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A Quintet of Quandaries: Five No-Go Theorems for Relational Quantum Mechanics

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Abstract

Relational quantum mechanics (RQM) proposes an ontology of relations between physical systems, where any system can serve as an ‘observer’ and any physical interaction between systems counts as a ‘measurement’. Quantities take unique values spontaneously in these interactions, and the occurrence of such ‘quantum events’ is strictly relative to the observing system, making them ‘relative facts’. The quantum state represents the objective information that one system has about another by virtue of correlations between their physical variables. The ontology of RQM thereby strives to uphold the universality and completeness of quantum theory, while at the same time maintaining that the actualization of each unique quantum event is a fundamental physical event. Can RQM sustain this precarious balancing act? Here we present five no-go theorems that imply it cannot; something has to give way.

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Notes

  1. Many worlds interpretations, while possibly in some sense ‘relational’, do not endorse the uniqueness of relative facts.

  2. Matthew Leifer calls them ‘Bell-Wigner mashups’ [30].

  3. For instance, a pertinent criticism of QBism’s claim to locality can be found in Sect. 3.2. of Ref. [37].

  4. Rovelli tends to eschew the terms “measurement”and “measurement outcome”in order to avoid the unwanted interpretational baggage that often accompanies these terms [43].

  5. Note that the quantum events relevant for determining a system’s state are not only interactions with the system itself, but also interactions with other systems that have previously interacted with it.

  6. For more about whether systems are real, see Rovelli’s remarks at the end of Sect. X in [43].

  7. Which variable gets assigned to which role depends on which of the two systems we take as the observer.

  8. Note that in light of RQM:5, the pointer variable here ought not to have any special status by virtue of being a “piece of paper”, i.e. something that you or I would intuitively recognize as a tool for keeping records. Other physical variables might serve equally well as apparatus pointers.

  9. See Ref. [45], p.8-9: “there is no half-a-measurement; there is probability one-half that the measurement has been made! [...] Imperfect correlation does not imply no measurement performed, but only a smaller than 1 probability that the measurement has been completed”.

  10. The first known derivation of this result is by Schrödinger; it has since been rediscovered and expanded upon by subsequent authors; see Refs. [46, 47] and citations therein.

  11. We acknowledge that Dorato already made this general point in Ref. [7]; we have just shored it up with mathematical arguments.

  12. A subscripted index on \(b^{(\mathcal {W})}\) would be desirable to maintain consistency of the notation, but we will scrupulously follow the notation of the original authors.

  13. Some readers might worry that this violates \(\mathcal {W}\)’s ‘freedom of choice’, but it is not at all clear that this is a legitimate complaint in RQM; after all, what if \(\mathcal {W}\) were just an asteroid, or an electron?

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Acknowledgements

Some ideas in this work were inspired by unpublished notes of Rüdiger Schack and Blake Stacey. I am grateful to Carlo Rovelli, Andrea Di Biagio and Federico Laudisa for answering many of my questions about the principles underlying RQM, although they probably disagree with the conclusions I have drawn from those principles. This work was supported in part by the John E. Fetzer Memorial Trust.

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  1. QBism Group, University of Massachusetts Boston, Boston, USA

    Jacques Pienaar

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Communicated by Carlo Rovelli

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Pienaar, J. A Quintet of Quandaries: Five No-Go Theorems for Relational Quantum Mechanics. Found Phys 51, 97 (2021). https://doi.org/10.1007/s10701-021-00500-6

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