Talk:Gauge theory/Archive 1

intro

• Can we have an intro paragraph for, say, the reader with only first year undergraduate physics, rather than this somewhat last-year mathematical physics student level.... also a roadmap for subjects to read about first (differential geometry, tensors...) ?

IMHO, first-year undergraduate level is rather a tough order -- but I am trying my best to divide this thing into an intro paragraph with the bare physical idea, a short historical note, and a section with a physics example - maybe SU(N) or something like that, followed by the heavy duty diffgeom. Would be nice to get some backup here - Amar 08:01, Jun 12, 2004 (UTC)

"If you can't explain it simply, you don't understand it well enough” - Albert Einstein. He also says you should be able to explain ideas in physics to any barmaid (not just those taking a physics degree!) —Preceding unsigned comment added by 79.65.253.30 (talk) 15:25, 13 August 2008 (UTC)

Help ! Images

• Just added an image that I drew to the classical theory section. Very evidently, it's kind of clunky - can someone with better artistic skills advise/add a better picture ? - Amar 09:57, Jun 19, 2004 (UTC)

what's a feynman diagram doing in the classical theory section? Feynman diagrams are objects in the quantum theory. and why not use LaTeX feynman diagrams instead of hand-drawn? -Lethe | Talk

The Yang-Mills action is NOT the most general gauge invariant action out there... Phys 05:48, 14 Aug 2004 (UTC)

I would like to rotate the diagram by 90 degrees. As this is a spacetime diagram, the current orientation depicts particles travelling backward in time ("downward" motion = time reversal). If the diagram is re-oriented, then the gauge interaction occurs at some known time (height above the x-axis) which how Feynman would have drawn this. Ancheta Wis 13:13, 28 Sep 2004 (UTC) Thinking about it, the particles need to be bosons, so the straight lines need to be wavy lines. What about the interaction with the gauge boson, now, does it make sense for a collision to occur, or is it simply a decay? Ancheta Wis 21:59, 28 Sep 2004 (UTC)

Re mathematical formalism

Maybe there should be some mention that the gauge group $G$ is the structure group of the vector or spinor bundle in question, whereas the group of gauge transformations is the group of $G$-bundle automorphisms, i.e. those bundle automorphisms whose induced isomorphism on the fibres lies within the gauge group $G$.

I would further welcome some examples of gauge groups and groups of gauge transformations for a few theories, e.g. Relativity or QED...

I have added a parragraph on gauge fixing, I wonder if it should be more detailed, jointly with some expansion on the classical invariances of a force field lagrangian (A.R.)

Maybe the abbreviation "rep" should be replaced by the full word "representation". 140.180.171.121 (talk) 23:21, 4 December 2007 (UTC)

Correction?

I think the expression for the Noether current(O(n) scalar theory) in the article is wrong. The current should not have the i index.I think the correct expression would be

$\ J_{\mu }^{a}=\imath \sum _{ij}\partial _{\mu }\varphi _{i}T_{ij}^{a}\varphi _{j}$

-Vatsa Jan 29, 2005

This article isn't consistent. It "defines" A with a factor of 1/g but has a 1/4g² coefficient for the Yang-Mills term. Phys 06:53, 3 Feb 2005 (UTC)

Correction

It's not true that the sections of a principal bundle form a group! So, this is not a good definition of the group of gauge transformations. I was confused about this for a long time myself. To get a group, you need the sections of the bundle associated to the principal bundle P by means of the adjoint action of G on itself. I don't have the energy to write a clear explanation of this for people who don't know this sort of stuff. But, I wanted to point out that the description of gauge transformations as sections of a principal bundle is wrong.

(I'll be amazed if this correction actually shows up;I have no clue how Wiki works. Sorry!)

- John Baez, February 3rd 2005

Definition of gauge symmetry

I followed the Gauge symmetry link from Magnetism, and it redirected here. At the end of the introduction I read Sometimes, the term gauge symmetry is used in a more general sense to include any local symmetry, like for example, diffeomorphisms. This sense of the term will not be used in this article. It doesn't however say what sense of the term is used in the article either here or at the couple of places it is later used. Hv 14:21, 16 July 2005 (UTC)

Also, 'gauge symmetry' should not be a redirect to 'gauge theory', as the two terms have different meanings (regardless of what anyone thinks the terms mean). There should be two separate articles. MP (talk) 15:43, 3 July 2006 (UTC)

Well, gauge "symmetry" is not a symmetry. This is suggested in the article, but then the usual inacurate physical langage is used. There is a great confusion in the physics language, especially in the more elemetary physics books. They speak of global symmetries and local(i.e. gauge) "symmetries". In (quantum) physics, symmetry is a transformation between physical states that preserves the expectation values of all observables O (in particular the Hamiltonian). S: |φ> → |ψ> = S|φ>; |<ψ|O|ψ>|²=|<φ|O|φ>|² .

The usual formulation of the physics theories uses fields, which sometimes are not physical quantities. Such are the gauge fields (fiber bundle connections for the mathematicians), which provide a redundant but convenient description of the physical degrees of freedom. The gauge (local) "symmetries" are a reflection of this redundancy. The physical quantities are certain equivalence classes of gauge fields.

An analogy can be made with the construction of the real numbers. We can use sequences of rational numbers that have the same limit. Of course, each real number is represented by infinitely many such sequences. We can choose a particular well defined sequence to be a representative of the real number. This corresponds to the procedure of gauge fixing in gauge theories.

