Talk:Introduction to gauge theory

Doing some rewriting

Note: This section refers to the Gauge theory article before it was split into two articles.

I agree with J M Rice and TStein that the article should be as accessible as possible to a general audience. In its present state it's a little disorganized and uneven, however, and I think some of the discussion in the lead is so loose and figurative that it's inaccurate. For example, the lead says "Gauge theory states that different reference frames can be used by different observers to measure the properties and parameters of particles and their interactions." This is incorrect, because a change of reference frame is a global symmetry transformation, whereas gauge transformations are local. The lead is also much, much too long and digressive. I'm going to try doing a little rewriting.--76.167.77.165 (talk) 18:09, 8 August 2009 (UTC)

I've rewritten the lead; deleted factually incorrect material from the lead; moved technical material out of the lead to later in the article; added a section on gauge symmetry in classical E&M; and rearranged the article somewhat so that less technical stuff comes earlier. Hope that helps. The level of technical sophistication in some of the later sections is still completely out of place, IMO. I'm not convinced that J M Rice is correct in saying that the authors of the overly technical material were arrogant or intent on showing off. I think the most likely explanation is that those editors honestly didn't know how to express the concept in nontechnical language. (E.g., Bakken didn't know how to explain what a gauge symmetry was without resorting to the concept of a Lagrangian.) Nevertheless, the final result is the same: much of the article is, as J M Rice says, "utterly useless" to the general reader. For example, the sections "Mathematical formalism" and "Classical gauge theory" simply do not belong here, because they are at the wrong level. Wikipedia:Make technical articles accessible has some useful ideas here. Not all articles have to be "accessible to a general audience," but all articles should be "accessible to the widest possible audience." There's nothing wrong with including extremely technical material like the "Mathematical formalism" and "Classical gauge theory" sections in Wikipedia; there is something wrong with including them in this article. This article is on a topic that can be explained to a wide audience, and therefore it should be written at a level that is intelligible to a wide audience. The discussion of the Yang-Mills theory is inevitably going to be much more technical; that's fine, but IMO that just means it belongs in an entirely different and more specialized article, aimed at a different audience. I would suggest spinning both of these sections off into their own articles; to 99% of all readers hitting this article, the absurdly fancy math is going to be completely unintelligible, and it will be a barrier rather than a positive aid to understanding.--76.167.77.165 (talk) 20:05, 8 August 2009 (UTC)

I've gone ahead and moved a lot of the overly technical material into Mathematical formalism of gauge theory.--76.167.77.165 (talk) 20:23, 8 August 2009 (UTC)

I've finished a fairly thorough rewrite. All the highly technical material is now in Mathematical formalism of gauge theory. I've removed the technical and cleanup tags.--76.167.77.165 (talk) 22:18, 8 August 2009 (UTC)

BTW, I normally edit as an anon these days, because I found that signing my user ID on my edits tended to get me too psychologically invested in the articles, driving me nuts. But just for the record, my name is Ben Crowell, and I can be contacted via http://www.lightandmatter.com/area4author.html . Since I don't normally log in to my WP account, and my IP is dynamically assigned, sending me messages via my user talk page on WP is not a reliable way to contact me.--76.167.77.165 (talk) 20:11, 9 August 2009 (UTC)

At least the old version was correct. Now the lead is full of meaningless and even wrong statements :(

The old version might have been "useless" for general reader, but this one is not even useless, it's misleading. Bakken (talk) 13:20, 10 August 2009 (UTC)

Could you be more specific about what you think is wrong?--76.167.77.165 (talk) 19:54, 11 August 2009 (UTC)

The lead stated (I have already corrected this) that a "general feauture of ... fields is gauge symmetry". This was not quite correct: it is the Lagrangian that is symmetric under certain transformation of the fields (called gauge transformation). That is, of course, if under "fields" you mean the electromagnetic potentials (as one should guess from the following discussion of the gauge transformation of the electric potential V). Now, the given definition of the gauge invariance, "gauge symmetry is a statement that any physical system can be changed in certain ways without violating the laws of physics", does not make any sense to me. Indeed, under a gauge transformation a "physical system" is not changed at all, not in any way, since all physical observables are gauge invariant. On the other hand one can change a physical system in all sorts of ways (for example: add an electron to it, accelerate it, scatter a photon against it) without violating the laws of physics. So, in my opinion the lead is just meaningless. Bakken (talk) 20:54, 11 August 2009 (UTC)

Thanks for your comments. I agree that your edit regarding the first point is an improvement. I disagree with you on the second point. In electromagnetism, for example, the gauge transformation changes the potentials V and A, and I don't think there's anything wrong with referring to that as a change in the physical system. It just depends on what is meant by "physical system," which is not really a term that has a widely accepted, precise definition. You're right that there are many types of changes to a physical system that would not violate the laws of physics. However, there are many changes that *would* violate the laws of physics, e.g., by creating a field configuration that violates Maxwell's equations. The article does not just define a gauge transformation as any change to the physical system, it defines it as a local transformation of the fields. Of course if you think it could be worded better, please go ahead, but it's never going to be possible to provide complete rigor without using math that's inappropriate for an encyclopedia that is supposed to be accessible to a general audience.--76.167.77.165 (talk) 22:33, 11 August 2009 (UTC)

Usually them term "physical system" implies the physical observables as opposed to the "mathematical model" of a physical system. The essence of a gauge transformation is that all observables of a physical system do not change under gauge transformation. Therefore calling gauge transformation a "change of a physical system" is in my opinion misleading. It is not a change of the physical system, it is a change of the mathematical model of a physical system, under which the physical system itself remains unchanged. Bakken (talk) 23:02, 11 August 2009 (UTC)

It looks like we just have to disagree on the meaning of "physical system," unless you have a reliable source that supports your specific interpretation of the term "physical system."--76.167.77.165 (talk) 23:37, 11 August 2009 (UTC)

Could you please give your own interpretation of the term "physical system", preferably with a reference to a "reliable source"? Well, actually, never mind, it was a joke... Bakken (talk) 07:18, 12 August 2009 (UTC)

Hi, Bakken -- I see you've made some edits to the lead. I think I'm going to revert them for the moment so we can discuss a little more here. The problems I see are: (1) The paragraph "When talking about gauge transformations..." uses technical and mathematical terminology that is not accessible to the general reader ("pseudovector," "partial derivative") (2) The same paragraph seems digressive to me. (2) The paragraph "Adding a constant to the electric potential..." duplicates the discussion in the following paragraph, and does it before the relevant concept has been explained properly for the general reader. (3) You seem to have a strong preference for the physical primacy of the E and B fields as opposed to the potentials V and A. That's fine as a philosophical stance, but it's not universally shared, e.g., Feynman argues for precisely the opposite stance in the Feynman lectures.--76.167.77.165 (talk) 23:37, 11 August 2009 (UTC)

