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Quantum indeterminacy and the eigenstate-eigenvalue link


Can quantum theory provide examples of metaphysical indeterminacy, indeterminacy that obtains in the world itself, independently of how one represents the world in language or thought? We provide a positive answer assuming just one constraint of orthodox quantum theory: the eigenstate-eigenvalue link. Our account adds a modal condition to preclude spurious indeterminacy in the presence of superselection sectors. No other extant account of metaphysical indeterminacy in quantum theory meets these demands.

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  1. That said, not all commentators believe that quantum theory must provide such examples (Glick 2017).

  2. A clarification is in order regarding our gloss on “metaphysical” indeterminacy as applied to quantum theory. When we say that such indeterminacy obtains independently of our linguistic and mental representations, we mean that it is not the result of, for example, semantic indecision or any other imprecision in our language or thought (Williams, 2008b). We do not mean that it is indeterminacy that obtains independently of our interpretation or mathematical formalization of quantum mechanics, e.g., Bohmian mechanics, spontaneous collapse theories, the Many Worlds interpretation, etc. (See Myrvold (2018, Sect. 4.2) and Lewis (2016, chap. 4) for a discussion of these interpretations and others.) Each of the latter takes a stance on how to correctly understand quantum theory, in particular the eigenstate-eigenvalue link—see Sect. 2.1 below—and thus represents a substantive position about what the world is ultimately like. On some of these interpretations there may be indeterminacy in the world itself—i.e., metaphysical indeterminacy—whereas on others there may not be.

  3. While it is not our intention in this essay to offer any conclusive argument in support of EEL, in Sect. 2.1 we will review the standard motivation for the principle and provide some evidence for our claim that EEL is in fact part of orthodox quantum theory. Not everyone agrees that EEL should be considered part of the orthodoxy: see Wallace (2012a, 4580; 2012b, 108; 2013, 215), but also Gilton (2016) for a rejoinder. Be this as it may, the fact that most recent accounts of quantum indeterminacy assume EEL is sufficient to warrant assuming it for present purposes.

  4. Barnes and Williams (2011) do not discuss quantum theory explicitly, although Williams does in an earlier review (2008b); the former has been influential for other accounts. Torza (2020, 4528–4529) endorses EEL in so many words, although not by name; Calosi and Wilson (2019, Sect. 2.1) do so, too, for the purposes of most of their project; and while Darby and Pickup do not mention EEL, their analysis of possible quantum indeterminacy begins from the orthodoxy about quantum theory that presupposes EEL, i.e., a standpoint prior to sophisticated interpretation (2021, 1686, 1689).

  5. The following compressed refresher assumes some familiarity with standard quantum theory. For more extended expositions, see Myrvold (2018, Sect. 2), Ismael (2015), and references therein.

  6. At most one of P and P can be the zero operator. In any case, P can be expressed as a real-weighted sum of non-zero projection operators.

  7. Fine actually uses the term “eigenvalue-eigenstate link,” but the meaning is the same.

  8. In his 1932 textbook, von Neumann focuses only on commuting projection operators Pk and Pj, i.e., ones that satisfy PkPj = PjPk.

  9. There are many traditions associated with quantum logic, some intended as research programs to solve the quantum measurement problem or to characterize quantum theory as non-classical probability theory, and others aimed instead at developing the formal structures of the logic for logicians’ purposes. We remain agnostic on all these goals, endorsing quantum logic only as a formalization of property ascription in orthodox quantum theory. See Bacciagaluppi (2009) for a conceptual-historical account of these controversies, which do not bear on our much more limited endorsement.

  10. It will suffice to confine ourselves to a propositional language. The syntax of the language is as usual. For details, including extensions to a first-order language, see, e.g., Dalla Chiara and Giuntini (2002, Sect. 2).