The fact that gauge fields are not physical degrees of freedom becomes very clear when we try to quantize them. Then we are forced to work in one way or another with the physical quantities by removing the redundancy (the gauge symmetry).

Another important illustration of the problem with the gauge “symmetries” is when we have anomalies. By definition these are symmetries, which exist in the classical system, but not in its quantum counterpart. Anomalies are something quite usual and also an experimental fact - e.g. the axial anomaly in the strong interactions. However, because gauge symmetries are not symmetries, gauge anomalies are not something that just complicates the quantum theory but something that kills it. I.e. there are no gauge "anomalies", because such theories don't exist. This is why having the exact relation between the number of flavors and quark colors in the Standard model is so important - otherwise there is a gauge anomaly and the theory does not exist. For the same reason string people live in 10 dimensions (26 for the purely bosonic string theorists). Only then the anomalies cancel.

Correction

The gauge field provided in the O(n) example is trivial (i.e., "pure gauge"). It is *not* the definition of a generic gauge field in the model. One needs some kind of Ansatz or field equation to specify the gauge field in terms of sources (I rather doubt the latter will be invoked), or else avoid saying "the gauge field is defined as" rather than "the gauge field transforms like [the appropriate expr]"...

- SH, September 13/14th 2005

Frame-dependent claims

A recent edit inserted

we choose a local frame (a local basis of sections) then we can represent this covariant derivative by the connection form A, a Lie algebra-valued 1-form which is called the gauge potential in physics and which is evidently not an intrinsic but a frame-dependent quantity. From this connection form we can construct the curvature form F, a Lie algebra-valued 2-form which is an intrinsic quantity,

Uhh, that's wrong I beleive, or misleading, or something, since I don't know what an "intrinsic quantity" is. For a non-abelian gauge theory, under the action of a gauge transformation φ, F transforms as

$F\to \phi F\phi ^{-1}$

and so naively F is frame-dependent as well. (The trace of F is not; the trace of F is intrinsic, its a Chern class). Can this wording be fixed? linas 00:17, 19 October 2005 (UTC)

linas, think of an instrinsic quantity as one that can be defined without reference to coordinates or bases. For example, a Riemannian metric is an intrinsic quantity, while, say, the trace of a Riemannian metric is not. What about connections and curvatures? Well, yes, they can definitely be defined without reference to coordinates on a G-bundle, so I would say the grandparent post is misleading for that reason: both are intrinsic. As I mention below, it is the local connection form (and connection coefficents) which rely on a trivialisation. Of course, if you're thinking of only the base space as "real" (as you might in physics, where the base space is spacetime), then I agree with the original poster: connection form is not intrinsic, curvature form is. -Lethe | Talk 01:17, 21 October 2005 (UTC)

I hope you can make sense of all the indices. 'a' is an abstract index, the rest are numerical indices. Let V be a vectorbundle and let E^n(V) be the vector-valued n-forms. Covariant derivative D : E^n(V) -> E^(n+1)(V) : s = s^i e_i |-> s^i (e)A_i_a. Now choose a different frame f_i = c_i^j e_j. s = z^i f_i = z^i c_i^j e_j = z^i c_i^j (e)A_j_a = z^i (f)A_i_a. A true vector valued form does not transform under coordinate transformations, only its components do. An invariant is (f)A_i_a f^i = (e)A_j_a c_i^j f^i = (e)A_j_a e^j which is a form, but this probably doesn't capture all information. I hope this explains why I called A "not intrinsic", which I used as a synonym for "not frame-independent". The geometric object which represent the covariant derivative is frame-dependent, not merely the components of that geometric object. This is not true for the curvature. --MarSch 10:38, 19 October 2005 (UTC)

I can't parse the formulas you gave. You seem to be saying that A is not a tensor on the base manifold, and that is correct. What I was trying to say is that F depends on the frame as well. F changes as you slide up and down the fibe. It also changes as you move from one coord patch to another (with φ the transition function as I gave above). Except in the abelian case, F is not frame-independent. linas 00:49, 20 October 2005 (UTC)

I dispute that F changes. Only its coordinates change. This is different for A which changes itself. You cannot specify the connection by giving the array of LAv forms A_i in g. You have to specify a frame for the vector bundle to fix it. Of course if you give A by specifying its coordinates in a given basis plus frame, then you will usually assume that this is the extra frame needed. But changing frames changes the array of geometrical objects A_i, which means that according to preference : either 1) A_i is not an array of LAv forms or 2) A_i is an array of LAv forms, but depends on a choice of frame. --MarSch 10:37, 20 October 2005 (UTC)

I don't know what an LAv form is. Do not confuse coord changes on the base manifold with gauge transformations. Let us for the sake of discussion keep the coords on the base manifold fixed and unchanging. Pick a section of the bundle. This fixes both A and F. To make a "local gauge transformation" is the same thing as saying that you will pick a different section of the bundle. Although picking a different section of a bundle is "kind of like" a "coordinate change" on the fiber, this is dangerous way of thinking about it, as it leads to confusion. A gauge transformation is more correctly visualized as a movement on the fibre, rather than a relabelling of the fiber. (for example: when I rotate a circle, do I rotate it, or the coordinates on it? I prefer to visualize the circle as fixed, and me moving on it. But that's just me.) The movement is given by the action of the gauge group.