Do your worst... At least I have saved the gauge theory article from your vandalism. Bakken (talk) 07:18, 12 August 2009 (UTC)

I don't think that kind of hostile language is productive. Please read Wikipedia:Assume_good_faith.--76.167.77.165 (talk) 15:54, 12 August 2009 (UTC)

Well, as the saying goes, the way to hell is paved with good faith... :( I tried to explain you that your definition of gauge invariance is meaningless, tried to correct your definition, but you simply deleted my contributions. Is it your good faith and constructive behaviour? Bakken (talk) 16:07, 12 August 2009 (UTC)

I didn't simply delete your contributions. I deleted them and explained why. I am open to further discussions about physics, especially if you can back up your statements with reliable sources.--76.167.77.165 (talk) 11 August 2009 (UTC) —Preceding unsigned comment added by 207.233.87.245 (talk)

When you say field you seem to refer to the electric and magnetic fields E and B rather than to the electromagnetic potentials V and A. However, the electric and magnetic fields E and B are gauge invariant: they are not changed under gauge transformation. On the contrary, the fields V and A are not gauge invariant and *do* change under gauge transformation. I just wanted to make sure you understand the difference when you talk about gauge transformations of the electromagnetic fields. Bakken (talk) 07:18, 12 August 2009 (UTC)

Yes, I do understand that. Thank you.--76.167.77.165 (talk) 15:54, 12 August 2009 (UTC)

archived old talk page contents

Since the article has recently been thoroughly rewritten by Ti-30X, me, et al., most of the old contents of the talk page are no longer relevant. I've archived them at Talk:Gauge_theory/Archive_1.--76.167.77.165 (talk) 00:43, 9 August 2009 (UTC)

new treatment of gauge bosons

I've added a new section on gauge bosons, which is my attempt to explain the concept of the calculation at http://en.wikipedia.org/wiki/Mathematical_formalism_of_gauge_theory#An_example:_Scalar_O.28n.29_gauge_theory without all the highfalutin math, and modulo any details about spin and polarization.--76.167.77.165 (talk) 20:14, 9 August 2009 (UTC)

Thanks for the invite

Hi, Ben. Thanks for inviting me to work on this page. I did not know it even existed, because my next project was going to be an Introduction to Gauge theory (symmetry) article. I was looking forward to it, but dreading it at the same time. Now I don't have to do it.

I know I have a few questions for you regarding this article, which I will get to in time. First, I want to say that I appreciate your response to Jim Rice's and TStein's critisms of the Gauge theory article, which I agree, is way too technical. Second, I am glad you know your stuff, and I feel more comfortable having checked out your web page.

Is my introduction still in existence in this article? And, it is interesting that you write in the opening discussion: " ' Gauge theory states that different reference frames can be used by different observers to measure the properties and parameters of particles and their interactions.' And you state that "This is incorrect, because a change of reference frame is a global symmetry transformation, whereas gauge transformations are local. The lead is also much, much too long and digressive."

I wrote that introduction partly from memory (a small part), and derived from an article. The article used "point of reference", rather than "frame of reference". Would it have made a difference if I used "point of reference" rather than "frame of reference". In addition, I expanded what was in the article, to make it more reader accessible. Did you like the example of the cone to express symmetry? And lastly - I hope it wasn't my lead that was too long and digressive. I added the very first part of the introduction, to the technical Gauge Theory, as an experiment and to get some feedback for only a few days. I wanted to show, that it was possible to present a lead that was accessible. Then I decided what the heck, I'll just see how long it stays there before the orginal author moves it or deletes it lol! I was going to use in my Introduction to Gauge theory, anyway, because I could always retrieve, cut and paste. Well, that's my story and I'm sticking to it! Ti-30X (talk) 16:42, 12 August 2009 (UTC)

Also, I have a tendency to misspell Guage. Ti-30X (talk) 16:42, 12 August 2009 (UTC)

Hi, Ti-30X -- Glad to see that you're interested in working on this article! I don't think there are any of your words left in the current intro, although certainly some of the same ideas are there. In my opinion the example about the cone was pedagogically useful, but out of place in this article, and particularly in the lead; the lead of an article should generally be only a few paragraphs, and should stay close to the main point. Rather than dulpicating the article Symmetry in physics, I think it makes more sense to link to it. Then the reader who isn't familiar with the more general concept of symmetry can click through to that article for more background.-Ben —Preceding unsigned comment added by 207.233.87.245 (talk) 22:24, 12 August 2009 (UTC)

cleanup tag

Hi, Bakken -- I see that you've added a cleanup tag, but it's not clear to me, without some explanation on the talk page, what kind of cleanup you think is needed. Wikipedia:Template messages/Cleanup suggests using one of the more specific cleanup tags whenever possible, rather than the general one, and I think that might be helpful here. I know we've had some discussion here on issues of accuracy, terminology, and reliable sources, but to me that seems separate from cleanup. To me, cleanup would typically imply a problem with organization, which I actually don't think applies here. I'm going to delete the cleanup tag, and ask that you explain here what you had in mind as requiring cleanup.--76.167.77.165 (talk) 20:24, 13 August 2009 (UTC)

ok: changed to "misleading" -- explanation is given below. Bakken (talk) 21:36, 13 August 2009 (UTC)

intro-rewrite tag

Hi, Bakken -- I see that you've added an intro-rewrite tag. This seems like a problematic situation, since you and I have not been able to agree on whether there is a problem with the intro. Maybe it will be helpful to get more editors involved so as to break the logjam. I would suggest that there are a couple of good ways we could try to resolve this: (1) The lead already cites several reliable sources, but if you have reliable sources that you think show there are problems with the lead, it would be helpful if you could state them here on the talk page. (2) We might want to try drafting a new lead in a scratch area such as Talk:Nontechnical introduction to gauge theory/intro-rewrite, so that all of us here could try to come to some kind of consensus rather than just getting into an edit war. The big problem I had with your previous edits to the lead was that they were disorganized and digressive; I think it's quite possible that we could converge on a version that all of us would be happy with in terms of accuracy as well as organization and concision.--76.167.77.165 (talk) 20:31, 13 August 2009 (UTC)

Well, in my opinion it is your lead that is disorganized and digressive -- it's simply a random bunch of largely irrelevant and inaccurate statements.

