I've read a few times that you can derive the Maxwell equations from knowing that U(1) exists, as this article implies, that every point in space has (among other things) the symmetry of a circle. But I've never seen this derivation. I assume it's either trivial or too complicated. Anyone has a link or thoughts on this?
In quantum mechanics the wave function Ψ is has complex values. If you multiply everything in the universe by -1, nothing changes because all the physical results use ΨΨ* (where * is the complex conjugation). You can also multiply everything by i or -i. Moreover by any other complex number of modulo 1 because ΨΨ* does not change. (The technical term for this is U(1) global gauge symmetry.)
But you can be more ambitious and want to multiply each point of the universe by a different complex number of modulo 1. ΨΨ* does not change but the derivatives of Ψ change and they are also important. (When you use the same complex number everywhere, the derivatives is just a multiple of the original derivative. When you use a different number in each point, it changes.)
The only way to fix the problem with the derivative is to add a new field A. When you and multiply each point of the universe by a different complex number of modulo 1, then A changes in a simple to calculate but not obvious way. The change in A fix the problem with the derivatives of Ψ.
So now the equations of the universe with Ψ and A don't change when you make this change. (The technical term for this is U(1) local gauge symmetry.) When you write carefully how a universe like this look like, the new field A is electromagnetism. (Actually, you can get the electric field and magnetic field using the derivatives of A.)
Too many more details in https://en.wikipedia.org/wiki/Quantum_electrodynamics#Mathem...
> I assume it's either trivial or too complicated.
It's trivial once you have 3 or 4 years studding Physics, but you will never understand how it is related to the magnets in your refrigerator. [There are like 5 simplification steps between U(1) and magnets in the refrigerator. Each one makes sense, but I can't see all of them together in my head.]