Hi Higgsy, Thanks for your comments. You wrote: Weber "... does not lead to the prediction of electromagnetic waves and the correct speed of light." Yes it does. The "old" or original interpretation of Weber did not include waves. It was an instantaneous action at a distance theory. But when I wrote Weber in terms of "fields", it looks like Maxwell's theory (in symbols). And when I introduced time retardation (following J. P. Wesley's method for introducing time retardation) Weber's theory gives waves with speed c. These are new results of my work, so I don't expect others to know about it all.
You wrote: "By definition the magnetic field remain unchanged under transformations of the vector potential by addition of the gradient of a scalar function, and the electric field remains unchanged under transformation of the scalar potential by the addition of the time derivative of a scalar function. So the classical theory of electromagnetism has this gauge freedom and is a gauge theory." Yes. And that, now, also holds for Weber's electrodynamics. So then why must we specify the Coulomb gauge (in Maxwell) for Gauss's law to be valid? If the gauge choice doesn't change the E field, then Gauss's law should remain valid regardless of the gauge specified. But Gauss's law, in Maxwell, is only valid in the Coulomb gauge.
And Ampere's Law doesn't hold mathematically. Take the mathematical definition of the vector potential function, A, and put it in left-hand side of Ampere's law
(\nabla x B = \nabla x (\nabla x A))
and you can't get the right hand side of Ampere's law. Is there some other gauge that makes this work, because it doesn't work in the Coulomb gauge. So, picking a gauge may make some of Maxwell's 4 equations mathematically correct (left-hand side equals right-hand side) but doesn't make all 4 Maxwell equations correct.
My approach, which is very simple, makes all 4 "corrected" Maxwell's equations correct for the current density and charge density functions.
You wrote: "Note that in QFT the magnetic vector potential has physical meaning in that it can be used to predict experimentally verifiable physical effects such as the Aharonov-Bohm effect." Thanks for reminding me. I meant to look at the Aharonov-Bohm effect in terms of Weber. Note that Weber's force law includes the Lorentz force law and other terms. Maybe these other terms give the Aharonov-Bohm effect.... I'll look at it.
You wrote: "As far as I am concerned the problem isn't whether there is gauge freedom in the theory or not, but whether the theory can be shown to be an accurate representation of experimental results, which test Maxwell's theory has passed with flying colours for more than 100 years." Yes, but those same tests also verifies Weber's electrodynamics. Note that Lorentz force law is contained in Weber's force law (as dominate terms). For experiments based physics, note that Ampere's force equation (not Ampere's law of Maxwell equations) was developed **from experimentation** by Ampere. (See Assis's book Ampere's electrodynamics, Graneau's book Ampere-Neumann Electrodynamics of Metals, Graneau's book Newtonian Electrodynamics, Wesley's book Selected Topics In Scientific Physics, and papers in the literature...) Why then isn't Ampere's force law in mainstream physics textbooks? It was developed from experiments... Oh, and Ampere's force law is also in Weber's force law. See http://www.weberelectrodynamics.com/Web ... Force.html
The typical experiment showing Lorentz force law alone is lacking are the so-called "exploding wire" experiments. This requires a longitudinal force along the current carrying wire to rip apart the wire. But the Lorentz force equation doesn't have a longitudinal force. Ampere's force equation (and hence Weber's force equation) does. There are other experiments that indicate the Lorentz force is not enough to describe all electrodynamics effects.
You wrote: "So I ask again, in your attempted derivation of Gauss's law using Maxwell's theory, which gauge are you working in?" I am not working in any particular gauge. I consider Maxwell's equations and potentials and Lorentz force and continuity equation as a single mathematical **system** of equations. Meaning that they **should** all be mathematically consistent with one another. They aren't. Just try (as I know you have) Gauss's law with the E field written in terms of grad Phi and @A/@t . It doesn't hold. Why should it? It was derived only for the electrostatic case (E = -grad Phi) and not for the electrodynamics case. If you want to use the Coulomb gauge, o.k. Then try to get Ampere's law to hold in the same Coulomb gauge. It doesn't work. (It all works with Weber's electrodynamics.)