Talk:Gauge covariant derivative

Relation to other derives

Isn't it so that the gauge covariant derivative and the covariant derivative of GR are a partition of the mathematical covariant derivative. In GR the vector bundle is the tangent bundle and the gauge group is GL(4, R), which could be reduced to SO(3, 1). The gauge cov. der. is for all vector bundles which are not the tangent bundle. Right? --MarSch 10:50, 21 October 2005 (UTC)

The term covariant derivative is often used to describe connections on other vector bundles as well. However, this is not the agreed upon usage for Wikipedia because covariant derivative has a very precise meaning in physics. Instead, the intrinsic description of a "gauge" covariant derivative can be found in the article Koszul connection (which is fairly standard nomenclature between mathematicians and physicists). Silly rabbit 15:48, 27 June 2006 (UTC)

Request

Hi linas,

Is there any way you would be willing to write something up on a gauge connection? Right now that page is just a redirect to gauge theory. A gauge connection (unless I'm mistaken) is like a connection form adapted to a particular gauge. It then behaves in a special way under gauge transformations. I know this can be viewed in the language of principal bundles to a certain extent. But I don't think that's the way physicists (or many mathematicians) view them.

The "functors":

(Gauge connection) + (gauge fields) -> (gauge covariant derivative)

(G-principal connection form) + (representation of G) -> (covariant derivative)

are "naturally equivalent".

Cheers, Silly rabbit 15:40, 27 June 2006 (UTC)

Hi, yes they are, and speaking from experience, one of the stumbling blocks is to match up the vocabulary and notation used in the physics gauge theory texts with the notation prefered by geometers (Cartan connection, connection form). I presumed most practicing physicists working in this area eventually learn both, and I understand why geometers don't like (and don't learn) the physics notation: its "dirty", suitable only if you actually have to do a calculation, not terribly intuitive, ugly if you have to state a general principle.

I'll have to think about how to present the two (or three, counting GR) notations side-by-side. The book "Gravition" by Wheeler, etal is the only text I know that does this; however its a frusrating read, its rather loose. I've been procrastinating on another promised task (for someone with a small Erdos number!, so I better get cracking!); so nothing before the 4 July weekend. linas 19:24, 27 June 2006 (UTC)

Perhaps I'm being glib. This article partially describes the physics notion, incompletely, in a slightly confused way. The physicists "gauge field" is the value of the connection on some (horizontal, of course) section of the fiber bundle. The physicists "local gauge transformation" is a description of what happens when one moves from one section to another. The word "local" is used to emphasize that the section can be choosen arbitrarily at every point (as long as its differentiable in the end). Physicists almost always assume the trivial bundle, so that sections are well-defined. A section is often chosen by "fixing a gauge", that is, the section is specified as those points on the bundle such that the connection is irrotational, or divergence free, or whatever. I still get confused by torsion but perhaps this can be an opportunity to ponder.

Hmm gauge transformation fixed gauge fixing a gauge choice of gauge section horizontal section vertical subspace choice of gauge non-Abelian gauge field gauge field field strength field curvature linas 20:00, 27 June 2006 (UTC)

I don't find that you're being glib. We need to expand connections out into the physics realm. So my tacked-on "natural equivalence" was more for the purposes of getting this kind of thinking on track. (See, it worked.) Anyway, yes "dirty" is good. Do your worst (if you decide to take up the gauntlet at all). Silly rabbit 00:36, 30 June 2006 (UTC)

Some need for some discussion

Someone reverted my recent edits but did not care to explain why they thought they were revert-worthy. Perhaps here, on the talk page, would be a good place to have such a discussion? I'm not sure what the objection could possibly be, so let me explain how I came to write the long intro section. I'd just reviewed and made minor corrections to other related articles, e.g. vertical bundle and exterior covariant derivative. Then I stumbled onto this article.

There are already several nice articles on WP on the guage covariant derivative, but this one stood out as (1) being the very simplest, least complicated of the set, and (2) it completely failed to explain what was going on. The first sentence was even wrong.

