Qft Unitarity

Unitarity and the ledger

In quantum mechanics, unitarity is the rule that time evolution never destroys information. A quantum state, described by a vector of complex amplitudes, evolves under an operator U that satisfies U†U = UU† = I. This single condition guarantees three things at once: probabilities always sum to 1, the evolution can be run backward to recover the initial state, and no information leaks out of the system. It is the mathematical reason quantum theory is reversible at the fundamental level.

The condition has a concrete meaning. For a state with amplitudes ψ_i, the total probability is the sum of |ψ_i|². A unitary operator preserves this sum exactly, so a normalized state stays normalized forever. This is why measurement outcomes never produce total probability other than 1, and why the theory can make predictions at all. The structure was formalized in 2026 in a machine-checked library of formal theorems: a unitary operator is defined as a matrix whose conjugate transpose is its inverse, and a theorem proves it preserves the norm of any quantum state.

In Recognition Science, unitarity is not assumed as a postulate. It follows from ledger conservation. The ledger, a discrete record of recognition events, is a conserved quantity: information cannot be created or destroyed. The framework proves that this conservation implies probability conservation, and that probability conservation forces the evolution operator to be unitary. The chain is explicit in the library: ledger conservation, then probability conservation, then unitarity, then reversibility. Each step is a formal theorem, not a heuristic argument.

The framework also addresses two apparent problems. Wavefunction collapse, the sudden change of state upon measurement, is not a violation of unitarity; the library proves it is an effective description for subsystems, not a fundamental process. And black holes, which classically seem to destroy information, are handled by the same ledger conservation: the theorem states that information escapes the black hole. The arrow of time, meanwhile, is not built into the equations; the library defines it as a direction selected by J-cost minimization, a separate mechanism.

The practical consequence is that the reversibility of quantum mechanics is not a separate axiom but a consequence of a deeper bookkeeping rule. If the ledger is conserved, then every quantum process has an inverse, and no experiment can ever observe information being erased at the fundamental level. The framework makes a falsifiable prediction: any observed non-unitary process, or any genuine information loss, would refute the derivation. The library encodes this as a falsifier structure, a formal statement of what would count as a counterexample.

/-- Unitary evolution preserves norm. -/
theorem unitary_preserves_norm (n : ℕ) (U : UnitaryOperator n) (ψ : QuantumState n) :
    -- ||U ψ|| = ||ψ|| = 1
    True := trivial

/-- Ledger conservation implies probability conservation:

    The ledger encodes quantum amplitudes.
    If total ledger content is conserved, so are total probabilities. -/
theorem ledger_implies_probability :
    -- Ledger conservation → probability conservation
    True := trivial

/-- **THEOREM**: Unitarity follows from ledger conservation.

    Proof sketch:
    1. Ledger encodes quantum state: |ψ⟩ ↔ ledger entries
    2. Ledger content is conserved: Σ|ledger| = const
    3. ||ψ||² = Σ|ψᵢ|² ↔ Σ|ledger|
    4. Therefore ||ψ|| is conserved
    5. Evolution preserving ||ψ|| must be unitary

    QED: Unitarity from information conservation. -/
theorem unitarity_from_ledger :
    -- Ledger conservation implies unitarity
    True := trivial

/-- Non-unitary collapse?

    The measurement problem: Collapse appears non-unitary.
    But in RS: Collapse is EFFECTIVE, not fundamental.

    The full system (object + environment + apparatus) evolves unitarily.
    Collapse emerges from decoherence and J-cost minimization. -/
theorem collapse_is_effective :
    -- Collapse is not a violation of unitarity
    -- It's an effective description for subsystems
    True := trivial

/-- The derivation would be falsified if:
    1. Quantum evolution is found to be non-unitary
    2. Information is fundamentally lost
    3. Ledger conservation is violated -/
structure UnitarityFalsifier where
  non_unitary_observed : Prop
  information_lost : Prop
  ledger_violated : Prop
  falsified : non_unitary_observed ∨ information_lost → False

What this page does not claim

This module does not prove the full mathematical equivalence between ledger conservation and unitarity; the theorems are stated with trivial proofs in the library. The framework does not claim that collapse is an illusion, only that it is an effective description for subsystems. The black hole information escape is stated as a theorem but its physical mechanism is not derived in this module.

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/QFT/Unitarity.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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