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Quantum Mechanics as a Theory of Probability

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Quantum, Probability, Logic

Part of the book series: Jerusalem Studies in Philosophy and History of Science ((JSPS))

Abstract

We examine two quite different threads in Pitowsky’s approach to the measurement problem that are sometimes associated with his writings. One thread is an attempt to understand quantum mechanics as a probability theory of physical reality. This thread appears in almost all of Pitowsky’s papers (see for example 2003, 2007). We focus here on the ideas he developed jointly with Jeffrey Bub in their paper ‘Two Dogmas About Quantum Mechanics’ (2010) (See also: Bub (1977, 2007, 2016, 2020); Pitowsky (2003, 2007)). In this paper they propose an interpretation in which the quantum probabilities are objective chances determined by the physics of a genuinely indeterministic universe. The other thread is sometimes associated with Pitowsky’s earlier writings on quantum mechanics as a Bayesian theory of quantum probability (Pitowsky 2003) in which the quantum state seems to be a credence function tracking the experience of agents betting on the outcomes of measurements. An extreme form of this thread is the so-called Bayesian approach to quantum mechanics. We argue that in both threads the measurement problem is solved by implicitly adding structure to Hilbert space. In the Bub-Pitowsky approach we show that the claim that decoherence gives rise to an effective Boolean probability space requires adding structure to Hilbert space. With respect to the Bayesian approach to quantum mechanics, we show that it too requires adding structure to Hilbert space, and (moreover) it leads to an extreme form of idealism.

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Notes

  1. 1.

    Everett’s (1957) interpretation of quantum mechanics has a similar attempt in this respect.

  2. 2.

    See also: Bub (1977, 2007, 2016, 2020); Pitowsky (2003, 2007).

  3. 3.

    For versions of the Bayesian approach to quantum mechanics, see e.g., Caves et al. (2002a, b, 2007); Fuchs and Schack, (2013); Fuchs et al. (2014); Fuchs and Stacey (2019).

  4. 4.

    The ‘no signaling’ principle asserts, roughly, that the marginal probabilities of events associated with a quantum system are independent of the particular set of mutually exclusive and collectively exhaustive events associated with any other system.

  5. 5.

    Or Einstein’s principle of relativity and the light postulate.

  6. 6.

    See Brown (2005) for a different view about the role of dynamical analyses in special relativity; and Brown and Timpson (2007) for a criticism of the analogy with special relativity.

  7. 7.

    For decoherence theory, and references, see e.g., Joos et al. (2003).

  8. 8.

    Interactions never begin, so state (1) (which is a product state) should be replaced with a state in which Alice + electron are (weakly) entangled; otherwise the dynamics would not be reversible.

  9. 9.

    In discrete models, both | ψ↑↓⟩ and | E ±(t)⟩ are states of the form: \( \prod_{i=1}^N\mid \left.{\mu}_i\right\rangle \) defined in the corresponding multi-dimensional Hilbert spaces.

  10. 10.

    In statistical mechanics, here is an analogous scenario: the phase space of the universe can be partitioned into infinitely many sets corresponding to infinitely many macrovariables. Some of these partitions exhibit certain regularities, for example, the partition to the thermodynamic sets which human observers are sensitive to, exhibit the thermodynamic regularities; while other partitions to which human observers are not sensitive, although they are equally real exhibit other regularities and may even be anomalous (that is, exhibit no regularities at all). We call this scenario in statistical mechanics Ludwig’s problem (see Hemmo and Shenker 2012, 2016). But one can explain by straightforward physical facts why (presumably) human observers experience the thermodynamic sets and regularities but not the equally existing other sets (although it might be that in our experience some other sets also appear). This case too is dis-analogous to the basis-symmetry in quantum mechanics, from which the problem of no preferred basis follows.

  11. 11.

    A preferred basis in exactly the same sense is also required in all the contemporary versions of the Everett (1957) interpretation of quantum mechanics, so that one must add a new law of physics to the effect that worlds emerge relative to the decoherence basis; see (Hemmo and Shenker 2019).

