Numerator: hbarPhi[mathcal L_rm G]

The passage around this formula

Call $\Phi$[$\mathcal L_{\rm G}$]=m\,$\mathbf b$$\cdot$$\mathbf v$/$\hbar$ the loop’s phase debt : what the representation charges for a loop the classical group considered free. The loop invariant $\mathbf b$$\cdot$$\mathbf v$ carries units of $\mathrm{m^2\,s^{-1}}$ , exactly the units of $\hbar$/$\text{mass}$ , so $\Phi$ is dimensionless as a phase must be, and mass is recoverable from it as $m_{\rm op}[\mathcal L_{\rm G}]=\frac{\hbar\,\Phi[\mathcal L_{\rm G}]}{\mathbf b\cdot\mathbf v}$. Because m enters only through the fixed central charge and $\mathbf b$,$\mathbf v$ are the loop’s own declared parameters, $m_{\rm op}$[$\mathcal L_{\rm G}$] does not depend on which inertial frame is used to describe the apparatus from outside: boosting the whole laboratory changes how an external observer labels…

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