However, as we shall see in chapter 9, the coupling of the electromagnetic field to fundamental charged matter (namely, charged fields) can be described only in terms of the potentials, not the field strengths. Furthermore, there are physically relevant situations where ##\vec E## and ##\vec B## do not contain all of the information about the electromagnetic field.
As an example, consider the region outside an infinite solenoid. Suppose that inside the solenoid, there is a nonvanishing, uniform magnetic field, but outside the solenoid, we have ##\vec E = \vec B = 0##. Since the region outside the solenoid is not simply connected, the fact that ##\vec E## and ##\vec B## vanish in that region does not imply that the potentials are gauge equivalent to zero there. Indeed, eq. (1.7) implies, via Stokes’s theorem, that when ##\vec B\ne 0## inside the solenoid, we have ##\oint \vec A \cdot d \vec l \ne 0## for any loop outside the solenoid that encloses it. (Note that ##\oint \vec A \cdot d \vec l## is gauge invariant, i.e., its value does not change under eq. (1.13).)
A quantum mechanical charged particle that stays entirely outside the solenoid will be affected by this vector potential, as it will produce a relative phase shift in the parts of the wave function that go around the solenoid in different directions, producing a physically measurable shift in the resulting interference pattern. This phenomenon, known as the Aharonov-Bohm effect, is sometimes attributed to the weirdness of quantum mechanics.
However, the effect has nothing to do with quantum mechanics - the same effect would occur for a classical charged field. And there is nothing weird about the effect, once one recognizes that the electromagnetic field is represented, at a fundamental level, by the potentials ##\phi, \vec A## (modulo gauge), not the field strengths ##\vec E, \vec B##.