory is speculative, which goes a long way towards explaining why it's so poorly integrated with the rest of the article. I would recommend hiving it off or deleting it entirely, and I'm not sure that I would recommend including a link to the new article in the main Yang-Mills article. (Full disclosure: I'm a math PhD student. I work on topics related to Yang-Mills theory. I found out about this issue via Woit's blog, and thought that since I know something abou the topic, I ought to speak up.) --A.J. Tolland —Preceding unsigned comment added by 198.129.67.74 (talk) 19:05, 26 February 2009 (UTC)

I also have read the section in some detail and would note that it also feels like original research. As a regular consumer of the Wiki information on math and physics I would prefer not to have to try to discern what is OR and what is not, particularly in an article as important as this one. That fact that Pra1998 is the author of this OR makes me even more skeptical that this information should remain. The references other than the OR itself (i.e. ref. 5 and 6) is Smilga's book on QCD but when one looks in that source the support for the information in this part of the article is non-existent. One can see from Frasca's blog where he is referencing a single aside comment from Smilga's book. See: http://marcofrasca.wordpress.com/2008/10/25/smilgas-choice-and-the-mapping-theorem/ Looking in the book one can find the referenced statement (page 13) which reads: "The solution (1.32) is a non-linear standing wave. By a Lorentz boost, one can obtain as well solutions describing nonlinear propagating waves. These solutions so not seem to have a particular physical significance, but maybe their meaning has not yet been unravelled." See: http://books.google.com/books?id=qkkYFaat_ZgC&pg=PR8&lpg=PR8&dq=Smilga+qft&source=bl&ots=DNd34MV2zG&sig=_blV-CZgAN_412g8IX3aIAGxf6o&hl=en&ei=Ht-mSe6TC8PQkAW8v83dDQ&sa=X&oi=book_result&resnum=1&ct=result#PPA13,M1 for the context. It is clear that Frasca's work and this section of the article are related to studying these solutions but it is also clear that his work is at this point speculative in nature and does not belong in this article. On a related idea, the so-called Smilga's Choice appears to be a term invented by Frasca, if you leave the article in or hive it then maybe he or you could find us a reference to this. OK, this is my first comment on wiki, I hope I did this correctly. Mbkmbk9 (talk) 19:48, 26 February 2009 (UTC)

WP:3RR warning

I've reverted to the pre-revert war state of the article. Beware of revert wars, as you may be blocked for it. Now that being said (I'm no admin, I'm just warning people that you could very well get banned for this), it is a bit sad that Mr. Woit simply did not explain his position in more details and gave up on the whole thing rather than explain to us how the Frasca/Smigma articles/books are not reliable when it comes to this topic (see his blog). Anyway, I left a message on his talk page, perhaps he'll come back an explain where Frasca got it wrong and give us some refs. Headbomb {^ταλκ_{κοντριβς} – WP Physics} 16:57, 26 February 2009 (UTC)

I'm not Peter, but I can give it a shot. The problem is not that the Smilga book is unreliable. The problem is that the material in the new section isn't sufficiently notable to be included in the main article on Yang-Mills theory. Yang-Mills is a large subject, and the main article can't reasonably cover every conceivable topic. The solutions from Smilga's book are solutions to the Yang-Mills equations, but they are not solutions that everyone who works with Yang-Mills theory needs to know about. (Most books on Yang-Mills don't mention these solutions.) The situation is further complicated because these solutions are presented in the context of some highly speculative research by Frasca (Pra1998), who authored the section. I would recommend deleting the section, as there's probably not enough notable material here to justify the effort required to extract it from the speculation and original research. --A.J. Tolland —Preceding unsigned comment added by 198.129.67.74 (talk) 19:51, 26 February 2009 (UTC)

Thanks AJ, couldn't have said it better myself.Peterwoit (talk) 20:00, 26 February 2009 (UTC)

Dear Headbomb,

Thank you very much for your intervention. People here do not even know how Wikipedia works. --Pra1998 (talk) 18:30, 26 February 2009 (UTC)

I don't really understand why Headbomb chose to intervene at all. As you admit yourself, this is not an area of your expertise.--Innerproduct (talk) 18:34, 26 February 2009 (UTC)

Regardless of who's right in this dispute, revert-warring without discussion is not the way to solve it. Headbomb quite properly restored The Wrong Version, and if further reverts happen without discussion here then page protection might be warranted. It seems to me that maybe the appropriate way to resolve this would be to set up a request for comment — someone who does know something about this subject want to set one up? —David Eppstein (talk) 19:36, 26 February 2009 (UTC)

I've restored to the previous version because there was a revert war, not because I supported it. Upon further review, there is definitely a conflict of interest here, or at the very least, enough grounds to suspect one as well as concerns of accuracy and original research (hence the tags on the article, were there concerns of conflict of interest and original research, only the disputed tag would've made its way). The physics project was notified, and apparently a tons of people came here from Peter Woit's blog. Everyone seems to agree that this section is at best non-notable, and at worse self-publishing. So the section should be removed from the article. Whether these people are aware of how wikipedia works or not is irrelevant, consensus is that should be removed. Headbomb {^ταλκ_{κοντριβς} – WP Physics} 00:03, 27 February 2009 (UTC)

The pre revert war is the one with Marco Frasca version, so I reverted to that version because otherwise people will NOT be able to judge the material properly. They will have to click on the history of the article, which is already extremely confusing. The dispute warning is enough to make sure one thinks that the information presented can be accurate or not, and is wainting for an evaluation on the talk page. If anyones think it's necessary, move the section for apreciation on the talk page, but please, do not delete it from the main article. Daniel de França (talk) 12:34, 27 February 2009 (UTC)

