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Quantum indeterminacy and the double-slit experiment

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  1. See, e.g., Darby (2010), Skow (2010), Bokulich (2014), Lewis (2016), Torza (2020), Calosi and Wilson (2018).

  2. Some suggest that certain interpretations of QM require revision of EEL as a principle connecting the quantum formalism to the having of a definite value of a given observable. For example, Albert and Loewer (1992) suggest that since collapse on the Ghirardi-Rimini-Weber (GRW) interpretation of QM leaves a lingering ‘tail’ of indeterminacy, EEL should be replaced by the Fuzzy Link (FL):

    (FL): A quantum system has a definite value v for a particular observable O iff the square projection of its state onto an eigenstate of O is greater than 1 – P, for some (suitably small) P.

    See also Lewis’s (2016) discussion of the ‘Vague Link’. As we note in our (2018), the need for such revision is controversial (see Frigg 2009) and also appears to be pragmatically motivated, in a way leaving seeming QMI intact. We later revisit whether GRW or other interpretations of QM are committed to QMI; at this point we aim simply to present the usual EEL-based motivations for QMI.

  3. Plausibly, superposition is the most general source of QMI, since in cases of incompatible observables \(O_1\) and \(O_2\), some eigenstates of \(O_1\) are superpositions of eigenstates of \(O_2\), and entanglement states are superposition states. Even so, just as the properties of colour, red, and blue are related (with the first being more general than the second and third) yet interestingly distinct, we similarly maintain that these sources of quantum MI are related yet interestingly distinct; see Calosi and Wilson (2018) for discussion.

  4. See Williamson (1994, Ch. 6), for relevant discussion.

  5. The following figures are taken from Barrett (2001, 4).

  6. Such a treatment is forced by the fact that if detectors are placed at each of the slits, the resulting pattern is just that associated with classical physics.

  7. This is a simplification. The initial wavefunction of each particle is an eigenstate of momentum; hence it has the form of a plane-wave \(|\psi \rangle = e^{ipx/\hbar }\), corresponding to a much more complicated superposition of position states. The simplification is frequently adopted in the literature (see, e.g., Barrett 2001), and for present purposes is harmless.

  8. As will become clear, our proposal is not that self-interference involves ‘multilocation’ of the sort recently defended, e.g., in Eagle (2016).

  9. As Johnsgard (1997) puts it: “The highly iridescent feathers of the hummingbird gorgets are among the most specialized of all bird feathers [...]. The colors do not directly depend on selective pigment absorption and reflection, as do brown and blacks produced by the melanin pigments of non-iridescent feathers. Rather, they depend on interference coloration, such as that resulting from the colors seen in an oil film or soap-bubble [...]. Put simply, red wavelengths are longer than those at the violet end of the spectrum and generally require films that are thicker or have higher refractive indices than those able to refract bluish or violet light. Thus, the optimum refractive index for red feathers is about 1.85; for blue feathers it is about 1.5 [...]. When an optical film is viewed from about, it reflects longer wavelengths than when viewed from angles progressively farther away from the perpendicular. Thus, a gorget may appear ruby red when seen with a beam of light coming from directly behind the eye, but as the angle is changed the gorget color will shift from red to blue and finally to black, as the angle of incidence increases (121–26).

  10. See Wilson (2013) for further discussion of this interpretation and its compatibility with several specific accounts of colour.

  11. To be sure, as is discussed in Wilson (2013), there are also understandings of the feather case which do not involve multiple relativized determination. For example (as a referee noted), one might accommodate the relativization in the feather case by taking the determinate colours as well as their associated determinables to be dyadic relations between objects and perspectives. This understanding strikes us as an unparsimonious and metaphysically inapropos way of treating what appears to be a singly instanced, unrelativized instantiation of the determinable colour, but again, for our purposes what is important is that an understanding in terms of multiple relativized determination makes sense. Here it may also be worth observing that properly metaphysical accommodation of the relativity at issue does not require that the perspectives (more generally, circumstances) be ‘built in’ to the properties at issue. On the contrary, in recent literature on perspectival facts this approach is commonly rejected; for example, Lipman (2016) argues that one should not account for what he calls “perspectival variance” by “saying that the apparent properties or relations merely turn out to have higher adicity—that these cases simply reveal a hidden argument place” (44), and Berenstain (2020) characterizes a perspectival fact as a fact expressed by a proposition whose truth value depends on the perspective of a particular observer, where the locus of relativization is the truth value of the proposition as opposed to a purportedly relational fact (or constitutive property). Nor is there any reason to think that non-relational conceptions of perspectival variance are not property metaphysical; see Evans (2020) for discussion as applied to the case of colour, in particular.

