$$\Phi[\mathcal L_{\rm G}]$$

Equation 49 · 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.

Nothing about this phase disturbs conservation of energy or momentum, and it is worth saying explicitly why. $\Phi$[$\mathcal L_{\rm G}$] does not come from a new term added to any Hamiltonian; it comes from how already-unitary translation and boost operators compose. Each operator in the product T($\mathbf b$)G($\mathbf v$)T(-$\mathbf b$)G(-$\mathbf v$) conserves probability on its own, and the product of unitaries is unitary regardless of whether the factors commute, so the loop as a whole still conserves probability and leaves every expectation value of energy and momentum exactly where an ordinary, non-extended calculation would put it. What changes is not a conserved quantity but the bookkeeping of…

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Equation 83 · 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.

The Galilei loop admits an exact, non-simulated numerical check of the whole chain, because $\Phi$[$\mathcal L_{\rm G}$] is precisely the recoil phase every stimulated-Raman atom interferometer already uses for its own calibration. Take a sodium atom, m=3.8175$\times$10^{-26}\,$\mathrm{kg}$ , driven by a two-photon stimulated-Raman transition on the 589\,$\mathrm{nm}$ line, effective wavevector $k_{\rm eff}$=2(2$\pi$/589\,$\mathrm{nm}$)=2.1335$\times$10^{7}\,$\mathrm{m^{-1}}$ [ 16 ] . The associated recoil velocity is $v_r$=$\hbar$ $k_{\rm eff}$/m=5.894\,$\mathrm{cm\,s^{-1}}$ . Hold the loop open for T=50\,$\mathrm{ms}$ , comparable to the short interrogation times of early stimulated-Raman interferometers, so the…

Meanings in this article

• $\Phi$: the single measured phase.

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