U(1) and QED

The first introductory paragraph implies that the U(1) of SU(2)xU(1) on electroweak theory is the U(1) of QED. This is not the case -the U(1) in electroweak is u(1) hypercharge. —Preceding unsigned comment added by 128.230.72.196 (talk) 17:04, 12 March 2009 (UTC)

Removed "Integrable solutions of classical Yang-Mills equations and QFT"

Per the arguments presented by everyone here, consensus is that this is original research, non-notable, and potentially self-publication, I have deleted this section. Headbomb {^ταλκ_{κοντριβς} – WP Physics} 00:14, 27 February 2009 (UTC)

See also Wikipedia:Suggestions for COI compliance.Headbomb {^ταλκ_{κοντριβς} – WP Physics} 01:01, 27 February 2009 (UTC)

Headbomb, do you think science is something decided by majority? Before an overwhelming number of people complaining, without a real understanding of the content, you removed it. The point here is that I am not a person who wrote a libel against a part of the scientific community becoming an instantaneous star, with anyhow a poor scientific curriculum, able to move a lot of people against a single one. If this is a serious project you were not.—Preceding unsigned comment added by Pra1998 (talk • contribs) 13:20, 27 February 2009 (UTC)

Pra1998, you would do well to read the Wikipedia guidelines on original research, notability, and conflict of interest, as Headbomb has already mentioned. There is no vendetta against you. Read and you will understand. - mako 00:57, 28 February 2009 (UTC)

The material you want to include is from 2006–2008, and so this material didn't have very much time to mature. There is strong opposition to including this in the YM article, for concerns of conflicts of interest, original research, non-notability. My personal opinion here is irrelevant, consensus is that this should not be included. If this is really notable, then you'll be able to find a review which cites Frasca's work positively which you may or may not be (although the evidence is pretty strong that you are indeed Frasca). Science is not decided by majority, true, but it also is not decided by lone rogues who do uses public forums of discussion to push their theories. Find a review which cites Frasca's work positively, and then it can be used in here. Otherwise, the consensus probably won't change that this is not worthy of inclusion. I'm not saying that Frasca's work is crap, or crank science or anything like that, I'm just saying that its place is not on Wokipedia. Note that most of your contribution to this article is deemed very acceptable (aka this part [2])

As far as libel, you may not have written one to become a "scientific star", but you certainly have no problem depicting Woit's curriculum as "being poor". If you have a problem with this material being removed, either find us a review citing Frasca positively, or take it to WP:RfC. See also Wikipedia:Suggestions for COI complianceHeadbomb {^ταλκ_{κοντριβς} – WP Physics} 01:00, 28 February 2009 (UTC)

It may be relevant to point out that one of the references cited in the disputed section [3] has a significant error in it, despite being published. Namely, in the proof of Theorem 1, the author is assuming that an extremum A for the Yang-Mills action for a special class of connections (namely those in which $A_{1}^{1}=A_{2}^{2}=A_{3}^{3}$ and all other components vanish) is necessarily an extremum for the Yang-Mills action for all other connections also, but this is not the case (just because $YM(A)\geq YM(A')$, for instance, for A' of this special form, does not imply that $YM(A)\geq YM(A')$ for general A'). Since one needs to be an extremiser (or critical point) in the space of all connections in order to be a solution to the Yang-Mills equations, the mapping provided in Theorem 1 has not been shown to actually produce solutions to the Yang-Mills equation (and I suspect that if one actually checks the Yang-Mills equation for this mapping, that one will not in fact get such a solution). Terry (talk) 20:32, 28 February 2009 (UTC)

Terry, the author assumes that exists a class of solutions that maps Yang-Mills action on the one of a scalar field. You can find that above solution is indeed a solution of Yang-Mills equations. Check Smilga book [4]. Instead to rely on questionable theoretical arguments, take Maple or Mathematica and check it. There is no claim about what you are saying --Pra1998 (talk) 11:27, 2 March 2009 (UTC)

Pra1998, I think you may be confusing the Yang-Mills action ${\frac {1}{4}}\int {\hbox{tr}}(F^{\mu \nu }F_{\mu \nu })$ with the Yang-Mills equations $D^{\mu }F_{\mu \nu }^{a}=0$. If one takes the ansatz $A_{1}^{1}=A_{2}^{2}=A_{3}^{3}=\phi$ suggested in the paper, with all other components zero, with $\phi$ obeying the $\phi ^{3}$ equation, then the Yang-Mills equations $D^{\mu }F_{\mu \nu }^{a}=0$ do not hold. For instance, the a=1, $\nu =2$ component of $D^{\mu }F_{\mu \nu }^{a}$ has a top order term of $\partial _{1}\partial _{2}\phi$ plus lower order terms, and this does not vanish for general solutions of the equation $\partial ^{\mu }\partial _{\mu }\phi =\phi ^{3}$.

I don't think Smilga's book makes the claim that every solution of the $\phi ^{3}$ equation maps to a solution to the Yang-Mills equation, as this paper does, but I would be interested to see a specific page number reference if I am mistaken. Terry (talk) 00:31, 3 March 2009 (UTC)

To put it another way: once one imposes a constraint on the fields (such as the ansatz $A_{1}^{1}=A_{2}^{2}=A_{3}^{3}$), then the Euler-Lagrange equations to the Yang-Mills functional acquire a Lagrange multiplier term, and the resulting critical points are no longer solutions to the original Yang-Mills equation, but to some modified version of this equation. The Frasca paper appears to ignore the Lagrange multiplier term completely. Terry (talk) 00:40, 3 March 2009 (UTC)

