Abstract
In Sect. 2.3.1, the measurement problem was formulated in the form of a trilemma. In this view, either (i) the wavefunction is not a complete description; or (ii) the time evolution is not a continuous unitary process; or (iii) measurements do not lead to well-defined results. The GRW theory described in Sect. 2.3.1 chooses alternative (ii); it adds a nonlinear term to the Schrödinger equation, which models a physical mechanism for the “actual” collapse of the wavefunction. The Copenhagen interpretation also denies a continuous time evolution which follows the Schrödinger equation; in contrast to the GRW theory, this process is however not given a realistic interpretation.
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Notes
- 1.
Bohm’s lack of knowledge of the earlier work is understandable if one is aware that de Broglie himself did not develop his theory further, but instead became a supporter of the “conventional” quantum theory. Only after reading Bohm’s publication of 1952 was his interest in these questions again aroused.
- 2.
We shall see that the lack of knowledge (and control) of the precise initial conditions plays an important role in the DBB theory. This aspect of the additional variables can indeed be considered to be “hidden”. Furthermore, the concept of “hidden variables” also refers to the fact that they do not occur in the standard interpretation.
- 3.
- 4.
Configuration space is of central importance even in conventional quantum theory, because the wavefunction is likewise defined on this space.
- 5.
There is, however, a decisive difference with respect to the hydrodynamic equation of continuity: While the mass density \(\rho _m\) is defined on real position space, the probability density \(\rho =|\psi |^2\) is a function on configuration space. A naive identification of \(|\psi |^2\) with a matter density thus appears to be impossible.
- 6.
In fact, the condition of being able to reproduce the statistical predictions of quantum mechanics does not fix the dynamics uniquely. In this sense, there are indeed infinitely many “de Broglie–Bohm-like” theories. In these theories, the individual trajectories do not follow Eq. (5.5), but they however reproduce the same statistics (Deotto and Ghirardi 1998).
- 7.
The equivalence to quantum mechanics presumes that all predictions can be uniquely described in terms of position coordinates—e.g. by “pointer positions” of a measurement apparatus.
- 8.
Speculations about “multiverses” change nothing in this situation—for there, also, as a rule any contact to the other “universes” is forbidden.
- 9.
The effective wavefunction \(\psi (x)\) of a subsystem with the variables x on configuration space, which belongs to the overall system \(\Psi (x,y)\), is defined as a part of the following decomposition: \(\Psi (x,y)=\psi (x)\Phi (y)+\Psi ^{\perp }(x,y)\). Here, \(\Phi \) and \(\Psi ^{\perp }\) have disjunct carriers, and the configuration of the environment (Y) lies in the carrier of \(\Phi \). For the overall system, one could think for example of subsystem \(+\) environment or, concretely, subsystem \(+\) measurement apparatus. The above decomposition occurs namely during a measurement interaction: If the configuration of the measurement setup corresponds to Y (this could be a particular “pointer position” of the measurement apparatus), the x system is guided by the wavefunction \(\psi (x)\). The remaining parts of \(\Psi \) are then irrelevant for the particle dynamics, and in this way, an “effective collapse” is described (cf. Sect. 5.1.5).
- 10.
Our treatment here could of course only roughly sketch the train of reasoning, and it suppresses many mathematical details. Thus, an impression of circularity may have (falsely) arisen: One postulates the \(|\Psi |^2\) distribution of the universe and obtains the \(|\psi |^2\) distribution of subsystems. See Dürr (2001, p. 201) for more on this topic.
- 11.
The following is naturally difficult to understand for those readers who are not well versed in mathematics. The decisive point is that the position (and velocity) of the Bohmian particles are mathematically determined by the wavefunction.
- 12.
In the case of \(\alpha \) decay, helium nuclei overcome the potential barrier at the surface of the decaying nucleus, although their energies, considered classically, are too small to permit this. In the case of nuclear fusion in the interior of the sun, hydrogen atoms combine to form helium. Here again—considered classically—the pressure and temperature are too low to overcome the repulsion of the positively charged hydrogen nuclei.
- 13.
This manner of speaking, “tunnelling” or “passing beneath”, is naturally to be understood as metaphorical, since the “height” of the potential barrier is not a spatial quantity, but rather an energy.
