Equation 37 · Part 1 · The Entry a Relabeling Cannot Write

Symbol W

W

W

What this part means

the dual of conjugation by.

Its job in the formula

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

Full expressionSymbol WArticle meaning

Where the article explains it

The dual of conjugation by W is conjugation by $W^\dagger$ : from the defining pairing, $\mathrm{Tr}$ [W $\rho$ $W^\dagger$ $O_c$ ] = $\mathrm{Tr}$ [ $\rho$ $W^\dagger$ $O_c$ W] holds for every $\rho$ , so $\Lambda^\dagger(O_c)$ = $W^\dagger$ $O_c$ 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 $\mathcal H_c$ = $\mathcal H_f$ — 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 $\Lambda(\rho)$ := W $\rho$ $W^\dagger$ 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, $H_c$ := W $H_f$ $W^\dagger$ .

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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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