Source
Di Biagio and Rovelli (2021), LIT-1666, §§1.1–1.3, pp. 2–6 of the journal text; read in NOTE-1226.
What was actually shown
The definitions (pp. 2–3). A value of L_S is a fact relative to F when an interaction makes a variable of F depend on it. Facts relative to one system obey the classical sum rule, P(b^(F)) = Σᵢ P(b^(F)|aᵢ^(F))P(aᵢ^(F)) (eq. 2), which RQM takes to hold always. Across systems the rule P(b^(W)) = Σᵢ P(b^(W)|aᵢ^(F))P(aᵢ^(F)) (eq. 3) “in general” fails, and its failure is interference. “If (3) holds, we say that the facts aᵢ^(F) are stable for W.”
The result (p. 4). For F entangled with an environment E, |ψ⟩ = Σᵢ cᵢ|F_ai⟩⊗|ψᵢ⟩, with 𝜖 = max_{i≠j}|⟨ψᵢ|ψⱼ⟩|², if (a) 𝜖 is small and (b) W does not interact with E, then W’s probabilities follow from the reduced state Σᵢ|cᵢ|²|F_ai⟩⟨F_ai| + O(𝜖), and eq. (3) holds up to O(𝜖) (eq. 8). “In the limit 𝜖 → 0, the variable L_F of the system F is exactly stable for W.” If L_S is entangled with a stable L_F, L_S is stable for W too (p. 5).
Relational (p. 5). “decoherence is not an absolute phenomenon, but a relational one: it depends on how the third system W interacts with the combined system F − E … Another system W′ that interacts differently with F − E might be able to detect interference effects.” This could have come out otherwise only if condition (b) were idle; it is not, since a W that measures F–E jointly sees the coherences the partial trace discards.
Stable without being a fact (pp. 5–6). “a fact stable for W is not necessarily a fact relative to W.” In a laboratory measurement, after the apparatus has decohered and before the experimenter reads it, L_S is stable for the experimenter and not a fact for her; after she reads the pointer it is “merely a stable fact for W” (p. 6).
Approximate (p. 6). “Stability is only approximate and relational. Approximate, because no fact is exactly stable for any finite 𝜖.”
A limit case without decoherence (footnote 2, p. 5). A fact relative to E with amplitude |c₁| near 1 is stable for a W that does not interact with E, to order η = 1 − |c₁|.
What this does not say
- That stable facts are absolute. The paper says the opposite: “one does not avoid relationalism by restricting to stable facts” (p. 7).
- That decoherence picks the outcome. The reduced state is nearly diagonal; which aᵢ occurred is not given by it, and the authors agree that “decoherence alone is not a solution of the measurement problem” (p. 10).
- A bound on the error. O(𝜖) is stated, not bounded. With 𝜖 defined as a squared overlap, the off-diagonal elements of the reduced state are bounded by |cᵢcⱼ|√𝜖; the paper does not discuss this. Partial access of W to E is not treated.
- That the same-context rule (eq. 2) is derived. It is RQM’s postulate (THEORY-621).
- What the systems are. S, F, E and W are given; how the world divides into them is not discussed.
- Anything about persistence in time. Stability here is a probability rule holding for a third system, not a record’s lasting or staying accessible.