I think this paper has a very misleading title, as the result it presents is not as remarkable as to be actually proof of what is known as Maxwell's equations (they were formulated by Heaviside btw.) Basically what the paper does is that it defines operators for ρ, j, E based on the hindsight knowledge of Maxwell's equations and non-relativistic equation of motion for a particle; then it shows that it is possible to have both sets as Heisenberg equations. That's definitely not something a physicist would call deriving Maxwell's equations. Most of the thing claimed to be derived in the title is actually defined/assumed, the result is merely that the Heisenberg formalism, the commutation relations and non-relativistic equation of motion do not seem incompatible. The speed of light in the result is there purely because Dyson knows the result he needs to get.
I disagree. I would call this a proof but not a derivation. In other words, each step follows logically from the previous ones, even if there's no way that you would discover the final equations this way if you didn't already know or guess them.
In general, this is part of what makes reading mathematical papers so difficult; the steps of the proofs are almost never written down in the order they were discovered. All of the false starts and a lot of the intuition is discarded to yield a shorter but sometimes totally mysterious path from the assumptions to the conclusion.
Right. It's not possible to prove Maxwell's equations from first principles, we only "know" they are true from experience. In particular, there's no logical reason divergence of B has to be zero; logically there could be magnetic monopoles, but experience seems to indicate there are not.
