Source
Belot (2011; EJPS 2012), LIT-100, pp. 10–13, read in full in NOTE-110.
What was actually shown
Setting. Non-relativistic Bohmian mechanics for n spinless particles in Euclidean 3-space, read as a primitive ontology of n worldlines. Wave functions are identified up to measure-zero differences and complex multiples (p. 3, fn. 8). The guidance law is eq. (1).
The dispositionalist reading. Belot introduces it himself (fn. 7; the property idea is credited to Monton 2006). The complete history is q(t) together with Φt, the system’s velocity dispositions, represented by the vector field X_t on configuration space that the guidance law derives from ψ(t) (pp. 10–11).
The counterexample (pp. 11–12). For a free particle in [0,1], every energy eigenstate φn(x) = sin(πnx) gives zero velocity everywhere under the guidance law. Infinitely many inequivalent states share one dispositional history. Their differences in energy, and the difference in which configurations are typical (φ1 against φ2), are effaced. Belot calls this “dismaying” (p. 12).
The rescue. Belot does not object to bare or holistic dispositions as such; content is the problem. He conjectures, crediting Goldstein, that the map ψ(t) ↦ X_t is injective for generic solutions (fn. 41), and proposes to accept, elsewhere, that there is no determinate fact about energy or typicality. The problem arises only for closed systems (fn. 42).
What could have come out otherwise. The map from states to velocity fields could have been injective on stationary states. The box’s eigenstates make it fail.
What this does not say
- That the dispositional reading is refuted. Its rescue rests on a genericity conjecture that is unproved either way (NOTE-110, C6, weak).
- That ψ is a law, or a field. Belot prices those readings separately and calls the outcome “inconclusive”: each option relies on contested metaphysics (§3). The costs of the law reading are in THEORY-669.
- That the reading also loses determinism by this route. That is a separate cost (fn. 39): (q, Φ) as the instantaneous state is not deterministic, possibly repaired by adding time-derivatives of Φ.
- Anything about GRW or relativistic primitive ontologies. The case is worked only for non-relativistic Bohmian mechanics.
Connections
- Dorato (LIT-176) puts intrinsic dispositions in events, and does not address this case (NOTE-105); that account is THEORY-656, and Belot’s box is its test.
- Gao (LIT-121) cites Belot against Esfeld et al.’s Bohmian dispositionalism (NOTE-140).
- Chen’s taxonomy of realist readings (LIT-204) takes its multi-field reading from this paper.