  12. Such an approach is reminiscent of the path integral formulation of QM (see Feynman and Hibbs 1965). As a referee observed, insofar as position and momentum are incompatible observables, quantum particles cannot (Bohmian mechanics aside) have perfectly well-defined trajectories. Correspondingly, our talk of possible particle trajectories should be understood as shorthand for one or other of the following three understandings. First, such talk may advert to possible classical trajectories—that is, trajectories that would be available if the particles were to behave classically. Second, such talk may advert to sequences of different spatial positions and regions that quantum particles can occupy without having definite position and momentum. Third, such talk may advert to approximations of classical trajectories. One way of developing this last strategy is along lines of Wallace’s (2008) remark that “If a system happens to be in a quasi-classical state \(|{\mathbf{q} }(t), {\mathbf{p }}(t)\rangle \otimes |\psi (t)\rangle \) [...] then its evolution will accurately track the phase-space point \(({\mathbf{q} }(t), {\mathbf{p} }(t))\)” (47); here particle trajectories can be taken to be represented by the quasi-classical evolution of the phase-space point \(({\mathbf{q} }(t), {\mathbf{p} }(t))\).

  13. As with the feather case, the suggestion here is that one can intelligibily understand the phenomenon of self-interference as involving multiple relativized determination, not that one must do so (indeed, we will shortly consider an alternative understanding). Again, what is important for our purposes is that one is not forced to endorse a metaphysical accommodation of the seeming self-interference in terms which do not involve metaphysical indeterminacy of one or other glutty variety, as on, e.g., a quantum variation on the relational proposal discussed in footnote 11. Here again, recent literature on perspectival facts (including Lipman 2016; Berenstain 2020; Evans 2020) is relevant, especially since much of this literature either focuses on or is intended to apply to cases of seeming perspectivalism in quantum mechanics.

  14. The degree-theoretic approach here is different from that in Smith and Rosen (2004); in particular, we reject three claims that Smith and Rosen accept, including that all fundamental properties are maximally precise, that MI involves an object’s being an ‘intermediate instance’ of a precise property, and that ‘fuzzy logic’ is the correct logic of MI. In particular, and unlike degree-theoretic approaches which depart from classical logic in allowing ‘degrees of truth’, our approach is that (as applied, e.g., to the quantum cases at hand) sentences of form ‘system s has value v of observable O’ are incomplete, and hence not truth-evaluable. Rather, it is sentences of form ‘system s has value v of observable O to degree n’ that are truth-evaluable, in line with both classical semantics and classical logic. This approach is in line with the more general supposition of determinable-based metaphysical indeterminacy as not inducing any propositional indeterminacy. See Wilson (2016) and Calosi and Wilson (in progress) for further details.

  15. By way of contrast, in the literature on multilocation, multiple exact location or position is had simpliciter.

  16. This suggestion can be made more precise by building on work by Wightman (1962) and developed by Pashby (2016), according to which any region of space \(r_i\) can be associated with a projection operator \({\hat{P}}_{r_i}\).

  17. In our (2018), we note certain problems with a gappy implementation of Determinable-based MI as applied to cases of superposition-based QMI. We direct the interested reader to that discussion.

  18. As Nina Emery pointed out, a glutty determinable-based approach also provides a basis for explaining what results when both slits are blocked.

  19. On a relational interpretation, certain fundamental properties of a system prior to interaction with other systems correspond to undetermined determinables; after interaction these properties may become determinate, relative to these other systems. Hence there appears to be fundamental QMI on a relational interpretation. On a modal interpretation, there is a distinction between the dynamical state and the value state, where (on the usual gloss) properties in the dynamical state are properties that a system might have, whereas properties in the value state are properties that a system has. One might reasonably suppose that a given system actually has the determinable properties associated with the merely possible properties in its dynamical state; if some of these undetermined determinables are fundamental, then there is fundamental QMI on a modal interpretation.

  20. See Calosi (2019) for a similar point, going beyond the quantum details.

  21. Relatedly, as Nina Emery pointed out, if particles in the double-slit experiment do not go through either slit (since not at all spatiotemporally located), then it is unclear why blocking both slits should change the result of the experiment, as in fact happens. Here again a glutty determinable-based approach comes out ahead, explanatorily speaking.

