I have found a beatiful post by Terry Tao, a Fields medallist, about gauge theories. See here for a worthwhile reading. This post is truly elucidating and so well written that I thought it was worthing a larger audience.
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I had read the first few paragraphs of that Terry Tao post and had concluded I wasn’t going to learn anything from it. On your suggestion I’ve looked again, and yes, it is a beautiful post. I like the terms “spent” and “bought”. However, his examples of broken gauge symmetries don’t seem similar to what a physicist thinks of. He seems to see them as methods in proving a result for a symmetry which a physicist would call unbroken.
I’m busily writing up a paper on the CKM and MNS mixing matrices. Terry’s article makes it clear that one can think of the choice of unitary matrix, that produces a given set of experimental data (i.e. the absolute magnitudes), is a form of gauge freedom. When we choose a particular parameterization we are spending that freedom.
And the example of the winds on the earth as defining a direction on each point of the earth, except for the hairy billiard ball theorem, reminds me of an analogous fact about spinors, one that is “one brick in the wall” in why I like density matrices over state vectors as far as quantum mechanics goes.
Suppose for each spin -1/2 spinor, you make a choice of the arbitrary complex phase. For example, you might choose the top component (i.e. the spin up component of the spinor) as real. Then it is a fact that your choice of complex phase cannot be continuous. In this the pure density matrices are more natural.
And before you ask I should add that the paper I’m writing will be sent in for the painful process of peer review. I still don’t have any great attraction to the process, but my co-author needs to “publish or perish”.