So, my first edit was to point out that he gauge covariant derivative is exactly the same thing as the exterior covariant derivative. Then I realized that perhaps this was too obtuse a statement, the kind where everyone accuses WP of being too abstract and too incomprehensible and too hard to understand. So then I thought I should simplify the statement. After simplifying it, I compared it to the rest of the article, and it was clear that even then, all of the other affine connection articles on WP were far more abstract and complex, as compared to this one. This article deals with the simplest-of-the-simple introductions to the connection. So I thought, rather than glomming it up with lots of complex equations, I thought it would be best to keep it very simple, and keep it accessible to young students. Thus the intro: it points out that there are more abstract, more beautiful approaches to the topic, but keeps the rest of this particular article very low-brow, very simple. Which seems like a good idea -- there needs to be someplace to start learning this stuff.

Is that a good enough explanation for what this is all about? 67.198.37.16 (talk) 02:08, 24 April 2016 (UTC)

See, for example, the "Request" discussion immediately above. It would be a very worthy project to create a "rosetta stone" that translates the notation used here to the standard notation used by mathematicians. I have not yet stumbled on any other article on WP that makes this "connection". It would be good to have, and this article might be an OK place for that. 67.198.37.16 (talk) 02:30, 24 April 2016 (UTC)

• Comment. This edit seems to be WP:Original synthesis. Other editors may wish to comment. Xxanthippe (talk) 06:47, 24 April 2016 (UTC).

Original synthesis? This is standard textbook material commonly taught in physics grad-school classes. See for example David Bleeker, "Gauge theory and variational principles." (1982) Or Misner, Thorne, Wheeler (1972) -- or lectures from I.M. Singer in the 1970's. There are surely many other books. If your professors are not at least mentioning this material in class, you should probably be looking around for better professors. This may seem like a combative statement, but seriously, if you think that this is "original synthesis", you really need to schedule some office time with your prof, and start asking questions about differential geometry. If he stalls, you need to escalate to the dean. You should not in any way have the impression that something new is being invented here: this book review sketches out the history, back to the 1930's, Hermann Weyl and Einstein himself, and the recognition of these ideas in the late 1950's, developement in the 1960's and the solidification and maturation of these ideas in the 1970's. It is the duty of your professors to teach you at least some of the basics of this stuff. 67.198.37.16 (talk) 16:04, 24 April 2016 (UTC)

Here, the bleeker book is so so old that its online as a PDF Its not a great book, because although its quite accurate, it fails to provide insightful commentary. But see for example, page 26 for the PFB, page 29 for the connection page 30 for the fundamental field, page 33 for the gauge potentials and electromagnetic field strength, chapter 3 for particle fields, page 43 for particle fields living in the associated bundle and hopping ahead .. chapter 6 for the Dirac fields, and page 81ff for spin structures and the Dirac eqn. 67.198.37.16 (talk) 16:15, 24 April 2016 (UTC)

Here, I'll just add these as refs to the main article. This article reads like an orphan raised by wolves; civilizing it is more work than I care to put in, but it certainly does need lots of loving care from someone. 67.198.37.16 (talk) 16:24, 24 April 2016 (UTC)

Gauge fields belong to _adjoint_ reps

I fixed this: "The gauge fields here belong to the fundamental representations of the electroweak Lie group U(1)⊗SU(2) times the color symmetry Lie group SU(3)" but someone reverted it back. Well. Check just about any other article (e.g gluon field) or any QFT book. Gauge boson fields are in adjoint rep. Fundamental reps of SU(n) are n-dimensional, while adjoint reps are (n^2-1)-dimensional. Now, how many gluons are there, 3 or 8? 213.175.37.10 (talk) 12:49, 20 December 2017 (UTC)

You are correct. JRSpriggs (talk) 20:30, 2 January 2023 (UTC)

Reversion of the edits on Gauge covariant derivative and its relation to Gaige transformations

