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Equation 36 · Part 3 · The Entry a Relabeling Cannot Write

Symbol W

Λ(ρ):=WρW†\Lambda(\rho) := W\rho W^\dagger
WW

What this part means

the dual of conjugation by.

Its job in the formula

W is an input to the expression that computes the quantity on the left.

Where the article explains it

The dual of conjugation by W is conjugation by W†W^\dagger : from the defining pairing, Tr\mathrm{Tr}[Wρ\rho W†W^\dagger OcO_c] = Tr\mathrm{Tr}[ρ\rho W†W^\dagger OcO_c W] holds for every ρ\rho , so Λ†(Oc)\Lambda^\dagger(O_c) = W†W^\dagger OcO_c W .

The passage around this formula

Before asking what a genuine coarse-graining exposes, the construction has to survive the case where nothing at all is discarded. Take Hc\mathcal H_c = Hf\mathcal H_f — no reduction in the space of states — and let the “coarse” description be nothing more than the same physics seen through a fixed, time-independent relabeling: a rotated spin basis, a renamed pair of levels, a change of which linear combination of states a lab calls “up.” Model this as Λ(ρ)\Lambda(\rho) := Wρ\rho W†W^\dagger for a fixed unitary W , and, because nothing has been thrown away, define the coarse Hamiltonian self-consistently as the same physical generator carried through that same relabeling, HcH_c := W HfH_f W†W^\dagger .

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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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