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Equation 28 · Part 2 · The Entry a Relabeling Cannot Write

subscript

HfH_f
subscript

What this part means

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

The passage around this formula

is a real number in units of energy for every fine state ρ\rho : how much the coarse description, evaluated through HcH_c pulled back along Λ\Lambda , disagrees with the fine description’s own HfH_f , for that specific state. Because expectation values determine a Hermitian operator uniquely — two Hermitian operators with the same trace against every density matrix are the same operator — the strongest statement follows at once: Δ\Delta EΛ(ρ)E_\Lambda(\rho) = 0 for every fine state ρ\rho if and only if DΛD_\Lambda = 0 as an operator identity, that is, if and only if Λ†(Hc)\Lambda^\dagger(H_c) = HfH_f exactly. Energy is preserved for every possible input under that one algebraic condition, never as a statistical…

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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

Open the illustrated subscripts: which member of a family? guide →

Sources cited in the article section

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