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It is only a remark about the interrogations contained in the second part of your question. You are mixing closed paths and open paths. For any closed path $C$ around the solenoid (taking $c = \hbar=1$ units), the phase difference is $\Delta \phi_C = q \, \Phi_B$, and this is independent of the choosen closed path (homotopy invariance). Separating the ...

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