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Multi-field and Bohm’s theory

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In the recent literature, it has been shown that the wave function in the de Broglie–Bohm theory can be regarded as a new kind of field, i.e., a “multi-field”, in three-dimensional space. In this paper, I argue that the natural framework for the multi-field is the original second-order Bohm’s theory. In this context, it is possible: (i) to construe the multi-field as a real-valued scalar field; (ii) to explain the physical interaction between the multi-field and the Bohmian particles; and (iii) to clarify the status of the energy–momentum conservation and the dynamics of the theory.

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  1. Note that the absolute square refers to the probability density of finding the particle in a given region if we perform a measurement on that region, which is different from the probability density for the particle’s being in a given region of space, independently from the measurement. Classical statistical mechanics is of the latter type, while quantum mechanics of the former type. Sometimes, this difference is overlooked in physics textbooks, making the conceptual structure of standard quantum mechanics less problematic than it really is.

  2. However, it is important to note that the Everett theory does not explain and describe physical processes at the quantum scale in the same way as the GRW and dBB theories do. In fact, while the latter have a primitive ontology of matter (matter density/flash or point-particles, respectively) and explain measurement outcomes in terms of these elements, a description of the same kind is missing in the Everett theory. In this theory, measurement-like interactions still have a crucial importance.

  3. Dürr, Goldstein and Zanghì do not speak in terms of information. However, there is a common idea between their nomological interpretation and Bohm’s informational view, i.e. the idea that the wave function is a mathematical abstract entity that dictates, describes, or encodes the dynamics of the Bohmian particles.

  4. The idea of the multi-field was originally proposed by Forrest (1988) in standard quantum mechanics. However, in that framework, the multi-field notion is extremely unintuitive and ontologically obscure. We can credit Belot (2012) with re-habilitating this option in Bohm’s theory.

  5. A more recent presentation can be found in Goldstein and Zanghì (2013).

  6. Historically, the original definition of the effective wave function in Bohm’s theory as the “collapsed” wave function for isolated subsystems is due to Bohm and Hiley (1987, sect. 4).

  7. The Hamiltonian is defined on all the phase space points of the system, i.e., it takes as variables not just the actual positions and velocities of the particles but all the possible positions and velocities. However, since this will not be relevant for the argument above, I prefer to keep a simple notation.

  8. The multi-field view, however, immediately solves the problem of configuration space by defining the wave function as a (new type of) field in three-dimensional space, while the problem of no-back reaction remains an open issue in this context, and will be discussed in Sect. 6.3.

  9. The Wheeler–deWitt equation indeed has many solutions, and to get to a unique solution one needs to add extra boundary conditions.

  10. One may still argue that the boundary conditions producing static universal wave function should be regarded as nomologically necessary. However, this would be physically odd, since the boundary conditions imposed on the cosmological models are usually set freely, and can vary from one model to another.

  11. I write the equation in the position basis here and, for the sake of simplicity, for a 1-particle system.

  12. See Callender (2015, sect. 2) for a clear exposition of the problems and Albert (2015, ch. 6) for a possible reply.

  13. See, e.g., Bacciagaluppi and Valentini (2009, Ch. 2).

  14. It is worth noting, however, that the amplitude R and the phase S are not independent terms, for they are dynamically coupled via the continuity equation (11).

  15. Two different approaches have been proposed in the literature to explain why the initial particle configuration of a Bohmian system is distributed according to \(|\psi |^2\): the typicality approach by Dürr et al. (1992) and the relaxation dynamical approach by Valentini (1991). A comprehensive and updated review of the two approaches is given by Norsen (2018).

  16. It is worth noting that the amplitude R has a double role in Bohm’s theory: on the one hand, it fixes the acceleration of the particles through the quantum potential Q (dynamical role) and, on the other hand, it determines the \(|\psi |^2\) probability distribution of the particles’ configuration (probabilistic role). Why R plays such a double role, and whether these two roles are connected at some other level, are open questions.

  17. For the derivation of the velocity formula from the continuity equation, see, e.g., Sakurai (1994, pp. 101–102).

  18. For a technical analysis of this point, see e.g. Goldstein and Struyve (2015).

  19. S is called “Aristotelian” potential because \(\nabla {S}\) determines the velocity of the particles.

  20. It is an interesting question, however, whether pointer states may be possibly correlated not only with real numbers (as it usually happens in a laboratory) but also with complex numbers. Thanks to an anonymous referee for this remark.

  21. The idea of splitting the multi-field into two scalar fields is discussed also in Chen (2018).

  22. See, e.g., Holland (1993, Ch. 6) for an approach of this kind.

  23. See e.g. Holland and Philippidis (2003).

  24. The term “mediated field” indicates the self-produced field of the charged particle.


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I want to thank Mario Hubert, Dustin Lazarovici, Matteo Morganti and Antonio Vassallo for valuable feedbacks and discussions on earlier drafts of this paper, and two anonymous referees for detailed and helpful comments. This work has been partially supported by the Austrian Agency for International Cooperation in Education and Research (OeAD-GmbH) and the Centre for International Cooperation & Mobility (ICM) through the Ernst Mach Grant (Reference Number: ICM-2018-10065).

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Romano, D. Multi-field and Bohm’s theory. Synthese 198, 10587–10609 (2021).

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