Numerator: im

The passage around this formula

That triviality is what makes the loop phase exact rather than approximate. Write X=-i$\mathbf b$$\cdot$$\mathbf P$/$\hbar$ and Y=-i$\mathbf v$$\cdot$$\mathbf K$/$\hbar$ . Their commutator is $[X,Y]=\frac{im}{\hbar}\,\mathbf b\cdot\mathbf v\,\mathbb 1$. Because [X,Y] is itself proportional to the identity, it commutes with both X and Y , so every higher term in the Baker–Campbell–Hausdorff expansion of $e^X$$e^Y$$e^{-X}$$e^{-Y}$ vanishes identically, at all orders, with no small-loop approximation required:

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