I just want to support the assertion that Terry is correct - and only a miracle could imply that the configuration under discussion is a solution to Yang-Mills equations despite the mistake. Solutions to Yang-Mills equations must be extrema of the action among all the configurations, not just within a limited subclass. Because there is no symmetry (of the action) under the transformation that locally flips the sign e.g. of $A_{1}^{1}-A_{2}^{2}$, by a space-dependent sign, there is no reason to expect that the functional derivative in the direction transverse to the $A_{1}^{1}(x)=A_{2}^{2}(x)=A_{3}^{3}(x)$ submanifold (of the configuration space) vanishes. Quite on the contrary: it almost never vanishes. It follows that new solutions to Yang-Mills equations haven't been found in this way. Incidentally, I would appreciate if Pra1998 didn't refer to himself as "the author" if the author is identical to Pra1998 because such terminology creates a false impression of neutrality.--Lumidek (talk) 07:03, 11 March 2009 (UTC)

One more comment added later. Discussions on Marco's blog [5] have made it clear that his configuration is not a solution to Yang-Mills equations, not even by a miracle. The discrepancy preventing him from solving the Yang-Mills equations of motion - from reducing them to the quartic scalar equations of motion multiplied by $\delta _{\nu }^{\prime a}$ (spatial components of the Kronecker delta) - can be summarized into the term $-(1-1/\alpha )\partial _{\nu }(\partial ^{\mu }A_{\mu }^{a})$. Marco or Pra1998 calls it a "gauge term". That's a nice nickname but it doesn't mean that the term can be neglected. Of course, the correct Yang-Mills equations are obtained for an infinite value of alpha and this term is nonzero. He can't set alpha to one, especially because this term is nonzero. More precisely, he can't be using the simplifications of the Lorentz gauge because his Ansatz doesn't satisfy the Lorentz gauge $(\partial ^{\mu }A_{\mu }^{a})=0$. So this method doesn't generate new non-trivial Yang-Mills equations from quartic scalar solutions, and one doesn't have to rely on Terry's observation that the "extreme action" arguments in Frasca's paper were not correctly proving the validity of the solution.

If you care, the wrong term is proportional to $\partial _{\nu }\partial _{a}\phi$ for Frasca's Ansatz $A_{1}^{1}(x)=A_{2}^{2}(x)=A_{3}^{3}(x)=\phi$ and it is the only nonzero term for $a\neq \nu$. If you want Frasca's method to be legitimate, you have to impose both Ansatz and the Lorentz gauge. But for his Ansatz, Lorentz gauge reduces to $\phi =$const and the remaining equations imply that $\phi$ must vanish. The zero solution is the only solution generated by Frasca's Ansatz. To save the work for Pra1998, Frasca now claims that the gauge field doesn't have any $A_{2}^{1}$ component because it is zero and equations of motion for degrees of freedom that vanish don't have to be satisfied. I am confident that everyone who has seen an apple fall, even when it was in the origin of coordinates two feet above the table, which includes all editors of Yang-Mills theory on Wikipedia except for Pra1998, will consider this question settled. --Lumidek (talk) 11:06, 11 March 2009 (UTC)

I was not aware of all this discussion about this question. It makes things surely interesting. Last paper by Frasca settles the question Frasca's preprint see also his blog. This means that the first comment by Terry was too harsh and surely the mapping theorem cannot be applied to all field configurations. The point is that Smilga's solutions are enough to prove original Frasca's paper right and Terry's scepticism about their non-existence wrong. In this paper there is no explicit derivation of such solutions that instead I have taken from Another Frasca's preprint.--Pra1998 (talk) 16:04, 16 March 2009 (UTC)

The preprint doesn't change anything about the mistakes pointed out above. And Pra1998 is the same person as Marco Frasca and it is dishonest to hide this fact repeatedly. --Lumidek (talk) 08:57, 23 March 2009 (UTC)

Lubos, I find dishonest your behavior. The preprint shows without doubt that such common solutions between Yang-Mills theory and scalar field theory do exist and this was what Terry asked for. Coming here to claiming something false is really dishonest. I would like to remember that you entered into Terry's blog, in the latest post Terry wrote at that time, in the comment area to put out your false arguments. Terry removed such upset entering. What is worst in all this is that you are proved repeatedly wrong in a lot of parts of the blogosphere but I have never seen you behaving honestly admitting your errors. So, be wise and turn back to the site you come from.--Pra1998 (talk) 09:34, 8 April 2009 (UTC)

"Do it yourself" isn't much of an argument here. If you have reviews or books giving positive ratings about the accuracy/soundness/pertinence of Frasca's work, then provide those, otherwise you're have nothing to stand on. If you can't find those, then this does not meet WP:NOTABILITY, regardless of whether or not the material is correct. Headbomb {^ταλκ_{κοντριβς} – WP Physics} 13:31, 2 March 2009 (UTC)

Dear Headbomb, you can find good reviews of some Frasca's works here [6], e.g. this MR2345223 (2008f:81084) and this MR2332380 (2008e:81089). If you belong to some recognized institution you should have access to this mathematical database. But here I just entered into this discussion area to answer a wrong affirmation by Terry, a claim that can be easily proved wrong with Maple or Mathematica. I have no interest to defend Frasca's work as you can see from my preceding interventions where I would have removed the refs without problem. The fact that you removed also Smilga's book, well, that is your choice. You removed just plain mathematics but it is your own right.Thank you anyway.--Pra1998 (talk) 20:31, 2 March 2009 (UTC)

That's a search engine. That's the equivalent of saying "if don't believe me, you can find good source here [7]". I'm not about to wade through possibly hundreds of hits (and even if it were ten hits), reading 50-100 pages documents looking for one which cites Frasca favourably. Headbomb {^ταλκ_{κοντριβς} – WP Physics} 00:06, 3 March 2009 (UTC)

