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Stochasticity and Bell-type quantum field theory


This paper critically discusses an objection proposed by Nikolić against the naturalness of the stochastic dynamics implemented by the Bell-type quantum field theory, an extension of Bohmian mechanics able to describe the phenomena of particles creation and annihilation. Here I present: (1) Nikolić’s ideas for a pilot-wave theory accounting for QFT phenomenology evaluating the robustness of his criticism, (2) Bell’s original proposal for a Bohmian QFT with a particle ontology and (3) the mentioned Bell-type QFT. I will argue that although Bell’s model should be interpreted as a heuristic example showing the possibility to extend Bohm’s pilot-wave theory to the domain of QFT, the same judgement does not hold for the Bell-type QFT, which is candidate to be a promising possible alternative proposal to the standard version of quantum field theory. Finally, contra Nikolić, I will provide arguments in order to show how a stochastic dynamics is perfectly compatible with a Bohmian quantum theory.

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  1. 1.

    The main ideas concerning Bell’s view on the ontology of physical theories can be found in Bell (1975).

  2. 2.

    Details concerning this methodology are contained in Allori et al. (2008, 2014) and Esfeld (2014).

  3. 3.

    In this paper I follow the interpretation of BM contained in Dürr et al. (2013b), which differs with respect to Bohm’s original version of the pilot-wave theory, where the wave function is considered a real physical field.

  4. 4.

    For an introduction to BM and to these foundational issues see Dürr et al. (2013b), Dürr and Teufel (2009), Bricmont (2016) and Bell (1987) and the fundamental papers Bohm (1952a, b).

  5. 5.

    In this regard the reader may refer to Barrett (2014). In particular, Dürr et al. (2004b) provides a clear analysis of the operator algebra in QM and BM and the unwelcome consequences of the identification between operators as observables.

  6. 6.

    This paper discusses also Bohmian QFTs implementing a field ontology.

  7. 7.

    For a detailed treatment of the measurement theory in BM see Bohm (1952b) and Dürr and Teufel (2009, Chapter 9). Since in his paper Nikolić neither proposes a new theory of measurement nor refers to a particular one, I suppose he tacitly assumes the standard Bohmian theory exposed in the mentioned references.

  8. 8.

    The support of the wave function is a region in configuration space where it has non-zero values.

  9. 9.

    This equation has been derived in Nikolić (2006). It is important to mention that (2) is equivariant if and only if we have probability distributions on \(\mathbb {R}^{4N}\) with density \(|\psi |^2\); this dynamical law fails to be equivariant considering probability distributions on \(\mathbb {R}^{3N}\) with density \(|\psi |^2\) setting all time variables to t. Plausibly, Nikolić may reply that his theory of particle creation and destruction is defined only in \(\mathbb {R}^{4N}\), circumventing this objection.

  10. 10.

    Within this theory bosons are not part of the ontology.

  11. 11.

    It is important to note that Bell himself repeatedly stressed that the choice of the beables for a given theory is not unique.

  12. 12.

    More precisely, the notation \([\dots ]^+\) considers only the positive part of the quantity between the squared brackets, setting the value equal to 0 whenever this quantity is negative; for details see Tumulka and Georgii (2005). Furthermore, the authors show that this is a special case of (4), i.e. the jump rate defined in Dürr et al. (2005).

  13. 13.

    In this regard it is important to underline different attitudes concerning Lorentz invariance in the context of Bohmian mechanics. One the one hand, there are exponents of the pilot-wave community e.g. Bohm, Valentini and Holland (among others) who think that a detailed microscopic description of the quantum mechanical regime must violate Lorentz invariance, but since it is not possible to have access to this description, there will occur no violation of the special theory of relativity; on the other hand, Dürr, Goldstein and Zanghì claimed that a genuine Lorentz invariant Bohmian theory is not in principle excluded. In both cases, however, the notion of absolute simultaneity must be recovered in order to define a guidance equation for the Bohmian particles, violating the spirit of special relativity (see Lienert 2011, sec. 4.1). For a detailed discussion the reader may refer to the following papers: Dürr et al. (1992, 2013a),Valentini (1991) and Butterfield (2007, sec. 7.1).

  14. 14.

    For details concerning identical particles in BM the reader should refer to Goldstein et al. (2005a, b). For the purpose of the paper it is sufficient to say that, being the particles identical, they are invariant under permutations: instead of having a given configuration of labeled particles, where position 1 is occupied by particle 1 and position n occupied by particle n, here we have a set of positions occupied by particles which could be permuted without affecting or modifying the particles’ configuration.

  15. 15.

    According to the theory, \(\Psi _t\) has the habitual double role: on the one hand, it guides the particles’ motion, on the other determines the statistical distribution of the particles’ positions.

  16. 16.

    This picture is taken from Dürr et al. (2004a).

  17. 17.

    For details see Dürr et al. (2005, Sections 6.1 and 6.2).

  18. 18.

    With this sentence my intention is not to claim that a Bohmian theory must necessarily implement a particle ontology: Bohm himself in the appendix of his Bohm (1952b) proposes a field ontology to extend his theory to electromagnetism. Furthermore, in literature exist many attempts to formulate a field ontology in the context of Bohmian QFTs, see Sec. 3 of Struyve (2010) for an overview.

  19. 19.

    For details on the similar structures of BM and GWR see Allori et al. (2008, 2014).


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I would like to thank Michael Esfeld, Anna Marmodoro, Davide Romano, Dustin Lazarovici, Mario Hubert, Olga Sarno and the anonymous referees for helpful comments on this paper. I am grateful to the Swiss National Science Foundation for financial support (Grant No. 105212-175971).

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Oldofredi, A. Stochasticity and Bell-type quantum field theory. Synthese 197, 731–750 (2020).

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  • Quantum field theory
  • Bohmian mechanics
  • Bell-type quantum field theory
  • Primitive ontology