Foundations of Physics

, Volume 49, Issue 2, pp 83–95 | Cite as

A Local \(\psi \)-Epistemic Retrocausal Hidden-Variable Model of Bell Correlations with Wavefunctions in Physical Space

  • Indrajit SenEmail author


We construct a local \(\psi \)-epistemic hidden-variable model of Bell correlations by a retrocausal adaptation of the originally superdeterministic model given by Brans. In our model, for a pair of particles the joint quantum state \(|\psi _e(t)\rangle \) as determined by preparation is epistemic. The model also assigns to the pair of particles a factorisable joint quantum state \(|\psi _o(t)\rangle \) which is different from the prepared quantum state \(|\psi _e(t)\rangle \) and has an ontic status. The ontic state of a single particle consists of two parts. First, a single particle ontic quantum state \(\chi (\vec {x},t)|i\rangle \), where \(\chi (\vec {x},t)\) is a 3-space wavepacket and \(|i\rangle \) is a spin eigenstate of the future measurement setting. Second, a particle position in 3-space \(\vec {x}(t)\), which evolves via a de Broglie–Bohm type guidance equation with the 3-space wavepacket \(\chi (\vec {x},t)\) acting as a local pilot wave. The joint ontic quantum state \(|\psi _o(t)\rangle \) fixes the measurement outcomes deterministically whereas the prepared quantum state \(|\psi _e(t)\rangle \) determines the distribution of the \(|\psi _o(t)\rangle \)’s over an ensemble. Both \(|\psi _o(t)\rangle \) and \(|\psi _e(t)\rangle \) evolve via the Schrodinger equation. Our model exactly reproduces the Bell correlations for any pair of measurement settings. We also consider ‘non-equilibrium’ extensions of the model with an arbitrary distribution of hidden variables. We show that, in non-equilibrium, the model generally violates no-signalling constraints while remaining local with respect to both ontology and interaction between particles. We argue that our model shares some structural similarities with the modal class of interpretations of quantum mechanics.


Retrocausality 3D wavefunctions Local causality Bell’s theorem Modal interpretations 



I would like to thank my thesis advisor Antony Valentini for stimulating discussions and helpful suggestions throughout this work, and for encouragement to work on retrocausality. I would also like to thank Ken Wharton for several helpful comments and discussions on an earlier draft of the paper.


  1. 1.
    Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  2. 2.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47(10), 777 (1935)ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Hall, M.J.: Local deterministic model of singlet state correlations based on relaxing measurement independence. Phys. Rev. Lett. 105(25), 250404 (2010)ADSCrossRefGoogle Scholar
  4. 4.
    Barrett, J., Gisin, N.: How much measurement independence is needed to demonstrate nonlocality? Phys. Rev. Lett. 106(10), 100406 (2011)ADSCrossRefGoogle Scholar
  5. 5.
    Price, H., Wharton, K.: A live alternative to quantum spooks (2015). arXiv:1510.06712
  6. 6.
    Pironio, S.: Random choices and the locality loophole (2015). arXiv:1510.00248
  7. 7.
    Brans, C.H.: Bell’s theorem does not eliminate fully causal hidden variables. Int. J. Theor. Phys. 27(2), 219–226 (1988)CrossRefGoogle Scholar
  8. 8.
    Hooft, G.: The fate of the quantum (2013). arXiv:1308.1007
  9. 9.
    Gallicchio, J., Friedman, A.S., Kaiser, D.I.: Testing Bells inequality with cosmic photons: closing the setting-independence loophole. Phys. Rev. Lett. 112(11), 110405 (2014)ADSCrossRefGoogle Scholar
  10. 10.
    Handsteiner, J., Friedman, A.S., Rauch, D., Gallicchio, J., Liu, B., Hosp, H., Kofler, J., Bricher, D., Fink, M., Leung, C.: Cosmic bell test: measurement settings from milky way stars. Phys. Rev. Lett. 118(6), 060401 (2017)ADSCrossRefGoogle Scholar
  11. 11.
    Cramer, J.G.: The transactional interpretation of quantum mechanics. Rev. Mod. Phys. 58(3), 647 (1986)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Price, H.: Time’s Arrow & Archimedes’ Point: New Directions for the Physics of Time. Oxford University Press, New York (1997)CrossRefGoogle Scholar
  13. 13.
    Aharonov, Y., Vaidman, L.: The Two-State Vector Formalism: An Updated Review. Time in Quantum Mechanics. Springer, Berlin (2008)zbMATHGoogle Scholar
  14. 14.
    Hall, M.J.: At the Frontier of Spacetime. The Significance of Measurement Independence for Bell Inequalities and Locality. Springer, Berlin (2016)Google Scholar
  15. 15.
    Sen, I.: Violating the Assumption of Measurement Independence in Quantum Foundations (2017). arXiv:1705.02434
  16. 16.
    Valentini, A.: Signal-locality, uncertainty, and the subquantum H-theorem. I. Phys. Lett. A 156(1–2), 5–11 (1991)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Valentini, A.: Signal-locality in hidden-variables theories. Phys. Lett. A 297(5–6), 273–278 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wood, C.J., Spekkens, R.W.: The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations require fine-tuning. New J. Phys. 17(3), 033002 (2015)ADSCrossRefGoogle Scholar
  19. 19.
    Valentini, A.: Astrophysical and cosmological tests of quantum theory. J. Phys. A 40(12), 3285 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Bell, J.S.: The theory of local beables. In: John S Bell on the Foundations of Quantum Mechanics. World Scientific, Singapore (2001)Google Scholar
  21. 21.
    Harrigan, N., Spekkens, R.W.: Einstein, incompleteness, and the epistemic view of quantum states. Found. Phys. 40(2), 125–157 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pusey, M.F., Barrett, J., Rudolph, T.: On the reality of the quantum state. Nat. Phys. 8, 475–478 (2012)CrossRefGoogle Scholar
  23. 23.
    Almada, D., Ch’ng, K., Kintner, S., Morrison, B., Wharton, K.: Are retrocausal accounts of entanglement unnaturally fine-tuned? (2015). arXiv:1510.03706
  24. 24.
    Price, H., Wharton, K.: Disentangling the quantum world. Entropy 17(11), 7752–7767 (2015)ADSCrossRefGoogle Scholar
  25. 25.
    Wharton, K.: Quantum states as ordinary information. Information 5(1), 190–208 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Lombardi, O., Dieks, D.: Modal interpretations of quantum mechanics. In: The Stanford Encyclopedia of Philosophy (2017)Google Scholar
  27. 27.
    Sutherland, R.I.: Causally symmetric Bohm model. Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Mod. Phys. 39(4), 782–805 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Norsen, T.: The theory of (exclusively) local beables. Found. Phys. 40(12), 1858–1884 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Norsen, T., Marian, D., Oriols, X.: Can the wave function in configuration space be replaced by single-particle wave functions in physical space? Synthese 192(10), 3125–3151 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Gondran, M., Gondran, A.: Replacing the singlet spinor of the EPR-B experiment in the configuration space with two single-particle spinors in physical space. Found. Phys. 46(9), 1109–1126 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Corry, R.: Retrocausal models for EPR. Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Mod. Phys. 49, 1–9 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Kinard Laboratory, Department of Physics and AstronomyClemson UniversityClemsonUSA

Personalised recommendations