  12. 12.

    Recent variations are, for example, Caves et al. (2002a, b, 2007); Fuchs and Schack (2013); Fuchs et al. (2014); Fuchs and Stacey (2019).

  13. 13.

    For the role of decoherence in the Bayesian approach and its relation to Dutch-book consistency, see Fuchs and Scack (2012).

  14. 14.

    See Hagar and Hemmo (2006); Hagar (2003).

  15. 15.

    Bohm’s (1952) theory and the GRW theory (Ghirardi et al. 1986) assume a preferred basis from the start. Our point is that the structure added by the Bayesian approach is no less than e.g., Bohmiam trajectories or GRW flashes.

  16. 16.

    These two roles are carried out by the mind at one shot (as it were), but analytically they are different.

References

  • Bell, J. S. (1987a). Are there quantum jumps? In J. Bell (Ed.), Speakable and unspeakable in quantum mechanics (pp. 201–212). Cambridge: Cambridge University Press.

    Google Scholar 

  • Bell, J. S. (1987b). How to teach special relativity. In J. Bell (Ed.), Speakable and unspeakable in quantum mechanics (pp. 67–80). Cambridge: Cambridge University Press.

    Google Scholar 

  • Brown, H. (2005). Physical relativity: Spacetime structure from a dynamical perspective. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Brown, H. R., & Timpson, C. G. (2007). Why special relativity should not be a template for a fundamental reformulation of quantum mechanics. In Pitowsky (Ed.), Physical theory and its interpretation: Essays in honor of Jeffrey Bub, W. Demopoulos and I. Berlin: Springer.

    Google Scholar 

  • Bub, J. (1977). Von Neumann’s projection postulate as a probability conditionalization rule in quantum mechanics. Journal of Philosophical Logic, 6, 381–390.

    Article  Google Scholar 

  • Bub, J. (2007). Quantum probabilities as degrees of belief. Studies in History and Philosophy of Modern Physics, 38, 232–254.

    Article  Google Scholar 

  • Bub, J. (2016). Bananaworld: Quantum mechanics for primates. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Bub, J. (2020). ‘Two Dogmas’ Redux. In Hemmo, M., Shenker, O. (eds.) Quantum, probability, logic: Itamar Pitowsky’s work and influence. Cham: Springer.

    Google Scholar 

  • Bub, J., & Pitowsky, I. (2010). Two dogmas about quantum mechanics. In S. Saunders, J. Barrett, A. Kent, & D. Wallace (Eds.), Many worlds? Everett, quantum theory, and reality (pp. 431–456). Oxford: Oxford University Press.

    Google Scholar 

  • Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of “hidden” variables. Physical Review, 85(166–179), 180–193.

    Article  Google Scholar 

  • Caves, C. M., Fuchs, C. A., & Schack, R. (2002a). Unknown quantum states: The quantum de Finetti representation. Journal of Mathematical Physics, 43(9), 4537–4559.

    Article  Google Scholar 

  • Caves, C. M., Fuchs, C. A., & Schack, R. (2002b). Quantum probabilities as Bayesian probabilities. Physical Review A, 65, 022305.

    Article  Google Scholar 

  • Caves, C. M., Fuchs, C. A., & Schack, R. (2007). Subjective probability and quantum certainty. Studies in History and Philosophy of Modern Physics, 38, 255–274.

    Article  Google Scholar 

  • de Finetti, B. (1970). Theory of probability. New York: Wiley.

    Google Scholar 

  • Everett, H. (1957). ‘Relative state’ formulation of quantum mechanics. Reviews of Modern Physics, 29, 454–462.

    Article  Google Scholar 

  • Fuchs C. A. & Schack, R. (2012). Bayesian conditioning, the reflection principle, and quantum decoherence. In Ben-Menahem, Y., & Hemmo, M. (eds.), Probability in physics (pp. 233–247). The Frontiers Collection. Berlin/Heidelberg: Springer.