For example, your statement "time translation symmetry, in which one offsets the times on all clocks by some consistent amount, say one hour, t→t + C; this does not result in any violation of the laws of physics" is inaccurate.

Indeed, people can offset their clocks by any random amounts, they can stop them, they can run them in the opposite direction -- this will not result in any violation of the laws of physics. You cannot violate laws of physics whatever you do with you clock.

And translation invariance is completely unrelated to gauge invariance. There are theories which are translation invariant but not gauge invariant. Bakken (talk) 21:30, 13 August 2009 (UTC)

If you take another look at the lead, you'll see that it specifically states that time translation invariance is not a type of gauge invariance: "For example, one symmetry of the laws of physics, which is not a gauge symmetry, is time translation symmetry,..." Notice the word "not" there. The point was to introduce a more familiar example of a continuous symmetry before contrasting it with gauge symmetry. Re the wording about "result in any violation of the laws of physics," I understand what you're getting at, although I don't see it as such a big issue. Do you want to propose a different wording? Again, I think it would be helpful if you could write a draft at Talk:Nontechnical introduction to gauge theory/intro-rewrite so that we can reach consensus rather than engaging in an edit war on the article itself.--76.167.77.165 (talk) 23:48, 13 August 2009 (UTC)

expert-subject tag

Hi, Bakken -- I see that you've added an { { expert } } tag. Since that tag is deprecated, I've replaced it with the more specific { { expert-subject|Physics } }. Although I disagree with your belief that the article is inaccurate as it stands, I would certainly see it as a positive thing if we could attract more involvement from people who are experts, since it might help us resolve the differences of opinion that have been making progress difficult.--76.167.77.165 (talk) 20:38, 13 August 2009 (UTC)

Yes, it is inaccurate: your very definition: "gauge symmetry, like other symmetries in physics, is a statement that any physical system can be changed in certain ways without violating the laws of physics", is inaccurate.

For example, the hydrogen atom (as an example of a physical system) does not "change in certain ways" under gauge transformations, not in any way, not at all. On the other hand the hydrogen atom can be changed in a number of ways, completely unrelated to gauge transformation, "without violating the laws of physics". Actually, anything happening to a hydrogen atom is according to laws of physics. The laws of physics can not be violated. Ever.

In a sense you definition is not even wrong: it simply meaningless.

I do not like your practice of deleting perfectly valid contributions of other people just because you disagree with them. Why would I want to collaborate with you if you destroy my work, which I put a certain effort into? Restore my definition, and then we could discuss it further. Bakken (talk) 21:10, 13 August 2009 (UTC)

Well, we seem to be at an impasse. (Re your example of the hydrogen atom, this is ground we've already covered; I've already pointed out to you that the article does not describe a gauge transformation is just any transformation, it describes it as a local, continuous transformation of the fields.) It will be helpful we can get some participation from other people. I've posted at Wikipedia talk:WikiProject Physics to see if anyone who doesn't already have a dog in this fight would like to step in and help out.--76.167.77.165 (talk) 23:51, 13 August 2009 (UTC)

Bakken, here are a few thoughts to try to illuminate what we seem to be disagreeing about. The core of the matter seems to be that you prefer a definition of gauge symmetry in terms of symmetries of the Lagrangian, as in the first sentence of Gauge theory: "In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a certain continuous group of transformations." This is perfectly fine, but it's not the only way to define it. For instance, Perkins, Introduction to High Energy Physics, p. 22 says: "In quantum mechanics, the phase of a fermion field (e.g., the wavefunction of an electron) is likewise arbitrary, and one could require the freedom to choose the phases of all fermion fields at all points in space-time in any way one pleases, without changing the physics. This local gauge invariance leads to conserved currents and to conservation of electric charge." In other words, he's describing gauge symmetry as one in which the fields can be changed in a certain way, but even after the change, the fields obey the laws of physics. There's this whole side-issue of whether it's (E,B) or (V,A) that should be considered the "fields" in E&M, but I think Perkins' example shows that this is really a side-issue. And in any case, there are certainly reliable sources that consider (V,A) to be more fundamental. For example, Feynman, Leighton, and Sands, The Feynman Lectures, II-15-7: "Is the vector potential merely a device which is useful in making calculations ... or is the vector potential a 'real' field?"...II-15-8: "...there are phenomena involving quantum mechanics which show that the field A is in fact a 'real' field in the sense that we have define it." He then describes the Aharanov-Bohm effect, concluding on II-15-12: "In our sense then, the A-field is 'real.' ... If we want to describe [the field's] influence not as action-at-a-distance, we must use the vector potential." So I think it's pretty clear that neither way of defining gauge invariance is incorrect. If you want to use the definition in terms of symmetries of the Lagrangian, then that's fine in Gauge theory, but it's just not going to work in a nontechnical article like Nontechnical introduction to gauge theory, since the nontechnical reader doesn't know what a Lagrangian is. I know you dislike the idea of speaking of "changing the fields," and you feel that it is (E,B) that is the experimental observable that deserves to be called the "field," not (V,A), but it's clear that your preferences are not universally accepted by reliable sources, and that many reliable sources have the opposite preference. What matters on Wikipedia is reliable sources.--76.167.77.165 (talk) 00:42, 14 August 2009 (UTC)

Since we're talking about how to present this idea to readers without a technical background, I thought it would be interesting to see if I could find any popular-level treatments that we could compare against. Roger Penrose's The Road to Reality turns out to present gauge invariance twice, once by each of the methods we've discussed. On p. 451: "...Ψ must be a complex field whose physical interpretation is, in some appropriate sense, insensitive to the replacement Ψ->e^iθΨ ... This replacement is referred to as an electromagnetic gauge transformation, and the fact that that the physical interpretation is insensitive to this replacement is called gauge invariance." Soon after that, he spends 19 pages explaining what Lagrangians are, and then he revisits gauge invariance from this new perspective. So it seems pretty clear to me that either way of defining gauge symmetry is valid, and the choice of one or the other is a matter of convenience or personal taste. Penrose's book is over 1000 pages, so he has the luxury of discussing the Lagrangian for 19 pages and then reintroducing gauge symmetry from the new perspective; we don't have that kind of space in a Wikipedia article.--76.167.77.165 (talk) 01:53, 14 August 2009 (UTC)

