substrate_healthy

formal source

  22  hamiltonian : Hamiltonian H
  23
  24/-- Substrate health corresponds to valid Hilbert space and unitary evolution. -/
  25def substrate_healthy (H : Type*) [RSHilbertSpace H] : Prop :=
  26  ∃ s : Substrate H, s.hamiltonian.toLinearOp.IsSelfAdjoint
  27
  28/-- Theorem: A healthy substrate exists for any valid Hilbert space. -/
  29theorem substrate_ok (H : Type*) [RSHilbertSpace H] : substrate_healthy H := by
  30  -- Use the existence theorem for normalized states
  31  have ⟨ψ, _⟩ := exists_normalized_state HS
  32  -- Construction of a default Hamiltonian (using identity as a bounded operator)
  33  let H_op : Hamiltonian HS := {
  34    toLinearOp := (identityObs HS).toLinearOp
  35    self_adjoint := (identityObs HS).self_adjoint
  36    bounded_below := ⟨1, fun ψ_norm => by
  37      -- ⟪I ψ, ψ⟫ = ‖ψ‖² = 1
  38      have h : (⟪(identityObs HS).op ψ_norm.vec, ψ_norm.vec⟫_ℂ) = (1 : ℂ) := by
  39        simp only [identityObs, ContinuousLinearMap.id_apply, inner_self]
  40        have h_norm := ψ_norm.norm_one
  41        -- ‖v‖ = 1 → ‖v‖² = 1
  42        have : (‖ψ_norm.vec‖ : ℂ) ^ 2 = (1 : ℂ) := by
  43          rw [h_norm]
  44          simp
  45        -- In inner product space, ⟪v, v⟫ = ‖v‖²
  46        rw [← @inner_self_eq_norm_sq ℂ]
  47        exact this
  48      rw [h]
  49      simp⟩
  50  }
  51  use ⟨ψ, H_op⟩
  52  exact (identityObs HS).self_adjoint
  53
  54end ILG
  55end Relativity