Abstract
Metaphysical indeterminacy in the context of quantum mechanics is often motivated by the eigenstate-eigenvalue link. However, the sparse view of Glick (Thought J Philos 6(3):204–213, 2017) illustrates why it has no such implications. Other links connecting quantum states and property ascriptions—such as those associated with the GRW theory—may introduce indeterminacy, but such indeterminacy may be viewed as merely representational and is susceptible to familiar treatments of vagueness. Thus, I contend that such links fail to provide a compelling motivation for quantum metaphysical indeterminacy.
To appear in V. Allori (ed.) Quantum Mechanics and Fundamentality: Naturalizing Quantum Theory between Scientific Realism and Ontological Indeterminacy, Springer Nature. _______
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Notes
- 1.
- 2.
A familiar example of the determinable-determinate relationship is red and scarlet. Scarlet is a particular way of being red, hence it is a determinate of the determinable red. Note that a property may be a determinate at one level of analysis, but a determinable at a “deeper” level—Venetian scarlet is a determinate of the scarlet determinable.
- 3.
- 4.
It’s not entirely clear what constitutes “orthodox” quantum mechanics. For my purposes here, I assume that it involves the eigenstate-eigenvalue link, the collapse postulate, and Born’s rule. See Wallace (2019) for a criticism of this view and Gilton (2016) for a defence of the role of the eigenstate-eigenvalue link in “orthodox” or “textbook” quantum mechanics.
- 5.
I will be challenging this inclination below. For now, a motivating idea might be that we can measure the particle’s x-spin, and when we do so, it will be found either up or down. So, the particle in question is the kind of thing that can possess a precise value of x-spin even if it doesn’t have one at the moment. This might incline one to regard it as possessing the x-spin determinable without a unique determinate of it (until it’s measured).
- 6.
Consider, for instance, category mistakes. The number two lacks a determinate mass, but this does not imply that its mass is indeterminate because it lacks the determinable as well.
- 7.
Of course, it’s often unclear what the “size” of a quantum system is. So, the relevant notion of a maximally precise location might be the smallest region to which the system can in principle be confined.
- 8.
Let x p denote a subregion of X that would be regarded as a maximally precise location for the particle. See previous footnote.
- 9.
Notice that there is nothing wrong with asking whether we will find the particle in the region x i ∈ X if we were to measure its position. To answer this question we use Born’s rule, not EEL.
- 10.
The quantum state will not be in a superposition of the operator invoked in EEL, but we may wish to describe the state in terms of a distinct operator with clear physical significance—in this case, an operator with |A〉 and |B〉 as eigenstates.
- 11.
Of course, critics of the sparse view may allege that the quantum state must be given some metaphysical analysis, and it’s here that metaphysical indeterminacy arises. Quantum Location is a denial of this demand. The property possessed by a system in a particular superposition of more precise position states is just that. Superpositions are novel kinds of determinates, not to be further analyzed in terms of the observables that appear in their arguments.
- 12.
Calosi and Wilson (2019) propose to replace EEL with DEEL, a principle that posits degrees of possession proportional to the coefficients of the arguments of a superposition. However, unlike EEL, DEEL is not part of the standard formulation of orthodox QM. Moreover, it goes beyond what is needed for orthodox QM to solve the measurement problem. See Sect. 21.3.
- 13.
This means that Quantum Location (and its generalization) is distinct from the glutty indeterminacy view of Calosi and Wilson (2019). The former takes a particular superposition to be a novel kind of determinate of the determinable associated with the operator that defines the basis. The latter takes a particular superposition to correspond to a plurality of determinates (each corresponding to eigenstates) each possessed to a degree less than 1. The present point is that the views are inequivalent because the quantum state contains more information than a weighted collection of determinates.
- 14.
- 15.
It may be thought that indeterminism alone implies indeterminacy, and indeed, some motivate metaphysical indeterminacy by consideration of the “open future” (Barnes & Cameron, 2009). However, the metaphysical status of the future is largely independent of whether there are stochastic laws of nature. The path from indeterminism to indeterminacy is not at all straightforward and inevitable.
- 16.
Armed with only EEL, the GRW theory is unable to secure determinate measurement outcomes because it precludes the assignment of reasonably precise positions to macroscopic systems like pointers that constitute such results. Again, the only position properties licensed by EEL are those that attribute a location in the region to which a system is strictly confined. Often this region will be no less than the entirety of space.
- 17.
\(\hat {P}_c\) is a projector onto the subspace of Hilbert space associated with being located in some region C including c.
- 18.
Of course, the GRW theory is a more “realist” interpretation than orthodox quantum mechanics, so one might think that we are owed more of a story about non-eigenstates. However, rejecting metaphysical indeterminacy doesn’t require silence about non-eigenstates. In addition to providing probabilities for various measurement outcomes, such states can also provide a basis for the attribution of novel properties along the lines of Quantum Location as outlined in Sect. 21.1.3.
- 19.
Lewis takes the GRW theory to involve indeterminacy that is both metaphysical and distinct from familiar non-quantum cases of indeterminacy. Here I treat the distinctness claim independently with the aim to rebut the claim that there is anything distinctively “quantum” about the indeterminacy introduced by the GRW links.
- 20.
For a defense of emergent quantum metaphysical indeterminacy, see Mariani (2021).
- 21.
Barnes doesn’t share Wilson’s determinable-based account of metaphysical indeterminacy, however, one may develop an understanding of the vague link that fits with the metaphysical supervaluationist account. In the present case of a position measurement in GRW, one could posit a candidate ersatz world corresponding to each location where the wavefunction has support. Then, the position of the system will be metaphysically indeterminate in that the truth of a proposition of the form the system is located in region R will differ between candidate ersatz worlds, hence it will be indeterminate whether the system is located in R.
- 22.
In the case of quantum field theory, Wallace advocates for spacetime state realism, which understands the universal quantum state in terms of density operators assigned to regions of spacetime. On this view, the fundamental ontology includes spacetime (regions) and properties corresponding to density operators (see Wallace & Timpson, 2010).
- 23.
For example, whether two events are simultaneous is indeterminate in special relativity. Such indeterminacy is naturally regarded as representational given that instants—global planes of simultaneity—are artifacts of our representation of spacetime. Analogously, if the Everettian multiverse is “a more useful description of an entity whose perfect description as a physical system lies (at least for the moment) beyond our ability to comprehend directly”(Wallace, 2002, p. 654), then any indeterminacy it engenders will be representational as well.
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Acknowledgements
Many thanks to Claudio Calosi, Sam Fletcher, Dana Goswick, Peter Lewis, Cristian Mariani, Alyssa Ney, Paul Teller, Jessica Wilson, and audiences at the Dartmouth Workshop on Quantum Indeterminacy, the University of California Davis, the Society for the Metaphysics of Science, and the California Quantum Interpretation Network.
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Glick, D. (2022). Quantum Mechanics Without Indeterminacy. In: Allori, V. (eds) Quantum Mechanics and Fundamentality . Synthese Library, vol 460. Springer, Cham. https://doi.org/10.1007/978-3-030-99642-0_21
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