If $\sigma (x)$ was one section of the bundle, and $\tau (x)$ was the other section, then $\phi (x)\in G$ is a gauge transformation, valued in the gauge group G, such that $\tau (x)=\phi (x)\cdot \sigma (x)$. Then on has $F(\tau (x))=\phi ^{-1}(x)F(\sigma (x))\phi (x)$ and $A(\tau (x))=\phi ^{-1}(x)d\phi (x)+\phi ^{-1}(x)A(\sigma (x))\phi (x)$. (I hope I got these formulas right, they're for "illustrative purposes only"). So as one moves on the fiber, both F and A change. There is no unique way to project F from the fiber to the base space, except in the Abelian case. linas 00:34, 21 October 2005 (UTC)

BTW I'm mystified why A is called a connection form, since it is an array of vector-valued 1-forms or equivalently a matrix of 1-forms in a non-invariant way as connection form also says.--MarSch 11:06, 19 October 2005 (UTC)

Historical usage. A is a form with respect to the base manifold; and not the total space. If you try to think of the fiber bundle as a manifold, then A is most definitely NOT a form on that. (and d+A is most definitly not a connection on that). linas 00:49, 20 October 2005 (UTC)

Linas, I'm not sure that that is correct. The connection form most definitely is a differential form on the total manifold. It is only when you consider local trivialisations that the connection form can be considered a differential form on the base manifold. And this depends on the local trivialisation (i.e. it is not guage invariant). This form is sometimes called the local connection form. -Lethe | Talk 00:51, 21 October 2005 (UTC)

g := T_e G. Perhaps Hom(g, g) is the Lie algebra A is valued in. I always assumed it would simply be g, but Hom(g, g) would make more sense. But is it a Lie algebra? --MarSch 11:45, 19 October 2005 (UTC)

Let me rephrase. A point in the Lie algebra tells you how to move a small distance in the Lie group; it does so by means of group-multiplication. So, when you move from one fiber in a fiber bundle to a nearby fiber, the lie-algebra tells you how to "rotate" the group slightly to get to the next fiber. A point in the lie algebra is an infinitesssimal automorphism of the lie group. Does it make sense now? linas 00:49, 20 October 2005 (UTC)

Jargon-based assured ignorance

If Wikipedia is to be accessible to reasonably educated people, it must be policed to assure that what passes for knowledge is not buried in the jargon of the priestly class. This article is one in a series of interconnected articles (I began with the Big Bang Timeline) which are completely useless to 99% of the world's educated people. If this practice were extended, for example, to articles on literature, only professors of literature would find them useful.

Jargon is used to exclude people from presumed knowledge, usually with the excuse that it is more efficient in conveying knowledge among experts. An encyclopedia seeks to convey knowledge to non-experts, so jargon must be left far behind (if it is to survive, it should be segregated from the encyclopedia in an "expert" site).

This article (and many of those connected to it) contributes, not to general education (the meaning and purpose of "encyclopedia") but to the preservation of a barrier between specialist and most of the world.

Jargon can always be learnt, if it is defined clearly. Jorbesch 20:29, 2 June 2006 (UTC)

Original poster here is a member of one of the world's most powerful elite classes: the class of people who think that they're incredibly special because they're being excluded and "kept down" by the elite classes, and are willing to display their superiority at every turn by reacting violently to the notion that they might be expected to know anything.

• * *

Uh? What?

Well, I happen to agree wholeheartedly with the original poster. The article is quite impenetrable - and I'm an astrophysicist! Encyclopaedia entries ought to leave readers with something more than a notion that what's being discussed is far too clever for them. Breathtaking "priestly jargon" is exactly what's on display here, and little attempt has been made to include the interested reader

Jargon, acronyms, and references made without associated context are indeed off-putting to a reader of highly-specialized and abstract information if that person is not immersed in the field of study. I agree with the sentiment of the original poster, but only if the author's intended audience was a group of people not already intimately familiar with gauge theory, supersymmetry, etc. In reading this, I assume that the intended audience was not the generalist but the specialist. For this audience, "jargon" serves as the professional shorthand necessary to communicate efficiently and concisely. Certainly there are individuals who use the jargon of their profession or special interest as an active exclusion of other individuals. But we shouldn't assume this is always the case. Leading-edge thought in Physics is becoming more and more popularized, and a large number of educated people without the requisit knowledge to understand the complete context of advanced topics such as those presented here are being exposed to them nevertheless. All the better, but if one wants to understand all of the context, ramifications, and issues surrounding, say, String Theory, one has a long path of knowledge and discovery to embark upon. This article is clearly meant for those who have "the context" and who rely on the jargon to discuss and advance the topic. It would be wonderful to have an article that provides a high-level treatment of gauge theory and spontaneous symmetry breaking for the educated lay-person - but I do not believe that was the intent of this article.WFN94 16:02, 19 May 2007 (UTC)

