what does it mean to say about the "classical electromagnetism" to be U(1) invariant?
Very nice thread - loved it.
Personally though I prefer the reverse - getting EM from local U(1) invariance of QM rather than asking what does it mean to be U(1) invariant.
It's even found in popular books eg Victor Stenger - The Comprehensible Cosmos - page 251 under Electromagnetism:.
He covers the other stuff too (symmetry breaking and all that), again at the populist level using just a little bit of math and calculus.
This is an advanced thread, but I mention it for those not advanced but would like a gentle introduction - you just need a bit of calculus.