I need your opinions. Why is the vector potential of a magnetic field important (or even necessary) to quantum mechanics? Why it has to be defined everywhere? Is there any fundamental reason you can think of?
My point of view:
If one wants to have gauge invariance then he needs to have a vector potential. In fact if we construct the QED Lagrangian as being invariant under U(1), then the vector potential is naturally introduced by the covariant derivative. Additionally, the gauge field is a dynamical variable which needs to be defined everywhere.
The Bohm-Aharonov effect shows that the vector potential (but not the gauge choice) is detectable and affect the probability distribution. Hence the vector potential is in some sense physical and need to be defined everywhere.