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Equation 45 · Part 4 · The Clock That Comes Back Wrong by Exactly Its Mass

Symbol b

mop[LG]=ℏ Φ[LG]b⋅v.m_{\rm op}[\mathcal L_{\rm G}]=\frac{\hbar\,\Phi[\mathcal L_{\rm G}]}{\mathbf b\cdot\mathbf v}.
bb

What this part means

b is one factor in the product that computes the quantity on the left.

Its job in the formula

b is one factor in the product that computes the quantity on the left.

The passage around this formula

…, exactly the units of ℏ\hbar/mass\text{mass} , so Φ\Phi is dimensionless as a phase must be, and mass is recoverable from it as mop[LG]=ℏ Φ[LG]b⋅vm_{\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 b\mathbf b,v\mathbf v are the loop’s own declared parameters, mopm_{\rm op}[LG\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 positions and velocities, but it does not change which…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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