Are we ready to remove this tag now?--Ben, 76.167.77.165 (talk) 01:25, 18 August 2009 (UTC)

you should remove both the tags straight away. the tag more reflects a somewhat arcane battle between two writers, than a general view of a problem with the page. The page is generally well written and nicely understandable

of course in any sensitive topic (not to mention cosmology, pure math, etc !!!) you will get hotheaded experts who feel minor issues are massive structural issues. these issues have nothing to do with general readership. Just remove the tag and if any of the battlers involved want to improve sentences in the article, just do so. there are NO EXISTING AUTHORS (I mean real aiuthors like wildly popular science writers, etc) who explain gauge theory clearly! Heh! so don't confuse your own arcane, internal 'religious" battles with general need for tags -- the article is nice, very good. get rid of the tags!

misleading tag

Bakken added a { { misleading } } tag to the article, but didn't explain on the talk page why he added it. I've deleted the tag, since I don't think the article is misleading.--76.167.77.165 (talk) 14:33, 14 August 2009 (UTC)

I did explain on the talk page (see my posts above) why I believe the article is misleading: the definition of gauge invariance is meaningless, the discussing of translation invariance is meaningless and digressing.Bakken (talk) 15:12, 14 August 2009 (UTC)

Actually the discussion of general relativity is also meaningless and digressing, but we shall deal with that later.Bakken (talk) 15:12, 14 August 2009 (UTC)

I've provided reliable sources to back up my assertion that the article is correct. You have not provided any reliable sources to back up your statements.--76.167.77.165 (talk) 15:19, 14 August 2009 (UTC)

The sources you refer to do not back up your assertions. You misinterpret your sources. Neither Perkins (which you refer to) nor any other physicist would define gauge invariance as "a statement that any physical system can be changed in certain ways without violating the laws of physics". Perkins, for example, defines gauge invariance as a property that "the energy of a system ... is invariant under arbitrary redefinitions of the global potential scale or gauge". This is a correct (although not complete) statement. Your statements are wrong.Bakken (talk) 18:20, 14 August 2009 (UTC)

Having just scanned the article, I was momentarily taken aback by the definition, but I think it correct, albeit unconventional. Normally GI is expressed in terms of invariance of the action, but since that amounts to invariance of the laws of physics, I don't really see the problem.--Michael C. Price ^talk 21:16, 14 August 2009 (UTC)

You cannot change a physical system with violations of the laws of physics. A physical system always follows the laws of physics. Imagine the following physical system: a hydrogen atom in the ground state -- how exactly do you "change it without violating the laws of physics" when switching from one gauge to another? What do you mean under "change of physical system"? Excitation? Exchange of one system with some other system? All these has no meaning. Compare, please, with Perkins (referred to by this vandal) -- "...invariant under ... redefinitions of the global potential".Bakken (talk) 21:37, 14 August 2009 (UTC)

Please, please, tell me how does your hydrogen atom "changes" when you change the gauge? Bakken (talk) 21:37, 14 August 2009 (UTC)

Simple, the fundamental fields all change, whilst leaving the measureable observables unchanged. See the Aharonov–Bohm effect, which has already been mentioned. Measurable does not equal physical, despite what Vienna's postivists try and tell you. Measurable is a subset of physical.--Michael C. Price ^talk 21:46, 14 August 2009 (UTC)

Ok, why don't we just agree on the following: non-measurable parameters of a theory can be changed in such a way, that all measurable quantities remain unchanged -- this is called gauge invariance. Any objections? Bakken (talk) 22:49, 14 August 2009 (UTC)

Fine, as long as we make it clear that all the fundamental fields are non-measurable.--Michael C. Price ^talk 22:59, 14 August 2009 (UTC)

Deal :) Bakken (talk) 23:43, 14 August 2009 (UTC)

Since there now seems to be a consensus on this issue, I've removed the { { misleading } } tag.--76.167.77.165 (talk) 23:27, 15 August 2009 (UTC)

Since we don't seem to be getting anywhere with this dispute, I've initiated the Wikipedia:Third opinion dispute resolution process.--76.167.77.165 (talk) 15:26, 14 August 2009 (UTC)

I popped in here from Third Opinion, but it looks like you have sorted it out together. I have de-listed this from Third Opinion. If there is a remaining conflict that I am not seeing, please do not hesitate to contact me. PGWG (talk) 16:40, 17 August 2009 (UTC)

Are we ready to remove the { { misleading } } tag now?--Ben, 76.167.77.165 (talk) 01:24, 18 August 2009 (UTC)

One dimensional thinking

The view expressed in a recent edit comment that:

gravitation in Einstein's general relativity is *not* a physical field (it is a curvature of space), it is completely dissimilar to matter fields and thterefore cannot be bunched together with them

is a mistake, since one model does not exclude another. Yes, the gravitatonal field can be regarded as geometrical, but that doesn't mean that it isn't physical. Also in physics different models are usually simultaneously valid. The gravitational field *can* be regarded as just another field. *And* it can be regarded as geometry. Both are true.

There's too much one-dimensional, absolutist thinking going on here. Stop pretending that one POV is the complete truth. --Michael C. Price ^talk 22:01, 14 August 2009 (UTC)

Why don't you listen? I said: in Einstein's general relativity. I do no say that Logunov's RTG, where gravitation is a spin-2 physical field like electromagnetic is wrong or anything. However, alternative theories are not generally accepted. In the generally accepted theory, Einstein's general relativity, the gravitational field is geometry of space. The following properties point:

1) Matter fields are on the right in the Eintsten's equation (in the matter energy-momentum tensor), the gravitational field is on the left (in the Einstein's tensor).

2) Gravitational field can be removed (locally) by a coordinate transformation, the matter fields can only be transformed, never removed.

3) Matter fields produce a symmetric energy-momentum tensor coupled to gravitational field. Gravitational field has a pseudo-tensor, not coupled to gravitational field.

Gauge invariance in the standard model is a simple thing. In general relativity it is not. Why do you want to bring in general relativity into a non-technical article on gauge invariance?

We had this discussion once did we not?

Can you give me an example of a gauge transformation in general relativity? —Preceding unsigned comment added by Bakken (talk • contribs) 23:01, 14 August 2009 (UTC)

Anyway, I do not object so strongly -- if you want gravitation in this article, ok, I put it back. Bakken (talk) 23:06, 14 August 2009 (UTC)

You're the one not listening. FYI I was talking about GR which everybody (except you, it seems) accepts as a gauge theory. Your little lecture abour tensors is completely irrelevant, and merely shows that you've digested not one single word of what I said about the validity of multiple models, but we've been through this before. (And please note my use of the word "model", again, which you seem to have misread as "theory".)