Xxanthippe reverted my edits on the Gauge covariant derivative on the ground that they have no references. I sympathise with the idea that proper references should be provided but I have just reread my edits and I can only say that IMO they are an improvement. One may argue that the introduction now contains too much information, and I can scale it down a bit, and in particular move the parallel transport to another section. However none of it is at all controversial or non standard. I also changed the part where the gauge covariant derivative is "derived" from gauge invariance. However that "derivation" was unclear and I think just wrong in the sense that it doesn't derive what it claims to derive. I have a degree in Physics a PhD on mathematics and published on mathematical gauge theory, and I think am in a positions to judge that. Thus, I wrote down something that is both correct (as should be easy to verify) and half the number of lines. I then explain how gauge transformations act on the gauge covariant derivative and that indeed they preserve Gauge covariant derivatives but not partial derivatives. The subtle thing here is the notion of "generalisation of partial derivative". Mathematicians know for about 100 years that the proper formulation of "generalisation of derivative" is satisfying a Leibnitz rule. Indeed this is how connections are defined mathematically and the straight forward translation to gauge covariant derivative language is what I wrote down.

I was busy with finishing touches when friends dropped in early for new-years eve celebrations, and perhaps should not have published yet, but thought (and still do)it was already much better than what was there so published things as was anyway. I have not reverted the revert yet, but I think I will if I get no reaction, and finish things up.

RogierBrussee (talk)

I have invited comment here Wikipedia talk:WikiProject Physics#Reversion of the edits on Gauge covariant derivative and its relation to Gaige transformations. Xxanthippe (talk) 02:58, 2 January 2023 (UTC).

Take it slow. Don't revert the revert. Wait for discussion here. If the changes are good, others will agree with you and either revert your edit back, or do something better with the material.

Reliable sources are core to what we do. One of the foundational policies of Wikipedia is Verifiability; it has to be possible for readers and editors to verify everything on Wikipedia, with reference to reliable sources. We don't rely on editors' assertions about their education or expertise. Everything must be verifiable. The advantage someone with a PhD has is access to and understanding of a richer variety of sources. Personal knowledge or experience is irrelevant except to the extent that it allows you to point to material that has been published, ideally in secondary sources.--Srleffler (talk) 03:58, 2 January 2023 (UTC)

Read wp:or and wp:v, anything you add must be supported by published wp:rs, and not what you know to be true. Also if it is "easy to verify" if should be east to provide a source supporting it. Slatersteven (talk) 12:08, 2 January 2023 (UTC)

I am rather puzzled by the revert and the responses here. There seems to be a complete misunderstanding about what WP:N and WP:OR says. If a statement is to be challenged, then be clear about what statement is being challenged, not just "edits", unless it is clear that a bunch of new claims have been inserted. In particular, these policies do not sanction challenges of copy-edits, which is what the reverted largely are. Requiring a source for a copy-edit would be ludicrous. —Quondum 17:01, 2 January 2023 (UTC)

I wouldn't characterize this as a copy edit. Perhaps citations wouldn't strictly be required in the lede, if the main text were adequately sourced, but the main text isn't adequately sourced, and the modified lede uses terminology (like "local frame" and "parallel transport") that the main text does not follow up on or elaborate. I don't think it's a particularly good edit overall. XOR'easter (talk) 17:29, 2 January 2023 (UTC)

Altering what something says is not a copy edit. For example, it alters the claim it is a variation, and instead implies it is stand-alone. Slatersteven (talk) 17:56, 2 January 2023 (UTC)

There seems to be a subtext here that I am missing. Nevertheless, I'll assume that because there is resistance from multiple people with expertise and my familiarity with the topic is marginal, that the objections have substance, e.g., that the introduced terms are not accepted as standard (or may not even be valid). That does not seem to make the original any better: someone making the reverse edit could be facing the same objection, namely that "variation" is undefined and not supported by the body. I guess the article may need attention. —Quondum 19:08, 2 January 2023 (UTC)

Well as it is "Gauge covariant derivative" it does seem to be in some way linked to covariant derivative. So what do RS say, is it a derivative, a variant an extension? Slatersteven (talk) 19:50, 2 January 2023 (UTC)

The covariant derivative in general relativity is a rather special example of a gauge covariant derivative.