Headbomb, I give you the exact refs. to look for. If you are not able to cope with scientific databases please ask somebody expert. This is not google, this is a database of American Mathematical Society, well known to people doing research, that gives reviews of papers after publication. Please, ask to a person being an active researcher in a recognized institution.--Pra1998 (talk) 07:56, 3 March 2009 (UTC)

Headbomb, don't fall for this nonsense. The Mathscinet review of the Prasca paper referred to in the deleted section isn't a review at all. It doesn't add any content or commentary. It simply quotes the introduction to Frasca's paper. —Preceding unsigned comment added by 99.48.55.131 (talk) 15:57, 3 March 2009 (UTC)

DOI

I have added some references but DOI numbers do not appear even if they seem correctly inserted. Any help?--Pra1998 (talk) 13:34, 5 May 2010 (UTC)

Thanks a lot to Headbomb!--Pra1998 (talk) 13:44, 5 May 2010 (UTC)

Vandalism

A user from wanadoo.fr added a disputable comment having no value for Wikipedia. I removed as vandalism. This is again the question if new physics results should go on Wikipedia producing such kind of effects. Let me know your view. But adding such a comment in the article rather than the discussion is pure vandalism.--Pra1998 (talk) 16:18, 6 November 2010 (UTC)

Anecdote without citation

An anonymous user introduced this:

"Prior to Yang-Mill's publication, Pauli had given a seminar on the same idea but did not publish because he did not believe it would work at the time. When Yang gave his talk, Pauli asked Yang a question which Yang could not answer. Yang's talk ended abruptly. Yang then avoided Pauli even though Pauli tried to reach Yang to discuss the idea in more detail. Yang's behavior led many scientists to question how much Yang had to do with the origination of the idea."

I think this kind of stories should be well supported by proper citations. Does anyone out there know this? —Preceding unsigned comment added by Pra1998 (talk • contribs) 08:43, 24 November 2009 (UTC)

[8] Please sign your posts. And if you think something need citations then tag don't delete. --Michael C. Price ^talk 10:29, 24 November 2009 (UTC)

Michael, sorry for omitting my signature. Just oversight. I tried to use but it did not seem to work in preview and so I have chosen the bad way.--Pra1998 (talk) 16:49, 24 November 2009 (UTC)

Yang had relationship breakdown with one of his co-workers, and never spoken to him since. Was it Mills or Lee, do you know? --Michael C. Price ^talk 22:16, 24 November 2009 (UTC)

If you take 't Hooft's book "50 years of Yang-Mills theory" you will find few lines written by Yang about Robert Mills. This recall does not appear to contain any rancor, rather these words praise Mills. This is recalled as an occasional but fruitful encounter. Instead, with Lee the situation was quite different as they collaborate for several years producing exceptional results in different areas of physics. Then, this cooperation stopped abruptly after Nobel prize was awarded to them. I cannot say more about this matter but I do not know any source that could corroborate relationship breakdown with Lee.--Pra1998 (talk) 10:36, 25 November 2009 (UTC)

Ah, Lee was it? I think it was some sort of priority dispute. New Scientist or Scientific American mentioned it briefly, with a photo of them both together (perhaps at the Nobel ceremony), commenting that they were still speaking to each other in those days, but not since.--Michael C. Price ^talk 10:48, 25 November 2009 (UTC)

Apparently the order of authors' names. So completely understandable. :-)

Sadly it would seem that the Nobel Prize also split apart the very successful partnership of Lee and Yang, the relationship eventually foundering in part over that occasional but trivial source of intense irritation to scientists who collaborate on research papers – the order of names on the paper. This was apparently particularly intense for the New Yorker article (Bernstein, 1962), and one suspects the final title, “A question of parity” is deeply ironic. Eventually things deteriorated so much that Lee and Yang, or perhaps that should be Yang and Lee, each published books describing entirely different histories of their collaborative papers (Regis, 1998). --Michael C. Price ^talk 00:57, 26 November 2009 (UTC)

't Hooft asked Yang to provide some materials for how he and Mills developed Yang Mills theory.  All Yang was able to provide for 't Hooft's " 50 years of Yang Mills theory" were a couple pages from his graduate student days...does that dovetail with above discussion that Yang had actually borrowed Pauli's unpublished idea and got away with it since nobody at that time thought Yang mills was anything significant?

Also, Yang co-published a paper shortly after his Yang Mills with T.D. Lee, where the key argument was anti (against) Yang Mills thoery. So, Yang himself was not convinced about the validity of Yang Mills theory at that time. —Preceding unsigned comment added by 163.166.135.44 (talk) 01:27, 24 December 2009 (UTC)

There are documents which show that wolfgang Pauli developed in 1953 the first consistent generalization of the five-dimensional theory of Kaluza, Klein, Fock and others to a higher dimensional internal space. Because Pauli saw no way to give masses to the gauge bosons, he refrained from publishing his results formally. Pauli gave talks on the subject and many physicists of the time, including Yang, debated Pauli's unplublished theory.(See "On Pauli's invention of non-abrlian Kaluza-Klein Theory in 1953") —Preceding unsigned comment added by 69.156.210.199 (talk) 13:50, 27 October 2010 (UTC)

About Pauli in 1953: apparently he compactified a 6-dimensional space and found SU(2) gauge theory (much like Kaluza and Klein toroidally compactified a 5-dimensional space and found U(1) gauge theory). However, Yang and Mills showed how requiring local gauge invariance severely restricts the terms in the Lagrangian. To me, these are very different things: on the one hand Pauli found one SU(2) gauge theory, and on the other hand Yang and Mills showed that it's basically the only reasonable one out there. We can debate forever from where Yang and Mills got their idea, but to me it's clear that the point here is the importance of local gauge invariance, and this is the point Yang and Mills made. —Preceding unsigned comment added by 132.206.126.18 (talk) 21:24, 15 February 2011 (UTC)

Unmanageable formulas

I'm a complete layman in the field, but IMO a lot of formulas are bloated and could be written more concisely in a coordinate-free manner and/or with some intermediate definitions. — Kallikanzarid^talk 18:47, 19 February 2011 (UTC)

Citations in Mathematical Overview

The lack of any citations in the Mathematical Overview section of the article seems like a bit of a problem; in particular it makes it almost impossible to use the article as a starting point for a review of the mathematics of Yang-Mills theory (as I currently am).