- 14.
The wavefunction is Gaussian, with its centre initially at 0.5 and a width (variance) of 0.05. The density of the trajectories between 0.66 and 0.68 was increased in order to be able to study the oscillatory behaviour within the barrier more precisely (see Dewdney and Hiley 1982).
- 15.
- 16.
In the excited states, where \(m \ne 0\), the azimuthal-angle \(\phi \) is time-dependent, and the Bohmian particle orbits around an axis (see Passon 2010, pp. 87f ). Note that also this motion does not correspond to Bohr’s atomic model, which is well known (and rebutted) in school physics.
- 17.
Formally speaking, we are considering here the superposition of several states of the overall system, consisting of a measurement object (\(\psi =\sum c_i\psi _i\)) and a measurement apparatus (\(\Phi _i\)). In the case that the measurement apparatus is initially at the position \(\Phi _0\), during the measurement interaction it undergoes a time evolution \(\psi \otimes \Phi _0 \rightarrow \sum c_i \psi _i\otimes \Phi _i\). Here, \(\Phi _i\) denotes the state of the apparatus after the measurement, on measuring the property which is associated with the state \(\psi _i\).
- 18.
At this point, it again becomes clear that the term “hidden variables” for the particle positions is misleading. It is just their un-hiddenness which qualifies them to describe the observable results of a measurement!
- 19.
The same is true of all other physical quantities. The particles in the de Broglie–Bohm theory have no properties besides their positions and their velocities. Even mass, momentum or charge cannot reasonably be attributed to the particles; think for example of quantum-mechanical interference experiments in which the influence of gravity or an electromagnetic interaction can (in principle) modify the wavefunction. Therefore, we have thus far avoided referring to the “Bohmian particles” as “electrons”, “atoms”, etc. However, in Holland (1993) as well as in Bohm and Hiley (1993), a possible spin variable is discussed. Our treatment here follows Bell (2001, pp. 5ff) and Dürr (2001).
- 20.
While categorial properties are associated with an object without any reference to its environment (e.g. “being round”), dispositions describe those properties which manifest themselves only in certain special contexts (e.g. “being fragile”).
- 21.
- 22.
Everett himself writes of his methodology: “The wavefunction is taken as the basic physical entity with no a priori interpretation. Interpretation comes only after an investigation of the logical structure of the theory. Here as always the theory itself sets the framework for its interpretation” (Everett 1957, p. 455).
- 23.
He models the “observer” by a physical system, in the concrete case a machine which has access to sensors and storage media.
- 24.
It is very questionable as to what extent this suggestion corresponds to Everett’s own understanding of the theory. Since Everett worked in the strategic planning department of the Pentagon after finishing his doctorate, and published no more work on quantum theory, this question can be answered only by consulting his sporadic correspondence and papers from his estate. These sources give the impression that Everett did not have a splitting up into different “worlds” in mind, whose definition would seem to make a connection with classical concepts. In some respects, the current version of the many-worlds interpretation, which we will discuss more detail in the following sections, appears to be more similar to Everett’s original conception. However, he did not categorically reject the language of DeWitt—especially since he was very grateful to the latter for the popularization of his ideas. See Barrett (2011), and the essay by Peter Byrne in Saunders et al. (2010), for more on this subject.
- 25.
This spacetime is subject to splitting only when the many-worlds idea is applied to theories of quantum gravitation.
- 26.
The ambiguity of the representation is the subject of the “biorthogonal decomposition theorem” (cf. Bub 1997, p. 151). The decomposition is unique if and only if all the components have different and nonzero coefficients.
- 27.
Note that there are no common eigenvectors of \(\sigma _x\) and \(\sigma _z\).
- 28.
The problem treated here thus occurs in other interpretations of quantum mechanics as well, and it shows that the measurement problem actually consists of two sub-problems: (i) The problem of the preferred basis and (ii) the problem of the definite outcome of a measurement. Within, e.g., the Copenhagen interpretation, however, (i) can be resolved by specifying the measurement setup (choice of direction).
- 29.
Deutsch mentions here (on p. 2) that he has taken up an idea of Everett’s, based on private conversations with him.
- 30.
