Aharonov–Bohm effect

Remarks

By Stokes' theorem, the phase is $q \Phi / \hbar$, where $\Phi$ is the magnetic flux through any surface spanned by $\gamma$, including the flux in the excluded region. Since $\nabla \times A = 0$, the phase depends only on the homotopy class of $\gamma$ in $\Omega$; it is the holonomy of the flat $U(1)$ connection given by $A$. A gauge change $A \mapsto A + \nabla \chi$ does not change it modulo $2\pi$ when $e^{iq\chi/\hbar}$ is single-valued.

Example: an infinite solenoid of radius $a$ along the $z$-axis carrying the flux $\Phi$. Outside it, in cylindrical coordinates, $A = \frac{\Phi}{2 \pi r} e_\theta$ for $r > a$. Then $(\nabla \times A)_z = \frac{1}{r} \frac{d}{dr} \left( r \cdot \frac{\Phi}{2 \pi r} \right) = 0$, while a loop once around the axis gives $\oint A \cdot dl = \frac{\Phi}{2 \pi r} \cdot 2 \pi r = \Phi$ and the phase $q \Phi / \hbar$.

Non-example: if $\Omega$ is simply connected, then $A = \nabla \chi$ on $\Omega$ for a function $\chi$, every loop integral vanishes, and there is no effect. The effect also vanishes when $q \Phi / \hbar$ is an integer multiple of $2\pi$.