I'm glad to see that after all that rhetoric you finally talked yourself around.

Please remove the "electron/positron field" stuff -- this is meant to be a nontechnical article, okay?

Also I think the earlier description of the electric and magnetic fields separately is correct. The E and B fields are gauge invariant whereas the electromagnetic field is NOT gauge invariant. The distinction here is critical. --Michael C. Price ^talk 07:18, 15 August 2009 (UTC)

May I call electron "an elementary particle"?. I understand that you want to reserve the use the term "electromagnetic field" for the potential A. All right, if you insist. I only wanted to underline that it has been shown hundreds years ago that the fields E and B are not separate entities but components of one single quantity -- the electromagnetic tensor $F^{\mu \nu }$. Bakken (talk) 08:07, 15 August 2009 (UTC)

I see that there's been quite a flurry of edits in the last 24 hours from Bakken and Michael C Price. I think some of the changes are good, but there are problems with others. Before doing any major edits, I'm going to make a list here of what I see as the good and bad changes. As far as I can tell, a lot of what's been going on has just been Bakken and Michael C Price reverting one another, but if I compare the article now with the article 24 hours ago, most of the differences are Bakken's edits.

Good

1. The lead is shorter and more focused. In particular, it fixes a problem with the old lead, which was that it discussed changes in the time coordinate twice, rather than flowing in a more logical progression in which each relevant topic was touched on once.
2. Bakken corrected the equation t->t+t^3, which had bad units.

Bad

1. The lead is much more vague than it used to be, so much so that it no longer communicates any clear idea of what a gauge symmetry is. Without one or two examples built into the lead, the reader is going to have no idea what this is about. Also, the lead no longer mentions that gauge transformations can be local, which is one of their defining characteristics. It also no longer links to symmetry in physics, which is a problem, since that is not a concept that most readers will know anything about.
2. Bakken has added a narrative description of disagreements between editors to the talk page: "There are, however, Wikipedia editors who believe that translation invariance should also be called gauge invariance on the ground, that translation invariance follows the same philosophy:..." This is inappropriate.
3. This is supposed to be a nontechnical article, but many of Bakken's edits introduce terminology and notation that are not understandable to the general reader. In particular, the section "Another example of a symmetry: the general covariance of Einstein's general relativity" uses terms like "covariance" and "covariant derivative" as well as tensor notation.

There have been some confusing edits and discussion as to whether t->t+C is a gauge transformation. My original version of the article gave this, outside the context of general relativity, as an example of a global symmetry that was *not* a gauge symmetry. Bakken seemed to have missed the word "not," and complained that it was inaccurate. I pointed out the word "not" on the talk page to him. Later, Michael C Price deleted the word "not." This is actually kind of a complicated issue. There are a whole bunch of global symmetries in physics that are clearly not gauge symmetries. Also, changes of coordinate are the gauge transformations of GR, but they are not the gauge transformations of other field theories, so outside of the context of GR, it's not really natural to talk about them as gauge transformations. Also, the example of t->t+C is global, not local, and a manifestly local example would be more relevant.

I'm going to go ahead and revert the article to its state as of yesterday, and then do some more edits to reincorporate the changes that Bakken made that, IMO, were positive.--76.167.77.165 (talk) 23:15, 15 August 2009 (UTC)

Yes, I deleted the word "not". Coordinate transformations are gauge transformations. Saying "only within GR" is bit of a red herring since it is the gauge transformation that defined GR (a la Hilbert). It's a bit meaningless to say that a coordinate transformation isn't a GT within, say, SU(3). That's a bit like saying that SU(3) isn't a GT within SU(2) -- meaningless and irrelevant.

This isn't just my opinion. All textbooks on GR will tell you that coordinate transformations are a type of gauge transformation. All symmetries -- local and global -- are gauge transformations by definition. --Michael C. Price ^talk 23:52, 15 August 2009 (UTC)

Hi, Michael C. Price -- I don't think I have any substantive disagreement -- or at most on a matter of nuance which is no big deal. However, Bakken has pointed out, correctly I think, that the older version of the lead was somewhat digressive, in the sense that it was unfortunate for the lead to talk about a change of time coordinate, shift topics, and then go back to another example of a change of time coordinate. That was my main reason for deleting one of the two such examples from the lead.--76.167.77.165 (talk) 00:46, 16 August 2009 (UTC)

I'm not looking at the article while it's being reverted back and forth, but in my opinion

1. the lead should talk about just one example, and that example should be the E and B fields (since that is what most non techs would have encountered) and explain how they remain invariant whilst the electromagnetic A field changes. This also allows us to mention local invariance, of course.
2. Other examples like translation invariance and general coordinate covariance, along with SU(2), SU(3) are best left for the main article.
3. Mention nonabelian in the main body, but not in the lead.
4. I agree that Bakken's language is generally less accessible and needs toning down and needs more links to accessible articles.

--Michael C. Price ^talk 01:13, 16 August 2009 (UTC)

I'll put an E&M example back in the lead.--76.167.77.165 (talk) 01:18, 16 August 2009 (UTC)

Okay, I've now undone the three things that IMO were bad, while attempting to retain the things that IMO were improvements.--76.167.77.165 (talk) 23:29, 15 August 2009 (UTC)

Bakken reverted my revert. Reverting again. Bakken, please discuss this on the talk page.--76.167.77.165 (talk) 23:37, 15 August 2009 (UTC)

I have discussed it with Michael C. Price and he agreed on my definition as long as I mention that all fundamental fields are not observable. Sorry, but you are in the minority here. Reverting back to the definition agreed upon by the majority. Bakken (talk) 23:42, 15 August 2009 (UTC)

Do not include me with you. I have agreed some points with you. And I have agreed some points with 76.167.77.165. But I have to say, that of the other two editors, you are the most pompous, opinionated and arrogant.--Michael C. Price ^talk 00:00, 16 August 2009 (UTC)

Thanks for good words. Yes, we did have some disagreements with you before. Agree though, that unlike this anonymous editor I at least try to preserve the contributions of other people. I only delete apparently wrong statements. Bakken (talk) 00:26, 16 August 2009 (UTC)

And I only include you in the vote for the textbook definition of gauge invariance. Not yet for anything else. We'll come to the rest later. Bakken (talk) 00:26, 16 August 2009 (UTC)

Bakken, you have not replied to the substantive points I raised. I've re-reverted again.--76.167.77.165 (talk) 00:42, 16 August 2009 (UTC)