This position is called “physicalism”. Physicalism (expressed in simplified form) asserts the metaphysical hypothesis that everything which exists is physical. It can be understood as a further development of materialism. In particular, it rejects any kind of dualism between physical and mental (“mind”) states. The relation between physical and mental states is not necessarily an identity, however. In the philosophy of the mind, the viewpoint is widespread that these two property areas are connected through a “supervenience relation”. The supervenience of A over B is understood to mean that (in “slogan” form) “no change in A is possible without a change in B”. This also permits speculations on a possibly non-reductionistic physicalism.
- 31.
One may consider the astute self-observation on which this conclusion is based not to be a particularly powerful tool for philosophical reflection. But for questions involving our conscious minds, it is however our only tool!
- 32.
Just how Barrett means this for a non-physicalistic conception of the mind remains unclear.
- 33.
The adjective “coherent” in the general vocabulary means “connected”. The physical-terminological expression “coherent” is usually applied to optics, and it describes, roughly speaking, the precondition that must be fulfilled by different wavetrains in order that they may be able to interfere with each other. Expressed non-technically, the decoherence program thus attempts to clarify the conditions and preconditions under which quantum states lose this “non-classical” property.
- 34.
This tridecompositional uniqueness theorem is valid under rather general conditions. The existence of the decomposition is by the way not guaranteed. The proof of this theorem can also be found in (Bub 1997, Sect. 5.5).
- 35.
For the adherents of the de Broglie–Bohm theory, the results of decoherence, for example, permit a more exact justification of the so-called effective collapse of the wavefunction (cf. Sect. 5.1.5).
- 36.
Expressed technically, one takes the trace of the density matrix over the degrees of freedom of the environment. This makes it (in the preferred basis) approximately diagonal. The off-diagonal elements are however just what give rise to the interference effects.
- 37.
Now and then at scientific conferences, surveys are conducted (not always with complete seriousness) about which interpretation of quantum theory is favoured by the conference attendees. Tegmark (1998) reports the result of such a survey at a workshop on quantum theory; it found that the Everett interpretation was the preferred alternative to the Copenhagen interpretation.
- 38.
This theorem was discovered by Neil Graham in 1970, during his doctoral work which was mentored by DeWitt. Already in 1968, James Hartle had proved an equivalent result (Hartle 1968).
- 39.
The many-minds interpretation buys the solution to the probability problem at the price of a substance dualism, which is accepted in modern philosophy of the mind by only a small minority of philosophers. This problem motivated Lockwood (1996) to suggest a variant of the many-minds interpretation, which dispenses with dualism and a probabilistic dynamics. Ironically, it is however controversial as to whether or not Lockwood’s theory permits a plausible probability interpretation at all (see Barrett 1999, pp. 206–211).
- 40.
The concept of “rationality” is used here in a very narrow or weak sense. Decision theory investigates logical limitations of the preferences and makes no claim to determine them with regard to content. A typical rationality requirement is the transitivity of preferences: If I prefer action A over action B, and B over C, then A must be preferred over C.
- 41.
Therefore, the Everett variant of the representation theorem makes an even stronger statement than its counterpart in classical decision theory. The latter determines the probability measure only relative to the corresponding preferences of the agent. There are however several such preferences which fulfil the rationality conditions!
- 42.
The concept of “probability” is thus again avoided here.
- 43.
In fact, variants of scientific realism are also possible which attribute a valid claim of truthfulness to certain theories, while the entities in question are not considered to be realistic (see Russell’s position in Hacking 1983, p. 27).
- 44.
Albert and Loewer (1988) formulate, in contrast, a dualistic position in their many-minds interpretation.
- 45.
Thus, we have here a conceptional similarity to the de Broglie–Bohm theory, which is not surprising if one casts a glance at the list of authors: with Valia Allori, Sheldon Goldstein, Roderich Tumulka and Nino Zanghì, we find here several prominent supporters of Bohmian mechanics.
- 46.
This difficulty was pointed out to Schrödinger by Hendrik Antoon Lorentz in a letter from March, 1926 (Jammer 1974, p. 31).
- 47.
In Sect. 5.2.2, we have already mentioned that this variant of the many-worlds interpretation is non-local. The problem of the preferred basis and the role of probability statements can likewise be treated differently in this theory.
- 48.
The description of the “effective collapse” of the wavefunction in addition profits from the results of the work on decoherence.
