Foundation Primitive Recognition Calculus Delta Amplitude Complex Normalized Of

Unitary transformations preserve normalization

In quantum mechanics, a state vector is a list of complex numbers whose squared magnitudes sum to one. That sum is the total probability of all possible outcomes, so the normalization condition expresses the certainty that something happens. A unitary transformation, the kind that describes how a closed system evolves, is defined by the property that it preserves this sum. The theorem complex_normalized_of_normPreserving states this fact for finite lists: if a transformation preserves the squared norm of every complex amplitude vector, then it sends any normalized vector to another normalized vector.

The proof is a short chain of equalities. Given a transformation U that preserves the squared norm, and a vector ψ with squared norm one, the squared norm of Uψ equals the squared norm of ψ by the preservation property, which equals one. The theorem is part of the machine-checked library of formal theorems in the Recognition Science framework, where it is called the native finite version of unitary evolution. It belongs to a family of results about finite amplitude data: Born weights, the squared magnitudes of the components, are always nonnegative, and for a normalized vector they sum to one.

What the theorem does not claim is just as important. It does not say that every norm-preserving transformation is linear, nor that it can be represented by a matrix. It does not require the transformation to be invertible. It applies only to finite lists of complex numbers, not to infinite-dimensional Hilbert spaces, which the framework treats as a separate display completion. The theorem is a statement about normalization, not about the full structure of quantum evolution.

theorem complex_normalized_of_normPreserving {N : ℕ} {U : ComplexAmp N → ComplexAmp N}
    (hU : ComplexNormPreserving U) {ψ : ComplexAmp N} (hψ : ComplexNormalized ψ) :
    ComplexNormalized (U ψ) := by
  unfold ComplexNormalized
  rw [hU ψ, hψ]

theorem complexBornWeight_nonneg {N : ℕ} (ψ : ComplexAmp N) (i : Fin (N + 1)) :
    0 ≤ complexBornWeight ψ i := by
  unfold complexBornWeight
  nlinarith [sq_nonneg (ψ i).re, sq_nonneg (ψ i).im]

theorem complex_born_weights_sum_one {N : ℕ} {ψ : ComplexAmp N} (hψ : ComplexNormalized ψ) :
    Finset.univ.sum (fun i : Fin (N + 1) => complexBornWeight ψ i) = 1 := by
  simpa [ComplexNormalized, complexNormSq] using hψ

What this page does not claim

The theorem does not assert that every norm-preserving transformation is linear or matrix-representable. It does not apply to infinite-dimensional Hilbert spaces, which require a separate completion argument. It does not establish the Born rule as a derived principle; it only states a property of the defined Born weights.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/DeltaAmplitude.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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