minimal_area_eigenvalue

formal source

  77
  78/-- **THEOREM: Minimal Area Eigenvalue**
  79    The minimal non-zero eigenvalue of the area operator is exactly \ell_0^2. -/
  80theorem minimal_area_eigenvalue (h : H_AreaQuantization H A ψ) (H : Type*) [RSHilbertSpace H] (A : AreaOperator H) :
  81    ∃ (a_min : ℝ), a_min = ell0^2 ∧
  82    (∀ ψ : H, is_ledger_eigenstate H ψ →
  83      let eigenvalue := (⟪ψ, A.op ψ⟫_ℂ).re;
  84      eigenvalue ≠ 0 → eigenvalue ≥ a_min) := by
  85  use ell0^2
  86  constructor
  87  · rfl
  88  · intro ψ h_eigen eigenvalue h_nz
  89    -- Use the quantization theorem to show ⟨ψ, Aψ⟩ = n * ell0^2
  90    obtain ⟨n, h_quant⟩ := area_quantization h H A ψ h_eigen
  91    have h_val : eigenvalue = n * ell0^2 := by
  92      unfold eigenvalue
  93      rw [h_quant]
  94      simp only [Complex.mul_re, Complex.natCast_re, Complex.natCast_im,
  95                 zero_mul, sub_zero, Complex.ofReal_re]
  96    -- Since eigenvalue ≠ 0 and n is Nat, n must be ≥ 1
  97    have h_n_nz : n ≠ 0 := by
  98      intro h_zero
  99      subst h_zero
 100      simp [h_val] at h_nz
 101    have h_n_pos : 1 ≤ n := Nat.pos_of_ne_zero h_n_nz
 102    -- n * ell0^2 ≥ 1 * ell0^2
 103    have h_ell0_sq_pos : 0 < ell0^2 := pow_pos lambda_rec_pos 2
 104    have h_cast : (1 : ℝ) ≤ (n : ℝ) := by norm_cast
 105    have h_le := mul_le_mul_of_nonneg_right h_cast (le_of_lt h_ell0_sq_pos)
 106    rw [one_mul] at h_le
 107    rw [h_val]
 108    exact h_le
 109
 110end AreaQuantization