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Hc:=WHfW†H_c := W H_f W^\dagger

Why this formula appears here

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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Hc:=WHfW†H_c := W H_f W^\dagger

Equation 38 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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 .

Meanings in this article

  • WW: the dual of conjugation by.
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