`probability`

definition

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module: IndisputableMonolith.QFT.SMatrixUnitarity
domain: QFT
line: 60 · github
papers citing: none yet

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IndisputableMonolith.QFT.SMatrixUnitarity on GitHub at line 60.

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formal source

  57def amplitude {n : ℕ} (S : SMatrix n) (i j : Fin n) : ℂ := S.matrix i j
  58
  59/-- Transition probability: |S_ij|². -/
  60noncomputable def probability {n : ℕ} (S : SMatrix n) (i j : Fin n) : ℝ :=
  61  ‖amplitude S i j‖^2
  62
  63/-! ## Unitarity and Probability Conservation -/
  64
  65/-- **THEOREM (Unitarity Inverse)**: S is invertible with S⁻¹ = S†.
  66    This is the other direction of unitarity.
  67
  68    For finite-dimensional spaces, S†S = I implies SS† = I.
  69    This is a standard result in linear algebra. -/
  70theorem s_inverse {n : ℕ} (S : SMatrix n) :
  71    S.matrix * S.matrix.conjTranspose = 1 := by
  72  -- Standard linear algebra: for square matrices, left inverse = right inverse
  73  -- Since S†S = I, S† is a left inverse of S, hence also a right inverse
  74  have h := S.unitary  -- S†S = I
  75  -- The key: S†S = I means S is an isometry on finite-dim space
  76  -- Isometries on finite-dim spaces are surjective (unitary), so SS† = I
  77  -- Using Mathlib's Matrix.mul_eq_one_comm for invertible matrices
  78  rwa [Matrix.mul_eq_one_comm] at h
  79
  80/-- **THEOREM (Probability Conservation)**: For any initial state i,
  81    the total probability of ending in any final state is 1.
  82
  83    Proof: From SS† = I (s_inverse), (SS†)_ii = Σ_j S_ij × conj(S_ij) = Σ_j |S_ij|² = 1.
  84
  85    Technical: requires careful handling of Complex.normSq/star/‖·‖² conversions. -/
  86theorem probability_conserved {n : ℕ} (S : SMatrix n) (i : Fin n) :
  87    (Finset.univ.sum fun j => probability S i j) = 1 := by
  88  unfold probability amplitude
  89  -- From s_inverse: SS† = I, so (SS†)_ii = Σ_j |S_ij|² = 1
  90  have h := s_inverse S

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