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On the Galilean Invariance of the Pilot-Wave Theory

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Abstract

Many agree that the pilot-wave theory is to be understood as a first-order theory, in which the law constrains the velocity of the particles. However, while Dürr, Goldstein and Zanghì maintain that the pilot-wave theory is Galilei invariant, Valentini argues that such a symmetry is mathematical but it has no physical significance. Moreover, some wavefunction realists insist that the pilot-wave theory is not Galilei invariant in any sense. It has been maintained by some that this disagreement originates in the disagreement about ontology, as Valentini, contrary to Dürr, Goldstein and Zanghì, has been taken to endorse wavefunction realism. In this paper I argue that Valentini’s argument is independent of the choice of the ontology of matter: it is based of the notion of natural kinematics for a theory, and the idea that the kinematics should match the dynamics. If so, I also argue that there are several reasons to dispute Valentini’s claim that the kinematical symmetries should constrain dynamical ones.

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Notes

  1. See also Skow [2].

  2. Belousek [5], Solè [6].

  3. Allori [7, 8].

  4. A common distinction is between epistemic and ontic views of the wavefunction. According to the former, the wavefunction is not objective, it is not part of the ontology of the theory but rather it represents the observer’s state of knowledge of a physical system. Instead, ontic approaches have in common the idea that the wavefunction represents some objective feature of reality, even if they disagree about what this feature is.

  5. Notice that these distinctions are closely connected with the debates concerning the notion of primitive ontology (see Allori [9]). The idea is that a theory is more explanatory if matter in it is represented by some low-dimensional (most likely three-dimensional) mathematical object. So, for what is relevant for this paper, the wavefunction, understood as a field in a high dimensional or abstract space (configuration space or the like), is not a suitable material ontology. This is the case even if it is, broadly speaking, part of the ontology of the theory in the sense that it represents an objective feature of reality. I do not think that introducing the notion of primitive ontology is necessary here, and I am afraid it will only generate more confusion. Thus, I will write about ontology of matter (instead of primitive ontology) to refer to the variables in the theory which represent material objects, and ontology in general to include also the variables representing other objective feature, such as the forces, the potentials, the Hamiltonians, the velocities, and so on. In any case, materialists with respect to the wavefunction, as defined above, are those who go against the primitive ontology approach, while non-materialists are more likely to be sympathetic to it.

  6. Forrest [10], Belot [11], Hubert and Romano [12].

  7. Maudlin [13].

  8. Dürr, Goldstein and Zanghì [14], Goldstein and Zanghì [15].

  9. Monton [16], Suàrez [17], Deckert and Esfeld [18].

  10. Valentini [4, 19] Belousek [5].

  11. Allori [20].

  12. See Chen [21] for more ways of understanding the wavefunction in quantum theory, including the pilot-wave theory.

  13. See Goldstein [26], Valentini [4, 19], Belousek [5], Solé [31].

  14. See, for starters, the contributions in Albert and Ney [27].

  15. See Allori Zanghi [28].

  16. Albert [29]. For time reversal invariance, see also Callender [30].

  17. Allori [7]

  18. For a proposal on how to make the theory time reversal invariant even in this context, see Struyve [31]. The proposal, however, requires a different understanding of the field the wavefunction is supposed to be represent. For the way in which the nomological approach deals with this, see Allori [7].

  19. As discussed in other papers, DGZ show that one can obtain the guidance law also through other considerations. Nonetheless, it is always clear that DGZ regard the theory as Galilei invariant, regardless of how the guidance equation is obtained (see also Dürr and Teufel [32]).

  20. This type of argument has been put forward in the case of time reversal by other authors as well: Earman [33], Malament [34], Arntzenius [35], Arntzenius and Greaves [36], Roberts [37, 38].

  21. See also Allori [20, 7, 39].

  22. See also Solè [6].

  23. See also Brown et al. [40].

  24. Belousek [5], Solè [6].

  25. Holland [25] chapter 4, Bricmont [41] chapter 5.

  26. Even if it is controversial that they are truly relativistic invariant. In any case, see Dürr et al. [42] and references therein.

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Acknowledgements

I am very grateful to Jean Bricmont, Tim Maudlin, Travis Norsen, Antony Valentini, and Roderich Tumulka for helpful discussions on the issues discussed in this paper. A special thanks to Aurélien Drezet for having organized this collection to honor the work of de Broglie and Bohm, and to the journal reviewer, whose insightful comments have certainly made this paper better than it was before.

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    Valia Allori

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Allori, V. On the Galilean Invariance of the Pilot-Wave Theory. Found Phys 52, 111 (2022). https://doi.org/10.1007/s10701-022-00631-4

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