Skip to main content

Derivations of the Born Rule

  • Chapter
  • First Online:
Quantum, Probability, Logic

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

Abstract

The Born rule, a cornerstone of quantum theory usually taken as a postulate, continues to attract numerous attempts for its derivation. A critical review of these derivations, from early attempts to very recent results, is presented. It is argued that the Born rule cannot be derived from the other postulates of quantum theory without some additional assumptions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Aharonov, Y., & Reznik, B. (2002). How macroscopic properties dictate microscopic probabilities. Physical Review A, 65, 052116.

    Google Scholar 

  • Aharonov, Y., & Vaidman, L. (1990). Properties of a quantum system during the time interval between two measurements. Physical Review A, 41, 11–20.

    Google Scholar 

  • Aharonov, Y., Bergmann, P. G., & Lebowitz, J. L. (1964). Time symmetry in the quantum process of measurement. Physical Review, 134, B1410–B1416.

    Google Scholar 

  • Aharonov, Y., Cohen, E., Gruss, E., & Landsberger, T. (2014). Measurement and collapse within the two-state vector formalism. Quantum Studies: Mathematics and Foundations, 1, 133–146.

    Google Scholar 

  • Aharonov, Y., Cohen, E., & Landsberger, T. (2017). The two-time interpretation and macroscopic time-reversibility. Entropy, 19, 111.

    Google Scholar 

  • Aicardi, F., Borsellino, A., Ghirardi, G. C., & Grassi, R. (1991). Dynamical models for state-vector reduction: Do they ensure that measurements have outcomes? Foundations of Physics Letters, 4, 109–128.

    Google Scholar 

  • Albert, D. Z., & Vaidman, L. (1989). On a proposed postulate of state-reduction. Physics letters A, 139, 1–4.

    Google Scholar 

  • Baker, D. J. (2007). Measurement outcomes and probability in Everettian quantum mechanics. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 38, 153–169.

    Google Scholar 

  • Barnum, H., Caves, C. M., Finkelstein, J., Fuchs, C. A., & Schack, R. (2000). Quantum probability from decision theory? Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 456, 1175–1182.

    Google Scholar 

  • Barrett, J. A. (2017). Typical worlds. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 58, 31–40.

    Google Scholar 

  • Boge, F. J. (2019). The best of many worlds, or, is quantum decoherence the manifestation of a disposition? Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 66, 135–144.

    Google Scholar 

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

    Google Scholar 

  • Brumer, P., & Gong, J. (2006). Born rule in quantum and classical mechanics. Physical Review A, 73, 052109.

    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, & reality (pp. 433–459). Oxford: Oxford University Press.

    Google Scholar 

  • Buniy, R. V., Hsu, S. D., & Zee, A. (2006). Discreteness and the origin of probability in quantum mechanics. Physics Letters B, 640, 219–223.

    Google Scholar 

  • Callender, C. (2007). The emergence and interpretation of probability in Bohmian mechanics. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 38, 351–370.

    Google Scholar 

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

    Google Scholar 

  • Caves, C. M., Fuchs, C. A., Manne, K. K., & Renes, J. M. (2004). Gleason-type derivations of the quantum probability rule for generalized measurements. Foundations of Physics, 34, 193–209.

    Google Scholar 

  • Chiribella, G., D’Ariano, G. M., & Perinotti, P. (2011). Informational derivation of quantum theory. Physical Review A, 84, 012311.

    Google Scholar 

  • Dawid, R., & Thébault, K. P. (2014). Against the empirical viability of the Deutsch–Wallace–Everett approach to quantum mechanics. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 47, 55–61.

    Google Scholar 

  • Deumens, E. (2019). On classical systems and measurements in quantum mechanics. Quantum Studies: Mathematics and Foundations, 6, 481–517.

    Google Scholar 

  • Deutsch, D. (1999). Quantum theory of probability and decisions. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 455, 3129–3137.

    Google Scholar 

  • Dürr, D., Goldstein, S., & Zanghi, N. (1992). Quantum equilibrium and the origin of absolute uncertainty. Journal of Statistical Physics, 67, 843–907.

    Google Scholar 

  • Elitzur, A. C., & Vaidman, L. (1993). Quantum mechanical interaction-free measurements. Foundations of Physics, 23, 987–997.