In particular, I'm looking for a citation for Pra1998's edit in 2009 adding the note about the lack of distinction between and upper and lower a indices; this fact seems to be taken implicitly in all books and articles I've found so far and was hoping someone would know a specific source explicitly stating that it is legitimate.

I've also made a post at the reference desk because, being new, I wasn't sure what the best channel was.

https://en.wikipedia.org/w/index.php?title=Wikipedia:Reference_desk/Science&oldid=648352928#Citation_for_Yang-Mills_Theory_-_Mathematical_Overview — Preceding unsigned comment added by Tjlr2 (talk • contribs) 18:09, 22 February 2015 (UTC)

Please, sign your comments. I have written to you in my talk area.--Pra1998 (talk) 18:44, 22 February 2015 (UTC)

Sorry, hadn't realised how to do that. Should I go back and sign the original now or leave it unsigned? — Tjlr2 (talk) 19:09, 22 February 2015 (UTC)

Science or math?

Is Yang-Mills theory science or math? They are not the same thing.

The lead says, "Yang–Mills theory seeks to describe the behavior of elementary particles..." which suggests that its developers have intended to provide a mathematical model for this aspect of the physical world - that is, they intend to provide a tool for elucidating a scientific theory of the world.

After providing a précis list of the symmetry groups of the Standard Model, in the 'History and theoretical description' section I read "This may be the reason why confinement has not been theoretically proven, though it is a consistent experimental observation." While as a lay person, I understand that confinement is in fact what has been observed, I don't at all understand what - in the context of a scientific theory of the world - can even be meant by "theoretically proven".

I know that it is common to refer to large parts of mathematics as the 'theory' of this or that, as the Theory of Numbers' and so forth and though I think this is not nonsensical, I am uncomfortable with this sort of construction appearing here, if, as in the lead, 'YM Theory' is intended to be an article describing a scientific theory of the natural world. Theorems are objects of mathematics. Theories - which are always contingent and so are unprovable - are objects of science.

In 1979, during his noted series of popular lectures given at the University of Aukland, New Zealand, Richard Feynman said

"We then have the following physical problem as I mentioned before - where does this number [1/α] come from? From experiment. I know. But a good theory would have that this thing is equal to one over two times pi times the cube root of three times six and so on, so that you know what it was, if you know what I mean.

"It's a number that has to be put in that nature has or so to speak, if you were religious you would say 'God has created that number.' But we would like to figure out if we can a little clue as to how He thinks, to make a number like this. For example, maybe that's... [pointing to a decimal on the chalkboard] why isn't that a four, there, you see?

[...]

"Now, that summarizes all of the problems associated with quantum electrodynamics. The most beautiful one is the coupling constant, one hundred thirty seven point... and so on and all good theoretical physicists put that up on their wall and worry about it. There is at the present time no idea of any utility for getting at that number. There have been from time to time suggestions but they didn't turn out to be useful. They would predict that the number was exactly a hundred and thirty seven... Well, the first idea was by Eddington, and experiments were very crude in those days... the number looked very close to a hundred and thirty six. So he proved, by pure logic, that it had to be a hundred and thirty six. Then it turned out that experiments showed that that was a little wrong - it was nearer a hundred and thirty seven. So he found a slight error in the logic and proved... 'pure logic' - it had to be exactly the integer, a hundred and thirty seven. It's not the integer. It's a hundred and thirty seven point oh three six oh. Every once in a while someone comes out and they find out that if they combine 'pi's and 'e's and twos and fives with the right powers and square roots, you can make that number.

"It seems to be a fact that's not fully appreciated by people who play with arithmetic that you'd be surprised how many numbers you can make by playing with 'pi's and twos and fives and so on. And if you haven't got anything to guide you except the answer, you can always make it come out even to several decimal places by suitable jiggling about. It's surprising how close you can make an arbitrary number by playing around with 'nice' numbers like 'pi' and 'e'.

"And therefore in the history of physics there [is] paper after paper [by] people who have noticed that certain specific combinations give answers which are very close in several decimal places to experiment except that the next decimal place of experiment disagrees with it. So it doesn't mean anything."

"Richard Feynman Video - The Douglas Robb Memorial Lectures - Part 4: New Queries"

Given at the University of Aukland, 1979

http://vega.org.uk/video/subseries/8

http://vega.org.uk/video/programme/48

~17:00

How is Feynman's discussion of the EM coupling constant not an example of the inappropriateness of imagining that proving a mathematical theorem or proving the consistency of some assertions in a symbolic calculus is the same thing as 'proving' an assertion about the world? Have I not argued sufficiently for the affinity between on the one hand claims for mathematical (geometric, topological etc.) derivations of α and on the other hand, hope for 'mathematical proof' that confinement is necessary?