Some topics require advanced terms to be accurate. Imagine that the average person only had a vocabulary of fifty words, would it, then, be reasonable to assert that everything could be explained in such terms? Of course not, it is equally unreasonable to expect that a topic such as Gauge Theory can be explained in a clear, accurate, and concise way to someone with no experience in the related areas; notice the word concise, given a book's length the topic could probably be introduced in a rudimentary way. Relatedly, I wonder why people assume that articles on things like Quantum Field Theory, Hilbert Spaces, Homological Algebra, etc. will be obvious and basic; consider, if you are looking such topics up, then you must have come across them somewhere; if you did not come across them in a technical piece of work, then you probably shouldn't assume you have the background to understand them. Many of the technical articles on wikipedia are bombarded by complaints about how technical they are, no one ever stops to realize that some topics are bound to be hard; perhaps the best solution would be to just omit these topics, better to have an encylopedia that everyone can understand than to have a much more complete one that has sufficent articles that everyone is somewhere confused? Finally, everyone always laments the technicality on physics and mathematics articles, but I never see anyone complaining about such things on philosophy articles; go read about type/token distinctions, The Dialectic, etc.; and nobody complains on the literary articles, but do you really get what is being said in the article on periphrasis? My point, no matter what area you are interested in, something in it is bound to be above your head; even when simplified to the simplest level at which it may still be accurate. Phoenix1177 (talk) 05:29, 4 July 2008 (UTC)

Not to labour on, but I just noticed two things that I must respond to. First, in response to, "...The article is quite impenetrable - and I'm an astrophysicist!" Why do you believe that being an astrophysicist is meaningful in this context? Astrophysics is not QFT, nor Gauge Theory. Would a computer programmer's opinon on the hardness of Homological Algebra be meaningful just because both areas make use of Category Theory? Second, I like how you say that this article was written for someone immersed in the field, a specialist in this area. It is incomprhensible how you justify your implicit assumption, "...since I don't understand it, it must be at the level of a specialist." that's rather arrogant. This article is not at the level of active research, nowhere near it...of course, this only stands to remphasize my point that considering the topic, the article is quite simple (as compared to what it would be full on, of course.) Phoenix1177 (talk) 05:41, 4 July 2008 (UTC)

Phoenix1177, I think it's quite reasonable for WFN94 to say that it's a problem if he's an astrophysicist and can't understand the article. That puts him at a higher level than 99.9% of the population, and WP articles are supposed to be written for the general reader. I have a PhD in physics, and my reaction to the impenetrable mass of specialized jargon and mathematics was pretty much the same as WFN94's. Anyhow, I hope that my recent edits, and the separation of the technical material into Mathematical formalism of gauge theory, will help to resolve this problem.--76.167.77.165 (talk) 00:33, 9 August 2009 (UTC)

Trivial Links

In the openin paragraph, there's a link to the Yang-Mills action. Yang-Mills action redirects back to Gauge theory. Is it expected that someone will eventually write a Yang-Mills action page, or should this be un-linked?

hmm

What's "locally" and "globally" refer to in the first sentence? Answering that might begin the road to a more clear article. If I understand this right, which I probably don't, something symmetrical should be the same after a symmetry transformation always regardless of what parts are transformed.

Clarification

The term "global transformation" (shorthanded "globally") means that the parameters of the applied symmetry transformation are the same in all space-time points (i.e. everywhere and at all times).

In contrast to this the term "local transformation" (and "locally", etc) indicates that the parameters of the transformation can (and in general do) depend on the space-time point A where the transformed field is evaluated/observed in.

For example, if the transformed field is a vector 3-dimensional ${U(x,y,z,t),V(x,y,z,t),W(x,y,z,t)}$, its global transformation would be rotation around the same axis ($N=(1,0,0)$) and for the same angle $φ$ in all space-time points $A=(x,y,z,t)$. Its local (= localized = gauge) transformation would be allowing for changes in either direction/axis (of rotation) or/and angle of rotation, i.e. both $N$ and $φ$ are now functions of the space-time location $A$: $N → N(x,y,z,t)$, $φ → φ(x,y,z,t)$.

There is too much information on this page

I only want to search for the gauge invariance in classical electrodynamics, but it redirects me to this page of gauge theory. I think it is necessary to separate the information into several entries, each containing more details. One certainly won't like a page titled "Physics" to cover materials from Newtonian Mechanics to Superstrings!

too general?

"Most powerful theories are described by Lagrangians which are invariant under certain symmetry transformation groups"

that is the first sentence, and i am thinking "WHAT POWERFUL THEORIES?" "no they are not" "i don't even know what lagrangians are..."

i don't think i am one who could fix this, but i think that first sentence needs to be cleaned up a bit. don't you think? —The preceding unsigned comment was added by 69.85.158.29 (talk) 09:05, 7 May 2007 (UTC). - BriEnBest 09:26, 7 May 2007 (UTC) sorry didn't sign before.

Gravity as a Gauge Theory ??

I think it should be pointed that Gravity is a Gauge theory and how could be applied to GR

This article needs a complete rewrite / break into pieces

If I have time in the future I will work on this, but several issues need to be addressed:

• Gauge invariance (gauge symmetry) is a more fundamental concept which is related, but logically distinct from that of a gauge theory.
• Yang Mills theory is a development significant enough to deserve its own article.
• The entire article is very disorganized, and the introduction needs to be expanded so that people can find links to articles which might give the required background.

