Abstract
The main goal of this chapter is to present “the statistical Copenhagen interpretation,” developed by the present author. This interpretation will be compared with the Bayesian interpretation of quantum mechanics, or a set of such interpretations, specifically nonrealist ones. The chapter also considers Bohr’s interpretation from this perspective and argues that Bohr appears to have ultimately inclined to a statistical view close to that of the statistical Copenhagen interpretation, which, however, expressly adopts a stronger form of the RWR principle, by placing quantum objects and processes not only beyond representation but also beyond conception, beyond thought. Following a brief introduction given in Sect. 4.1, Sect. 4.2 is devoted to the statistical Copenhagen interpretation, and Sect. 4.3 to the discussion of Bohr’s position concerning the statistical vs. probabilistic (Bayesian) view of quantum mechanics.
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Notes
- 1.
In addition to Khrennikov’s interpretation cited in Chap. 1 (Khrennikov 2012; Plotnitsky and Khrennikov 2015), I might note a compelling statistical interpretation proposed in (Allahverdyan et al., 2013). I mention these two interpretations primarily because, although ultimately realist, they exhibit instructive affinities with SCI, as discussed in (Plotnitsky and Khrennikov 2015). There are other statistical interpretations of quantum mechanics, which cannot be considered within my scope here.
- 2.
- 3.
Such views may, and Jaynes’s and De Finetti’s do, deviate from each other in other respects. Bayes’s theorem itself (Bayes himself proved a version of it), which relates the conditional and marginal probabilities of two random events, according to a definite very simple formula, and is the origin of Bayesian interpretations, is general and valid in the frequentist approach as well. Bayes’s formula or theorem relates the conditional and marginal probabilities of events A and B, where B has a nonzero probability of occurrence:
$$ P\left(A\Big|B\right)=\frac{P\left(B\Big|A\right)P(A)}{P(B)}. $$Here P(A) is the prior probability or marginal probability of A (also known as the “prior”), which does not take into account any information concerning B; P(A|B) is the conditional probability of A, given B, called the posterior probability, which depends on the value of B; P(B|A) is the conditional probability of B given A; and P(B) is the prior probability of B. The theorem may be seen as describing how one’s beliefs about observing A are updated by having observed B.
- 4.
See (Mermin 2016, pp. 232–248) on the difference between QBism and Bohr, even if not in “the spirit Copenhagen,” which is not the language or concept used by Mermin, who may or may not see QBism in this spirit. Mermin does suggest some affinities between QBist and Bohr’s views, albeit without exploring Bohr’s views in any depth, in particular, as concern the nature of Bohr’s concept of reality as “reality without realism,” and thus the RWR principle. This is a juncture of Bohr’s thought at which Bohr is on some points closer to QBism and on others departs more radically from it than Mermin suggests. Mermin primarily (correctly) stresses the difference between QBism’s emphasis on subjectivity vs. Bohr’s view of measurement as objective, as considered above, albeit, again, without really examining Bohr’s concept of objectivity, for example, in relation to the disciplinary practice of physics. Mermin does see QBism in terms of the statistical vs. Bayesian view of quantum probability. I might add that, while QBists are quite right to critique the (naïve) conceptions and requirement of objectivity or naïve realism, still common in the scientific community, it appears to me that the concept of subjectivity (and certain concepts associated with it, such as and in particular “belief”) are used by them uncritically, rather than analytically examined. Perhaps, it will come in time.
- 5.
Of course, in the Bayesian scheme of things, there may be events to which one cannot assign probabilities, but not in general, which is the opposite of what takes place in SCI, where such assignments are only possible sometimes but are impossible in general.
- 6.
This type of Bayesian prior is found in T. Stoppard’s play Rosencrantz and Guildenstern are Dead, in a deliberate contrast to or a parody of these characters’ prior in Hamlet, where they, naively, bet on the future with certainty, as opposed to Hamlet himself. Hamlet weighs his Bayesian bets with great deliberation and changes them with new information, which he assiduously seeks, to gain more certainty on his bets throughout the play. And yet he often bets wrong as well. Games of probability, especially of the Bayesian variety, are quite complex in Shakespeare, and one can find all types of priors in them.
- 7.
I am indebted to J-Å. Larsson for suggesting this last qualification.
- 8.
There is a curious nuance to Einstein’s assessment in question. It is true that this is, as Bohr says, an assessment to the effect that “the quantum-mechanical description is to be considered merely as a means of accounting for the average behavior of a large number of atomic systems.” But it also based in Einstein’s belief that one should be able derive the inner workings of individual quantum systems even on experimental grounds. It replies to the following rhetorical question: “Is there really any physicist who believes that we shall never get an inside view of these important alterations [due to individual perturbations] in the single system, in their structure and their causal connections, and this regardless of the fact that these single happenings have been brought so close to us, thanks to the marvelous invention of the Wilson [cloud] chamber and the Geiger counter?” (Einstein 1936, p. 375). Einstein’s question is not unreasonable. It is not easy to believe that it is impossible to ever get an inside view, also literally in the sense of intuition [Anschaulichkeit], concerning the inner workings of individual quantum systems, and establish an Einsten-complete theory of them, even if not derive this theory from quantum mechanics as a statistical theory of ensembles. As discussed in Chap. 1, Einstein came to doubt this possibility, a doubt expressed on the same occasion as well (Einstein 1936, p, 361). As also discussed in Chap. 1, one is likely to need a theory different from quantum mechanics in order to achieve this, if quantum phenomena will ever allow us to do so. Einstein’s “dream” was, again, an ontologically classical-like field theory (Einstein 1936, p. 378, 1949a, pp. 83–85).
- 9.
In certain situations, such as those of the EPR type, as considered in Chap. 3, we can, for all practical purposes, predict certain quantities exactly (at least in the case of discrete variables), but this is never true in full rigor either. There is always a nonzero probability that the object in question will not be found where it is expected to be found at the moment of time for which the prediction is made. As noted earlier, unlike the Bell-Bohm version of the EPR experiment for spin (at stake in Bell’s theorem), the actual experiment proposed by EPR, dealing with continuous variables, cannot be physically realized, because the EPR-entangled quantum state is not normalizable. As indicated in Chap. 3, there are experiments (e.g., those involving photon pairs produced in parametric down conversion) that statistically approximate the idealized entangled state constructed by EPR for continuous variables. These experiments are consistent with the present argument. They also reflect the fact that the EPR thought experiment is a manifestation of correlated events for identically prepared experiments with EPR pairs, which can in this regard be understood on the model of the Bell-Bohm version of the EPR experiment. In any event, there are quantum experiments, such as, paradigmatically, the double-slit experiment, in which the assignment of probabilities to the outcomes of individual events is difficult or even all but impossible to assume.
- 10.
For a discussion of Einstein’s ensemble view of quantum mechanics and complexities it involves, see (van Dongen 2010, 174–181), which, however, only considers the situation in terms of Einstein-completeness (in the present terminology).
- 11.
The so-called cosmic landscapes theories could be considered from this perspective.
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Plotnitsky, A. (2016). The Statistical Copenhagen Interpretation. In: The Principles of Quantum Theory, From Planck's Quanta to the Higgs Boson. Springer, Cham. https://doi.org/10.1007/978-3-319-32068-7_4
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