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Equation 5 · Part 6 · The Bit Comes Back Before the Bearing

superscript

∣ψ(g)⟩=e−igJ^/ℏ ∣ψ(0)⟩,g∈[0,2π).|\psi(g)\rangle = e^{-ig\hat J/\hbar}\,|\psi(0)\rangle, \qquad g \in [0,2\pi).
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What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

The passage around this formula

Into this old hole go two systems. The first, A , is a two-dimensional qubit prepared in an arbitrary, unknown state and coupled to nothing the hole’s own symmetries track — a spectator, chosen so that its recovery is the cleanest possible test of the Hayden-Preskill mechanism on its own terms. The second, C , is a compass: a physical carrier prepared in a state |ψ(g)\psi(g)⟩\rangle that transforms under a one-parameter group of rotations generated by a Hermitian charge J^\hat J , the same charge the hole’s own angular momentum is built from, ∣ψ(g)⟩=e−igJ^/ℏ ∣ψ(0)⟩,g∈[0,2π)|\psi(g)\rangle = e^{-ig\hat J/\hbar}\,|\psi(0)\rangle, \qquad g \in [0,2\pi). g is an angle: dimensionless, periodic, and physically read as “which way the compass points” relative to the hole’s spin axis. J^\hat J…

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An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the surrounding passage

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