Alternately, I understand that this observation is supported by a thirty-five-year-old perspective from a man with a complicated attitude toward 'villozovy' and so on. Is Feynman's view now considered dowdy? Or still just inconvenient? Rt3368 (talk) 17:03, 25 August 2015 (UTC)

The interplay between theoretical physics and mathematics has a long history and has its share of controversies. Mathematicians often work hard to rigorously justify analytic leaps made by theoretical physicists, who are often unconcerned with mathematical rigor. An earlier example where mathematicians and physicists struggled futilly for centuries to prove something observationally obvious is Stability of the Solar System. The give and take can be productive for both sides. Proof that Yang–Mills theory does not invariably lead to confinement might suggest a new direction for experiments, for example. And all too often that hard work gets us nowhere in terms of real world insights. But your argument is with the theoretical physics community, not Wikipedia, which only attempts to fairly summarize their work, and that community clearly considers a proof of confinement important. --agr (talk) 22:29, 25 August 2015 (UTC)

Very well. But I think that Wikipedians ought to be able to explain the meaning of phrases that appear in Wikipedia's articles. In the 'History and theoretical description' section I read

"This may be the reason why confinement has not been theoretically proven..."

What is meant by the phrase "theoretically proven"? Rt3368 (talk) 18:27, 28 August 2015 (UTC)

Physics is an experimental science and whenever people do experiments get back numbers. So, mathematics is the language used by nature and physics must use it. Theoretical physics is that part of physics that, by using mathematics, tries to explain what the outcomes of an experiments are and, sometimes, to provide other tests for experimental physicists. E.g., Higgs, using a mathematical model, postulated a new particle. On 2012, physicists at LHC observed it with the expected properties (except for the mass that was not possible to forecast but it will be with other mathematical models). So, also confinement is observed in all experiments where particles interacting through the strong force are observed, as at LHC or in cosmic rays. But, notwithstanding we have the correct mathematical model quantum chromodynamics that uses Yang-Mills theory, nobody was able to solve the equations to prove confinement so far. This is the meaning of "theoretically proven". I hope this is enough.--Pra1998 (talk) 20:58, 28 August 2015 (UTC)

No, it's not, actually.

As a lay person, I'm yet familiar with the narrative you describe, understand it at a lay person's level and don't dispute the events or tentative conclusions. I have no interest in unorthodox theoretical interpretations of these experimentalists' results.

In the context of the article, "theoretically proven" might mean

"proven by someone's theory of the word 'proof'"; or

"proven, using in some way some scientific theory of the world"; or

"theorematically proven"; or

"shown to be a consequence of a mathematical model, which model otherwise more or less informs a scientific theory of the world"; or

something else.

The first of these seems unlikely to have been intended. The second of these is impossible, since a theory of the world is always contingent and incapable of offering proof of anything. Something like the third or the fourth meaning above seems most likely but the matter is opaque.

Here's a pretty good proof, you'll allow:

Theorem: "There is NO largest prime number."

Proof:

Let S be a non-empty finite sized set of prime numbers.

Consider P, the product of all members of S.

Consider Q = P + 1.

Q must be evenly divisible by a prime number.

Q divided by any member of S must have remainder of 1.

Ergo, there must be a prime number not a member of S.

Q.E.D.

Proofs are objects of courts of law and of the tautological theorEMs of mathematics. They are never objects of the always contingent theorIEs of scientific inquiry. I say that "theoretically proven" joins other oxymoronic phrases such as "scientific proof" that at least in English have no meaning at all and convey no useful information or description. They're littered throughout discussions of science in popular media - including in articles in Wikipedia - and they obfuscate intended meaning. They are due to inattentive thinking and inattentive habits of expression and they proliferate because no one objects to their inadequacy. Rt3368 (talk) 15:05, 2 September 2015 (UTC)

Your points just say to me that you miss completely what physics is and what physicists do. I invite you to take a look at that article. This could help you before to cope with a highly specialistic article.--Pra1998 (talk) 16:23, 2 September 2015 (UTC)

That's a pretty empty response. I understand perfectly well what physics is and I understand also when an individual is utterly unfamiliar with the philosophy of science, to the extent that their expertise in a narrow field obscures the most straightforward ideas and trades enlightenment for obscurantism. It's a malady endemic in the sciences, and at Wikipedia. Rt3368 (talk) 01:05, 11 September 2015 (UTC)

No, you don't. You are utterly unfamiliar with physics and are lost in your obscure and empty analysis of what science is. Please, move on to a more proper place to discuss. Thanks.--Pra1998 (talk) 07:27, 11 September 2015 (UTC)

Would "This may be the reason why confinement has not been proven within Yang Mill theory..." make it more clear?--agr (talk) 19:57, 28 August 2015 (UTC)

That would be better. I think I would choose (trying to maintain some brevity) something like

"This may be the reason why confinement has not yet been proven to be a consequence of the Yang-Mills mathematical model..."

Alternately a simple change to "theorematically proven" - as long as this is a mathematically orthodox statement in the sense that a demonstration of the confinement requirement can be identified as a proof of some stated theorem - might convey the same meaning in a less explicit way.

In contrast, I believe a subsequent sentence,

"Proof that QCD confines at low energy is a mathematical problem of great relevance..."

is descriptive and correct as it stands, since it very rightly identifies that the truth status of a mathematical assertion that's part of QCD is - as it states - "of great relevance". Rt3368 (talk) 15:05, 2 September 2015 (UTC)

Scaling?

At this date (6/11/2017), it's written "{\displaystyle [A]=[L^{\frac {2-D}{2}}]} [A]=[L^{\frac {2-D}{2}}]" . I think it's 2-\frac{D}{2}, but I could be wrong (my own computation gives this power, and I also think it agrees with the next result, contrary to the given power) — Preceding unsigned comment added by 134.157.64.191 (talk) 18:23, 6 November 2017 (UTC)

Peter Woit is back with an edit war

I think that, in agreement with Woit's ideas we should remove all the section. The paper he is questioning is regularly published in a prestigious journal and is a collaboration with a reputable physicist.--Pra1998 (talk) 21:52, 27 April 2018 (UTC)

This page keeps on being vandalized by Peter Woit or some of his sockpuppets. This should be ended.--Pra1998 (talk) 14:09, 29 April 2018 (UTC)

