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TERM-024: noncommutativity, of events and of operations

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Properties5
StatusActive
Version3
Tagsquantum-foundations, probabilistic-modeling
Linepragmatic-transport
Used byCLAIM-041 — Context-sensitive interpretation, order effects and noncommuting framing operations do not by themselves establish nonclassical probability or quantum contextuality, CLAIM-052 — Interpretive framing operations generally do not commute, and this can be investigated without committing to quantum probability, CLAIM-073 — Demonstrating that interpretive framings do not commute establishes that the judgements cannot be described by one ordinary joint probability distribution, CLAIM-063 — A transport that approximately intertwines each framing operation approximately preserves their commutator, so the degree to which interpretive perspectives interfere can itself be part of what a translation preserves; on observed responses that degree is the order effect, which the intertwining defects bound only with a readout defect added and which the data identify only through observational distortion, CLAIM-117 — Pragmatic fidelity requires preserving the dynamics of interpretation, not only final judgements: a good transport approximately intertwines source and target framing operations under a correspondence anchored to independently measured distinctions, CLAIM-127 — Conditioning on a communicative situation, intervening on it and intervening on what the interpreter is told are three different operations, so a framing intervention should index the empirical model, e_C^a = P(Y_C | do(a)), not enter its cover as one more context, CLAIM-373 — If a map approximately intertwines two operators, with defects E_A = ΦA − A′Φ and E_B = ΦB − B′Φ, then Φ[A,B] − [A′,B′]Φ = A′E_B + E_A B − B′E_A − E_B A, so the commutator is carried to within (‖A‖ + ‖A′‖)‖E_B‖ + (‖B‖ + ‖B′‖)‖E_A‖

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Definition

Two senses, distinguished at A30 in answer to the owner’s U13 (CLAIM-073):

  1. Of events. Classical events are sets and their indicators commute; quantum projectors need not. “For projective measurements, noncommuting observables generally lack a joint measurement with the same sharp marginals.”

  2. Of operations on probability states. Classical stochastic maps can fail to commute (CASE-001).

  3. Order dependence in observed responses (A40 §6.1): “Distinguish noncommutativity of state transformations from order dependence in observed responses”, measured by Δ_(A,B) = d(P(Y | A→B), P(Y | B→A)), where “the observed difference may arise from framing effects, memory, learning, task demands, or measurement disturbance”. The first two senses are properties of a model; this one is data.

  4. Of string concatenation (A43 §2): “Syntactic noncommutativity is trivial”: (c‖a)‖b ≠ (c‖b)‖a “is merely a fact about string concatenation”. A substantive claim needs a difference in induced behaviour, separated from “artifacts of positional encoding, attention masks, recency, and instruction hierarchy”. A48 §7.2 kept three levels (concatenation, effective operations, observed response distributions; “only the latter two have potential explanatory significance”). C1 Appendix C kept three tiers: “(i) syntactic noncommutation of prompt concatenation, (ii) noncommutation of inferred stochastic transition operators, and (iii) incompatibility of measurement observables in a quantum representation. A test of (i) never alone establishes (ii); even (ii) does not entail (iii).” The manuscript keeps (ii) and (iii) and does not mention concatenation.

What it is not

Order dependence in judgements is evidence for the second sense, not the first. A30: “observing order dependence is sufficient to establish noncommutativity of the effective operations, but not sufficient to establish nonclassical probability.” The manuscript (§9) keeps the distinction.

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