# What is the importance of vector potential not being unique?

For a magnetic field we can have different solutions of its vector potential. What is the physical aspect of this fact? I mean, why the nature allows us not to have an unique vector potential of a field?

Note that even the effect that is usually cited as showing the potentials being "physical", the Aharanov-Bohm effect, does not make the potential unique. The quantity that is relevant is the integral of the vector potential $A$ along a closed loop $\gamma$, and if we denote the region inside $\gamma$ as $U$, we have $\int_\gamma A = \int_U B$ by Stokes' theorem, so what really matters here is the flux through the loop, not the specific value of the potential. And one has to close the loop to observe a phase difference (or, well, maybe not always, but the phase is still only dependent on the flux, not on a gauge-variant potential value). In any case, this is a quantum effect. In the classical theory, the potential is definitely not "physical" in the sense of being measureable.