Magnetic Flux Quantization from Gauge Symmetry

For simplicity, let's develop equations for regions with zero B field. We can choose a gauge transformation so that $\vec{A}'=0$ and hence $\vec{A}=\vec{\nabla}f$. This will only be physically interesting for $\vec{B}$ nonzero in other regions.

$$\bgroup\color{black}\int\limits_{\vec{r}_0}^{\vec{r}}d\vec{r}... ...c{r}}d\vec{r}\cdot\vec{\nabla}f =f(\vec{r})-f(\vec{r_0})\egroup$$

If we choose $f$ so that $f(\vec{r_0})=0$, then we have

$$\bgroup\color{black}f(\vec{r})=\int\limits_{\vec{r}_0}^{\vec{r}}d\vec{r}\cdot\vec{A}\egroup$$

Now consider two different paths from $\vec{r}_0$ to $\vec{r}$.

$$\bgroup\color{black}f_1(\vec{r})-f_2(\vec{r})=\oint d\vec{r}\... ...vec{\nabla}\times\vec{A} =\int d\vec{S}\cdot\vec{B}=\Phi\egroup$$

Now $f$ is not a physical observable so the $f_1-f_2$ does not have to be zero, but, $\psi$ does have to be single valued.

$$\begin{eqnarray*} & \psi_1=\psi_2 \\ & \Rightarrow e^{-i{e\over\hbar c}f_1}=e... ...f_2)=2n\pi \\ & \Rightarrow \Phi=f_1-f_2={2n\pi\hbar c\over e} \end{eqnarray*}$$

Magnetic flux is observed to be quantized in a region enclosed by a superconductor. however, the fundamental charge seen is $2e$.