Abstract
Recent philosophical discussions about metaphysical indeterminacy have been substantiated with the idea that quantum mechanics, one of the most successful physical theories in the history of science, provides explicit instances of worldly indefiniteness. Against this background, several philosophers underline that there are alternative formulations of quantum theory in which such indeterminacy has no room and plays no role. A typical example is Bohmian mechanics in virtue of its clear particle ontology. Contrary to these latter claims, this paper aims at showing that different pilot-wave theories do in fact instantiate diverse forms of metaphysical indeterminacy. Namely, I argue that there are various questions about worldly states of affairs that cannot be determined by looking exclusively at their ontologies and dynamical laws. Moreover, it will be claimed that Bohmian mechanics generates a new form of modal indeterminacy. Finally, it will be concluded that ontological clarity and indeterminacy are not mutually exclusive, i.e., the two can coexist in the same theory.
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Notes
For the sake of simplicity, in this paper I will use “indeterminacy” and “indefiniteness” as synonyms. Although not technically precise, this will not cause issues for the proposed arguments. For details cf. Torza (2023) p. 1.
For space reasons, in this essay I will focus exclusively on Bohmian mechanics. For a discussion of MI within GRW theories the reader may refer to Mariani (2022).
Lewis (2022) seems to endorse such conditions. I thank Referee 2 for pointing this out.
Such conditions can be trivially extended to any other property. Indeed, Lewis proposes also an example involving the energy of Bohmian particles, cf. Lewis (2016), p. 103.
The example of classical macroscopic objects is obvious but useful: although we know that they are ontologically reduced to their microscopic constituents, we believe that objects like tables, chairs, mountains etc., do exist in the world. In this case reduction does not amount to elimination, but it is conservative.
This view about spin has been employed by the present author in Oldofredi (2020a) to show that BM as defined in Dürr et al. (2013) generates a distributive lattice of propositions. Although that essay draws a conclusion valid for the mentioned account of BM, it does not necessarily reflect my current opinion concerning the status and role of spin in pilot-wave theory.
It is worth noting that interpreting spin as a real attribute of the Bohmian particles does not automatically entail indeterminacy as shown in Dewdney et al. (1987). Moreover, Lewis suggests ways to escape indeterminacy in Bohm’s approach, as for instance employing retrocausality. It is outside the scope of the present paper to analyze the latter proposal.
For a recent discussion of Bohmian models with spin and their application to quantum chemistry cf. Lombardi and Fortin (2024).
More on this issue in the next section.
To this specific regard, it should be underlined that Deutsch (1996) and Brown and Wallace (2005) argued that Bohmian hidden parameters are explanatorily irrelevant and thus the casual approach could have been reduced to a many worlds interpretation in denial. However, Lewis (2007) and Valentini (2010) convincingly answered these objections against the pilot-wave theory. I will not consider this discussion in what follows being not strictly relevant to my argument.
In Bohm and Hiley (1993) the authors seem to realize that a full-fledged realist interpretation of \(\psi \) as a physical field may entail the presence of indeterminacy. Indeed, in their joint work they proposed a new interpretation of the wave function as a carrier of information. The selected wave packet then contains what the authors call “active” information—that is actually guiding and determining the motion of the particle—whereas the empty packets represent only inactive or potential information that can be neglected, playing no dynamical role for the particles’ trajectories. In this manner there are no real empty waves spreading and interacting in space.
Repetita iuvant: In this paper my aim is to defend this argument from common objections, not to show that is necessarily valid. Clearly, one may reject one or more assumptions contained in Lewis’ arguments, as for instance the branch-relative fuzzy or the branch-relative vague links.
Contrary to Bohm’s theory, in BM the wave function has a nomological character in the sense specified by Goldstein and Zanghì (2013). Thus, it is not considered physically real as well as the quantum potential, which loses its status of genuine quantum force.
As the authors remark, they do not question the uniqueness of solutions for a given velocity field.