The covariant derivative in general relativity is the Levi Civita connection $\nabla$ on the tangent bundle, written out with respect to a coordinate tangent frame $\{\partial _{\mu }\}_{\mu =0..3}$ coming from local coordinates $\{x^{\mu }\}_{\mu =0..3}$, while the gauge covariant derivative is a general connection on a vector bundle written out with respect to a general local frame for the bundle (i.e. a basis of fields $\{e_{a}(x)\}_{a=1..N}$with respect to which one can write a general field $\phi (x)$ as a linear combination $\phi (x)=\phi ^{a}(x)e_{a}(x)$ with internal index $a$), and independently, local coordinates $x^{\mu }$ on space time.

Hence for the covariant derivative in GR the internal indices are the same as the space time indices, i.e. if the covariant derivative of a vector field $a=a^{\nu }\partial _{\nu }$ in GR is

$(\nabla _{\mu }a)^{\nu }=(\partial _{\mu }a^{\nu }+\Gamma _{\mu \lambda }^{\nu }a^{\lambda })$

then in the language of the gauge covariant derivative the Christoffel symbol is a gauge potential with

$\Gamma _{\mu \lambda }^{\nu }=(A_{\mu })_{\lambda }^{\nu }$.

Coordinate changes give rise to changes in the tangent frame field, hence to gauge transformations on the tangent bundle, "acting on the $\lambda ,\nu$ indices", but because they also changes the partial derivative, i.e. "acts on the $\mu$ index", a coordinate transformation looks a bit different than the action of a gauge transformation. If you express the $\nu ,\lambda$ indices with respect to a 4-bein $\{e_{m}=e_{m}^{\mu }\partial _{\mu }\}_{m=1..4}$ i.e. write $a=a^{m}e_{m}$ and

$(\nabla _{\mu }a)^{n}=(\partial _{\mu }a^{n}+\Gamma _{\mu \ell }^{n}a^{\ell })$

a change in 4-bein $e'_{m}=\Lambda _{m}^{n}e_{n}(x)$ gives gauge transformations which are exactly the gauge transformations that you get from changing the frame $e_{a}$ in the general case.

All of this is standard differential geometry. RogierBrussee (talk) 11:58, 3 January 2023 (UTC)

That does not really help, can you not say (simply) how it related to "standard" covariant derivative? Slatersteven (talk) 17:44, 4 January 2023 (UTC)

I'll concur with what Srleffler and Slatersteven said above and add that the revert shouldn't be taken personally. We all want encyclopedia articles that explain math and physics; nobody has a problem with experts providing expertise, but the nature of this project and the platform it is built upon impose certain constraints that are unlike the requirements on other technical writing. XOR'easter (talk) 17:41, 2 January 2023 (UTC)

• Fucking hell (pardon my French). Could we avoid driving off all the domain experts by beating them over the head with rules and regulations? Their expertise is incredibly valuable and much appreciated. See WP:AGF and WP:DONTBITE, which were specifically written to prevent this kind of experienced-editor toxicity. No, politely asking for sources isn't toxic, but being demanding towards volunteers donating their free time (especially experts, who could earn hundreds of dollars an hour with said free time, and yet choose to donate it here) is incredibly self-defeating. Nor is rudeness remotely appropriate towards these most valuable users.

Expertise is not Googleable. Most mathematics papers can only be understood after years of study. It's astonishing to think that we non-experts can get it right by Googling for half an hour, and reading a source sideways (often merely an abstract). Sometimes, we manage to represent the sources accurately. But all scientific or mathematical articles still need proper expert scrutiny. Otherwise, how can we know we understood the cited material properly, and summarized it accurately? Pause, and ask yourselves: how are these articles supposed to improve, if the experts are driven off, and everyone left (including me) is just a Googling shmuck? We ought to treat experts with a little bit of gratitude. We want their help.