    Google Scholar 

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

    Google Scholar 

  • Farhi, E., Goldstone, J., & Gutmann, S. (1989). How probability arises in quantum mechanics. Annals of Physics, 192, 368–382.

    Google Scholar 

  • Finkelstein, J. (2003). Comment on “How macroscopic properties dictate microscopic probabilities”. Physical Review A, 67, 026101.

    Google Scholar 

  • Galley, T. D., & Masanes, L. (2017). Classification of all alternatives to the Born rule in terms of informational properties. Quantum, 1, 15.

    Google Scholar 

  • Ghirardi, G.C., Rimini, A., & Weber, T. (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review D, 34, 470–491.

    Google Scholar 

  • Gill, R. (2005). On an argument of David Deutsch. In M. Schürmann & U. Franz (Eds.), Quantum probability and infinite dimensional analysis: From foundations to applications (QP-PQ Series, Vol. 18, pp. 277–292). Singapore: World Scientific.

    Google Scholar 

  • Gleason, A. M. (1957). Measures on the closed subspaces of a Hilbert space. Journal of Mathematics and Mechanics, 6, 885–893.

    Google Scholar 

  • Goldstein, S., & Struyve, W. (2007). On the uniqueness of quantum equilibrium in Bohmian mechanics. Journal of Statistical Physics, 128, 1197–1209.

    Google Scholar 

  • Greaves, H. (2004). Understanding Deutsch’s probability in a deterministic multiverse. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 35, 423–456.

    Google Scholar 

  • Groisman, B., Hallakoun, N., & Vaidman, L. (2013). The measure of existence of a quantum world and the Sleeping Beauty Problem. Analysis, 73, 695–706.

    Google Scholar 

  • Hardy, L. (2001). Quantum theory from five reasonable axioms. arXiv preprint quant-ph/0101012.

    Google Scholar 

  • Hartle, J. B. (1968). Quantum mechanics of individual systems. American Journal of Physics, 36, 704–712.

    Google Scholar 

  • Hemmo, M., & Pitowsky, I. (2007). Quantum probability and many worlds. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 38, 333–350.

    Google Scholar 

  • Kent, A. (2015). Does it make sense to speak of self-locating uncertainty in the universal wave function? Remarks on Sebens and Carroll. Foundations of Physics, 45, 211–217.

    Google Scholar 

  • Kwiat, P., Weinfurter, H., Herzog, T., Zeilinger, A., & Kasevich, M. A. (1995). Interaction-free measurement. Physical Review Letters, 74, 4763–4766.

    Google Scholar 

  • Landsman, N. P. (2008). Macroscopic observables and the Born rule, I: Long run frequencies. Reviews in Mathematical Physics, 20, 1173–1190.

    Google Scholar 

  • Landsman, N. P. (2009). Born rule and its interpretation. In D. Greenberger, K. Hentschel, & F. Weinert (Eds.), Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy (pp. 64–70). Heidelberg: Springer.

    Google Scholar 

  • Lewis, P. J. (2007). Uncertainty and probability for branching selves. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 38, 1–14.

    Google Scholar 

  • Lewis, P. J. (2010). Probability in Everettian quantum mechanics. Manuscrito: Revista Internacional de Filosofía, 33, 285–306.

    Google Scholar 

  • Li, P., Field, G., Greschner, M., Ahn, D., Gunning, D., Mathieson, K., Sher, A., Litke, A., & Chichilnisky, E. (2014). Retinal representation of the elementary visual signal. Neuron, 81, 130–139.

    Google Scholar 

  • Masanes, L., Galley, T. D., & Müller, M. P. (2019). The measurement postulates of quantum mechanics are operationally redundant. Nature Communications, 10, 1361.

    Google Scholar 

  • McQueen, K. J., & Vaidman, L. (2018). In defence of the self-location uncertainty account of probability in the many-worlds interpretation. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 66, 14–23.

    Google Scholar 

  • Norsen, T. (2018). On the explanation of Born-rule statistics in the de Broglie-Bohm pilot-wave theory. Entropy, 20, 422.

    Google Scholar 

  • Pitowsky, I. (1989). Quantum probability-quantum logic. Berlin: Springer.

    Google Scholar 

  • Pitowsky, I. (1998). Infinite and finite Gleason’s theorems and the logic of indeterminacy. Journal of Mathematical Physics, 39, 218–228.

