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.]
If A and B are zero then F is zero and you can forget the second part of the right hand part of the equation. Also, you must replace D with ∂.
Now you have equation of electrons and positrons that move in a universe that has no electromagnetism. The important part is that in that equation, the only variable is ψ(t,x,y,z) that appears twice, the rest of the things written there are just derivatives, constants or indices.
Now you can turn on B(t,x,y,z) that represents the external field, so the electrons and positions move in a more interesting patterns, but they don't "see" each other. Again, the only variable is ψ(t,x,y,z).
Now you do the trick with a local U(1) symmetry, and the only way to do the trick is to add a new variable A(t,x,y,z). But you must use the complete equation because as explained in Wikipedia F(t,x,y,z) is calculated using the derivatives of A(t,x,y,z). So now you have two variables ψ(t,x,y,z) and A(t,x,y,z).
So you "add a new field A" to the list of variables that appear in the right hand of the equation, or to be more precise, you get a new equation that has one additional variable A.
Typo or rhetoric?
A collectible, either way ;-)
Putting in some of those keywords will find plenty of detailed notes online, like
http://www.physics.usu.edu/torre/Classical_Field_Theory/Lect...
or more quick overviews like
https://mugndonut.wordpress.com/2018/03/31/symmetries-yield-...