Akriasas (talk) 18:47, 14 December 2007 (UTC)

I'd like to add that I agree this article needs a lot of revision. The ideas being discussed here are of great importance in mathematics and physics, so it would be nice to see a discussion that is more accessible. Obviously, a lot of work has been put into this, but if Akriasas is willing to edit it to provide greater clarity, that would be great. —Preceding unsigned comment added by 194.94.224.254 (talk) 13:21, 6 February 2008 (UTC)

Intuitive example of local gauge symmetry

To go with the electrical ground example. Consider the apparent sizes of distant objects seen when stood on a plane. These are the differences between the bearings of the sides/ends of the object, but bearings are defined from an arbitrary zero (North) that can be varied from point to point (e.g. Magnetic or True North) with no effect on angular sizes. That's a bit wordy, can anyone shorten it?

172.201.128.223 (talk) 23:34, 9 February 2008 (UTC)SB.

Big conceptual mistake at the beginning of the article

The following sentence:

"In a gauge theory the requirement of global transformations is relaxed such that the Lagrangian is required to have merely local symmetry. "

is wrong. The requirement of LOCAL symmetry is much more strict than the requirement of GLOBAL symmetry. In fact, a global simmetry is just a local symmetry whose group's parameters are fixed in space-time. My written English isn't very good, so I don't feel confident in editing the article. Is anybody willing to do that?

Ciao,

Guido —Preceding unsigned comment added by Coccoinomane (talk • contribs) 19:30, 20 February 2008 (UTC)

Thank you, I will do it. Masterpiece2000 (talk) 07:45, 23 February 2008 (UTC)

Splitting off another page for Gauge Invariance (Physics)

This article is redirected to from gauge invariance. I have been working on an article (see User:TStein/Gauge_invariance) that deals with gauge invariance from a lot more basic perspective. It has a lot of problems that I need to fix yet, but I am getting close enough to done that I was hoping I could get a few eyes on it.

known problems

• needs references (lots of references)
• need to write last sections
• too much explanation of Lagrangians and other stuff

Right now I am looking for general ideas about what to keep and what to toss and what needs to be done before I can publish it. (If it can be posted.) This is my first attempt at a new page here, so I am not confident what to reference here. The line between common knowledge and what needs siting is still fuzzy to me (in the wikipedia context at least). TStein (talk) 21:11, 13 June 2008 (UTC)

New summary; please help trim/organize

I wrote what I hope is a more accessible summary a few days ago. I just rolled back a well-meaning attempt to restore the previous intro content (which I think was not very valuable, and in places rather misleading). If other people disagree, I won't roll back a second time. But I would love to see someone inspired to organize this article's content more suitably for an encyclopedia and replace the bits specific to Yang-Mills, QED, etc. with references to the appropriate articles. Michael K. Edwards (talk) 01:18, 20 September 2008 (UTC)

Thanks to everyone who improved this article

I have not looked at this article for perhaps 9 months or so, but from the point of the view of an physics prof. who teaches undergraduate E&M, it has improved dramatically from how I remembered it. I still don't understand most of it but I now feel like I could figure it out if I had enough time and were able to chase down all the links.

Keep up the good work.

TStein (talk) 19:14, 23 January 2009 (UTC)

Correction?

I think that the transformation rule for A (the ninth equation down in the "An example: Scalar O(n) gauge theory" section) should have a "plus sign" and not a "minus sign" before the second term on the r.h.s. Without this correction, two terms in the transformation that should cancel don't cancel. Comparison with equation 10.9 of "Quantum Field Theory" By V. P. Nair seems to corroborate this assertion. —Preceding unsigned comment added by 69.116.11.132 (talk) 22:39, 31 March 2009 (UTC)

Utterly useless article by show-offs

A previous poster made the point a couple of years ago, and nothing has changed. The Wikipedia guideline is that articles be accessible to the general reader. From the first sentence to the last, this article makes a mockery of this guideline. Someone even has had the cheek to accuse the original poster of elitism. Is he serious? Alas, I think he is.

There is a reason for the arrogance, which applies not only to this but to most of the scientific articles. Wikipedia is a playground for pedants, usually adolescents who want to "publish" their school papers, and others in need of validating their self-esteem. Thus, we get articles of value only to other adolescents. This behavior is related to articles on celebrities, which have been turned into fan sites by those seeking to indulge their enthusiasms. (The gracious comment by TStein notwithstanding, I doubt that the articles are of interest to professors, whose just regard for Wikipedia ranges from strong reservation to contempt.)

Gauge Theory, or at least the gauge principle, is extremely important to physics. It should first be stated in laymen's terms, then the history of its development -- the whens, whos, wheres and hows -- should be presented as simply and concisely as befits an encyclopedia, with the technical showcase relegated to Talk.

The general reader comes to the article to find out what Gauge Theory is all about. It's obvious that, despite a number of appeals here, the editors of the article are too lazy, too intent on showing off, to post anything useful. I also suspect that they simply are not capable of explaining their arcane subject in plain language; if they are, then I challenge them to do it.

J M Rice (talk) 15:49, 4 June 2009 (UTC)

Let's talk about the first sentence. Which "layman's terms" are we allowed to use? Gauge theories are theories of fields. They are based on Lagrangians. A gauge theory is a field theory where the Lagrangian is invariant under gauge transformation. How else could one formulate this? Bakken (talk) 20:18, 4 June 2009 (UTC)

"Allowed to use"? Sorry for the despotic tone. Can't Lagrangian be broken down to a phrase? I'm reading Lee Smolin's The Trouble with Physics, where he discusses extensively gauge theory without ever resorting to "Lagrangian".