- 49.
Bohr however saw no sort of causal relationship here; instead, he compared the influence of a measurement on its outcome with the connection between the frame of reference and the observations within the special theory of relativity.
- 50.
This “pairing” is intended to illustrate that even within the Copenhagen interpretation, a complete description of the physical world with reference to the wavefunction alone is not possible. The classic textbook of Landau and Lifschitz formulates this relation in a particularly pointed way: “Quantum mechanics thus occupies a rather remarkable position among physical theories: It contains classical mechanics as a limiting case, and at the same time, it requires this limiting case for its own justification” (Landau and Lifschitz 2011, p. 3).
References
Albert, David. 2010. Probability in the Everett picture. In Saunders, et al., 355–368.
Albert, David, and Barry Loewer. 1988. Interpreting the many-worlds interpretation. Synthese 77: 195–213.
Allori, Valia, Sheldon Goldstein, Roderich Tumulka, and Nino Zanghì. 2011. Many worlds and Schrödinger’s first quantum theory. British Journal for the Philosophy of Science 62: 1–27.
Bacciagaluppi, Guido. 2002. Remarks on spacetime and locality in Everett’s interpretation. In Non-locality and Modality, ed. T. Placek, and J. Butterfield., NATO Science Series Dordrecht: Kluwer.
Bacciagaluppi, Guido, and Antony Valentini. 2009. Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference. Cambridge: Cambridge University Press.
Ballentine, Leslie. 1971. Can the statistical postulate of quantum theory be derived? - a critique of the many-universes interpretation. Foundations of Physics 3 (2): 229.
Barrett, Jeffrey A. 1999. The Quantum Mechanics of Minds and Worlds. Oxford: Oxford University Press.
Barrett, Jeffrey A. 2011. Everett’s pure wave mechanics and the notion of worlds. European Journal for the Philosophy of Science 1: 277–302.
Bartels, Andreas. 2007. Wissenschaftlicher Realismus. In Wissenschaftstheorie, ed. A. Bartels, and M. Stöckler. Paderborn: Mentis.
Bell, John S. 1980. de Broglie–Bohm, delayed-choice, double-slit experiment, and density matrix. In Bell et al., 2001: 94.
Bell, Mary, Kurt Gottfried, and Martinus Veltman (eds.). 2001. John S. Bell on the Foundation of Quantum Mechanics. Singapore: World Scientific.
Bohm, David. 1952. A suggested interpretation of the quantum theory in terms of ‘hidden’ variables. Physical Review 85: 166 (1st part); 180 (2nd part).
Bohm, David. 1953. Proof that probability density approaches \(|\psi |^2\) in causal interpretation of the quantum theory. Physical Review 89: 458.
Bohm, David, and Basil J. Hiley. 1993. The Undivided Universe. London: Routledge.
Bohr, Niels. 1935. Quantum mechanics and physical reality. Nature 136: 1025–1026.
Bricmont, J. 2016. Making Sense of Quantum Mechanics. Heidelberg: Springer.
Brown, Harvey, and David Wallace. 2005. Solving the measurement problem: De Broglie-Bohm loses out to Everett. Foundations of Physics 35: 517.
Bub, Jeffrey. 1997. Interpreting the Quantum World. Cambridge: Cambridge University Press.
Cushing, James T. 1995. Quantum tunneling times: a crucial test for the causal program? Foundations of Physics 25: 296.
Daumer, Martin, Detlef Dürr, Sheldon Goldstein, and Nino Zanghì. 1996. Naive realism about operators. Erkenntnis 45: 379–397.
de Broglie, Louis. 1927. La structure atomique de la matière et du rayonnement et la méchanique ondulatoire. Journal de Physique VI 8: 25.
Deotto, Enrico, and GianCarlo Ghirardi. 1998. Bohmian mechanics revisited. Foundations of Physics 28: 1.
Dewdney, Chris, and Basil J. Hiley. 1982. A Quantum potential description of one-dimensional time-dependent scattering from square barriers and square wells. Foundations of Physics 12 (1): 27–48.
DeWitt, Bryce S., and Neil Graham (eds.). 1973. The Many-Worlds Interpretation of Quantum Mechanics. Princeton: Princeton University Press.
