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Hc=Hf\mathcal H_c = \mathcal H_f

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=Hf\mathcal H_c = \mathcal H_f

Equation 35 · 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 .

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Hc=Hf\mathcal H_c = \mathcal H_f

Equation 51 · 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.

Passive relabeling is one degenerate case; the other is simpler still. Let Λ\Lambda be the identity channel, Hc\mathcal H_c = Hf\mathcal H_f , HcH_c = HfH_f : no relabeling, no discarding, the coarse description simply is the fine one. Then trivially DΛD_\Lambda = 0 . Nothing in the paragraphs above required that; it follows on inspection. But the triviality is the point, because this is the operator-language shadow of the oldest and most secure fact the subject has. A system evolving under its own Hamiltonian, with nothing coarse-grained away and nothing relabeled, conserves that Hamiltonian’s own expectation value exactly, because the Hamiltonian generates its own time evolution and commutes with…

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Hc=Hf\mathcal H_c = \mathcal H_f

Equation 106 · 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.

A third regime sits between the first two and needs its own treatment rather than being folded into either. Nothing here is discarded — Hc\mathcal H_c = Hf\mathcal H_f still — but the relabeling itself now carries explicit time dependence, because the “coarse” observer describes the same system from a frame in motion relative to the frame that defines HfH_f : a rotating platform, a driven interaction picture, a magnet ramping in time. Let U(t) be the corresponding time-dependent unitary and |ψ\psi'(t)⟩\rangle := U(t)|ψ(t)\psi(t)⟩\rangle , with |ψ(t)\psi(t)⟩\rangle solving iℏ\hbar\, d|ψ\psi⟩\rangle/dt = HfH_f|ψ\psi⟩\rangle . Differentiating the product and using U†U^\dagger U = I gives

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