What's wrong with the definition of gauge invariance as it stands now? Bakken (talk) 00:49, 16 August 2009 (UTC)

Don't you see that I preserve basically all your content while you keep deleting mine? Bakken (talk) 00:49, 16 August 2009 (UTC)

I explained above, at point number 1 under "bad," what I think is wrong with the definition you've written in your version of the lead.--76.167.77.165 (talk) 00:52, 16 August 2009 (UTC)

In all textbook, including Perkins which you refer to, gauge invariance is defined similar to what is now written in the lead: different configurations of non-observable fields (like V and V+C in your example) result in identical observables (in your case -- reading on your voltmeter). Unlike your unfortunate sentence this one can not be misinterpreted, can it? Bakken (talk) 01:27, 16 August 2009 (UTC)

If you take a look at what I wrote under 1 above, I wasn't complaining about that. I was complaining that it needed at least one example, and that it wasn't specific enough about gauge transformations being local. Does the present version work for you? If so, can we delete the intro-rewrite tag?--Ben, 76.167.77.165 (talk) 01:35, 16 August 2009 (UTC)

Yes, I am satisfied with the present lead. Bakken (talk) 01:55, 16 August 2009 (UTC)

Okay, I'll delete the intro-rewrite tag.--Ben, 76.167.77.165 (talk) 02:16, 16 August 2009 (UTC)

Why don't you get a name? How should I call you? Bakken (talk) 01:27, 16 August 2009 (UTC)

My name is Ben. I used to edit as User:bcrowell, but I munged that account for reasons described on my user page.--76.167.77.165 (talk) 01:35, 16 August 2009 (UTC)

Thank you, Ben. I am really impressed with your decision to stop the war. I'll try to be equally constructive.

It looks like we're coexisting pretty well now in terms of our recent edits to the article. I look forward to working amicably with you in the future.--Ben, 76.167.77.165 (talk) 02:16, 16 August 2009 (UTC)

Excellent -- I can start reading the article again. --Michael C. Price ^talk 02:18, 16 August 2009 (UTC)

But how does gauge theory relate to particle physics?

Wow, the disputes here are hot and heavy. I think they reflect the difficulty of the project. Simplification is just as difficult as formulation, isn't it?

The main article Talk was "retired," along with my last comments there. It may be that they were too inane to be addressed or just that the discussion was retired before any further comments could be made. So let me take up a few of the points I thought might be constructive.

My main suggestion was that a nontechnical article should relate gauge theory to current news from particle physics. My jumping off point was the competition between Fermilab and Cern's LHC to detect the Higgs boson, which gauge theory predicts. (I see a Gauge Boson section. Does this relate at all to Higgs boson?) I also suggested explaining gauge theory in relation to symmetry and spontaneous symmetry breaking and the Higgs field. Also suggested mention of how gauge theory elegantly predicts SU(5) breaking, which prediction has failed experimentally, i.e. after over twenty years of trying no one has detected proton decay. This is my impression of what physicists are doing with gauge theory. Like, "This is gauge theory, and this is what we do with it. We use it to predict the existence of particles which point to a GUT." I see lots of discussion of what gauge theory is but nothing about its history. What gave rise to it? (Weyl's mathematical model?) Who was involved in its development and application? (Glashow, Weinberg-Salam, Englert-Brout-Higgs, etc.) How important is it? How has it been applied? For example, how is Maxwell's theory explained by gauge force? How does gauge theory segue from field to particle? Yang-Mills?

Yes, some subjects are indeed inaccessible to the layman, and it's quite difficult to take the disparate points above and turn them into a coherent narrative, but gauge theory is so important that I think it's worth the effort. I take my cue from Lee Smolin:

The gauge principle is best understood in terms of...a symmetry.... That all the properties of a force can be determined by knowing the symmetries is one of the most important discoveries of twentieth-century physics. This idea is what is meant by the gauge principle.

The discovery that all three forces [electromagnetic, weak, strong] are expressions of a single unifying principle -- the gauge principle -- is the deepest accomplishment of theoretical particle physics to date.... The standard model is the result of decades of hard, often frustrating experimental and theoretical work by hundreds of people. It was completed in 1973 and has held up for thirty years against a wide array of experiments. We physicists are justly proud of it.

Maybe the article should take it from there.

J M Rice (talk) 10:58, 15 August 2009 (UTC)

Yes, the history of the subject is illuminating. I suggest another article (cf History of string theory). I could certainly help write it.--Michael C. Price ^talk 11:22, 15 August 2009 (UTC)

Oops, I was still editing when you commented. I added some examples of whos and whats that might be useful. Do you think the history should be spun off? It would be nice if we could work the history into the non-technical article to give it a narrative framework, which would address the accessibility issue. J M Rice (talk) 11:42, 15 August 2009 (UTC)

I would favour a history spin-off, one reason being that it would supplement both the technical and nontechnical articles. (Otherwise we would have to write a technical and nontechnical history.) I think just explain gauge theories here, and deal with the history elsewhere. --Michael C. Price ^talk 17:07, 15 August 2009 (UTC)

Sounds good. Maybe a summary of gauge theory in the history intro. As you can see from the sturm und drang here, that's a hard enough assignment as it is. Let me take a crack at it, then let the experts shoot it down. J M Rice (talk) 17:45, 15 August 2009 (UTC)

PS -- Michael, yes, a spin-off would be fine. Like you say it would service both technical and non-technical articles. Don't you think that, ideally, the non-technical and technical articles each should have short historical intros appropriate to their respective levels? Likewise, the history articles would have a brief, non-technical introduction, explaining what gauge theory is, before proceeding to the history proper. -- JMR (Log-in server problem) 70.165.243.44 (talk) 18:48, 16 August 2009 (UTC) J M Rice (talk) 01:04, 17 August 2009 (UTC)

Generally I favour linking to explanations rather than duplicating material, so I would prefer we just link to the history article rather than try to write multiple synopses of it. Also history of science topics are generally accessible to various levels of expertise -- providing they are heavily linked. (Again cf history of string theory.)--Michael C. Price ^talk 23:16, 16 August 2009 (UTC)

I agree with J M Rice that the article should have more particle theory in it. I'm just more comfortable with the E&M and GR stuff, so that's what I wrote the most about. Explaining SU(2) and SU(3) to nontechnical readers would be cool, but probably hard to do well. J M Rice, please knock yourself out -- it would be great to have more particle physics!--76.167.77.165 (talk) 23:34, 15 August 2009 (UTC)

