Electromagnetism is called a gauge theory because the gauge symmetry actually defines the theory. It turns out that the weak and the strong interactions are also gauge theories and, in some sense, have the next simplest possible gauge symmetries after the one in Electromagnetism. We will write our standard gauge transformation in the traditional way to conform a bit better to the textbooks. There are measurable quantum physics consequences of this symmetry. We can derive the quantization of magnetic flux by calculating the line integral of in a field free region.
Magnetic flux is excluded from the superconducting region.
We have shown then, that the flux going through a ring of superconductor is quantized. Flux is observed to be quantized but the charge of the particle seen is 2 e . This is due to the pairing of electrons inside a superconductor. The Aharanov Bohm Effect brings us back to the two slit diffraction experiment but adds magnetic fields. The electron beams travel through two slits in field free regions but we have the ability to vary a magnetic field enclosed by the path of the electrons.