= Discrete gauge cancellation in two-path interference
{title2=$W_C=\prod_{(i,j)\in C}U_{ij}$}
A four-vertex model proves the essential <gauge invariance> without solving an <Electron> scattering problem. Use two directed paths $S\to U\to D$ and $S\to L\to D$. Assign an edge phase $a_{ij}$ and a complex unit link $U_{ij}=e^{ia_{ij}}$ to each directed edge, with $U_{ji}=U_{ij}^{-1}$. These links represent discretized magnetic <line integrals> multiplied by $q/\hbar$. Under a <gauge transformation> with arbitrary vertex phases $\chi_i$,
$$
a'_{ij}=a_{ij}+\chi_j-\chi_i,\qquad
P_1=U_{SU}U_{UD},\qquad P_2=U_{SL}U_{LD}.
$$
Both path products acquire the same factor $e^{i(\chi_D-\chi_S)}$. Their <gauge-invariant> ratio is the closed-loop <holonomy>
$$
W_C=P_1P_2^*=\exp\!\left(i[a_{SU}+a_{UD}-a_{SL}-a_{LD}]\right)=e^{i\delta}.
$$
Every intermediate vertex phase cancels. For any fixed nonnegative real amplitudes $a,b$, the <interference> quantity $|aP_1+bP_2|^2$ is unchanged too. A gauge-dependent open-path phase consequently cannot be interpreted as an observable by itself.
For an exact example, measure phases in units of $\pi$ and order edges as $(SU,UD,SL,LD)$. Choose edge phases $(0,0,0,1/2)$ and vertex phases $(\chi_S,\chi_U,\chi_L,\chi_D)=(1/3,-1/4,2/5,1/7)\pi$. The transformed edge phases are
$$
(a'_{SU},a'_{UD},a'_{SL},a'_{LD})
=\left(-\frac7{12},\frac{11}{28},\frac1{15},\frac{17}{70}\right)\pi.
$$
The upper and lower sums become $-4\pi/21$ and $13\pi/42$; their difference remains $-\pi/2$. With amplitudes $a=1$, $b=1/2$, the intensity remains $5/4$. Sweeping one physical loop phase through $2\pi$ instead produces extrema $9/4$ and $1/4$, agreeing with the <interferometric visibility> example above.
\Image[/aharonov-bohm-flux-fringes-and-gauge-cancellation.png]
{title=Flux moves fringes while vertex gauge phases cancel}
{description=Original scalar two-path calculation with $I_1=1$ and $I_2=1/4$. Left: carrier fringes at three chosen enclosed fluxes, in units of the <Electron> period $h/e$. Right: the exact four-edge example before and after a vertex <gauge transformation>; open-path phase sums change together and their difference is fixed. Curves are illustrative model intensities, not an experimental interferogram.}
{source=aharonov-bohm-flux-fringes-and-gauge-cancellation.py}
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