Abstract
As mentioned in Chap. 2, different quantum theories without observers have been suggested and we shall present one now which is particularly well worked out: Bohmian mechanics—a quantum theory of particles in motion.
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Notes
- 1.
Foundations of Physics 12, 989 (1982). Reprinted in Bell, J. S. Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, Cambridge, 2004.
- 2.
Quoted from G. Bacciagaluppi and A. Valentini, Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, Cambridge University Press, Cambridge, 2009, p. 433.
- 3.
For more details see, e.g., D. Drr and S. Teufel, Bohmian Mechanics. The Physics and Mathematics of Quantum Theory. Springer, 2009.
- 4.
We must take the notion of macroscopic disjointness with a grain of salt, since it will only be true in an approximate sense (although to an excellent approximation), e.g., in the sense of L 2, which means that wave functions are close if their |Ψ|2-measures are close. In view of Born’s law (see Sect. 4.2), this sense of closeness is the empirically relevant one. In short,
$$\displaystyle \begin{aligned} \varPsi\approx\tilde{\varPsi}\Longleftrightarrow P^\varPsi\approx P^{\tilde{\varPsi}}\,. \end{aligned}$$ - 5.
Although macroscopic disjointness is clearly a vague notion, it is nevertheless sufficiently intuitive to be practical. To understand macroscopically disjoint y-supports, a configuration space picture might be helpful.
- 6.
See S. Goldstein und W. Struyve, On the uniqueness of quantum equilibrium in Bohmian mechanics. Journal of Statistical Physics 128 (5), 1197–1209 (2007).
- 7.
Actually, it suffices to require that every member of the ensemble have effective wave function φ i, from which the product structure of the ensemble wave function then follows. See, for example, D. Dürr and S. Teufel, Bohmian Mechanics. The Physics and Mathematics of Quantum Theory. Springer, 2009.
- 8.
Note that the subsystem coordinates X 1, …, X M are coarse-graining functions of the universal coordinates Q, where the universal configuration space is the fundamental Ω space in the sense of probability theory.
- 9.
D. Dürr, S. Goldstein, and N. Zanghì, Quantum equilibrium and the origin of absolute uncertainty. In: Quantum Physics Without Quantum Philosophy, Springer, 2013.
- 10.
Concerning its role at the beginning of quantum physics, de Broglie said in the Solvay conference Électrons et Photons, Paris 1928, p. 111: Il semble un peu paradoxal de construire un espace de configuration avec des coordonnées de points qui n’existent pas. (We leave the translation to the reader.) It should be noted that the configuration space of identical particles is an old concept, sitting right at the heart of Boltzmann’s statistical mechanics. But in classical physics the topology of the configuration space plays little role, contrary to the situation in quantum physics, as we shall see. In fact, it is the entanglement of the wave function that changes the picture.
- 11.
To be precise, we still need to prove that in a representation either γ τ = 1, ∀τ, or γ τ = −1, ∀τ. This is true because the permutation group can in fact be generated by elements of the form τ ∘ τ 0 ∘ τ for a fixed τ 0.
- 12.
For further details we refer to D. Drr, S. Goldstein, J. Taylor, R. Tumulka, and N. Zanghì, Topological factors derived from Bohmian mechanics, Ann. H. Poincar 7, 791–807 (2006), reprinted in D. Drr, S. Goldstein, and N. Zanghì, Quantum Physics Without Quantum Philosophy, Springer, 2013.
- 13.
The German word often used here is “unanschaulich”.
- 14.
For more on this, see D. Dürr and S. Teufel, Bohmian Mechanics. The Physics and Mathematics of Quantum Theory. Springer, 2009.
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Dürr, D., Lazarovici, D. (2020). Bohmian Mechanics. In: Understanding Quantum Mechanics . Springer, Cham. https://doi.org/10.1007/978-3-030-40068-2_4
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