  11. For the rest of this section and the next, in which, respectively, we describe quantum logic and then present our own theory of quantum indeterminacy based on that logic, we will largely leave any ‘determinate(ly)’ qualifiers tacit. That is because, as we are about to see, in quantum logic there is no difference between truth and determinate truth (nor any difference between extension and determinate extension); any indeterminacy amounts to a simple lack of either truth or falsity. So there is no harm in leaving out the qualifier in this context, and doing so will simplify our exposition. We will reinstate the qualifiers, as necessary, when we turn to competing theories of quantum indeterminacy in Sect. 4.

  12. We will use “property ascription” to refer to both propositional and sentential ascriptions of properties. The context should make clear which is intended on any given occasion.

  13. In fact, the disjunction of just those sentences among p1, …, pn that lack a truth value will be true.

  14. They all follow from the fact that the (closed) subspaces of any Hilbert space form an ortho-complemented lattice (Birkhoff and von Neumann 1936).

  15. This marks a critical difference between quantum logic and many-valued Boolean logic. The latter is semantically non-classical but maintains all classical equivalences (Rasiowa and Sikorski 1963).

  16. Dalla Chiara and Guintini (2002) call this the “Herbrand-Tarski property.” The absence of this property for standard quantum logic interestingly parallels the same absence for logics of standard supervaluationist semantics for languages that include a determinacy operator (Fine 1975, 290).

  17. Actually, states that are linear combinations of states in different superselection sectors are possible, but they are mixed states rather than the pure states to which we have confined attention here. We return to the topic of indeterminacy for mixed quantum states in the concluding section.

  18. More precisely, they may be defined only over dense subsets of some subspace of the total Hilbert space, but this distinction only makes a difference for infinite-dimensional Hilbert spaces.

  19. At least propositions do not admit of imprecision in the way that sentences can (e.g., by semantic indecision). A proposition might be called imprecise if it (precisely) represents an imprecise property or object. But such imprecision, and any indeterminacy it might give rise to, is ultimately rooted in the world, not in any representation thereof.

  20. This is clear from Calosi and Wilson (2019), Darby and Pickup (2021), and Torza (2020). Barnes and Williams (2011) never explicitly apply their theory to quantum indeterminacy. But it’s reasonable to assume that they agree with the thought that, if there’s any quantum indeterminacy, superpositions provide examples of such indeterminacy (Williams, 2008b, 765). Finally, as we will see in a moment, Calosi and Wilson probably would object to our description of the relevant indeterminacy as indeterminacy regarding “whether e is ↑x.” But they certainly agree that there is indeterminacy in the present case (if there is any quantum indeterminacy at all).

  21. With some harmless simplifications, this characterization is taken verbatim from Wilson’s official statement of the view (2013, 366).

  22. Wilson writes of an electron in a superposition state of being ↑x and being ↓x that its state

    is an eigenstate of the operator O corresponding to: Do you have a spin? Hence the system has the property of having a spin. But [its state] is not an eigenstate of the operator O* corresponding to: are you x-spin ↑? So the system does not have the property of having x-spin ↑; nor does it have the property of having x-spin ↓. Here the property corresponding to O acts as the determinable and the property corresponding to O* acts as the determinate, with the system having the determinable but not any of the corresponding determinates. (2013, 371; emphasis ours).

    That having x-spin ↑ and having x-spin ↓ are the only determinates mentioned of the determinable of interest pragmatically implies that SPINx is the determinable represented by O that Wilson has in mind.

  23. CW in fact consider two other sorts of glutty cases: (i) those in which the determinates are not relatively instantiated in the same individual but in different individuals in the same state of affairs; and (ii) those in which the determinates are instantiated to some positive degree, but not fully. We set aside the first sort because CW do not endorse it in any case and the second because CW acknowledge that it would require replacing EEL with a degree-theoretic version (2019, 2621).

  24. See also the passage from Calosi and Wilson (2019, 2601) quoted later on in this subsection.

  25. In the glutty case, one can demonstrate an inconsistency between EEL and CW’s characterization even without appealing to their commitment to classical logic. If our example is a glutty case, e instantiates both being ↑x and being ↓x. In particular, it follows that e is ↑x. But recall that for CW, there is no literal sense in which it can be indeterminate whether an object instantiates a given property. It either (simply, determinately) does or it (simply, determinately) doesn’t. Therefore we can conclude that in our example, e (simply, determinately) is ↑x. But from this it follows, by EEL, that e is in an eigenstate of the projection operator representing ↑x, contradicting the assumption of the example that it is not.

