$$\langle H\rangle$$

Equation 54 · 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.

where H is the total energy operator [ 2 ] . The right-hand side is not proportional to the identity. It is proportional to H , an operator with a spectrum, different on every energy eigenstate. Repeating the loop calculation of the previous section on a state with sharp energy $\langle$ H$\rangle$ gives a phase $\langle$ H$\rangle$\,$\mathbf b$$\cdot$$\mathbf v$/($\hbar$ $c^2$) that varies from state to state. A central extension, by definition, must give the same phase to every vector in the representation; a state-dependent phase is not a central extension, it is an ordinary consequence of ordinary dynamics, and it can be removed by working with true, non-projective unitary representations of the Poincaré…

Meanings in this article

• $H$: the total energy operator [ 2 ].

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Equation 10 · Evolutionary Physics

A Horizon Is a Toll Booth, Not a Loophole

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

where $\langle$ H$\rangle$ is the mean energy in the evolving state and $E_0$ is the ground-state energy of H [ 2 , 1 ] . Inverting this gives a maximum rate of orthogonal state changes per unit proper time, 2($\langle$ H$\rangle$-$E_0$)/$\pi$$\hbar$ , and Giovannetti, Lloyd, and Maccone showed that for a periodic evolution this rate integrates cleanly: a system with mean energy $\langle$ H$\rangle$-$E_0$ above its ground state can pass through at most

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