Discussion, edit warring. Comment

Could we please discuss the passages and parts regarding known or unknown quantities which seem to inflame our anonymous editor so much? I'm afraid I don't know much about theoretical physics, but I'm willing to try to learn. — Javert2113 (talk) 15:05, 29 April 2018 (UTC)

Sorry, forgot to add @Pra1998: thoughts? — Javert2113 (talk) 15:11, 29 April 2018 (UTC)

It is quite simple. Wherever this guy reads "Frasca" he aims to remove the material without mercy. The point is that the cited paper is on the same foot of the others in that section and in the preceding one and so, based on his principles, we should remove both of them. The paper in question is published in one of the most important journals of particle physics, Physics Letters B (It is the one where CERN published its work on the discovery of the Higgs particle). Besides, it is a collaboration with one of the Editors of the European Physical Journal C that, as importance, is on the same foot of the other.--Pra1998 (talk) 15:13, 29 April 2018 (UTC)

There is no need to deal with the complex scientific issues here. Pra1998=Marco Frasca, and my understanding is that Wikipedia policy does not allow people to add references to their own work to Wikipedia pages. Peterwoit (talk) 14:50, 30 April 2018 (UTC)

Commuting versus non-commuting gauge fields

I would like to add a new section to this article, with the above section title, following the section "Quantization." My proposed section is below. Are there any objections or suggestions, prior to my doing so?

Studying the physics of Yang-Mills gauge theory requires understanding what happens to Maxwell’s electrodynamics, and U(1) quantum electrodynamics (QED), when Maxwell’s commuting (abelian) gauge fields ${{A}^{\mu }}$ become non-commuting (nonabelian) gauge fields ${{G}^{\mu }}={{\tau }_{i}}G_{i}^{\mu }$ covariantly transforming, for example, under the compact simple Yang-Mills gauge group SU(N) with NxN Hermitian generators ${{\tau }_{i}}={{\tau }_{i}}^{\dagger }$ and a commutator $\left[{{\tau }_{i}},{{\tau }_{j}}\right]=i{{f}_{ijk}}{{\tau }_{k}}$ typically normalized such that ${\text{tr}}\left({{\tau }_{i}}^{2}\right)={\tfrac {1}{2}}$ for each $i=1...{{N}^{2}}-1$. Whereas electrodynamics is a linear theory in which the gauge fields to not interact with one another, Yang-Mills theory is highly nonlinear with mutual interactions amongst the gauge fields.

In flat spacetime, in classical electrodynamics, a gauge-invariant field strength ${{F}^{\mu \nu }}$ is related to the gauge fields ${{A}^{\mu }}$ by:

${{F}^{\mu \nu }}={{\partial }^{\mu }}{{A}^{\nu }}-{{\partial }^{\nu }}{{A}^{\mu }}$.

This may also be written more generally as ${{F}^{\mu \nu }}={{D}^{\mu }}{{A}^{\nu }}-{{D}^{\nu }}{{A}^{\mu }}$ using the gauge-covariant derivative ${{D}^{\mu }}={{\partial }^{\mu }}-ig{{A}^{\mu }}/\hbar c$, because the commutator $\left[{{A}_{\mu }},{{A}_{\nu }}\right]=0$. With ${{c}^{2}}{{\mu }_{0}}{{\varepsilon }_{0}}=1$ and Coulomb constant ${{k}_{\text{e}}}=1/4\pi {{\varepsilon }_{0}}$, the classical Maxwell equation for electric charge strength is:

$c{{\mu }_{0}}{{j}^{\nu }}={{\partial }_{\sigma }}{{F}^{\sigma \nu }}=\left({{g}^{\mu \nu }}{{\partial }_{\sigma }}{{\partial }^{\sigma }}-{{\partial }^{\nu }}{{\partial }^{\mu }}\right){{A}_{\mu }}$,

which spacetime-covariantly includes Gauss’ electricity and Ampere’s current laws. The classical equation for magnetic charge strength is

$c{{\mu }_{0}}{{p}^{\sigma \mu \nu }}={{\partial }^{\sigma }}{{F}^{\mu \nu }}+{{\partial }^{\mu }}{{F}^{\nu \sigma }}+{{\partial }^{\nu }}{{F}^{\sigma \mu }}=0$,

which spacetime-covariantly includes Gauss’ magnetism and Faraday’s induction laws. The zero in the monopole equation and thus the non-existence of magnetic monopoles (setting aside possible Dirac charge quantization) arises from the flat spacetime commutator of ordinary derivatives being $\left[{{\partial }_{\mu }},{{\partial }_{\nu }}\right]=0$. In integral form, the Gauss’ magnetism law component of the above becomes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \;\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 0} , whereby there is no net flux of magnetic fields across closed spatial surfaces. (Note: The point of various “bag models” of QCD quark confinement, is that there is similarly no net flux of color charge across the closed spatial surfaces of color-neutral baryons.)

Summing the four-gradient ${{\partial }_{\nu }}$ with the above electric charge strength, we readily obtain:

$c{{\mu }_{0}}{{\partial }_{\nu }}{{j}^{\nu }}={{\partial }_{\nu }}{{\partial }_{\sigma }}{{\partial }^{\sigma }}{{A}^{\nu }}-{{\partial }_{\nu }}{{\partial }_{\sigma }}{{\partial }^{\nu }}{{A}^{\sigma }}=0$,

which is the continuity equation governing the conservation of electric charge. This becomes zero, once again, because of flat spacetime commutator $\left[{{\partial }_{\mu }},{{\partial }_{\nu }}\right]=0$.

