Skip to main content

Advertisement

Log in

A generalized definition of Bell’s local causality

  • Published:
Synthese Aims and scope Submit manuscript

Abstract

This paper aims to implement Bell’s notion of local causality into a framework, called local physical theory, which is general enough to integrate both probabilistic and spatiotemporal concepts and also classical and quantum theories. Bell’s original idea of local causality will then arise as the classical case of our definition. First, we investigate what is needed for a local physical theory to be locally causal. Then we compare local causality with Reichenbach’s common cause principle and relate both to the Bell inequalities. We find a nice parallelism: both local causality and the common cause principle are more general notions than captured by the Bell inequalities. Namely, Bell inequalities cannot be derived neither from local causality nor from a common cause unless the local physical theory is classical or the common cause is commuting, respectively.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Notes

  1. We note that our definition of a local physical theory does not embrace models beyond the Tsirelson bound. In order to incorporate also such models (Popescu–Rohrlich box) one should generalize the net of local algebras to a net of order-unit vector spaces. See (Summers and Werner 1987a) and (Popescu and Rohrlich 1994).

  2. For the sake of uniformity we slightly changed Bell’s notation and figure.

  3. For a similar approach to local causality using \(\sigma \)-algebras see (Henson 2013); for a general definition of local causality via completely positive maps and for a comparison of the two approaches see our (Hofer-Szabó and Vecsernyés 2015).

  4. Finding a common cause for a correlation does not mean to provide the most detailed description for the physical situation; it simply means that at this coarser level of description correlations can be causally accounted for. For an opposing view see Uffink (1999) and Henson (2005).

  5. For an argument to use noncommuting common causes in causal explanation of quantum correlations see (Hofer-Szabó and Vecsernyés 2013a, b). For a criticism of noncommuting common causal explanation see Cavalcanti and Lal (2014) and for an answer to this see our (Hofer-Szabó and Vecsernyés 2015).

References

  • Bell, J. S. (2004). Speakable and unspeakable in quantum mechanics. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Butterfield, J. (1995). Vacuum correlations and outcome independence in algebraic quantum field theory. In D. Greenberger & A. Zeilinger (Eds.), Fundamental problems in quantum theory (pp. 768–785). New York: New York Academy of Sciences.

    Google Scholar 

  • Butterfield, J. (2007). Stochastic Einstein locality revisited. British Journal for the Philosophy of Science, 58, 805–867.

    Article  Google Scholar 

  • Brunetti, R., Fredenhagen, K., & Verch, R. (2003). The generally covariant locality principle: A new paradigm for local quantum field theory. Communications in Mathematical Physics, 237, 31–68.

    Article  Google Scholar 

  • Cavalcanti, E. G., & Lal, R. (2014). On modifications of Reichenbach’s principle of common cause in light of Bell’s theorems. Journal of Physics A, 47, 424018.

    Article  Google Scholar 

  • Earman, J., & Valente, G. (2014). Relativistic causality in algebraic quantum field theory. International Studies in the Philosophy of Science, 28(1), 1–48.

    Article  Google Scholar 

  • Haag, R. (1992). Local quantum physics. Berlin: Springer.

    Book  Google Scholar 

  • Halvorson, H. (2007). Algebraic quantum field theory. In J. Butterfield & J. Earman (Eds.), Philosophy of physics (Vol. I, pp. 731–922). Amsterdam: Elsevier.

    Chapter  Google Scholar 

  • Henson, J. (2005). Comparing causality principles. Studies in History and Philosophy of Modern Physics, 36, 519–543.

    Article  Google Scholar 

  • Henson, J. (2013). Non-separability does not relieve the problem of Bell’s theorem. Foundations of Physics, 43, 1008–1038.

    Article  Google Scholar 

  • Hofer-Szabó, G., & Vecsernyés, P. (2012a). Reichenbach’s common cause principle in AQFT with locally finite degrees of freedom. Foundations of Physics, 42, 241–255.

  • Hofer-Szabó, G., & Vecsernyés, P. (2012b). Noncommuting local common causes for correlations violating the Clauser-Horne inequality. Journal of Mathematical Physics, 53, 12230.

