Confusion on Maxwells equations and Gauge Transformations I know a little bit about electrodynamics but I don't understand the validity of Gauge Transformations. 
In particular I am confused on how the theory can be consistent among different gauges, in addition I don't understand what is "real" versus a byproduct of mathematics.  
Are the electric field and magnetic fields "real"? While potential is a mathematical construct?
Why are these field "real" versus taking the scalar and vector potentials to be real? 
In addition when one uses the Coulomb gauge versus the Lorenz gauge in deriving radiation - how do we know they are consistent? (I think I read some slides saying the Coulomb gauge actually gave the wrong results). 
What's going on? 
 A: What is "real" in electrodynamics is what you measure. The fields can be measured without ambiguity due to their influence on charges and currents, and so these are the things that are real. Maxwell's equations directly reference these fields and contain only physical information.
Potentials, on the other hand, are defined through the relations $\vec{E} = -\nabla\Phi - \partial\vec{A}/\partial t$ and $\vec{B} = \nabla\times\vec{A}$, subject to the constraints imposed by Maxwell's equations. This is entirely a trick that we use to hopefully make life easier.
Now given these relationships, there is an obvious ambiguity in $\vec{A}$, where we can take $\vec{A}' = \vec{A} + \nabla\Lambda$ for any differentiable function $\Lambda$ and still get the same magnetic field. With this, we can also see that if we take $\Phi' = \Phi - \partial\Lambda/\partial t$, then our electric and magnetic fields in terms of the potentials are the same as they were before. These are gauge transformations. Due to the fact that the fields don't change, they are independent of our gauge, and so any physical consequences--as they depend on the fields--must be consistent in all gauges.
In general, knowing that deriving formulas in different gauges is consistent can be done by actually going in and verifying that they give the same fields by hand, or trusting that, because the fields are by construction invariant under gauge transformation, unless an error was made somewhere the answers will be consistent.
EDIT
With respect to the Aharonov-Bohm effect, yes in quantum mechanics things get a little stickier and the potentials have a more important role, however it still remains that anything measurable must be gauge invariant. And the Aharonov-Bohm effect is gauge invariant, since the phase difference between paths can be written in terms of the magnetic flux, which is something physical.
To be as explicit as possible: imposing a gauge condition cannot have any physical effect. These gauge degrees of freedom represent redundancies in describing the system using potentials, and so "fixing the gauge" by imposing some condition doesn't change the system, it just gets rid of these redundancies. If you change a gauge and see a measurable difference, you either did something wrong, or your theory isn't actually gauge invariant.
A: The theory of electromagnetism, and physics in general, is a model of reality. It is not reality itself so we cannot claim reality for the E and B fields.
Having dealt with the philosophy, one may ask if E and B are enough to account for electromagnetism. My answer to this is no. The Aharonov-Bohm effect cannot be described in terms of E and B. More generally, we are unable to get around the four potential in quantum mechanics. It is impossible to describe electromagnetic or photon spin as a separate property using E and B. It is impossible to implement the Lorenz gauge in quantum mechanics without "breaking the gauge", as the trick is called. The Noether conservation laws that follow from the standard Lagrangian, which is expressed in E and B, cannot themselves be expressed in terms of E and B without an unwarranted modification. The conservation laws of gauge invariant electromagnetism exhibit numerous paradoxes.
A: Of course they're real - as real as (say) velocity is. Like velocity, its measurement is dependent on the observer frame; only here, "frame" doesn't just mean a speed reference or a location in space to use as an origin for a coordinate grid or a reference time to use as time 0, but also a gauge; specifically, the ground for potential. Just because something is relative doesn't mean it doesn't exist. Otherwise, you'd be saying that velocities don't exist because they're relative ... or that phase doesn't exist, because it's also relative ... when (in fact) phase is practically everything as you can see in this demo here.
A Scalographic Demo
The phase (color coded) is, itself, what determines the scalogram.
https://www.youtube.com/watch?v=OugT7uGGtNg
[Also: in quantum representations, in case anyone forgot, the electromagnetic potentials show up as coefficients for the expression of the phase.]
Contrary to what you may see in some folklore, sometimes even propagated by physicists themselves (who should know better!), both the magnetic potential $$ and the electric potential $φ$ have always been present in Maxwell's formulation of electromagnetic theory from the very first day - meaning: even the papers he wrote, predating the treatise on electromagnetic theory (e.g. his 1864 "A Dynamic Theory of the Electromagnetic Field"). They were not put there as afterthoughts by later people. Maxwell called $$ the "Electromagnetic Momentum" and was accurate in that call, because the canonical form of the force and power law for an electric charge $e$ with momentum $$ and energy $H$ can be written as
$$\frac{d}{dt}( + e) = -∇U, \hspace 1em \frac{d}{dt}(H + eφ) = \frac{∂U}{∂t}, \hspace 1em U(t,,) = e(φ(t,) - ·(t,)).$$
So, it plays the same role with respect to momentum, that the electric potential plays with respect to energy - it is a "potential momentum" per unit charge.
It was only in the later part of the 19th century, during the Wars of the Diadochi period, following Maxwell's early demise, as his successors carved up his realm between one another, that you saw people (meaning: Hertz) trying to push the potentials away as if they were some kind of fiction and trying to write everything in terms of the field strengths alone. It is partly due to the misguided efforts of these later 19th century successors that the notion entered the folklore that potentials somehow weren't real.
A clear hall-mark of that being the wrong approach is that you can't write down any action principle. Everything's in the wrong places. You'd have field strengths as fundamental variables, so then what order would they appear as in, in the Lagrangian?
Dirac and Bohm, in the 20th century, put a stop to the "potentials are fiction" meme with their respective analyses (to say nothing of Weil).
Even today, Physicists have not fully learned that lesson or to pay attention to clues like that, because the same thing is still happening - with the Dirac equation and its action principle. There, too, you find an odd-ball Lagrangian that is totally zeroed out on-shell - which is a dead giveaway that you have order 0 and order 1 terms mixed up together in the wrong place.
Unbeknownst apparently to them all, this too can be written in terms of "fermionic" potentials, with the spinor field being the field strengths of the potentials. When written this way, the Lagrangian becomes quadratic and can even be factored as a perfect square, similar to what happens with the Maxwell-Lorentz or Yang-Mills Lagrangians. And here, too, there is a gauge condition for the potentials; and there could be some interesting physics here that has yet to be explored.
So, one should also be talking about fermionic potentials, as well; not just gauge and electromagnetic potentials; and, if anything, people have (even to the present day) yet to realize the full potential of potentials.