Bakken, please see the lead of the article as presently written for an example of how one can formulate an explanation of gauge theory without referring to Lagrangians.--76.167.77.165 (talk) 19:53, 8 August 2009 (UTC)

Not every article has to be completely accessible to everyone. There are many technical articles in wikipedia that are very useful to those whom they serve. I agree that there should also be a less technical description of gauge theory. The lead probably needs to be made a little less technical. In my opinion, as a physicist, there is too much Math jargon for my comfort, especially in the lead.

On the other hand, it is VERY hard to make articles like these accessible. It requires a very good understanding of the material (reciting a catch phrase is much easier then understanding what it means.) It also requires understanding what the average person trying to read the article is coming from. (This is perhaps the hardest step.) Finally, it requires an ability to translate this into good english. And oh yeah, this also has to be done while other editors are trying to make every sentence technically correct and complete and more useful to the technical audience. I know something about gauge invariance from a general interest and from teaching EandM, but I know that fixing this article is out of my league for instance.

After all this effort, when someone comes along and complains it make editors who work hard to try and make it more accessible throw up there hands in frustration. This is especially true when the complaint contains no real new information to help fix the problem.

One thing that might help is if a separate page on gauge invariance or an introduction to gauge theory was created. I have a very crude start to gauge invariance that I have been working on in my user space User:TStein/Gauge invariance. It is not near where it should be, though and I have not had time to fix it up. TStein (talk) 22:13, 4 June 2009 (UTC)

Your introductory article looks useful and rather extensive. Why not post it and thereby invite some assistance? Brews ohare (talk) 20:31, 7 June 2009 (UTC)

I agree that some subjects, especailly of a technical nature, can't be accessible to everybody. And sorry to come across as insisting that articles should be like "______ for Dummies". I meant the educated layman, someone who knows basic science, so you don't have to expain what protons and neutrons are, but maybe fermions and bosons. Gauge theory may be intrinsically recondite, but I think its importance -- it's arguably the keystone of particle physiscs -- demands an effort to present it in plain language, without resort to terms, like Lagrangian, which themselves require looking up, where the layman is confronted with yet another inscrutable explanation. In the old Britannica, Einstein explained relativity in layman's terms without dumbing it down (I remember he used Lorentz transformation). I think Einstein could explain it because he understood it completely. I wonder if gauge theory is difficult to explain simply, because it's difficult even for scientists to wrap their heads around.

About being a complainer, I understand what you mean. But in this instance it's not the complainer's job to "bring new information". I won't presume to lecture on how to write, and I certainly can't expound on the subject, since I'm here to learn not to teach. Maybe I share a common impression that academics who would rather talk to each other than the students.

A few suggestions. Right now, the public is reading the big projects at CERN (LHC -- it broke) and Fermilab to confirm the Higgs particle. This is what we hear is going on in particle theory. We need to undertand why. Why is this important? There's no mention of Higgs here. Why is Higgs-gauge important? There's no mention of symmetry breaking. How about a history of SU(5), a beautiful model set up by applying gauge theory and how it fell, when no proton decay was detected? I think understanding gauge theory, or at least the gauge principle, is key to the layman's undertanding of how today's particle physics works and also a bit about what particle physics debates.

J M Rice (talk) 05:54, 3 August 2009 (UTC)

Added content to the Introduction

I have added content to the introduction. Please do not roll back, or remove this content for few days, because I have asked a couple of people to look at it. I would like them read it and offer an opinion, as to whether an introduction, which includes this makes, the article more accessible. I have not modified any other section or content in this article, because I am sure it is valuable. Ti-30X (talk) 14:33, 31 July 2009 (UTC)

Very useful, a good start. Absolutely agree it should not be rolled back. The only changes should be to simplify further or to make additions in the same style. J M Rice (talk) 22:10, 2 August 2009 (UTC)

Merger Proposal

I don't think we can write much about Gauge symmetry that can't be included in the general article about Gauge theory
SPat ^talk 03:55, 31 October 2009 (UTC)

Pure gauge

The section Gauge theory#Pure gauge was merged here from Pure gauge in November 2008 with no discussion, having been a separate article before. However, it doesn't really seem to belong here, being a brief snippet on a too specialized subtopic. Perhaps move it into Gauge fixing? Or write more on it (well, that should anyway be done) and promote it into a separate article again? Currently its importance is not anyhow evident. — Pt _(T) 12:25, 20 December 2009 (UTC)

Definition

What about global gauge invariance? —Preceding unsigned comment added by Paranoidhuman (talk • contribs) 22:15, 17 March 2010 (UTC)

Order of sections

I swapped the order of the sections to place the more general "Description" of gauge theories, before the specific mathematical examples in "Classical gauge theory". I found the transition in level of description be jarring when read in the original order.

I don't think that this breaks any topical dependencies. Eric Drexler (talk) 03:06, 17 May 2010 (UTC)

Removal of the cleanup tag

I've removed the cleanup tag that had been on the page since last August. In between, much of the article has been reworded, the old complaints are mostly yet older than the tag was and, in my humble opinion, are not directly applicable anymore. There are no specific active cleanup proposals on the current talk page. If you feel there is still any need for such a prominent tag, then feel free to put it back, of course! — Pt _(T) 21:25, 4 June 2010 (UTC)

Is the gauge really a function?

in this edit, an anonymous editor added:

A gauge is a function introduced into a field equation to produce a convenient form of the equation but having no observable physical consequences. The transformations between possible gauges (called gauge transformations) must form a Lie group...

