Equation 42 · Part 1 · The Clock That Comes Back Wrong by Exactly Its Mass

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\mathrm{m^2\,s^{-1}}

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What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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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

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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