In quantum electrodynamics, the charge density becomes related to the Dirac wavefunctions $\psi$ for individual fermions by ${{j}^{\nu }}=e{\overline {\psi }}Q{{\gamma }^{\nu }}\psi$ where $e$ is the electric charge strength related to the running "fine structure" coupling ${{\alpha }_{e}}\left(\mu =0\right)=1/137.036...$ by ${{k}_{\text{e}}}{{e}^{2}}=\hbar c{{\alpha }_{e}}$, and $Q=-1,+{\tfrac {2}{3}},-{\tfrac {1}{3}}$ for the electron, up and down fermions, and their higher-generational counterparts. Meanwhile the propagators for the individual photons which form the gauge fields are obtained by inverting the electric charge equation and converting from configuration into momentum space using the substitution $i\hbar {{\partial }^{\mu }}\to {{q}^{\mu }}$ and the $+i\varepsilon$ prescription. Because the charge equation is not invertible without taking some further steps, it is customary to utilize the gauge condition ${{\partial }_{\sigma }}{{A}^{\sigma }}=0$ to obtain

${{A}_{\alpha }}={{\hbar }^{2}}c{{\mu }_{0}}{\frac {-{{g}_{\alpha \nu }}}{{{q}_{\sigma }}{{q}^{\sigma }}+i\varepsilon }}{{j}^{\nu }}$

which includes the photon propagator up to a factor of $i$. Alternatively, one can introduce a Proca mass by hand into the charge equation. Then, ${{\partial }_{\sigma }}{{A}^{\sigma }}=0$ is no longer a gauge condition but a requirement to maintain continuity (charge conservation), and with $i\hbar {{\partial }^{\mu }}\to {{k}^{\mu }}$ we arrive at the inverse:

${{A}_{\alpha }}={{\hbar }^{2}}c{{\mu }_{0}}{\frac {-{{g}_{\alpha \nu }}+{{k}_{\nu }}{{k}_{\alpha }}/{{m}^{2}}}{{{k}_{\sigma }}{{k}^{\sigma }}-{{m}^{2}}+i\varepsilon }}{{j}^{\nu }}$

which includes a massive vector boson propagator up to $i$. Of course, adding a mass by hand destroys renormalizability, so it is necessary to find a way that this can be restored.

In Yang-Mills Gauge Theory, ${{A}^{\mu }}\to {{G}^{\mu }}={{\tau }_{i}}G_{i}^{\mu }$ becomes a non-commuting gauge field, $\left[{{G}_{\mu }},{{G}_{\nu }}\right]=\left[{{\tau }_{i}},{{\tau }_{j}}\right]{{G}_{i}}_{\mu }{{G}_{j}}_{\nu }=i{{f}_{ijk}}{{\tau }_{k}}{{G}_{i}}_{\mu }{{G}_{j}}_{\nu }\neq 0$, and the field strength therefore graduates to the gauge-covariant, not gauge-invariant:

${{F}^{\mu \nu }}={{\tau }_{k}}{{F}_{k}}^{\mu \nu }={{D}^{\mu }}{{G}^{\nu }}-{{D}^{\nu }}{{G}^{\mu }}={{\partial }^{\mu }}{{G}^{\nu }}-{{\partial }^{\nu }}{{G}^{\mu }}-ig\left[{{G}^{\mu }},{{G}^{\nu }}\right]={{\tau }_{k}}\left\{{{\partial }^{\mu }}G_{k}^{\nu }-{{\partial }^{\nu }}G_{k}^{\mu }+g{{f}_{ijk}}{{G}_{i}}^{\mu }{{G}_{j}}^{\nu }\right\}$.

With A replaced by G, it will be seen that this contains the equation $F_{\mu \nu }^{a}=\partial _{\mu }G_{\nu }^{a}-\partial _{\nu }G_{\mu }^{a}+gf^{abc}G_{\mu }^{b}G_{\nu }^{c}$ from the Mathematical overview above. Using differential forms, this may be written as the curvature $F=dG+G\wedge G$ arising from the gauge connection, see at pages 1 and 2. The non-linearity of Yang-Mills gauge theories becomes apparent if one uses the above to advance the source-free Lagrangian from the Mathematical overview to:

${\mathcal {L}}_{\mathrm {gf} }=-{\tfrac {1}{2}}{\text{Tr}}\left({{F}^{\mu \nu }}{{F}_{\mu \nu }}\right)=-{\tfrac {1}{4}}{{F}_{i}}^{\mu \nu }{{F}_{i}}_{\mu \nu }=-{\tfrac {1}{4}}{{\partial }^{[\mu }}{{G}_{i}}^{\nu ]}{{\partial }_{[\mu }}{{G}_{i}}_{\nu ]}-{\tfrac {1}{2}}g{{f}_{ijk}}{{\partial }^{[\mu }}{{G}_{i}}^{\nu ]}{{G}_{j}}_{\mu }{{G}_{k}}_{\nu }-{\tfrac {1}{4}}{{g}^{2}}{{f}_{ijk}}{{f}_{ilm}}{{G}_{j}}^{\mu }{{G}_{k}}^{\nu }{{G}_{l}}_{\mu }{{G}_{m}}_{\nu }$,

which includes three- and four-gauge boson interaction vertices.

Yang-Mills gauge theory differs from the abelian gauge theory of U(1) electrodynamics, by the mathematical and physical consequences of what happens when the gauge fields go from commuting to non-commuting in this way. PatentPhysicist (talk) 17:34, 21 February 2021 (UTC)

I think this could be a valuable addition to the article. Chanacya (talk) 18:41, 21 February 2021 (UTC)

PS: I noticed that this is a level 5 vital priority article, but only B class rated. I have studied Yang-Mill gauge theories for over 15 years, and would like to try to contribute to raising this. PatentPhysicist (talk) 02:51, 22 February 2021 (UTC)

I just added the section as written above, to the main article. PatentPhysicist (talk) 18:54, 23 February 2021 (UTC)

References