Geometric phase in weak measurements

Core claim

The central claim is the identity $\arg P_w^B(A,C) = \Delta(A,B,C)$, where $P_w^B(A,C)=\langle C|B\rangle\langle B|A\rangle/\langle C|A\rangle$ is the weak value of the projector onto $B$ in an ensemble pre-selected in $|A\rangle$ and post-selected in $|C\rangle$, and $\Delta(A,B,C)=\arg\langle A|C\rangle\langle C|B\rangle\langle B|A\rangle$ is the Pancharatnam geometric phase of the geodesic triangle $A,B,C$. Multiplying numerator and denominator of the weak value by $\langle A|C\rangle$ leaves a positive real denominator, so the argument of the weak value is exactly the triangle phase. Combining this identity with the weak-measurement pointer shifts $\delta q = \kappa\sigma^2\,\mathrm{Im}\,P_w^B$ and $\delta p = -\hbar\kappa\,\mathrm{Re}\,P_w^B$ gives the main result $\Delta(A,B,C) = -\arctan\left(\frac{\hbar}{\sigma^2}\frac{\delta q}{\delta p}\right)$, with no need for a phase relation between $|A\rangle$ and $|C\rangle$. The paper also shows that the geometric phase of any discrete path $A_1,\dots,A_n$ is a sum of such triangle phases, and that the weak value diverges exactly when the post-selected state becomes orthogonal to the pre-selected state, which is precisely where the geometric phase is undefined.