  26. In bra-ket notation, σx =|↑x〉〈↑x|−|↓x〉〈↓x|. For more on this notation, see Ismael (2015, Sect. 2).

  27. Strictly speaking, the fact that the state of e is not in the range of A only allows us to infer that e does not determinately possess SPINx. But we know that for CW this is no different from saying that e (simply) lacks SPINx.

  28. See Wolff (2015) for a detailed discussion of the prospects of treating spin as a determinable. Though Wolff does presume EEL in her essay, she does not consider the problem we outline above.

  29. It may be worth noting that our second criticism would apply just as well were we to take, for instance, having (non-zero) spin or having ½-spin as the relevant determinable instead of SPINx. Consider the former (as similar reasoning applies to the latter). For reasons that parallel those above regarding SPINx, the projection operator representing having spin would have to be one which acts as the identity on the spin-½ superselection sector. Therefore, no matter what state e is in, it will have the determinable having spin, as expected. However, for any direction d, being d and being d will both count as determinates of this determinable. This is also to be expected (since being d is a way of having spin). But now CW face exactly the same problems in the case of SPINx. Because there is always some direction d such that e is in the eigenstate of one of the operators corresponding to being d and being d, it will always have one of the determinates of having spin. It follows that having spin cannot be a part of any witness to gappy indeterminacy. And because there is always some direction d such that e is in a superposition of being ↑d and ↓d, it follows that having spin is always a part of a witness to glutty indeterminacy.

  30. It won’t do, for instance, to suppose that SPINd and SPINd′, despite having identical extensions, may be distinguishable (when d ≠ d′) as distinct determinates of the spin-½ determinable referenced in the previous footnote. That’s because this supposition is incompatible with a characteristic feature of the relationships between determinables and their determinates as described by Wilson (2017, Sect. 2.1): The determinates of a determinable are supposed to be more specific ways of being the determinable. But for every direction d, any way of being spin-½ is a way of being SPINd, because the extensions of the two are the same. So in fact being SPINd is not a more specific way for something to be spin-½.

  31. The label “actuality” comes from Williams (2008a). In other work BW use different terminology.

  32. Penumbral connections are a certain type of necessary connection between concepts or properties, the determinacy of which (one might think) ought to be maintained even in the face of indeterminacy in the instantiation of the individual concepts or properties. For instance, “If Susan and Martha have the same net worth, then Susan is wealthy iff Martha is wealthy” is a penumbral connection that (on this line of thought) should come out determinate even when it’s indeterminate whether Susan is wealthy. See Fine (1975) for more on penumbral connections.

  33. One might find it odd to think of EEL as being determinately true, since EEL, being an interpretive principle, has a different status than propositions about the properties of quantum systems. On the other hand, if it weren’t determinately true, it would have to be indeterminate or determinately false, and neither of those seems compatible with our assumption of its truth. In any case, in Sect. 4.3.4 below we raise an objection to Darby and Pickup’s account that does not rely on the determinacy of EEL, and which applies equally to BW’s view. See the end of Sect. 4.3.4 for relevant discussion.

  34. The following objection also applies to Akiba’s (2004) theory, which is structurally similar to BW’s.

  35. Technically, in order to infer this conclusion via EEL, we need to assume that, at w, it is determinate that e is both ↑x and ↑y. It’s easy to establish the second of these in our argument. We simply strengthen our assumption (d) to: (d*) it is determinate that it is determinate that e is ↑y. This assumption is just as plausible as (d) in our scenario, and for exactly the same reasons. BW’s characterization then returns the result that it is determinate that e is ↑y in every actuality, and thus at w in particular. See Barnes and Williams (2011, Sect. 6) for details. It is not as obvious, however, that we are entitled to assume that, at w, it is determinate that e is ↑x. No matter. We still know, for the reasons given above, that e is ↑x at w. And this at least implies that e is not in the range of the operator corresponding to being ↑y, which is all that we need to derive a contradiction at w.

