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b⋅v\mathbf b\cdot\mathbf v

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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

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Symbol v

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Published contexts (2)

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b⋅v\mathbf b\cdot\mathbf v

Equation 41 · Evolutionary Physics

The Clock That Comes Back Wrong by Exactly Its Mass

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

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

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b⋅v\mathbf b\cdot\mathbf v

Equation 69 · Evolutionary Physics

The Clock That Comes Back Wrong by Exactly Its Mass

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Before pointing this construction at an interferometer, one methodological trap needs naming. A single measured phase Φ\Phi at one fixed loop area cannot, by itself, distinguish m from an unknown additive offset baked into the apparatus — any constant phase shift from an uncalibrated pulse timing, an unaccounted path-length difference, or a stray field looks identical to a rescaling of b\mathbf b⋅\cdotv\mathbf v . The estimator that survives this is a derivative, not a snapshot:

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