rateAction

formal source

  34  residualAction rot
  35
  36/-- Rate action A = -ln(sin θ_s) from eq (4.7) of Local-Collapse -/
  37noncomputable def rateAction (rot : TwoBranchRotation) : ℝ :=
  38  - Real.log (Real.sin rot.θ_s)
  39
  40/-- Rate action is positive for θ_s ∈ (0, π/2) -/
  41lemma rateAction_pos (rot : TwoBranchRotation) : 0 < rateAction rot := by
  42  unfold rateAction
  43  apply neg_pos.mpr
  44  have ⟨h1, h2⟩ := rot.θ_s_bounds
  45  have hsin_pos : 0 < Real.sin rot.θ_s :=
  46    sin_pos_of_pos_of_lt_pi h1 (by linarith : rot.θ_s < π)
  47  -- sin θ < 1 for 0 < θ < π/2
  48  have hsin_lt_one : Real.sin rot.θ_s < 1 := by
  49    have hx1 : -(π / 2) ≤ rot.θ_s := by linarith
  50    have hlt : rot.θ_s < π / 2 := h2
  51    have : Real.sin rot.θ_s < Real.sin (π / 2) :=
  52      sin_lt_sin_of_lt_of_le_pi_div_two hx1 le_rfl hlt
  53    simpa [Real.sin_pi_div_two] using this
  54  exact Real.log_neg hsin_pos hsin_lt_one
  55
  56/-- Born weight from rate action: exp(-2A) = sin²(θ_s) -/
  57theorem born_weight_from_rate (rot : TwoBranchRotation) :
  58  Real.exp (- 2 * rateAction rot) = (Real.sin rot.θ_s) ^ 2 := by
  59  unfold rateAction
  60  -- exp(-2*(-log(sin θ))) = exp(2 log(sin θ))
  61  have ⟨h1, h2⟩ := rot.θ_s_bounds
  62  have hsin_pos : 0 < Real.sin rot.θ_s :=
  63    sin_pos_of_pos_of_lt_pi h1 (by linarith : rot.θ_s < π)
  64  calc Real.exp (- 2 * (- Real.log (Real.sin rot.θ_s)))
  65      = Real.exp (2 * Real.log (Real.sin rot.θ_s)) := by ring_nf
  66      _ = Real.exp (Real.log ((Real.sin rot.θ_s) ^ 2)) := by
  67        congr 1