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Can the Ontology of Bohmian Mechanics Consists Only in Particles? The PBR Theorem Says No

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Abstract

The meaning of the wave function is an important unresolved issue in Bohmian mechanics. On the one hand, according to the nomological view, the wave function of the universe or the universal wave function is nomological, like a law of nature. On the other hand, the PBR theorem proves that the wave function in quantum mechanics or the effective wave function in Bohmian mechanics is ontic, representing the ontic state of a physical system in the universe. It is usually thought that the nomological view of the universal wave function is compatible with the ontic view of the effective wave function, and thus the PBR theorem has no implications for the nomological view. In this paper, I argue that this is not the case, and these two views are in fact incompatible. This means that if the effective wave function is ontic as the PBR theorem proves, then the universal wave function cannot be nomological, and the ontology of Bohmian mechanics cannot consist only in particles. This incompatibility result holds true not only for Humeanism and dispositionalism but also for primitivism about laws of nature, which attributes a fundamental ontic role to the universal wave function. Moreover, I argue that although the nomological view can be held by rejecting one key assumption of the PBR theorem, the rejection will lead to serious problems, such as that the results of measurements and their probabilities cannot be explained in ontology in Bohmian mechanics. Finally, I briefly discuss three \(\psi\)-ontologies, namely a physical field in a fundamental high-dimensional space, a multi-field in three-dimensional space, and RDMP (Random Discontinuous Motion of Particles) in three-dimensional space, and argue that the RDMP ontology can answer the objections to the \(\psi\)-ontology raised by the proponents of the nomological view.

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Notes

  1. Throughout this paper, the word “ontic” or “ontology” denotes only material ontology which does not contain laws of nature unless otherwise stated. This is consistent with the use of the word in the PBR theorem and relevant literature. Note that some Bohmians use the word “ontology” to denote a general ontology which contains also laws of nature [3, 33].

  2. Note that Bohmian mechanics is also compatible with primitivism about laws as suggested by Maudlin [33]. I will discuss whether my analysis applies to this view additionally.

  3. Romano [38] already noticed that the metaphysical status of effective wave functions in Bohmian mechanics is not well defined within the nomological view of the universal wave function, although he did not give an analysis of the issue.

  4. Here it is worth pointing out that several authors have recently argued that the ontological models framework and the PBR theorem have certain limitations, and in particular, they do not apply to Rovelli’s [39] relational quantum mechanics, which employs ontic states dealing with relational properties but regards the wave function as a computational device encoding observers’ information [35, 36]. However, these limitations do not affect my analysis of the implications of the PBR theorem for the nomological view of the wave function.

  5. It can be readily shown that different orthogonal states correspond to different ontic states based on the ontological models framework. Thus the proof given here concerns only nonorthogonal states.

  6. If the properties of Bohmian particles are only position and velocity as usually thought, then it will be obvious that the effective wave function does not represent a physical property of these particles, and thus the nomological view of the universal wave function is not compatible with the ontic view of the effective wave function. But, as Esfeld et al [18] thought, there may exist other properties of Bohmian particles besides position and velocity, such as nonlocal interactions between these particles, so that the effective wave function of a subsystem encodes the nonlocal influence of other particles on the subsystem. The purpose of my following analysis is to exclude this possibility.

  7. In this sense, it is sometimes said that the effective wave function is quasi-nomological. However, as noted by Romano [38], it is still not clear what “quasi-nomological” means. This paper does not aim to clarify the quasi-nomological status of effective wave functions but aims to argue that no matter what “quasi-nomological” means, the nomological view is inconsistent with the result of the PBR theorem.

  8. It has been recently argued that the ontological models framework is wrong in representing quantum theories [8]. My opinion is that this claim is debatable (see also [30]). Even if the ontological models framework is wrong, a realist can still prove the result that two orthogonal states correspond to two distinct ontic states (e.g. by an analysis of the Mach–Zehnder interferometer where the same ontic state cannot result in two different definite interference results). This result is enough for us to argue against the nomological view; on this view, two orthogonal states such as two energy eigenstates of a particle in a box may correspond to the same ontic state, i.e. the two Bohmian particles being in these two states may be at rest in the same position in the box. Besides, it is worth noting that the question of whether Bohmian mechanics is consistent with the ontological models framework has been discussed by several authors [14, 19, 32]. Here I will focus on the issue of whether the nomological view of the wave function is consistent with the ontological models framework.

  9. Note that when the effective wave function is not nomological but related to the state of reality, the second assumption of the ontological models framework should not be revised this way but keep unchanged, since the complete ontic state \(\lambda\) already includes all parts of the state of reality (see also [14, 15, 32]).

  10. Similarly, the proof that different orthogonal states correspond to different ontic states can also be blocked by the revised assumption.

  11. This is possible as can be seen from the previous example. In the example, the effective wave function of the measured system is an energy eigenstate in a box. The ontic state of the measured system, which is represented by the position of its particle in the box, may be the same for different energy eigenstates. Moreover, the ontic states of the measuring devices are supposed to be the same before different measurements.

  12. There is also a similar example which can reveal the problem of the nomological view without referring to measurements. Suppose there are two electrons which have different effective wave functions but whose Bohmian particles have the same position and velocity initially. Then, Bohmian mechanics will predict that these two Bohmian particles may have different positions and velocities at a later time. Since no laws of motion can lead to that two identical particles, which have the same ontic state initially, have different ontic states later, it seems that there cannot be only particles in ontolgy in Bohmian mechanics. I will analyze this issue in more detail in future work.

  13. According to the RDMP interpretation of the wave function, a quantum system is composed of particles with mass and charge, which undergo random discontinuous motion in three-dimensional space, and the wave function represents the propensities of these particles which determine their motion, and as a result, the state of motion of particles is also described by the wave function. At each instant all particles have a definite position, while during an infinitesimal time interval around each instant they move throughout the whole space where the wave function is nonzero in a random and discontinuous way, and the probability density that they appear in every possible group of positions in space is given by the modulus squared of the wave function there.

  14. The proponents of wave function realism may insist that the high-dimensional configuration space is a real, fundamental space, while the three-dimensional space we perceive is somehow illusory. However, this leads to the well-known problem of how to explain our three-dimensional impressions (see [34], chap.8 for the latest analysis of this problem).

  15. It seems that even for wave function realists and multi-field ontologists this objection is not a very strong argument either. They may claim that the physical entity represented by the wave function is a new type of entity and it is not subject to back reaction (see, e.g. [31, 38]).

  16. However, global existence and uniqueness for the Bohmian dynamics can be proven [5, 40].

  17. Certainly, this consistency is still obtained by the postulate that the RDMP dynamics is in agreement with Born’s distribution. But it is arguable that one unified postulate is better than two separate postulates. Moreover, there are plausible arguments supporting the RDMP ontology (see [22,23,24]).

  18. Even if the Bohmian particles are assumed to have mass and charge, these properties have no causal efficacy required in ontology such as deviating the pointer of a measuring device. By contrast, the mass and charge of a particle in the RDMP ontology have such required causal efficacy.

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Acknowledgements

I am grateful to Valia Allori, Eddy Keming Chen, Hrvoje Nikolić, Tim Maudlin and Howard Wiseman for helpful discussion.

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    Shan Gao

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Gao, S. Can the Ontology of Bohmian Mechanics Consists Only in Particles? The PBR Theorem Says No. Found Phys 53, 91 (2023). https://doi.org/10.1007/s10701-023-00731-9

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