For simplicity, let's develop equations for regions with zero B field . We can choose a gauge transformation so that and hence . This will only be physically interesting for nonzero in other regions. If we choose f so that , then we have Now consider two different paths from to . Now f is not a physical observable so the f 1 - f 2 does not have to be zero, but, does have to be single valued.
Magnetic flux is observed to be quantized in a region enclosed by a superconductor.