I'll give it a shot. The nexus between gauge theory and particle physics is where the rubber hits the road, where abstract means concrete. Apparently that nexus is Yang-Mills. Hope I can wrap my head around that one and somehow incorporate it briefly. --- JMR (log-in server problem) 70.165.243.44 (talk) 18:48, 16 August 2009 (UTC) J M Rice (talk) 01:04, 17 August 2009 (UTC)

Quit the reverts -- both of you

You both know when you're making edits that will be reverted. Talk first, then edit article, OK? Othewise it is too childish for other editors to bother with. --Michael C. Price ^talk 00:52, 16 August 2009 (UTC)

I stand corrected. But I would really like to get a substantive discussion going here on the talk page of the three issues I raised above. I would also like to point out that I did discuss on the talk page.--76.167.77.165 (talk) 00:56, 16 August 2009 (UTC)

Agreed, but I think you and Bakken are both a bit quick to start editting after talking; Listening and discussing are important too. (Although if someone doesn't respond then the later is moot, as I'm sure you will quite rightly point out.)--Michael C. Price ^talk 02:04, 16 August 2009 (UTC)

redundancies and contradictions

The subsection "An example of gauge transformation: adding a constant to electrostatic potential" is now redundant, since it just repeats an example given in the lead. I suggest simply deleting it, but I'd like to hear Bakken's opinion.--Ben, 76.167.77.165 (talk) 01:52, 16 August 2009 (UTC)

The subsection "An example of a symmetry in a physical theory: translation invariance" and the subsection "Another example of a symmetry: the invariance of Einstein's field equation under arbitrary coordinate transformations" contradict one another. The former essentially says that coordinate transformation is not a gauge symmetry of GR, while the latter says that it is. I'm going to merge these two into one as a first step, and then try to iron out the contradictions. --Ben, 76.167.77.165 (talk) 01:52, 16 August 2009 (UTC)

Don't merge, they are (sufficiently) different things, although the contradictions must be removed, of course. Translation invariance goes back before Galileo, velociy boosts to him. Whereas general covariance only came with Hilbert/Einstein/Weyl.--Michael C. Price ^talk 02:00, 16 August 2009 (UTC)

Okay, I have them in separate sub-subsections now. Your point about translation invariance dating back to much earlier in history is a good one. I'll try to reword it a little so that that flows better and comes out more naturally.--Ben, 76.167.77.165 (talk) 02:08, 16 August 2009 (UTC)

Both of the above subsections are redundant with the ones under "In classical physics." I'm going to merge them.--76.167.77.165 (talk) 01:53, 16 August 2009 (UTC)

Give people time to respond. Perhaps I should have said, talk, listen then edit :-) --Michael C. Price ^talk 02:00, 16 August 2009 (UTC)

You are of course absolutely correct.--Ben, 76.167.77.165 (talk) 02:26, 16 August 2009 (UTC)

more edits to lead

I've done some more edits to the lead, most of which I hope will be noncontroversial. One possible exception is that I reverted Michael C. Price's replacement of "V and A" with "A." This article is for nontechnical readers, and they won't know about four-vectors. This is also why I reinserted "voltage" in parens after the first mention of V; more people know about "voltage" than "potential." The rest of the article also uses (V,A), rather than a single 4-vector A.--Ben, 76.167.77.165 (talk) 01:23, 18 August 2009 (UTC)

They may know about "voltage" but is that the same thing as the electric potential(?). According to electric potential they are only the same in electrostatics. Whilst it is nontechnical it has to be accurate. I don't think it is a problem introducing new terms to the audience, provided we don't try to baffle them with the maths. Just saying there is an electromagnetic potential and it changes doesn't strike me as too heavy.--Michael C. Price ^talk 09:01, 18 August 2009 (UTC)

I don't think "voltage" is typically used at all in texts that get to the level of sophistication where you have to worry about these distinctions. Some words have precise definitions ("mass," "force"), others don't ("love," "Mach's principle"). Some, like "voltage," fall in between. I don't think it's an error to use a broad, imprecise term in a broad, imprecise way. I'd prefer to use language that Joe the Plumber is likely to be able to connect with.--Ben,76.167.77.165 (talk) 04:25, 21 August 2009 (UTC)

You haven't actually addressd my point about "accuracy", which is not the same as "imprecise". "Voltage" is not used in those texts because it is the wrong word, which you can confirm by following the links. I do not believe that we have to sacrifice accuracy in a nontechnical article.--Michael C. Price ^talk 05:15, 21 August 2009 (UTC)

I don't see the word "voltage" used at all in electromagnetic potential or Electric potential, except in a "see also" link. Is there a specific statement you had in mind in one of those two articles?--Ben,76.167.77.165 (talk) 15:54, 21 August 2009 (UTC)

Although the insertion of the word "voltage" is controversial, please leave the separate notation of V and A. As I've already pointed out, this is a nontechnical article, and nontechnical readers don't know about four-vectors. This is an entirely separate point from the issue of the word "voltage." Throughout the article, the symbol V is being used to mean the timelike component of the four-potential.--Ben,76.167.77.165 (talk) 16:12, 21 August 2009 (UTC)

I don't see the word "voltage" used at all in electromagnetic potential or Electric potential -- exactly, so why do you insist on using it? And note the sentence This generalized electric potential cannot be simply interpreted as the ratio of potential energy to charge, however.. --Michael C. Price ^talk 17:48, 21 August 2009 (UTC)

Your edits assume that gauge symmetry = local gauge symmetry, which is not the case since there are global gauge symmetries. Forces come from local symmetries not global symmetries.--Michael C. Price ^talk 09:01, 18 August 2009 (UTC)

We have a set of gauge transformations A, of which a proper subset B is global, and another proper subset C is not global. You seem to be interpreting "local" as meaning C, whereas I intended "local" to mean A. This is one of those trivialities like worrying about whether "curve" includes straight lines. I don't really care if you want to reword it, but the lead should definitely explain that gauge transformations *can* be non-global.--Ben,76.167.77.165 (talk) 04:25, 21 August 2009 (UTC)

I'll reword it then, making that point.--Michael C. Price ^talk 05:09, 21 August 2009 (UTC)

Incidentally I propose deleting about 99% of the Aharonov–Bohm effect section since the article itself has been much improved recently, and we don't need all those details here. Any objections?--Michael C. Price ^talk 09:25, 18 August 2009 (UTC)

Objection, objection! Actually I don't really understand what you mean here. Could you explain more? I consider this section the heart of the article.--Ben, 76.167.77.165 (talk) 04:09, 21 August 2009 (UTC)

