Equation 36 · The Entry a Relabeling Cannot Write
What does this equation mean?
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol Lambda
Lambda is part of the quantity the equation computes from the expression on the right.
Symbol ρ
ρ is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
Symbol W^dagger
agger is an input to the expression that computes the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
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What the article says around this equation
Before asking what a genuine coarse-graining exposes, the construction has to survive the case where nothing at all is discarded. Take = — 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 := W 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, := W .
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