Equation 45 · Part 3 · The Clock That Comes Back Wrong by Exactly Its Mass

Symbol Phi

m_{\rm op}[\mathcal L_{\rm G}]=\frac{\hbar\,\Phi[\mathcal L_{\rm G}]}{\mathbf b\cdot\mathbf v}.

\Phi

What this part means

dimensionless as a phase must be, and mass is recoverable from it as [displayed formula].

Its job in the formula

Phi occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Full expressionSymbol PhiArticle meaning

Where the article explains it

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 $m_{\rm op}[\mathcal L_{\rm G}]=\frac{\hbar\,\Phi[\mathcal L_{\rm G}]}{\mathbf b\cdot\mathbf v}$ .

The passage around this formula

…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…

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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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