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Is it possible to find a gauge in which the vector potential outside the solenoid (with axis along $z$-axis) is made equal to zero everywhere? If so, wouldn't the phase difference in the Aharanov-Bohm experiment be equal to zero in that gauge when two electron beams interfere anywhere on the screen?

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I did not follow the "holonomy bit" of the answer by Chiral Anomaly because I do not know the topological aspects of the AB effect. Leaving that aside, I get the sense from his answer that it is not possible to find a single gauge that would cover the entire XY plane. Please help me understand that.

Suppose, $\Phi_B$ denotes the flux through an infinitely long solenoid. A valid choice of the vector potential is $${\vec A}=\frac{\Phi_B}{2\pi r^2}(-y\hat{x}+x\hat{y}).$$ Now, choosing a scalar function $$\alpha(\vec r)=-\frac{\Phi_B}{2\pi}\phi,$$ (where $\phi$ is the angle in the XY plane in the plane polar coordinates, $0\leq \phi<2\pi$) and defining a new vector potential ${\vec A}'=A+\nabla\alpha$, we seem to trivially make ${\vec A}'$ vanish everywhere. I can understand that this is wrong because it makes ${\vec A}$ vanish everywhere and thus ${\vec B}$ too. Please point out what's wrong with this gauge.

Solidification
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This answer assumes the question is asking about an infinitely long solenoid, so that the magnetic field outside the solenoid is zero even though it's nonzero inside.

In any contractible region of space where the magnetic field is zero, we can choose the gauge so that the vector potential is zero. But the space outside an infinitely-long solenoid is not contractible. We can choose a gauge so that the vector potential is zero almost everywhere outside the solenoid, but it must remain nonzero somewhere so that the holonomy around the solenoid is nonzero. The holonomy is gauge-invariant, as is interference pattern produced when an electron passes around both sides of the solenoid.

Alternatively, we could cover the space outside the solenoid with two overlapping patches, each of which is contractible by itself. Then we can use different gauge transformations in each patch to make the respective vector potentials zero in each patch. The nonzero holonomy is encoded in the transition functions that relate the vector fields in the two patches where they overlap.

Related: Small confusion about the Aharonov-Bohm effect

Chiral Anomaly
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  • I have edited the question to offer a gauge in which the transformed vector potential is made to vanish. What's wrong with this gauge? – Solidification Mar 12 '21 at 05:39
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    @mithusengupta123 The angle $\phi$ is either multivalued or non-differentiable. If you walk continuously around the solenoid, $\phi$ either changes by $2\pi$ (multivalued) or jumps discontinuously by $2\pi$ (non-differentiable). When we talk about gauge transformations, it's understood that we're using single-valued differentiable functions. – Chiral Anomaly Mar 12 '21 at 14:28
  • @mithusengupta123 In this example, that's why we need to use more than one patch: the "transition function" I mentioned relates a single-valued $\phi$ in one patch to a single-valued $\phi$ in the other. Or, if we allow $\phi$ to jump discontinuously, then $A$ is nonzero where it jumps, which is a singular limit of the first case I mentioned. – Chiral Anomaly Mar 12 '21 at 14:28
  • So the point is, in short, we have to define ${\vec A}$ differently in two different patches such that it is single-valued and differentiable everywhere. I have one more question. I think what makes the space outside the solenoid non-contractible is not the presence of the solenoid itself (i.e. with zero current) but the presence of a solenoid carrying a nonzero current. But I am not able to justify this. Can you help? – Solidification Mar 15 '21 at 14:44
  • @mithusengupta123 When I referred to the space outside the solenoid, that's all I meant: the part of space that excludes where the solenoid is. That part of space is non-contractible whether or not a current is flowing in the solenoid, and whether or not fields are present or absent anywhere. It's just geometry, not physics. But the reason we choose to consider that part of space is that when a current is flowing, the magnetic field is only zero outside the solenoid, and that's where we're trying to transform the vector potential to zero. – Chiral Anomaly Mar 15 '21 at 17:11