In my understanding, specifying a gauge means adding another equation to accompany the field equation(s). Thus, the gauge is an equation and not a function. E. g. the Lorenz gauge is $\partial _{\mu }A^{\mu }=0$. Another viewpoint often employed is adding a gauge fixing term to the lagrangian (see $R_{\xi }$ gauges in particular), calling the term a function would be a very loose use of the word (a function from where to where?). Its effect on the field equations is not as trivial as "just adding a function", it rather follows the idea of lagrange multipliers. — Pt _(T) 14:26, 2 July 2010 (UTC)

I agree with you. Have just rephrased the sentence. Bakken (talk) 16:15, 3 July 2010 (UTC)

History of this article

The history of this article may be a little confusing, so I'll give a brief explanation. There was a long discussion, archived above, about problems with accessibility to the general reader. In August 2009, I spun off the technical stuff into Mathematical formalism of gauge theory, leaving the nontechnical material in Gauge theory. Bakken then renamed the articles so that that Gauge theory was the technical article, with Nontechnical introduction to gauge theory being the nontechnical stuff. Confusingly, the old edit history of Gauge theory is now all at in the edit history of Nontechnical introduction to gauge theory.--76.167.77.165 (talk) 15:28, 10 August 2009 (UTC)

People seem to think that Gauge Theory is difficult. Bah! What's really confusing is the history of this article;) YohanN7 (talk) 22:45, 15 October 2009 (UTC)

Haven't the histories been merged here already? The history of Nontechnical introduction to gauge theory is basically empty now and this page's history seems coherent and complete. So, shouldn't the {{copied}} template be deleted from this talk page? — Pt _(T) 20:50, 21 February 2010 (UTC)

Nontechnical introduction to gauge theory was subsequently renamed to Introduction to gauge theory, and the history of that page is where all the pre-2009 activity for the Gauge theory article can be found. –Henning Makholm (talk) 21:50, 29 September 2010 (UTC)

Redlinked "gauge transform"

Could the red link "Gauge transform" be replaced by the green link "gauge tranformation"? Puzl bustr (talk) 21:13, 13 September 2009 (UTC)

(This used to be in the middle of the above discussion, but it clearly is a separate comment. As of this writing, the redlink that existed in September 2009 has indeed been replaced as suggested by Puzl bustr. –Henning Makholm (talk) 22:02, 29 September 2010 (UTC))

Gauge Transformation

Could somebody explain why the current gauge transformation formula $A'_{\mu }=GA_{\mu }G^{-1}-{\frac {1}{g}}[\partial _{\mu },G]G^{-1}$ is better? The formula without a commutator seemed perfectly correct to me (I'm talking about the last change). Terminus0 (talk) 21:43, 29 November 2009 (UTC)

It amounts to the same thing, but I agree that the former expression was less confusing. --Michael C. Price ^talk 23:49, 29 November 2009 (UTC)

It's not clear to me that it amounts to the same thing; there's an extra factor of 2. I understand the commutator notation to mean $[\partial ,G]G^{-1}=(\partial G-G\partial )G^{-1}=(\partial G)G^{-1}-G\partial G^{-1}=2(\partial G)G^{-1}$ (last equals due to $0=\partial I=\partial (GG^{-1})=(\partial G)G^{-1}+G\partial G^{-1}$). I have changed it back to the commutator-less notation, which is correct according to my rederivation –Henning Makholm (talk) 22:45, 29 September 2010 (UTC)

Argh, scratch that! I was trying to evaluate the commutator between things of different rank (understanding $G$ as an n×n matrix and $\partial$ as an operator on such matrices), which Is Not Allowed. It is indeed true that

$[\partial _{\mu }^{n},G]G^{-1}=(\partial _{\mu }^{n\times n}G)G^{-1}$

where $\partial _{\mu }^{n}$ is a differential operator on n-vectors, the commutator is computed in the ring of linear operators on n-vector fields, and $\partial _{\mu }^{n\times n}$ is a differential operator on n×n matrices. It is also true that

$[\partial _{\mu }^{n\times n},G{-}](G^{-1})=(\partial _{\mu }^{n\times n}G)G^{-1}$

where the commutator is an operator on n×n matrices which operates on $G^{-1}$, and $G{-}$ denotes left multiplication by $G$. But either of these readings is too subtle to expect a random reader of the article to figure out, especially given that they don't seem to be more useful than the right-hand side. –Henning Makholm (talk) 03:06, 10 October 2010 (UTC)

Gauge Symmetry

The page on Gauge Symmetry is unreadable and should just be deleted with a comment -- see gauge theory. — Preceding unsigned comment added by Stevesimon2 (talk • contribs) 16:00, 24 July 2011 (UTC)

Pronouniciation

Can we have one of those pronunciation guides, please? — Preceding unsigned comment added by 82.41.201.163 (talk) 11:15, 23 November 2011 (UTC)

Gauge Invariant

There is no explanation of "Gauge Invariant", even though the article on "Gauge Invariant" redirects here. — Preceding unsigned comment added by 173.13.158.102 (talk) 23:26, 18 December 2011 (UTC)

