Abstract
It is often claimed that there are three “realist” versions of quantum mechanics: the de Broglie-Bohm theory or Bohmian mechanics, the spontaneous collapse theories and the many worlds interpretation.
We will explain why the two latter proposals suffer from serious defects coming from their ontology (or lack thereof) and that the many worlds interpretation is unable to account for the statistics encoded in the Born rule. The de Broglie-Bohm theory, on the other hand, has no problem of ontology and accounts naturally for the Born rule.
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Notes
- 1.
Unless the wave function happens to be an eigenstate of some operator.
- 2.
Technically, it means that the operators associated to those properties or “observables” commute., More precisely, the Bell-Kochen-Specker result implies that, if \(\mathcal H\) is a Hilbert space of dimension at least four, and if \({\mathcal A}\) is the set of self-adjoint operators on \(\mathcal {H}\), there does not exist a map \(v: {\mathcal A} \to R\) such that:
-
(1)
\(\forall O \in { {\mathcal A}}\),
$$\displaystyle \begin{aligned} v(O) \;\;\mbox{is an eigenvalue of} \;\;O. \end{aligned}$$ -
(2)
\(\forall O, O' \in { {\mathcal A}}\) with [O, O′] = OO′− O′O = 0,
$$\displaystyle \begin{aligned} v(O O')=v(O) v(O'). \end{aligned}$$
Obviously, there cannot be a statistical distribution of maps that do not exist.
-
(1)
- 3.
This idea is emphasized in Tumulka (2018) Sect. 5.1.
- 4.
Superposition refers to sums such (17.1) (Note of J.B.).
- 5.
- 6.
- 7.
For more on all this, see e.g. Maudlin (2019) (Chap. 6).
- 8.
The famous quote “The Lord God is subtle, but malicious he is not” is due to Einstein.
- 9.
This problem is similar to the one posed by postulating a lower degree of existence for low probability worlds, see Sect. 17.2.1.
- 10.
- 11.
In Norsen (2014), the author explains why the pure wave function ontology is solipsism “for all practical purposes” because the world of the pure wave function ontology has a status similar to the idea that we are just brains in a vat, with our brains manipulated from the outside in just such a way as to make our illusory conscious experiences what they are. For more discussion of this ontology, see the collection of essays edited by Ney and Albert (2013).
- 12.
- 13.
In actual fact, in GRW theories, wave functions do not collapses exactly to 0, but to a very small value. This raises additional problems but we will not discuss them.
- 14.
Our presentation of the de Broglie-Bohm theory follows the one of Bell (2004) and of Dürr et al. (2013) rather than the one of Bohm (1952a,b). This approach is actually close to the original one of de Broglie, see Bacciagaluppi and Valentini (2009).
Many expositions of the de Broglie–Bohm theory are available, see, e.g., Albert (1994) or Tumulka (2004) for elementary introductions and Bacciagaluppi and Valentini (2009), Bohm and Hiley (1993), Bricmont (2016), Dürr and Teufel (2009), Dürr et al. (2013), Goldstein (2013), Towler (2009) for more advanced ones. There are also pedagogical videos made by students in Munich, available at: https://cast.itunes.uni-muenchen.de/vod/playlists/URqb5J7RBr.html.
- 15.
For example, one could “measure” the spin of a particle in a given direction, with a Stern-Gerlach apparatus, starting with exactly the same initial wave function and initial position for the particle, but with two different orientations of the gradient of the magnetic field in the apparatus, and obtain two opposite results; thus no “spin property” of the particle has been measured; see Daumer et al. (1996), Bricmont et al. (2019) for more details.
- 16.
See Maudlin (2011) (Chap. 10) for a similar discussion of the GRWm and GRWf ontologies, as opposed to the de Broglie–Bohm ontology.
- 17.
Some people might argue that cutting the box in two already collapses the wave function. Since we discuss here gedanken experiments, it is not easy to determine what happens. But here is a modification of that experiment that avoids this objection: imagine that the box is inside a long cylinder, stretching from New York to Tokyo if you wish, and that we remove the two sides of the box perpendicular to the long side of the cylinder. Then the wave function will spread itself in the cylinder and the particle can be detected later on one side or the other of the cylinder.
- 18.
For other criticisms of the GRWf theory, in particular in what sense is it really Lorentz invariant, see Esfeld and Gisin (2014).
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Acknowledgements
I thank two referees for useful comments, Sheldon Goldstein and Tim Maudlin for many discussions on Bohmian mechanics and the many-worlds theories, and specially Valia Allori for illuminating exchanges on the topics discussed here.
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Bricmont, J. (2022). Ψcat alive and Ψcat dead Are not Cats! Ontology and Statistics in “Realist” Versions of Quantum Mechanics. In: Allori, V. (eds) Quantum Mechanics and Fundamentality . Synthese Library, vol 460. Springer, Cham. https://doi.org/10.1007/978-3-030-99642-0_17
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