I dunno of a good article as an introduction. As a grad student, I found Terry Tao's explanation [1] rather helpful, but of course it has a strongly mathematical flavor.
[1] https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/
Also, a textbook is revealed on what seems to be this exact topic. Does this seem right? https://physicstoday.scitation.org/doi/pdf/10.1063/PT.3.2421
(Which I realize is the least helpful possible reply.)
The textbook Peskin&Schroeder is standard in this field. "If" you can find a PDF online, there's a chapter introducing gauge theories (the chapter on yang-mills) that has a reasonably clear explanation near the beginning, I think.
> “Gauge theory” is a term which has connotations of being a fearsomely complicated part of mathematics – for instance, playing an important role in quantum field theory, general relativity, geometric PDE, and so forth. But the underlying concept is really quite simple:
Then goes on to write an incomprehensible novella length explanation.
I like Tao, but this is a problem in 'pure' maths. I feel the issue is exacerbated by the fact that if you can follow along with Terry then you likely will be unable to recognise the issue.
A good counter point is this series by Timothy Gowers developing a proof of "Pingala's Determinant".
When Gowers finds the proof he states:
> Since this clearly has determinant 1 and this clearly has determinant 1, so does their products, ie. Pascal's[sic] determinant is 1. Could have done that, but that would have been uh, well depends on what you find interesting, but that would have been really sort of rabbit out of the hat, look what a billiant clever mathematician I am. I can just sort of produce this fabulous identity out of nowhere and it's got a nice simple proof that also came out of nowhere. What I want to emphasize is these sort of proofs don't come out of nowhere.
Timestamped to quotation: https://youtu.be/m8R9rVb0M5o?t=886
The series is great and will show anyone outside the field of pure maths that even pure mathematicians use numerical reasoning while trying to reason about a subject.
Not all people will agree that Tao's exposition is unclear. I linked it precisely because I found it more clear than other, "more physical" explanations (as a graduate student in physics, not math). When I was trying to understand gauge symmetry, it was that article that finally made things click for me. And, of course, many people I know would disagree, and point to other expositions as superior. That's the point: different people will not agree on what is well-written and what is incomprehensible.
You state "even pure mathematicians use numerical reasoning while trying to reason about a subject". Yes, some pure mathematicians do. Others use graphical reasoning. Others tend to lean purely on algebraic manipulations. Others tend to "reason" by analogy. And I'm sure there are other modes of thinking with which I'm too poorly acquainted to name. The result is that it's perfectly reasonable for people to disagree, quite strongly, about what is comprehensible and what is not, particularly people coming from different intellectual traditions.
It's a tangential nitpick, but I can't resist:
> [...] novella length explanation
As an undergrad I wanted to learn differential geometry, so I went to the library to pick a diff geo textbook. Well, I didn't want a difficult one, so I decided to pick the shortest book I could find --- surely that would be the easiest textbook to read? Haha.
As far as I'm concerned, the length of Tao's explanation is a strong merit.
> I feel the issue is exacerbated by the fact that if you can follow along with Terry then you likely will be unable to recognise the issue.
On the topic of "heterogeneity" it would seem to follow from my saying "If you can follow along with Terry" that I recognise some people are capable to find his explanation clear.
My concern is providing that link as educational stead insightful.
On the heterogeneity of pure mathematicians it is very frustrating to see someone try to claim, in a public forum where people unfamiliar with the subject can read, that a Mathematician uses anything other than "numerical reasoning while trying to reason about a subject", and I feel it causes great harm to people outside of mathematics.
You say, "some use graphical reasoning", well where did their graphical intuitions come from? Are they born with it? Or perhaps they were taught graphical reasoning by way of numerical reasoning and now are able to just use the graphical reasoning on new problems.
This is part of what I was trying to express by referencing Gowers. Using identities to reason about a problem is fine, and arguably necessary, but introducing identities when teaching without explanation is befuddling and I argue deters learning.
The op said "I'd like to learn more about symmetry", you used the word "introduction" and Tao used the word "simple".
If the op said, "I went through a physics program in undergrad and grad school and I feel like I have all the mathematical pieces but I'm still struggling to put them all together... can anyone recommend a resource that avoids getting bogged down in prerequisite explanations and just focuses on the relations."
Then I would have scrolled past thinking, yeah that Tao article is perfect.
If you would have said, "After years of higher education this article made it click for me in ways multiple professors failed to but you'll need an equivalent amount of the numerical examples from my education of the various topics used to understand this."
Then I would have scrolled past thinking, yeah Tao does a great job of illuminating concepts.
If Tao would have used any other word than "simple".
Then I would have left out the "novella" jab. Though now I'm tempted to ask what about the short textbook was more difficult, because in my similar library experiences the short books were the ones that just had equations and identities and left out any of the numerical reasoning. Is that what you discovered when you Haha'd at that little textbook?
I worry for Mathematics when someone says, "Hey I think I might be interested in that thing..." and the response they recieve is a bludgeon of latin characters, and the dubious unqualified follow up "well other people get it".
It is now doubly hard for that person to explore their interest because they need to both find their own resources that actually teach the material, as well as decide whether its even worth it if this is the community they are working to become part of.