  36. There’s actually a more basic problem for BW. Once we’ve allowed EEL to be true at all actualities, we’re forced to conclude that at w e is in an eigenstate of being spin-up in the x direction. But then w determinately misrepresents reality after all, since in reality e is in an eigenstate of being spin-up in the y direction (and presumably is determinately so). We present the argument in the way we do above because it compares better with similar criticisms in the literature, which we discuss later on.

  37. Torza really has two objections about the retreat to impossible worlds. The first is the one discussed above concerning matters of ontological parsimony, and which we think misses the mark. The second is very similar to what we claim is the real problem, that it renders certain contradictions indeterminately true. So on this we are in agreement with Torza.

  38. (e) and (f) are de re attributions of indeterminacy, since they are saying, of a particular state, that it is determinate that e is in that state (and in no other). This seems (more than) fair to stipulate as part of a scenario for the purposes of our argument. Still, it is worth noting that our objection to DP would go through using only weaker, de dicto propositions:

    (e*) It’s determinate that e is in an eigenstate (with eigenvalue 1) of the projection operator corresponding to being spin-up in the y direction,

    and mutatis mutandis for (f).

  39. Just as above (see note 35), to infer this conclusion via EEL, we (arguably) need to suppose that, at s, it is determinate that e is ↑x, and it is not obvious what entitles us to this assumption. But as before, this does not affect the force of our argument. For we do know that e is ↑x at s. And from this we can at least infer that e is not in an eigenstate of the operator corresponding to being ↑y,, which is all that we need to derive the contradiction in our argument above.

  40. Here’s an analogy. Suppose we embrace counterpart theory (regardless of whether we’re modal realists). We are considering a possible world w and want to know who, if anyone, at w is Saul Kripke. We should apply counterpart theory. That’s not because counterpart theory is true at w; it’s because counterpart theory is part of our background theory that we can and should apply to various worlds in order to know what’s true in them.

  41. For the remainder of this section, we will understand “possible worlds” in a neutral way, so that, for instance, both BW’s complete worlds and DP’s partial situations count as possible worlds.

  42. If it is indeterminate whether e is ↑x, then it will also be indeterminate whether e is x, which, at least according to quantum logic, is equivalent to saying that it is indeterminate whether e is not ↑x.

  43. Torza (personal correspondence) has recently modified his view to apply to propositions rather than sentences. Assuming propositions are representational, as we understand them in our theory, then this does not make his view any more reductionist than before (just as our account is not reductionist). However, Torza has also suggested to us that he is understanding propositions in a non-representational way, e.g., as facts. In that case maybe he is providing a genuine reduction after all. Here let us just note one difficulty in moving from propositions (understood representationally) to facts: so-called “truth-value” gaps in facts amount to gaps in the existence of facts: no fact that p and no fact that not-p. And this requires a general ontology of negative facts irreducible to the non-existence of positive facts.

  44. See Myrvold (2018, Sect. 4.2) and Lewis (2016, chap. 4) for brief descriptions of these interpretations and references to further literature, especially the latter for their connection with indeterminacy.

  45. See van Fraassen (1991, chap. 7.3) and references therein for some of the difficulties.


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We would like to thank the audience at the 2019 Quantum Indeterminacy Workshop at Dartmouth College (where we presented under the title, “Little Shallows in the Depths: Indeterminacy in Quantum Theory,”) and three anonymous referees for their comments, which led to substantial improvement in our arguments.

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Correspondence to Samuel C. Fletcher.

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Fletcher, S.C., Taylor, D.E. Quantum indeterminacy and the eigenstate-eigenvalue link. Synthese 199, 11181–11212 (2021).

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  • Indeterminacy
  • Eigenstate-eigenvalue link
  • Quantum logic
  • Superselection