Forced by one wave, one history and no signalling; carried to single histories
Part III: The Theory
The most successful rule in physics — and nobody knows where it comes from.
Max Born proposed this rule in 1926: the probability of finding a quantum system in a particular state equals the square of the wave function's amplitude. It has been one of the most stringently tested rules in physics.
But in standard quantum mechanics, the Born rule is simply postulated. It's added as an axiom, with nothing more fundamental beneath it.
Postulate it as an axiom
Build it into the noise law by construction
Forced: by one wave, one history, support certainty, no signalling and continuity — Zurek's argument in Barnum's form
Six premises. Five are things ACT already says; one is new.
A record's chance is one function of the wave and the record, the same function whatever the basis. Nothing else exists for it to depend on.
Exactly one record is actual, so the chance of "this record or that one" is the sum.
Records are pointer sectors; a joint record with zero amplitude has zero chance. Anchoring selects among components and creates none.
A record's chance here does not change when a record is made elsewhere — for every division of the world into here and elsewhere. The Bell requirement is its spacelike case.
Any record can be refined by a further apparatus.
A record's chance changes continuously with the wave. This one is new.
From these the weights are |ψ|², by Zurek's envariance argument in Barnum's no-signalling form (the flagship, Appendix A). Each alternative rule fails a named premise: |ψ|⁴ renormalized fails no signalling (a distant record shifts local statistics by up to 0.2); a basis-dependent rule fails one law; a rule that is never certain fails support certainty; a rule that treats irrational weights apart fails continuity.
With the weights forced, descent is the check that the single-history law and the master equation agree.
ACT does not produce |ψ|² from a mechanism. In the nondemolition model — a record that leaves the recorded variable undisturbed — the first-event process is outcome-neutral (a hypothesis of Paper I, Prop. 2): the hazard h = Φ̇irr says when a record is released and nothing about which. Requiring the runs, averaged, to agree with the decoherence the ensemble shows — ensemble agreement — then forces the anchored weights to be the populations pₖ = ⟨k|ρ|k⟩ (Paper I, Prop. 2). The weights descend from the ensemble, and they agree with the weights the six premises force.
Given the ensemble's populations, which standard quantum mechanics supplies, here is what each alternative rule for single runs would do:
Runs averaged would not reproduce the decoherence they must sum to
A nonlinear reweighting — and a distant choice could shift local statistics
The weights the ensemble already carries; outcome-neutral release leaves them untouched
The no-signalling check is no longer a check: it is premise 4. Any other weighting would let a distant choice shift local statistics, and that is exactly what premise 4 forbids.
A system in state |ψ⟩ = α|A⟩ + β|B⟩ has two possible outcomes, with |α|² + |β|² = 1, and standard populations p_A = |α|², p_B = |β|².
Coupled to the bath, the ensemble state loses its off-diagonal terms in the pointer basis: ρ → |α|²|A⟩⟨A| + |β|²|B⟩⟨B|. The populations pₖ = ⟨k|ρ|k⟩ are untouched. Standard physics; no assumption about |ψ|² or field-energy density is introduced.
In the nondemolition model the first event comes at hazard h = Φ̇irr, survival S = e−Φirr. The hazard says when a record is released; it says nothing about which. The stopped process gives ρ(t) = S ρcoh + (1 − S) Σₖ Pₖ ρ(0) Pₖ.
Runs not yet anchored plus runs anchored must average to the decoherence the ensemble shows. That holds only if anchored runs fall with the populations: P(A) = |α|², P(B) = |β|² (Paper I, Prop. 2). The conditioned weights form a martingale — a process whose expected value never drifts, so the weight a run carries at the end is the weight the ensemble gave it at the start — and the same weights carry to each single history.
The ensemble's populations descend to each run and agree with the forced weights. What remains is ACT's one primitive, that one record history is actual, and the continuity premise.
After decoherence the reduced state is diagonal in the pointer basis:
Write the anchored ensemble as a stopped process. Two facts make the weights Born, with no |ψ|² or field-energy input (Paper I, Prop. 2):
Single histories: the conditioned weights pk(t) form a bounded martingale, so the ensemble's weights descend to each run. No signalling, now a premise rather than a check: a nonlinear reweighting would let a distant choice shift local statistics; only the Born weights pass. What remains is ACT's one primitive, that one record history is actual, and continuity.
Where does the randomness come from, and is it truly random?
What is stochastic is the actual record history: which records are released, and when. The event law supplies their probability measure; ACT's one primitive is that one such history is actual.
It is tempting to say "the randomness comes from environmental noise" — but a stochastic unraveling is only a representation of the same reduced dynamics, and different unravelings share it. So the objective content is not "noise is the source of probability"; it is the actual record history, over global dynamics that are never modified. Which unraveling is physical is fixed by the record architecture: the selection theorem says fragment records support only localization-type conditioning, never coherent momentum kicks.
The actual record history X[0,t] — which records are released, and when. The measure is fixed by the state and the bath's own instrument.
ACT adds one commitment — one history of marks is actual — over an unchanged global evolution. That no event leaves two marks is a theorem for the systems the companion papers treat (Paper II's record-algebra class; Paper I, which works without that result, states it as a commitment), and a further commitment beyond them.
What is random is which history is actual, and when its records are released. Nothing in the bath is a die.
One wave, one history, no record without a component, no signalling, refinement, continuity: |ψ|² is forced;
the martingale descent is the consistency check.
Not from nothing — one primitive remains: one record history is actual.
Next: Lecture 7 — Resolving the Paradoxes