# Why is the electromagnetic four-potential $A_{\mu}$ not an observable?

Why within classical field-theory the electromagnetic four-potential (usually $A_{\mu}$) not an observable?

In classical mechanics we don't have problems with energy measurements and in quantum mechanics we talk all the time about the Hamiltonian which "is the observable of energy".

So why is $A_{\mu}$ not also considered as an observable? If we don't have a smart way to measure it how can we be 100% certain that tomorrow some dude won't figure out how to do it?

This confuses me greatly, especially with respect to the Aharonov–Bohm effect which is within quantum mechanics "a way to measure $A$".

The non-observable nature of $A$ seems to be the reason why we "gauge" it in order to do all kinds of stuff. That is why I'm interested in it.