  22. The case of the infinite square well models a particle moving in one dimension inside a small region with impenetrable barriers, associated with the following potential:

    $$V(x) = {\left\{ \begin{array}{ll} 0, {\text {if}} \; 0 \le x \le a\\ \infty , {\text {otherwise}} \end{array}\right. }$$

    The particle is free to move in the potential V(x) except at the two ends (\(x=0\) and \(x=a\)), where an infinite force prevents it from escaping.

  23. To be clear, we take the suggestion that a superposition of position states is itself a position state to be independently intuitively plausible given the properties at issue, as opposed to following from a general principle to the effect that a superposition of states of a given type will also be of that type. As a referee points out, the general principle appears to be false (e.g., a superposition of z-spin properties will not be a z-spin property); but our purposes require only that a superposition specifically of position states is plausibly itself a position state.

  24. Indeed, key to the determinable-based approach to MI is that determinables are not reducible to disjunctions or other constructions of determinates; see Wilson (2013), citing arguments for such irreducibility in Wilson (2012) and elsewhere.

  25. For example, the determinable momentum is associated with the operator \({\hat{p}} = -i\hbar \frac{\partial }{\partial x}\); its eigenfunctions are plain waves \(|\psi \rangle = e^{ipx/\hbar }\) with eigenvalues \(p = \hbar k\). The determinable spin of a \(\frac{1}{2}\)-particle along a given direction \(\alpha \) is associated with the general operator

    $$\begin{aligned} {\hat{S}}_{\alpha } = \begin{pmatrix} z &{} x -iy \\ x + iy &{} -z \end{pmatrix} \end{aligned}$$

    of which the Pauli matrices for spin in the xy and z directions, having eigenvalues \(\pm \frac{1}{2}\), are specific examples. And so on.

  26. On these non-classical understandings, quantum disjunctions, unlike classical disjunctions, are not equivalent to existential statements.

  27. See also Calosi (2019) for reasons to think that position should not be regimented in disjunctive terms.

  28. This claim presupposes that the operative logic is classical, which some deny for quantum contexts. But first, note that Torza’s original objection also rests on the endorsement of classical logic—hence it was that we challenged 3 and 4 rather than 1 and 2. And second, if one endorses quantum logic, then nothing prevents one from identifying a quantum determinable with a quantum disjunction of determinates, since the logic will allow that such a disjunction can be true without any of its disjuncts being true, and will also rule out a disjunction’s being equivalent to an existential statement. So while endorsing quantum logic would block our specific complaint here, doing so would also undercut the original argument from revisionism against the determinable-based account. Thanks to a referee for discussion.

  29. One might still want to hear more about how to express the having of position, insofar as the associated predicate will still need to be a formula in one free variable—namely, one that can be predicated of a particle x just in case x has position. Here we suggest that the proponent of determinables can and should avail themselves of predicates for relevant determinables as well as relevant determinates in their language. One such predicate will be DP, representing the maximally unspecific determinable position; the property of x having position can then be represented by the formula DP(x). Hence in a language with higher-order quantification and identity, using ‘D’ as a variable ranging over maximally unspecific determinables, ‘DP’ as a predicate representing (the property of having) position, and ‘M’ as a predicate representing (the property of being a) material object, that all material objects have position can be expressed as follows: \(\exists D ((D = DP) \wedge \forall x (M(x) \rightarrow DP(x)))\).

  30. As Nina Emery observed, similar remarks may attend to recently popular high-dimension ontologies for QM (e.g., wave-function realism à la Albert 1996), to the extent that the attraction of such views reflects concerns that QMI could not be reconciled with more familiar particle ontologies.


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Correspondence to Jessica Wilson.

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Thanks to participants at the 2019 Dartmouth Quantum Indeterminacy Workshop, and special thanks to Nina Emery, Michael Miller, and an anonymous referee for this journal, for valuable comments. Calosi acknowledges the support of the Swiss National Science Foundation (SNF), project number PCEFP1\(\_\)181088, and Wilson acknowledges the support of the Social Sciences and Humanities Research Council (SSHRC), fellowship number 435-2017-1362.

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Calosi, C., Wilson, J. Quantum indeterminacy and the double-slit experiment. Philos Stud 178, 3291–3317 (2021).

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  • Quantum indeterminacy
  • Metaphysical indeterminacy
  • Double-slit experiment
  • Determinables
  • Supervaluationism