The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region inaccessible to the particle. Classically, such a nonlocal effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle and the electromagnetic vector potential
A, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics
A is in some sense more ``fundamental'' than
B in spite of the former being gauge dependent and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB effect can be fully accounted for by considering only the gauge-invariant
B field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge-invariant formulation is that we must give up locality---the AB phase for the particle will arise from the change in the action for the
B field in the region inaccessible to the particle.
The electric Aharonov-Bohm effect is usually explained by means of the scalar potential, which is present in a multiply connected region of space time where no electric field acts on the charged particle. An alternative interpretation is given in terms of the particle's polarization field penetrating the region occupied by the electric field but inaccessible to the particle itself. Similarly, the magnetic Aharonov-Bohm effect is usually explained by means of the vector potential, which is present in a multiply connected region of space where no magnetic-induction field acts on the charged particle. An alternative interpretation of this effect is given in terms of the particle's magnetization field penetrating the region occupied by the magnetic-induction field but inaccessible to the particle itself. Our interpretations of the electric and magnetic Aharonov-Bohm effects are both developed by using the multipolar form of the Lagrangian instead of the minimal-coupling form in the Feynman path integral.