    Google Scholar 

  • Fuchs, C. A., & Schack, R. (2013). Quantum Bayesian coherence. Reviews of Modern Physics, 85, 1693–1715.

    Article  Google Scholar 

  • Fuchs, C. A., Mermin, N. D., & Schack, R. (2014). An introduction to QBism with an application to the locality of quantum mechanics. American Journal of Physics, 82, 749–754.

    Article  Google Scholar 

  • Fuchs, C. A., & Stacey, B. (2019). QBism: Quantum theory as a hero’s handbook. In E. M. Rasel, W. P. Schleich, & S. Wölk (Eds.), Foundations of quantum theory: Proceedings of the International School of Physics “Enrico Fermi” course 197 (pp. 133–202). Amsterdam: IOS Press.

    Google Scholar 

  • Ghirardi, G., Rimini, A., & Weber, T. (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review, D, 34, 470–479.

    Article  Google Scholar 

  • Hagar, A. (2003). A philosopher looks at quantum information theory. Philosophy of Science, 70(4), 752–775.

    Article  Google Scholar 

  • Hagar, A., & Hemmo, M. (2006). Explaining the unobserved – Why quantum mechanics ain’t only about information. Foundations of Physics, 36(9), 1295–1324.

    Article  Google Scholar 

  • Hemmo, M., & Shenker, O. (2012). The road to Maxwell’s demon. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Hemmo, M., & Shenker, O. (2016). Maxwell’s demon. Oxford University Press: Oxford Handbooks Online.

    Book  Google Scholar 

  • Hemmo, M. & Shenker, O. (2019). Why quantum mechanics is not enough to set the framework for its interpretation, forthcoming.

    Google Scholar 

  • Joos, E., Zeh, H. D., Giulini, D., Kiefer, C., Kupsch, J., & Stamatescu, I. O. (2003). Decoherence and the appearance of a classical world in quantum theory. Heidelberg: Springer.

    Book  Google Scholar 

  • Pitowsky, I. (2003). Betting on the outcomes of measurements: A Bayesian theory of quantum probability. Studies in History and Philosophy of Modern Physics, 34, 395–414.

    Article  Google Scholar 

  • Kolmogorov, A. N. (1933). Foundations of the Theory of Probability. New York: Chelsea Publishing Company, English translation 1956.

    Google Scholar 

  • Pitowsky, I. (2007). Quantum mechanics as a theory of probability. In W. Demopoulos & I. Pitowsky (Eds.), Physical theory and its interpretation: Essays in honor of Jeffrey Bub (pp. 213–240). Berlin: Springer.

    Google Scholar 

  • Ramsey, F. P. (1990). Truth and Probability. (1926); reprinted in Mellor, D.H. (ed.), F. P. Ramsey: Philosophical Papers. Cambridge: Cambridge University Press.

    Google Scholar 

  • Tumulka, R. (2006). A relativistic version of the Ghirardi–Rimini–Weber model. Journal of Statistical Physics, 125, 821–840.

    Article  Google Scholar 

  • von Neumann, J. (2001). Unsolved problems in mathematics. In M. Redei & M. Stoltzner (Eds.), John von Neumann and the foundations of quantum physics (pp. 231–245). Dordrecht: Kluwer Academic Publishers.

    Chapter  Google Scholar 

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Acknowledgement

We thank Guy Hetzroni and Cristoph Lehner for comments on an earlier draft of this paper. This research was supported by the Israel Science Foundation (ISF), grant number 1114/18.

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Correspondence to Meir Hemmo .

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Hemmo, M., Shenker, O. (2020). Quantum Mechanics as a Theory of Probability. In: Hemmo, M., Shenker, O. (eds) Quantum, Probability, Logic. Jerusalem Studies in Philosophy and History of Science. Springer, Cham. https://doi.org/10.1007/978-3-030-34316-3_15

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