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The preferred basis problem in the many-worlds interpretation of quantum mechanics: why decoherence does not solve it

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Abstract

We start by very briefly describing the measurement problem in quantum mechanics and its solution by the Many Worlds Interpretation (MWI). We then describe the preferred basis problem, and the role of decoherence in the MWI. We discuss a number of approaches to the preferred basis problem and argue that contrary to the received wisdom, decoherence by itself does not solve the problem. We address Wallace’s emergentist approach based on what he calls Dennett’s criterion, and we compare the logical structure of Wallace’s argument that the Hilbert space structure and the unitary dynamics should be considered a complete description of the physics of the universe with the logical structure of the EPR argument against the completeness of quantum mechanics. Then, we consider the nature of so-called “high level emergent facts” and Dennett’s criterion in Wallace’s approach and we discuss the implications of its non-reductive nature. Finally, we conclude that (contrary to the received wisdom) the MWI is not a straightforward interpretation of pure quantum mechanics that doesn't add anything to it. Whether or not it is more parsimonious than other quantum mechanical theories (such as the GRW theory and Bohm’s theory) depends on the details of the additions.

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Notes

  1. There are many versions of the MWI; see Everett’s (1957) ‘relative-state’ formulation; and later versions, e.g.: DeWitt (1970), Zeh (1970, 1973, 2001), Deutsch (1985), Saunders (1995), Vaidman 1998, 2018), Tegmark 1998, Wallace (2012), and Carroll and Singh (2018). Our arguments in this paper apply to all the versions.

  2. It seems to us that essentially the same point arises if one replaces quantum mechanics with quantum field theory and applies the MWI to the latter without adding additional structure.

  3. Strictly speaking, interactions never begin, so state (1) (which is a tensor product state) should be replaced with a state in which Alice + electron are (weakly) entangled; otherwise the dynamics would not be reversible. Here we use the standard presentation and ignore this problem.

  4. Except for a set of worlds of small measure, by the standard quantum mechanical ‘absolute squared’ measure. This is not a trivial point, but it is part of the question of whether the probabilistic aspects of quantum mechanics can be recovered in the MWI, which we don’t address here.

  5. See Everett (1957); DeWitt (1970); Albert (1992); Deutsch (1999); Hemmo and Pitowsky (2007); Wallace (2003, 2012); Vaidman (2018); Brown and Ben-Porath (2019).

  6. Another example: In Bohm’s theory the wavefunction has a double role. It determines the motion of the particle via the guiding equation and it also determines the probability distribution via the psi-squared law. These two roles are independent of each other.

  7. For decoherence theory, and references, see e.g., (Joos et al., 2003; Wallace 2012; Zeh 2001; Zurek 1981; 1993; Zurek, Habib and Paz 1993).

  8. In the literature, the decoherent states might be some narrowly peaked Gaussians in both position and momentum space; see Zurek, Habib and Paz (1993).

  9. The 'random' probability distribution over the states of the environment requires defining a measure over the state space. Questions arise with respect to the choice of this measure (e.g., see Hemmo & Shenker, 2012, Ch. 8; Hemmo & Shenker, 2015a), since infinitely many measures may be defined on the state space, and different choices will yield different probability distributions. This is a major issue in the foundations of statistical mechanics, which we think has implications also with respect to the way decoherence is understood in quantum mechanics. But for the purposes of this paper, we set this issue aside.

  10. As we said, there are bases in the Hilbert space of Alice (the so-called Schmidt basis) that exactly diagonalise the reduced state at each time. But these bases are unstable and change abruptly under the decoherence evolution near degeneracy points, and so they cannot account for our experience.

  11. See Bacciagaluppi and Hemmo (1996); Bacciagaluppi (2000). This is one of the reasons that the so-called modal interpretations of quantum mechanics failed.

  12. Using coarse-graining to describe our experience of so-called quasi-classical states is not a solution, since it repeats the problem: some coarse-grained observables decohere and some don't; It is a fact that our experience singles out the decohered ones; the task is to explain this fact from the Hilbert space structure and the unitary dynamics.

  13. We thank an anonymous reviewer for raising this point.

  14. It seems to us that Rovelli’s relational approach (see Laudisa & Rovelli, 2021) is in this direction.

  15. Assuming that all the structures in the Hilbert space correspond to real worlds might encounter Kochen-Specker-type contradictions, due to the contextuality of quantum mechanics. But we set this issue aside.

  16. This is called Ludwig’s problem in (Hemmo and Shenker 2016). For overviews and discussion of the foundations of statistical mechanics, see (Sklar, 1993; Albert, 2000; Uffink, 2007; Frigg, 2008; Hemmo & Shenker, 2012).

  17. The preference for localised states is in line with some models of decoherence; see (Zurek et al., 1993).

  18. EPR’s locality condition is essentially that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past: “But on one supposition we should, in my opinion, absolutely hold fast: the real factual situation of the system S2 is independent of what is done with the system SI, which is spatially separated from the former.” (Einstein, 1949, p. 85). The condition was meant to express causality in the sense of special relativity; see Bell (1964) who formulated this idea mathematically essentially as a condition of statistical independence of the marginal probabilities.

  19. EPR’s criterion of reality is considered by many to be analytic (see Maudlin, 2014).

  20. EPR’s conclusion was that a completion of quantum mechanics requires additional variables that “were to restore to the theory causality and locality.” (Bell, 1964, p. 195). Such a theory, however, cannot be Lorentz invariant (see Bell, 1964, p. 199). We set aside these issues.

  21. These arguments are based on results in the foundations of statistical mechanics and focus on one of the central motivations for non-reductive approaches to the special sciences called multiple-realizability. They are not versions of Kim’s (1993, Ch. 17, 18) so-called causal exclusion argument. We cannot go into more details for lack of space.

  22. In the present context, it is important to notice that the relation between statistical mechanics and thermodynamics, where successful, is one of strict reduction, in both the classical and quantum domains (see for the classical case Hemmo & Shenker, 2012, 2019a, 2019b; 2022; for the quantum case, Shenker 2020).

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Acknowledgements

We thank anonymous reviewers for very detailed, helpful and valuable comments and criticism. This research was supported by the Israel Science Foundation, grant numbers 1148/2018 and 690/2021.

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Israel Science Foundation (ISF), Grants Number: 1148/18 & 690/2021.

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  1. Philosophy Department, University of Haifa, 199 Aba Khoushy Ave. Mount Carmel, 3498838, Haifa, Israel

    Meir Hemmo

  2. Program in History and Philosophy of Science, Hebrew University of Jerusalem, 91905, Jerusalem, Israel

    Orly Shenker

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  1. Meir Hemmo
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Correspondence to Meir Hemmo.

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Hemmo, M., Shenker, O. The preferred basis problem in the many-worlds interpretation of quantum mechanics: why decoherence does not solve it. Synthese 200, 261 (2022). https://doi.org/10.1007/s11229-022-03713-y

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