multiplication

The passage around this formula

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: $T(\mathbf b)\,G(\mathbf v)\,T(-\mathbf b)\,G(-\mathbf v)=e^{[X,Y]}=\exp\!\left(\frac{im}{\hbar}\,\mathbf b\cdot\mathbf v\right)\mathbb 1$. The structure is the same one that makes an Aharonov–Bohm phase or a Berry-phase holonomy nonzero: a closed path in a classical parameter space picks up a phase set by a curvature that lives on that space, even though nothing classical distinguishes the endpoint from the start. Here the “curvature” is the commutator [$K_i$,$P_j$] itself, and the “charge” that couples to it is mass. The analogy is a guide to intuition, not a…

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