Global/Local Symmetry

Why is it said that local symmetry is a stricter constraint? To me, in the usual mathematical notion of strict or weak, local symmetry is a weaker notion than global symmetry--and hence more general--by simple virtue of the fact that global symmetry implies local symmetry whereas the other direction is false in general. — Preceding unsigned comment added by 66.213.36.2 (talk) 18:22, 13 March 2012 (UTC)

An example: Scalar O(n) gauge theory

Shouldn't it read (change in bold):

The gauge field is an element of the Lie algebra, and can therefore be expanded as

$\ A_{\mu }=\sum _{a}A_{\mu }^{a}T^{a}$

There are therefore as many linearly independent gauge fields as there are generators of the Lie algebra. — Preceding unsigned comment added by Dieterich (talk • contribs) 14:15, 16 April 2012 (UTC)

Mathematical Language

This article is clearly written by a physicist. While the author tries to translate some of it from physics to math (and it is much appreciated), there are a lot of translating errors. I would change them myself, but no one seems to like my edits. — Preceding unsigned comment added by K3thomps (talk • contribs) 16:24, 19 June 2012 (UTC)

Hi K3thomps, I understand your point about inaccurate language, please feel free to change wherever you think you can make improvements. There shouldn't be any problem as long as all content is verifiable. If you need any help with finding or added references, let me know. Remember, Be Bold!
SPat ^talk 18:30, 19 June 2012 (UTC)

Mathematical Language Again

The section Gauge theory#Mathematical Formalism needs to build a bridge between the physicists and the mathematicians formulation of Gauge Theory. It doesn't do that.

For example, the section begins with "... Mathematically, a gauge is just a choice of a (local) section of some principal bundle...". Exactly how does this relate to the (excellent) examples given earlier? Specifically, how does the examples constitute the ingredients of a principal bundle? What are the manifolds involved? What are the maps between them? Etcetera, etcetera ...

Where do things live?

Please make this section relate to the rest of the article. I am sure that the whole article would benifit enormously. YohanN7 (talk) 08:23, 21 August 2012 (UTC)

WP:SOFIXIT?

Quantum gauge theory: expand or merge?

The section Gauge theory#Quantization of gauge theories should link to Quantum gauge theory as the main article. Right now the link is only amidst the "See also" section and in a general template. However, there is nothing to link to! The Quantum gauge theory page has truly nothing to add the present section in this article nor the quantization page. So I'd encourage a complete rewrite of Quantum gauge theory to systematize and summarize the many methods. If that is deemed unnecessary, the page should rather be merged here. — Pt _(T) 04:48, 22 December 2009 (UTC)

I've now added the link. It does no harm, at least… But the sad situation with Quantum gauge theory has not yet improved anyhow. — Pt _(T) 05:59, 3 September 2012 (UTC)

The 2012-10-10 edit

I'm not sure the edit is helpful. For one thing, it's not clear that local gauge invariance is stricter than global gauge invariance. I'd say it's the other way around. Also, while Lagrangians are central objects in practice, they are perhaps not of foundational nature in quantum field theory (see e.g. weinberg, QFT vol 1). I don't say the edit is bad, just that I'm not sure, especially when it comes to the stricness part. YohanN7 (talk) 13:54, 11 October 2012 (UTC)

Since any Lagrangian with local gauge invariance also has global gauge invariance (assuming that spacetime is topologically trivial so that there are no topological shenanigans), the class of Lagrangians with local gauge invariance is strictly smaller than the class of Lagrangians with global gauge invariance.TR 06:50, 12 October 2012 (UTC)

The meaning of "gauge"

The word "gauge" has many meanings. What does it mean as it appears in "gauge theory"? Is it "[measure] theory", or "[measuring] theory" or "[thickness] theory" etc? — Preceding unsigned comment added by 144.122.104.211 (talk) 20:15, 29 May 2013 (UTC)

O(n) Result Changed

I believe there was an erroneous misplacement of factors of i when calculating the interaction Lagrangian found at the end of the O(n) gauge theory section. I changed it but would appreciate someone checking this as well, and perhaps even finding a reference who completes the calculation independently. Brent Perreault (talk) 20:47, 30 October 2013 (UTC)

Local symmetry group

Should Local symmetry group link here? 70.247.161.12 (talk) 14:11, 28 April 2014 (UTC)

section "2.1 Global and local symmetries"

How is that an appropriate title for this section? At no point does this section make the slightest attempt to define what a global or local symmetry is, or what the difference is between them. In fact, neither the word "global" nor the word "local" ever appears in it, except to say that the idea can be generalised to include local as well as global symmetries. Quite what that's actually supposed to mean is left entirely unclear. — Preceding unsigned comment added by 131.169.205.156 (talk) 15:34, 2 February 2015 (UTC)

section "2.2 Gauge Fields"

This needs to be rewritten. You cannot begin the first paragraph of a new section using vague pronouns instead of specific subjects. The topic is confusing enough, there's no reason to make the text indecipherable. "The "gauge covariant" version of a gauge theory accounts for this effect by..." What effect? Michael McGinnis (talk) 21:50, 15 September 2016 (UTC)

semi-gauge field

A semi-gauge field has a finite range of transformative invariance. — Preceding unsigned comment added by 2A02:587:410D:7800:A54E:6F32:59F7:49A1 (talk) 15:58, 30 September 2016 (UTC)