I mean that the Aharonov–Bohm effect article itself is much improved, so that we can have just a much shorter summary here. The details of the experiment are irrelevant, although the implications are relevant. So just state the implications here and link to the article for the more technical details. --Michael C. Price ^talk 05:04, 21 August 2009 (UTC)

The Aharonov–Bohm effect is extremely technical. A reader who's operating at the level of this article is not going to be able to understand that one at all.--Ben,76.167.77.165 (talk) 15:56, 21 August 2009 (UTC)

A misleading statement again

"Under a gauge transformation which jointly alters both the potentials and the wavefunctions of all electrically charged particles, such as the electron, no change occurs in E or B." It sounds like you needed to change some wave-functions for E and B to remain unchanged. This is wrong. You don't need to change any wave-functions for E and B to remain unchanged under the gauge transformation $A_{\mu }\rightarrow A_{\mu }+d_{\mu }f$. Moreover, in classical electro-dynamics there are no wave-functions at all, still there are gauge transformations and gauge invariance. —Preceding unsigned comment added by Bakken (talk • contribs) 11:14, 18 August 2009 (UTC)

My statement was not wrong; stop presenting incomplete justifications. You know that the transform of A is not the whole gauge transformation. --Michael C. Price ^talk 05:28, 21 August 2009 (UTC)

I didn't say "statement was wrong", I said "statement was misleading". Your way of arguing is called "red herring". Classical electrodinamycs is also a gauge theory, but there are no wave-functions there, only currents. Your statement excluded classical electrodynamics from gauge theories, which was misleading. Bakken (talk) 07:52, 21 August 2009 (UTC)

I see the word "wrong" in your statement, not the word "misleading". Nevermind. --Michael C. Price ^talk 10:31, 21 August 2009 (UTC)

My statement is in the subtitle: "a misleading statement again" -- there is no word "wrong", there is a word "misleading". The word "wrong" in my explanation refers not to your statement, but to the logical conclusion directly following from your statement: that a change in the wave-function is needed for the E and B to remain unchanged. Bakken (talk) 11:00, 21 August 2009 (UTC)

Which it is in modern physics, which is what the lead is talking about. --Michael C. Price ^talk 11:34, 21 August 2009 (UTC)

The substantive issue here is that you and Ben seem to be assuming that the lead should only be about classical physics. --Michael C. Price ^talk 10:31, 21 August 2009 (UTC)

I think Michael C. Price added the part about "and the wavefunctions of all electrically charged particles." I agree that it's unnecessary, since the lead is just referring to the classical theory. I don't think it's misleading, just unnecessary.--Ben. 04:13, 21 August 2009 (UTC) —Preceding unsigned comment added by 76.167.77.165 (talk)

That *was* misleading, precisely because it is "unnecessary" for E and B. It is like adding "and changing all neutrinos to antineutrinos" -- that will also be technically not wrong, but it will be misleading. Bakken (talk) 07:52, 21 August 2009 (UTC)

Where does the lead say it is talking about only classical physics? And bear in mind that gauge transformations, historically a la Weyl, only referred to quantum systems. If we are going to distinguish between classical and quantum physics in the lead that is going to complicate the lead considerably. --Michael C. Price ^talk 05:28, 21 August 2009 (UTC)

Upon reflection I think we need a short statement in the lead to the effect that gauge transformations are of no physical significance in classical physics.--Michael C. Price ^talk 10:26, 21 August 2009 (UTC)

Then you first have to define what a "physical significance" is. Classical electrodynamics *is* a gauge theory. Calling it "physically insignificant" sounds a bit strange to me. For example, as Ben has noted, gauge invariance is connected to charge conservation. Indeed gauge fields can only couple to conserved currents, otherwise the gauge invariance gets broken. Is this not "physically significant" to you? Bakken (talk) 11:10, 21 August 2009 (UTC)

By physical significance I refer to the reality of the fundamental gauge fields, as demonstrated by the Aharonov–Bohm effect.

--Michael C. Price ^talk 11:30, 21 August 2009 (UTC)

And how exactly does this make gauge invariance in electrodynamics "insignificant"? Bakken (talk) 11:48, 21 August 2009 (UTC)

Because you can work directly from the gauge invariant fields for everything, including conserved currents.--Michael C. Price ^talk 11:52, 21 August 2009 (UTC)

Sorry, but I don't get what your statement is... And since I don't think it's physically significant anyway (as most of your statements), we can just as well stop here... :) Sorry for confrontation, but you started first. Eye for an eye. Bakken (talk) 11:59, 21 August 2009 (UTC)

Actually most editors would see your section title as confrontational, but not you, I guess.... --Michael C. Price ^talk 14:59, 21 August 2009 (UTC)

No, not me -- in my view it is a correct and meaningful subtitle. Bakken (talk) 15:28, 21 August 2009 (UTC)

Yes, that's what I thought you'd say. --Michael C. Price ^talk 17:54, 21 August 2009 (UTC)

Michael C Price is of course correct that the lead need to be technically correct, and should not assume classical E&M when the context makes it sound like it could be quantum field theory. The lead seems more or less OK to me in its present state, except that we're drifting more and more toward advanced ideas and abstraction rather than nontechnical ideas and concrete examples. The average nontechnical reader could probably have understood the example we used to have in the lead of V->V+C. That same reader is extremely unlikely to be able to get the drift of the current version of the lead. I'm going to rewrite a little bit so that the classical stuff comes first, then the quantum stuff comes in after that.--Ben,76.167.77.165 (talk) 16:02, 21 August 2009 (UTC)

I give up. I leave you two to fight it out. Cheers. --Michael C. Price ^talk 17:54, 21 August 2009 (UTC)

So long and thanks for your input. And come again. Bakken (talk) 18:30, 21 August 2009 (UTC)

Kudos

I don't see how you can have an edit war about such a trivial thing, but the article itself is gold, first actual explanation I see. By the way, this "we cannot tell" business reminds me of the natural numbers and the Peano axioms. We like to think that + and * are "just" operations on |N, but this is not true, without + and * there is no |N, only a faceless chain of one faceless whatever after another (the Peano axioms are totally overrated IMHO). We can call the first chain element "0" or "1" or "Wikipedia" or "Quantum", it does not matter, we cannot tell unless we have "+", since 0 is the neutral element of +. + and * make the naturals out of faceless set dust. --88.74.130.189 (talk) 06:38, 26 October 2009 (UTC)