Equation 36 · Part 2 · A Horizon Is a Toll Booth, Not a Loophole

Symbol τ

\tau_{\rm ML}(\tau) = \pi\hbar/[2(\langle H(\tau)\rangle-E_0(\tau))]

\tau

What this part means

a time, the integral has action units.

Its job in the formula

τ is part of the quantity the equation computes from the expression on the right.

Full expressionSymbol τArticle meaning

Where the article explains it

$N_\perp$ [ $\Gamma$ ] is a real-valued, non-negative functional of a timelike worldline in a fixed background spacetime together with a stated local energy convention; because the integrand is an energy and d $\tau$ is a time, the integral has action units and $N_\perp$ is dimensionless.

The passage around this formula

The dossier 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( $\tau$ ) 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 $N_\perp$ [ $\Gamma$ ] as a legitimate partition into instantaneously static Levitin-Toffoli steps is a quasi-static one: over one local orthogonalization time $\tau_{\rm ML}(\tau)$ = $\pi$ $\hbar$ /[2( $\langle$ H( $\tau$ ) $\rangle$ - $E_0(\tau)$ )] ,…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the article section

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