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Equation 12 · Decoherence: The Quiet Selection That Makes the World Look Solid

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Δx≈10−3\Delta x \approx 10^{-3}

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Δx\Delta x

Symbol Δ x

Δ x is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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≈

≈

Approximately equal to; the equality is not exact.

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change

change

Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.

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superscript

superscript

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.

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Reading a decoherence time off this table only requires the exponent above: a superposition separated by Δ\Delta x decays with a characteristic time t ∼\sim 1/(Λ\Lambda\, Δ\Delta x2x^2) . Take the largest grain delocalized over roughly its own diameter, Δ\Delta x ≈\approx 10^{-3} centimetres, and the numbers in the first column turn into times measured in seconds only in the single, almost pathologically clean case of the cosmic background radiation acting alone — about one second. Every other row is not a slower version of the same story; it is a different order of magnitude of story entirely. Ordinary air brings that same superposition down to roughly 10^{-30} seconds, and a superposition…
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Reading a decoherence time off this table only requires the exponent above: a superposition separated by Δ\Delta x decays with a characteristic time t ∼\sim 1/(Λ\Lambda\, Δ\Delta x2x^2) . Take the largest grain delocalized over roughly its own diameter, Δ\Delta x ≈\approx 10^{-3} centimetres, and the numbers in the first column turn into times measured in seconds only in the single, almost pathologically clean case of the cosmic background radiation acting alone — about one second. Every other row is not a slower version of the same story; it is a different order of magnitude of story entirely. Ordinary air brings that same superposition down to roughly 10^{-30} seconds, and a superposition delocalized over a full centimetre — genuinely macroscopically distinct positions — decoheres in air in roughly 10^{-36} seconds, a span of time with no operational meaning at all next to any timescale on which a physicist, or a chair, does anything. Even Joos and Zeh’s own “large molecule” column, sized for something in the mass range of a C60 cage rather than a visible speck of dust, still shows scattering off ordinary air overwhelming scattering off the cosmic background by thirty orders of magnitude — which is exactly why the 1999 and 2019 interferometry results opening this article had to be run in high vacuum, on a molecular beam, and nowhere near open air [ 7 ] [ 9 ] [ 10 ] . Joos and Zeh’s own language for this asymmetry is understated: an equilibrated environment such as thermal radiation “is able to destroy (or dislocalize) interference terms,” and the mechanism at work is “the sensitivity of macroscopic systems upon their environment” [ 7 ] . The arithmetic is what makes that sentence load-bearing rather than merely qualitative.

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