Abstract
Einstein made several attempts to argue for the incompleteness of quantum mechanics (QM), not all of them using a separation principle. One unpublished example, the box parable, has received increased attention in the recent literature. Though the example is tailor-made for applying a separation principle and Einstein indeed applies one, he begins his discussion without it. An analysis of this first part of the parable naturally leads to an argument for incompleteness not involving a separation principle. I discuss the argument and its systematic import. Though it should be kept in mind that the argument is not the one Einstein intends, I show how it suggests itself and leads to a conflict between QM’s completeness and a physical principle more fundamental than the separation principle, i.e. a principle saying that QM should deliver probabilities for physical systems possessing properties at definite times.
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Notes
Einstein’s letter to Schrödinger of June 19, 1935 (EA 22 047). The German text is reproduced in [1, pp. 69, nos. 7, 9, 71] and [4, pp. 131–132]. Note that both sources present only the first part of Einstein’s discussion; the second part (but not the first translated here) is quoted in [3, pp. 179, nos. 19, 20, 180]. My translation follows Fine’s with two emendations (‘tasteless, so-to-speak’ for ‘sozusagen abgeschmackt’ and ‘overcome’ for ‘beikommen’); the italics reproduce Einstein’s original emphases.
See e.g. Popper ([8, p. 69]). Probably, Popper was the first to express this view but it later became the orthodox view about QM probabilities. Thus, Butterfield rightly says that “the orthodox interpretation of a quantum state is as a catalogue of probabilistic dispositions (‘Born-rule probabilities’)” ([9, p. 115]).
Indeed all three assumptions are endorsed by approaches to quantum gravity like [10]. Rovelli’s generalization of QM explicates a generalized Born Rule that produces probabilities for spacetime events occupying a time-interval \(\Delta \hbox {t}\) . For \(\Delta \hbox {t}\rightarrow 0\) that rule reproduces the Born Rule of ordinary QM but with a time-reference such that the probabilities produced are of the form (viii) (see [10, pp. 172–173]).
See recently [14, pp. 26, 56].
Explicitly, there are either states \(\vert \Phi \) (t*) \(>\)with \(\vert \Phi \) (t*) \(\ge \) U (t*) \(\vert \Psi \) (t) \(>\)or states \(\vert \) a (t\(^{\# })>\) with \(\vert \) a\(_\mathrm{k}\) (t\(^{\prime }\)) \(\ge \) U (t\(^{\prime }\)) \(\vert \) a (t\(^{\# }) >\) (or states of both kinds), where t \(<\) t* \(<\) t\(^{\# } <\) t\(^{\prime }\). Assuming that there is a state \(\vert \Phi \) (t*)\(>\), there certainly is an evolution such that \(\vert \langle {\mathrm{a}_\mathrm{k}} (\mathrm{t}^{\prime })\vert \Psi (t) \rangle \vert ^{2}\ne \) \(\vert \langle {\mathrm{a}_\mathrm{k}} ({\mathrm{t}^{\prime }})\vert \Psi ({\mathrm{t}^{*}}) \rangle \vert ^{2}\), e.g. an evolution with a time-dependent external potential (and analogously for \(\vert \) a (t\(^{\# })>\) and \(\vert \) a\(_\mathrm{k}\) (t\(^{\prime }\)) \(>)\).
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Held, C. Einstein’s Boxes: Incompleteness of Quantum Mechanics Without a Separation Principle. Found Phys 45, 1002–1018 (2015). https://doi.org/10.1007/s10701-014-9845-6
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DOI: https://doi.org/10.1007/s10701-014-9845-6