No-signaling in topos formulation and a common ontological basis for classical and non-classical physical theories

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Marek Kuś

Abstract

Starting from logical structures of classical and quantum mechanics we reconstruct the logic of so-called no-signaling theories, where the correlations among subsystems of a composite system are restricted only by a simplest form of causality forbidding an instantaneous communication. Although such theories are, as it seems, irrelevant for the description of physical reality, they are helpful in understanding the relevance of quantum mechanics. The logical structure of each theory has an epistemological flavor, as it is based on analysis of possible results of experiments. In this note we emphasize that not only logical structures of classical, quantum and no-signaling theory may be treated on the same ground but it is also possible to give to all of them a common ontological basis by constructing a “phase space” in all cases. In non-classical cases the phase space is not a set, as in classical theory, but a more general object obtained by means of category theory, but conceptually it plays the same role as the phase space in classical physics.

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References

Aaronson, S., 2004. Is quantum mechanics an island in theoryspace? arXiv:quant-ph/0401062 [Online]. Available at: [visited on 30 November 2020].

Bell, J.S., 1964. On the Einstein-Podolsky-Rosen paradox. Physics, 1, pp.195–200.

Birkhoff, G. and von Neumann, J., 1936. The logic of quantum mechanics. Annals of Mathematics, 37, pp.823–843.

Cirel’son, B.S., 1980. Quantum generalizations of Bell’s inequality. Letters in Mathematical Physics [Online], 4(2), pp.93–100. Available at: https://doi.org/10.1007/BF00417500 [visited on 30 November 2020].

Clauser, J.F., Horne, M.A., Shimony, A. and Holt, R.A., 1969. Proposed experiment to test local hidden-variable theories. Physical Review Letters, 23(15), pp.880–884.

Döring, A. and Isham, C.J., 2008a. A topos foundation for theories of physics: I. Formal languages for physics. Journal of Mathematical Physics [Online], 49(5), p.053515. Available at: https://doi.org/10.1063/1.2883740 [visited on 30 November 2020].

Döring, A. and Isham, C.J., 2008b. A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory. Journal of Mathematical Physics [Online], 49(5), p.053516. Available at: https://doi.org/10.1063/1.2883742 [visited on 30 November 2020].

Döring, A. and Isham, C.J., 2008c. A topos foundation for theories of physics: III. The representation of physical quantities with arrows. Journal of Mathematical Physics [Online], 49(5), p.053517. Available at: https://doi.org/10.1063/1.2883777 [visited on 30 November 2020].

Döring, A. and Isham, C.J., 2008d. A topos foundation for theories of physics: IV. Categories of systems. Journal of Mathematical Physics [Online], 49(5), p.053518. Available at: https://doi.org/10.1063/1.2883826 [visited on 30 November 2020].

Flori, C., 2013. A First Course in Topos Quantum Theory, Lecture Notes in Physics 868. Berlin; Heidelberg: Springer.

Flori, C., 2018. A Second Course in Topos Quantum Theory, Lecture Notes in Physics 944. Cham, Switzerland: Springer.

Gleason, A.M., 1957. Measures on the closed subspaces of a Hilbert space. Journal of Mathematics and Mechanics [Online], 6(6), pp.885–893. Available at: [visited on 25 November 2020].

Goldblatt, R., 2014. Topoi: the Categorial Analysis of Logic. Amsterdam: Elsevier Science.

Gutt, J. and Kuś, M., 2016. Non-signalling boxes and Bohrification. arXiv:1602.04702 [math-ph, physics:quant-ph] [Online]. Available at: [visited on 30 November 2020].

Heunen, C., Landsman, N.P. and Spitters, B., 2009. A topos for algebraic quantum theory. Communications in Mathematical Physics [Online], 291(1), pp.63–110. Available at: https://doi.org/10.1007/s00220-009-0865-6 [visited on 30 November 2020].

Heunen, C., Landsman, N.P., Spitters, B. and Wolters, S.A., 2011. The Gelfand spectrum of a noncommutative C*-algebra: a topos-theoretic approach. Journal of the Australian Mathematical Society [Online], 90(1), pp.39–52. Available at: https://doi.org/10.1017/S1446788711001157 [visited on 30 November 2020].

Kuhn, T.S., 1970. The Structure of Scientific Revolutions. 2nd ed., enl, International Encyclopedia of Unified Science vol. 2, no. 2. Chicago; London: University of Chicago Press.

Popescu, S. and Rohrlich, D., 1994. Quantum nonlocality as an axiom. Foundations of Physics [Online], 24(3), pp.379–385. Available at: https://doi.org/10.1007/BF02058098 [visited on 30 November 2020].

Tylec, T.I. and Kuś, M., 2015. Non-signaling boxes and quantum logics. Journal of Physics A: Mathematical and Theoretical [Online], 48(50), p.505303. Available at: https://doi.org/10.1088/1751-8113/48/50/505303 [visited on 30 November 2020].

Tylec, T.I. and Kuś, M., 2018. Ignorance is a bliss: Mathematical structure of many-box models. Journal of Mathematical Physics [Online], 59(3), p.032202. Available at: https://doi.org/10.1063/1.5027205 [visited on 30 November 2020].

Wolters, S.A., 2013. A comparison of two topos-theoretic approaches to quantum theory. Communications in Mathematical Physics [Online], 317(1), pp.3–53. Available at: https://doi.org/10.1007/s00220-012-1652-3 [visited on 30 November 2020].