I'm currently reading the book Quantum Paradoxes: Quantum Theory for the Perplexed by Aharonov and Rohrlich and in chapter 4 they show the existence of AB effect. I'm already familiar with the explanation through path integral, but they do it in a different manner.
First, they show how the wave function changes under gauge transformation. I have no problem to understand the electric version of the effect. But, in the magnetic case, they consider the following:
- We have a conducting cylinder rotating with constant angular velocity, with a charged wire inside. The magnetic field outside cylinder is zero, but since we have magnetic flux inside it, the Vector potential cannot be zero everywhere. We can find a vector potential for this setup as $$ \vec A = {\Phi_B \over 2\pi} \nabla \theta $$ and this vector potential can be obtained through a gauge transform of null field $\vec A = 0$ by the gauge function $\Lambda = {\Phi_B\over 2\pi} \theta$.
- Since the function $\Lambda$ is not single-valued, it is not a proper gauge transformation everywhere. But being multi-valued, we can use it as a gauge transform in separate regions, so they apply it in left side and right side of cylinder.
- if $|\Psi_L^0\rangle$ is the wave function in the left side without vector potential, with the vector potential it would be $$ |\Psi_L\rangle = e^{ie\Phi_B\theta/2\pi\hbar c} |\Psi_L^0\rangle $$
- by the same reasoning, if $|\Psi_R^0\rangle$ is the wave function in the right side without vector potential, with the vector potential it would be $$ |\Psi_R\rangle = e^{ie\Phi_B\theta/2\pi\hbar c} |\Psi_R^0\rangle $$
- if the state before the solenoid is $1/\sqrt{2}(|\Psi_L^0\rangle+|\Psi_R^0\rangle)$, after it would be
$$ {1\over \sqrt{2}}(|\Psi_L^0\rangle+e^{ie\Phi_B/\hbar c}|\Psi_R^0\rangle) $$ Ignoring an overall phase and considering that the phase difference between left and right partial waves are $2\pi$.
My question is: why the phase difference between left and right waves are $2\pi$, since it seems that the gauge transformation seems to be the same for both functions?
I thought to consider the multi-valued function $\theta$ as the explanation, but if we define theta being zero in the left side, it will value $\pi$ in the right side, not $2\pi$, as claimed in the last point.
I also thought that there is a typo when defining the gauge transformation for the left and right sides, but if it's the case, how should it be the right transformation?
