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Equation 37 · Part 3 · A Horizon Is a Toll Booth, Not a Loophole

Symbol H

∣ddτln⁡ ⁣[⟨H(τ)⟩−E0(τ)]∣≪2[⟨H(τ)⟩−E0(τ)]πℏ.\left|\frac{d}{d\tau}\ln\!\left[\langle H(\tau)\rangle - E_0(\tau)\right]\right| \ll \frac{2\left[\langle H(\tau)\rangle-E_0(\tau)\right]}{\pi\hbar}.
HH

What this part means

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

Its job in the formula

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

The passage around this formula

…for this paper flags something the two equations above do not, by themselves, settle: the periodic-evolution bound above was derived for a Hamiltonian with a genuinely time-independent spectrum. A real computation drives H(τ\tau) through a sequence of different instantaneous Hamiltonians, one per gate, and nothing guarantees in advance that a chain of instantaneously applied bounds sums to a valid bound on the whole chain. The condition that licenses treating N⊥N_\perp[Γ\Gamma] as a legitimate partition into…

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A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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Sources cited in the surrounding passage

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