# Need of vector potential in quantum mechanics

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:

1. 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.

2. 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.

• I can't comment on the mathematical requirements. I would certainly agree with your second point. The interesting question in my mind is, wether there is a classical effect that allows to detect the vector potential in the same way as the Bohm-Aharonov effect does? I am not aware of that at this moment. If there is none, then 2 is a unique requirement for the quantum case. If there is a classical effect, however, then the necessity for the vector potential just propagates into the quantum realm and the structural argument seems the stronger one. – CuriousOne Dec 13 '14 at 16:30
• – Qmechanic Dec 13 '14 at 16:32