We have already studied many aspects of gauge invariance in electromagnetism and the corresponding invariance under a phase transformation in Quantum Mechanics. One point to note is that, with our choice to ``treat each component of as an independent field'', we are making a theory for the vector field with a gauge symmetry, not really a theory for the field .
Recall that the gauge symmetry of Electricity and Magnetism and the phase symmetry of electron wavefunctions are really one and the same. Neither the phase of the wavefunction nor the vector potential are directly observable, but the symmetry is.
We will not go over the consequences of gauge invariance again here, but, we do want to use gauge invariance to simplify our equations.
Maxwell's equation is
There is still substantial gauge freedom possible. The second derivative of is set by the Lorentz condition but there is still freedom in the first derivative which will modify . Gauge transformations can be made as shown below.
Jim Branson 2013-04-22