It is worth noting that Deotto and Ghirardi (1998) discuss which formal requirements must be imposed to \(v^{\psi }_{A_k}(Q)\) in order to guarantee the existence of trajectories and the validity of the continuity equation. These technical details, however, are not needed for the purpose of the present discussion.
Cf. Fankhauser and Dürr, 2021, p. 18 for similar views.
One may certainly reply that such indeterminacy is only representational, claiming that it affects how BM represents reality but not reality per se. This criticism, however, would apply to any account of quantum indeterminacy because each of them seems to rest on the following implicit premise: if QM were the correct description of the microphysical world, then from its formalism, theorems, etc., one can derive the presence of various forms of indeterminacy, e.g. indeterminacy of identity and/or indeterminacy of observables. Moreover, it is usually argued, given that QI is neither semantic, nor epistemic, then it must be metaphysical. Trivially, without such an implicit assumption, i.e. rejecting the hypothesis that QM provides an approximately true representation of reality, no account of QI would be metaphysically salient (cf. Skow, 2010). The same reasoning applies in the case of Bohmian mechanics.
This fact will be analyzed in more detail in Section 4.
For a discussion of Nikolić’s perspective in the context of Bohmian quantum field theory cf. Oldofredi (2020).
Bell’s original model introduces a discrete lattice to model spacetime, the beables of this framework are fermion number densities defined at each node of the lattice.
If one takes into consideration only a single species of particles, e.g. the electrons, the configuration space is
$$\begin{aligned} \mathcal {Q}=\bigcup _{N=0}^\infty \mathcal {Q}^{[N]}_{e^-}, \end{aligned}$$where \(\mathcal {Q}^{[N]}=\mathbb {R}^{3N}/\text {permutations}\). Describing more than a single particle species, the total configuration space is the Cartesian product of the involved particle sectors. The example proposed by the authors is the configuration space of Quantum Electro-Dynamics, which involves three different species of particles: electrons, positions and photons.
Notably, although there are many possible choices in order to describe the jump rates, the authors provide a rigorous proof concerning the existence of a unique “minimal” jump rate, in the sense that at any time t only one between these two jumps \(q_1 \rightarrow q_2\) and \(q_2 \rightarrow q_1\) is allowed.
The results on the global existence, coherence and uniqueness of the jump rates in BTQFT can be found in Dürr et al. (2005) and Tumulka and Georgii (2005). In the former essay it has been shown that if the particle configuration \(Q(t_0)\) is chosen randomly with distribution \(|\Psi (t_0)|^2\), then at any later time t it will be distributed with density \(|\Psi (t)|^2\). This result is the extension of equivariance in the context of regularized QFT.
Here we do not want to conflate indeterminism with indeterminacy, which are logically distinct notions. However, it is our aim to show that there exist connections between the two.
It is outside the scope of the present paper to study whether multiple forms of indeterminacy coexist in the same pilot-wave theory. For instance, we can safely conclude that the indeterminacy of BTQFT cannot be extended to the other Bohmian frameworks analyzed here, as well as one may similarly claim that the sort of indeterminacy present in Nikolić’s QFT is not to be found either. More interesting and complex are the cases involving the reality of spin and modal indeterminacy. To see whether the latter applies to other Bohmian theories and whether spin generates indeterminacy also in BM and its extensions to QFT will be left for future research.
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Acknowledgements
I warmly thank the Editor of this Special Issue Cristian Mariani for the useful conversations about this paper. I have to thank Antonio Vassallo for his valuable remarks on the nature of empty waves in pilot-wave theory and Olga Sarno for helpful comments on previous draft of the paper. Moreover, my gratitude goes to the two anonymous referees, their constructive criticisms and suggestions greatly improved the present essay. Finally, I am grateful to the Fundaç\(\tilde{\textrm{a}}\)o para a Ciência e a Tecnologia (FCT) for financial support (Grant no. 2020.02858.CEECIND).
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Oldofredi, A. Unexpected quantum indeterminacy. Euro Jnl Phil Sci 14, 15 (2024). https://doi.org/10.1007/s13194-024-00574-9
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DOI: https://doi.org/10.1007/s13194-024-00574-9