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Equation 3 · Part 7 · Decoherence: The Quiet Selection That Makes the World Look Solid

superscript

ρ(x,x′,t)=ρ(x,x′,0) exp⁡ ⁣[−Λ(x−x′)2t]\rho(x, x', t) = \rho(x, x', 0)\, \exp\!\left[-\Lambda (x - x')^2 t\right]
superscript

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

Joos and Zeh made this quantitative in 1985 by deriving a genuine, non-phenomenological master equation for the reduced density matrix of an object’s centre-of-mass position under repeated recoil-free scattering [ 7 ] . For the off-diagonal elements of that density matrix — the very quantity whose survival or destruction is the entire question of macroscopic superposition — their result, in the short-time, many-collisions regime, takes the form of a simple exponential decay in the separation between the two positions being superposed: ρ(x,x′,t)=ρ(x,x′,0) exp⁡ ⁣[−Λ(x−x′)2t]\rho(x, x', t) = \rho(x, x', 0)\, \exp\!\left[-\Lambda (x - x')^2 t\right]. Here Λ\Lambda is what Joos and Zeh call the localization rate: a single number, with units of inverse length squared per unit time, built…

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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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