    Google Scholar 

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

    Google Scholar 

  • Pitowsky, I. (2006). Quantum mechanics as a theory of probability. In Physical theory and its interpretation (pp. 213–240). Berlin/Heidelberg: Springer.

    Google Scholar 

  • Popescu, S., & Rohrlich, D. (1994). Quantum nonlocality as an axiom. Foundations of Physics, 24, 379–385.

    Google Scholar 

  • Rae, A. I. (2009). Everett and the Born rule. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 40, 243–250.

    Google Scholar 

  • Saunders, S. (2004). Derivation of the Born rule from operational assumptions. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 460, 1771–1788.

    Google Scholar 

  • Saunders, S. (2010). Chance in the Everett interpretation. In S. Saunders, J. Barrett, A. Kent, & D. Wallace (Eds.), Many worlds?: Everett, quantum theory, & reality (pp. 181–205). Oxford: Oxford University Press.

    Google Scholar 

  • Saunders, S., & Wallace, D. (2008). Branching and uncertainty. The British Journal for the Philosophy of Science, 59, 293–305.

    Google Scholar 

  • Schlosshauer, M., & Fine, A. (2005). On Zurek’s derivation of the Born rule. Foundations of Physics, 35, 197–213.

    Google Scholar 

  • Sebens, C. T., & Carroll, S. M. (2016). Self-locating uncertainty and the origin of probability in Everettian quantum mechanics. The British Journal for the Philosophy of Science, 69, 25–74.

    Google Scholar 

  • Squires, E. J. (1990). On an alleged “proof” of the quantum probability law. Physics Letters A, 145, 67–68.

    Google Scholar 

  • Tappenden, P. (2010). Evidence and uncertainty in Everett’s multiverse. British Journal for the Philosophy of Science, 62, 99–123.

    Google Scholar 

  • Tappenden, P. (2017). Objective probability and the mind-body relation. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 57, 8–16.

    Google Scholar 

  • Towler, M., Russell, N., & Valentini, A. (2011). Time scales for dynamical relaxation to the Born rule. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468, 990–1013.

    Google Scholar 

  • Vaidman, L. (1998). On schizophrenic experiences of the neutron or why we should believe in the many-worlds interpretation of quantum theory. International Studies in the Philosophy of Science, 12, 245–261.

    Google Scholar 

  • Vaidman, L. (2002). Many-Worlds interpretation of Quantum mechanics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/cgi-bin/encyclopedia/archinfo.cgi?entry=qm-manyworlds

    Google Scholar 

  • Vaidman, L. (2012). Probability in the many-worlds interpretation of quantum mechanics. In Y. Ben-Menahem & M. Hemmo (Eds.), Probability in physics (pp. 299–311). Berlin: Springer.

    Google Scholar 

  • Vaidman, L. (2016). All is ψ. Journal of Physics: Conference Series, 701, 012020.

    Google Scholar 

  • Vaidman, L. (2019). Ontology of the wave function and the many-worlds interpretation. In O. Lombardi, S. Fortin, C. López, & F. Holik (Eds.), Quantum worlds: Perspectives on the ontology of quantum mechanics. (pp. 93–106). Cambridge: Cambridge University Press.

    Google Scholar 

  • Valentini, A., & Westman, H. (2005). Dynamical origin of quantum probabilities. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461, 253–272.

    Google Scholar 

  • Van Wesep, R. A. (2006). Many worlds and the appearance of probability in quantum mechanics. Annals of Physics, 321, 2438–2452.

    Google Scholar 

  • Wallace, D. (2007). Quantum probability from subjective likelihood: Improving on Deutsch’s proof of the probability rule. Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 38, 311–332.

    Google Scholar 

  • Wallace, D. (2010). How to prove the Born rule. In S. Saunders, J. Barrett, A. Kent, D. Wallace (Eds.) Many worlds?: Everett, quantum theory, & reality (pp. 227–263). Oxford University Press.

    Google Scholar 

  • Zurek, W. H. (2005). Probabilities from entanglement, Born’s rule p k = |ψ k|2 from envariance. Physical Review A, 71, 052105.

    Google Scholar 

Download references

Acknowledgements

This work has been supported in part by the Israel Science Foundation Grant No. 2064/19.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Lev Vaidman .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Vaidman, L. (2020). Derivations of the Born Rule. 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_26

Download citation

Publish with us

Policies and ethics