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Let me clarify my problem, we all know what Aharonov-Bohm effect is, where we treat the vector potential $A_\mu$ as a classical field.

Calculate this effect under classical quantum mechanics using wave function is easy, but how should I explain the "phase cumulation" after I quantize $A_\mu$, and calculate it by QED?

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Because E = B = 0 outside of an infinite solenoid the associated Aharanov-Bohm vector potential is unaffected by EM quantization performed in the Coulomb gauge, leaving the traditional single-particle Schrodinger / QM analysis intact.

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