  • Hofer-Szabó, G., & Vecsernyés, P. (2013a). Noncommutative common cause principles in AQFT. Journal of Mathematical Physics, 54, 042301.

  • Hofer-Szabó, G., & Vecsernyés, P. (2013b). Bell inequality and common causal explanation in algebraic quantum field theory. Studies in the History and Philosophy of Modern Physics, 44(4), 404–416.

  • Hofer-Szabó, G., Rédei, M., & Szabó, L. E. (2013). The principle of the common cause. Cambridge: Cambridge University Press.

  • Hofer-Szabó G. (2015). Local causality and complete specification: A reply to Seevinck and Uffink. In U. Mäki, I. Votsis, S. Ruphy, & G. Schurz (Eds.), Recent Developments in the Philosophy of Science: EPSA \(_\mathit{13}\) Helsinki (pp. 209–226). Switzerland: Springer.

  • Hofer-Szabó, G., & Vecsernyés, P. (2015). On Bell’s local causality in local classical and quantum theory. Journal of Mathematical Physics, 56, 032303.

  • Norsen, T. (2009). Local causality and completeness: Bell versus Jarrett. Foundations of Physics, 39, 273.

    Article  Google Scholar 

  • Norsen, T. (2011). J.S. Bell’s concept of local causality. American Journal of Physics, 79, 12.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Rédei, M. (1997). Reichenbach’s common cause principle and quantum field theory. Foundations of Physics, 27, 1309–1321.

    Article  Google Scholar 

  • Rédei, M., & Summers, J. S. (2002). Local primitive causality and the common cause principle in quantum field theory. Foundations of Physics, 32, 335–355.

    Article  Google Scholar 

  • Rédei, M. (2014). A categorial approach to relativistic locality. Studies in History and Philosophy of Modern Physics, 48, 137–146.

    Article  Google Scholar 

  • Reichenbach, H. (1956). The direction of time. Los Angeles: University of California Press.

    Google Scholar 

  • Ruetsche, L. (2011). Interpreting quantum theories. Oxford: Clarendon Press.

    Book  Google Scholar 

  • Seevinck, M. P., & Uffink, J. (2011). Not throwing out the baby with the bathwater: Bell’s condition of local causality mathematically ’sharp and clean’. In D. Dieks, W. J. Gonzalez, S. Hartmann, Th Uebel, & M. Weber (Eds.), Explanation, prediction, and confirmation the philosophy of science in a European perspective (Vol. 2, pp. 425–450). Dordrecht: Springer.

    Chapter  Google Scholar 

  • Summers, S. J., & Werner, R. (1987a). Bell’s inequalities and quantum field theory, I: General setting. Journal of Mathematical Physics, 28, 2440–2447.

    Article  Google Scholar 

  • Summers, S. J., & Werner, R. (1987b). Bell’s inequalities and quantum field theory, II: Bell’s inequalities are maximally violated in the vacuum. Journal of Mathematical Physics, 28, 2448–2456.

    Article  Google Scholar 

  • Summers, S. J., & Werner, R. (1988). Maximal violation of Bell’s inequalities for algebras of observables in tangent spacetime regions. Annales de l’Institut Henri Poincaré - Phys. Théor., 49, 215–243.

    Google Scholar 

  • Summers, S. J. (1990). On the independence of local algebras in quantum field theory. Reviews in Mathematical Physics, 2, 201–247.

    Article  Google Scholar 

  • Summers, S. J. (2009). Subsystems and independence in relativistic microphysics. Studies in History and Philosophy of Modern Physics, 40, 133–141.

    Article  Google Scholar 

  • Uffink, J. (1999). The principle of the common cause faces the Bernstein paradox. Philosophy of Science, 66, 512–525.

    Article  Google Scholar 

Download references

Acknowledgments

This work has been supported by the Hungarian Scientific Research Fund OTKA K-100715 and K-108384, and the National Research, Development and Innovation Office, K-115593.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gábor Hofer-Szabó.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hofer-Szabó, G., Vecsernyés, P. A generalized definition of Bell’s local causality. Synthese 193, 3195–3207 (2016). https://doi.org/10.1007/s11229-015-0925-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11229-015-0925-8

Keywords

Navigation