What's crazy is, the previous lead was also unsourced! Policies do not state that all uncited additions should be immediately reverted. That would be highly disruptive and would lead to a block. The fact is, while WP:BRD and WP:BURDEN give editors great leeway to revert edits, you |should still only revert when you have reason to believe the addition makes the article worse. Leads do not need to be cited. If you're an expert who disagrees with the change, then revert and discuss. If you're a nonexpert with no specific objection, why revert? What are we supposed to discuss? Procedural crap? For all scientific and mathematical articles, the appropriate thing to do is to ask a second expert to proofread it. That's it. Yes, the lead should straightforwardly summarize the body. But this is a technical subject, and we have no confidence the summary is accurate unless checked by experts. Well, we have one right here!

This is somewhat of a systemic problem on Wikipedia. WP:SYSTEMICBIAS also encompasses the overwhelming lack of expertise of Wikipedia editors.

Rant over. DFlhb (talk) 17:47, 3 January 2023 (UTC)

Unsourced material should be removed, not replaced with other unsourced material. Wikipedia is not a geometry text book, it is an encyclopedia that should be understandable to the nonexpert. If you need a degree to understand a page, it has failed. We need people who are experts but can also make themselves understandable to nonexperts.Slatersteven (talk) 17:47, 4 January 2023 (UTC)

For the present topic, I'd be content with "partially understandable to third- or fourth-year undergraduates". XOR'easter (talk) 16:08, 6 January 2023 (UTC)

Seems like the best course of action to me. BTW, I do apologize to both of you for the tone of my rant; and I'm glad the subsequent discussion is focusing on specific issues with the content, rather than generalities like my comment. Best, DFlhb (talk) 16:15, 6 January 2023 (UTC)

No offense taken at my end. :-) Being an "expert" of some kind myself, and having hung around the math/physics side of Wikipedia for a while, I've noticed that we have a seemingly endless stream of pages that are under-sourced or even completely unreferenced, some of which have been that way for years. Unfortunately, people with expertise writing without bothering to provide pointers into the literature is a contributing factor. Sure, it's a pain to dig up footnotes for ideas that seem obvious, and it's not like physicists and mathematicians are ever actually taught to write in a way that is comprehensible to people who don't already know what's being said, but the result is unsatisfying all the same.

For my own part, I keep textbooks within easy reach, and I try to look up what they say about a topic and where they say it before I add material here. Even readers who know a lot about a subject can benefit from exact page numbers, after all. XOR'easter (talk) 17:18, 6 January 2023 (UTC)

The edits that got reverted were these. They didn't cite any sources, but the content that they changed did not have sources either (at least not inline). As far as I can see, no-one in the discussion above has any source-based objections to the changes. So what is actually getting debated here? – Uanfala (talk) 12:18, 5 January 2023 (UTC)

Apparently we're debating whether things like this:

the minimally-coupled gauge covariant derivative is defined as [...math equation here...]

are better than that:

the gauge covariant derivative $D_{\mu }$ of a Dirac spinor field $\psi$ of charge $q$ is defined as [...math equation here...] Since this is a minimal replacement of $\partial _{\mu }$ with $D_{\mu }$ it is called minimal coupling.

and claiming that the sole problem is not providing inline citations. Do we need sources for this change? I'm not sure that this introduces any new facts, but if so, maybe one of these will do: [1][2][3][4] (Do I know that any of these verify these changes? No. I just know that they contain matching keywords.)

If it were easy in the software to revert changes to this paragraph but keep the changes to that one, or if each line had been changed in a different week, I wonder how many of these changes would have been accepted without any complaint. WhatamIdoing (talk) 02:23, 6 January 2023 (UTC)

Yes, that introduces a new fact (the reason why a particular manipulation is called "minimal coupling" — a term which could well sound like it means something else). Containing matching keywords is not a good place to start finding references for mathematics and physics articles. The first link is relevant but doesn't spell out the claim in need of support. The second and fourth links are to copies of books that are so mangled by unknown technical issues that the mathematics is illegible. The third is topical but insufficiently detailed (in fact, equation-free). XOR'easter (talk) 16:01, 6 January 2023 (UTC)