Deutsch, David. 1985. Quantum theory as a universal physical theory. International Journal of Theoretical Physics 24 (1): 1–41.
Deutsch, David. 1996. Comment on Lockwood. British Journal for the Philosophy of Science 47 (2): 222–228.
Deutsch, David. 1999. Quantum theory of probability and decisions. Proceedings of the Royal Society of London A 455: 3129–3137.
Dürr, Detlef. 2001. Bohmsche Mechanik als Grundlage der Quantenmechanik. Berlin: Springer.
Dürr, Detlef, Sheldon Goldstein and Nino Zanghì. 1992. Quantum equilibrium and the origin of absolute uncertainty. Journal of Statistical Physics 67: 843. arXiv:quant-ph/0308039 (the page numbers refer to those in the arXiv version).
Dürr, Detlef, Sheldon Goldstein, and Nino Zanghì. 1996. Bohmian mechanics and the meaning of the wave function. In Experimental Metaphysics - Quantum Mechanical Studies in Honor of Abner Shimony, ed. R.S. Cohen, M. Horne, J. Stachel, et al. Boston Studies in the Philosophy of Science Dordrecht: Kluwer.
Dürr, Detlef, and Dustin Lazarovici. 2012. Quantenphysik ohne Quantenphilosophie. In Philosophie der Physik, ed. M. Esfeld. Berlin: Suhrkamp.
Elby, Andrew, and Jeffrey Bub. 1994. Triorthogonal uniqueness theorem and its relevance to the interpretation of quantum mechanics. Physical Review A 49 (5): 4213.
Everett, I.I.I., Hugh. 1957. “Relative state” formulation of quantum mechanics. Review of Modern Physics 29 (3): 454.
Greaves, Hilary. 2004. Understanding Deutsch’s probability in a deterministic multiverse. Studies in History and Philosophy of Modern Physics 35: 423.
Greaves, Hilary. 2007. Probability in the Everett interpretation. Philosophy Compass 2 (1): 109–128.
Greenberger, Daniel, Klaus Hentschel, and Friedel Weinert (eds.). 2009. Compendium of Quantum Physics. Dordrecht: Springer.
Hacking, Ian. 1983. Representing and Intervening. Cambridge: Cambridge University Press.
Hartle, James. 1968. Quantum mechanics of individual systems. American Journal of Physics 36: 704–712.
Heisenberg, Werner. 1959. Physik und Philosophie. Frankfurt/M.: Ullstein.
Hemmo, Meir, and Itamar Pitowsky. 2007. Quantum probability and many worlds. Studies in History and Philosophy of Modern Physics 38: 333–350.
Hiley, Basil J. 1999. Active information and teleportation. In Epistemological and Experimental Perspectives on Quantum Physics, ed. D. Greenberger, et al. Dordrecht: Kluwer Academic Publishers.
Holland, Peter R. 1993. The Quantum Theory of Motion. Cambridge: Cambridge University Press.
Jammer, Max. 1974. The Philosophy of Quantum Mechanics. New York: Wiley.
Landau, Lev D. and Jewgeni M. Lifschitz. 2011. Lehrbuch der theoretischen Physik (Vol. 3) Quantenmechanik. Unedited reprint of the 9th edition, 1986. Frankfurt/Main: Harri Deutsch.
Lockwood, Michael. 1996. ‘Many minds’ interpretations of quantum mechanics. British Journal for the Philosophy of Science 47: 159–188.
Madelung, Erwin. 1926. Eine anschauliche Deutung der Gleichung von Schrödinger. Naturwissenschaften 14 (45): 1004.
Maudlin, Tim. 1995. Three measurement problems. Topoi 14 (1): 7.
Maudlin, Tim. 2010. Can the world be only wavefunction? In Saunders, et al., 2010: 121–143.
Mermin, David. 1990. Simple unified form for the major no-hidden variables theorems. Physical Review Letters 65: 3373.
Möllenstedt, Gottfried, and Claus Jönsson. 1959. Elektronen-Mehrfachinterferenzen an regelmäßig hergestellten Feinspalten. Zeitschrift für Physik 155: 472–474.
Myrvold, Wayne C. 2003. On some early objections to Bohm’s theory. International Studies in the Philosophy of Science 17 (1): 7–24.
