Symbol i

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Represent the loop $\mathcal L_{\rm G}$ on a Hilbert space. A spatial translation by $\mathbf b$ is the unitary T($\mathbf b$)=$\exp(-i\mathbf b\cdot\mathbf P/\hbar)$ , built from the momentum operator $\mathbf P$ ; a Galilei boost by $\mathbf v$ is G($\mathbf v$)=$\exp(-i\mathbf v\cdot\mathbf K/\hbar)$ , built from the boost generator $\mathbf K$ . The classical loop is T($\mathbf b$)G($\mathbf v$)T(-$\mathbf b$)G(-$\mathbf v$)=$\mathbb 1$ . The operator loop is not automatically the identity, because $\mathbf K$ and $\mathbf P$ need not commute as operators even when translations and boosts commute as group elements.

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