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Published equation contexts

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}

Why this formula appears here

Call Φ\Phi[LG\mathcal L_{\rm G}]=m\,b\mathbf b⋅\cdotv\mathbf v/ℏ\hbar the loop’s phase debt : what the representation charges for a loop the classical group considered free. The loop invariant b\mathbf b⋅\cdotv\mathbf v carries units of m2 s−1\mathrm{m^2\,s^{-1}} , 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…

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mopm_{\rm op}

Symbol m_rm op

mrm_rm op is part of the quantity the equation computes from the expression on the right.

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LGL_{\rm G}

Symbol L_rm G

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

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vv

Symbol v

v occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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ℏ Φ[LG]\hbar\,\Phi[\mathcal L_{\rm G}]

Numerator: hbarPhi[mathcal L_rm G]

The complete quantity above the fraction bar.

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b⋅v\mathbf b\cdot\mathbf v

Denominator: mathbf b × mathbf v

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

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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}.

Equation 45 · Evolutionary Physics

The Clock That Comes Back Wrong by Exactly Its Mass

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Call Φ\Phi[LG\mathcal L_{\rm G}]=m\,b\mathbf b⋅\cdotv\mathbf v/ℏ\hbar the loop’s phase debt : what the representation charges for a loop the classical group considered free. The loop invariant b\mathbf b⋅\cdotv\mathbf v carries units of m2 s−1\mathrm{m^2\,s^{-1}} , 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…

Meanings in this article

  • Φ\Phi: dimensionless as a phase must be, and mass is recoverable from it as [displayed formula].
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