Pagonis, Constantine, and Rob Clifton. 1995. Unremarkable contextualism: dispositions in the Bohm theory. Foundations of Physics 25 (2): 281.
Passon, Oliver. 2004. What you always wanted to know about Bohmian mechanics but were afraid to ask. Physics and Philosophy 3.
Passon, Oliver. 2010. Bohmsche Mechanik, 2nd ed. Frankfurt/Main: Harri Deutsch.
Philippidis, Chris, Chris Dewdney, and Basil J. Hiley. 1979. Quantum interference and the quantum potential. Il Nuovo Cimento 52 B: 15.
Saunders, Simon. 1998. Quantum mechanics, and probability. Synthese 114: 373.
Saunders, Simon, Jonathan Barrett, Adrian Kent, and David Wallace (eds.). 2010. Many Worlds? Everett, Quantum Theory, & Reality. Oxford: Oxford University Press.
Schlosshauer, Maximilian. 2005. Decoherence, the measurement problem, and interpretations of quantum mechanics. Review of Modern Physics 76: 1267–1305.
Schrödinger, Erwin. 1926. Quantisierung als Eigenwertproblem (Vierte Mitteilung). Annalen der Physik 81: 109–139.
Tegmark, Max. 1998. The Interpretation of quantum mechanics: many worlds or many words. Fortschritte der Physik 46: 855–862.
Teufel, Stefan, and Roderich Tumulka. 2005. Simple proof for global existence of Bohmian trajectories. Communications in Mathematical Physics 258: 349–365.
Tumulka, Roderich. 2007. The ‘unromantic pictures’ of quantum theory. Journal of Physics A: Mathematical and Theoretical 40: 3245–3273.
Vaidman, Lev. 1998. On schizophrenic experiences of the neutron or why we should believe in the many-worlds interpretation of quantum theory. International Studies in the Philosophy of Science 12: 245–261.
Vaidman, Lev. 2008. Many-worlds Interpretation of quantum mechanics. In The Stanford Encyclopedia of Philosophy, ed. E.N. Zalta, (Fall 2008 Edition).
Valentini, Antony. 1991. Signal-locality, uncertainty, and the subquantum H-theorem. Physics Letters A 156 (1–2): 5 (I); Physics Letters A 158 (1–2): 1 (II).
Valentini, Antony. 2004. Universal signature of non-quantum systems. Physics Letters A 332: 187–193.
Valentini, Antony, and Hans Westman. 2005. Dynamical origin of quantum probabilities. Proceedings of the Royal Society of London A 461: 253–272.
Wallace, David. 2003. Everettian rationality: defending Deutsch’s approach to probability in the Everett interpretation. Studies in the History and Philosophy of Modern Physics 34: 415–438.
Wallace, David. 2005. Language use in a branching universe. http://philsci-archive.pitt.edu/2554/.
Wallace, David. 2010. The Everett interpretation. In The Oxford Handbook of Philosophy of Physics, ed. Batterman (in preparation, cited from the preprint version at http://philsci-archive.pitt.edu).
Zeh, Hans Dieter. 1970. On the interpretation of measurement in quantum theory. Foundations of Physics 1: 69–76.
Zurek, Wojciech H. 1981. Pointer basis of quantum apparatus: into what mixture does the wave packet collapse? Physical Review D 24: 1516–1525.
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Exercises
Exercises
-
1.
The de Broglie–Bohm theory is frequently called a theory of “hidden variables”. This term implies the criticism that the theory introduces in principle unobservable quantities into its description. Write a brief dialogue between an advocate of the de Broglie–Bohm theory and a supporter of the Copenhagen interpretation, in which the former defends the theory against this criticism and accuses the “Copenhagen” advocate of actually making this error herself. In the course of this debate, additional arguments pro and contra could be introduced!
-
2.
Explain why within the de Broglie–Bohm theory, the uncertainty relation \(\Delta x \cdot \Delta p \ge \frac{\hbar }{2}\) is not violated!
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3.
Compare the solutions to the measurement problem in the de Broglie–Bohm and the Everett interpretations. Give examples of structural similarities and differences between them.
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Passon, O. (2018). No-Collapse Interpretations of Quantum Theory. In: The Philosophy of Quantum Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-78356-7_5
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