Foundations of Physics

, Volume 45, Issue 10, pp 1341–1350 | Cite as

Quantum Theory as a Critical Regime of Language Dynamics

Article

Abstract

Some mathematical theories in physics justify their explanatory superiority over earlier formalisms by the clarity of their postulates. In particular, axiomatic reconstructions drive home the importance of the composition rule and the continuity assumption as two pillars of quantum theory. Our approach sits on these pillars and combines new mathematics with a testable prediction. If the observer is defined by a limit on string complexity, information dynamics leads to an emergent continuous model in the critical regime. Restricting it to a family of binary codes describing ‘bipartite systems,’ we find strong evidence of an upper bound on bipartite correlations equal to 2.82537. This is measurably different from the Tsirelson bound. The Hilbert space formalism emerges from this mathematical investigation as an effective description of a fundamental discrete theory in the critical regime.

Keywords

Reconstruction of quantum theory Algebraic coding theory Critical phenomena Continuity Tsirelson bound  Bipartite correlations 

References

  1. 1.
    Adler, S.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1995)MATHGoogle Scholar
  2. 2.
    Barrett, J.: Information processing in generalized probabilistic theories. Phys. Rev. A 75, 032304 (2007)CrossRefADSGoogle Scholar
  3. 3.
    Bell, J.: On the Einstein–Podolsky–Rosen paradox. Physica 1, 195–200 (1964)Google Scholar
  4. 4.
    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. Phys. 37 823–843 (1936) (Reprinted in J. von Neumann, Collected Works, Pergamon Press, Oxford, 1961, Vol. IV, pp. 105–125)Google Scholar
  5. 5.
    Bohr, N.: Atomic Theory and the Description of Nature. Cambridge University Press, Cambridge (1934)MATHGoogle Scholar
  6. 6.
    Bohr, N.: Atomic Theory and Human Knowledge. Wiley, New York (1958)Google Scholar
  7. 7.
    Chiribella, G., d’Ariano, G.M., Perinotti, P.: Informational derivation of quantum theory. Phys. Rev. A 84, 012311 (2011)CrossRefADSGoogle Scholar
  8. 8.
    Christensen, B.G., et al.: Detection-loophole-free test of quantum nonlocality, and applications. Phys. Rev. Lett. 111, 130406 (2013)CrossRefADSGoogle Scholar
  9. 9.
    Cirel’son, B.S.: Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4(2), 93–100 (1980)MathSciNetCrossRefADSGoogle Scholar
  10. 10.
    Clifton, R., Bub, J., Halvorson, H.: Characterizing quantum theory in terms of information-theoretic constraints. Found. Phys. 33(11), 1561–1591 (2003)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dakić B., Brukner, Č.: Quantum theory and beyond: Is entanglement special? In: Halvorson, H. (ed) Deep Beauty: Understanding the Quantum World through Mathematical Innovation, pp. 365–392, Cambridge University Press, Cambridge (2011). arXiv:0911.0695
  12. 12.
    El-Showk, S., et al.: Solving the 3D ising model with the conformal bootstrap. Phys. Rev. D 86, 025022 (2012). arXiv:1203.6064
  13. 13.
    El-Showk, S., et al.: Solving the 3D ising model with the conformal bootstrap ii. C-minimization and precise critical exponents. J. Stat. Phys. 157, pp. 869–914 (2014). arXiv:1403.4545
  14. 14.
    Everett, H.: “Relative state” formulation of quantum mechanics. Rev. Mod. Phys. 29, 454–462 (1957)MathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Gorelik, G., Frenkel, V.: Matvei Petrovich Bronstein and Soviet Theoretical Physics in the Thirties. Birkhäuser, Boston (1994)MATHCrossRefGoogle Scholar
  16. 16.
    Grinbaum, A.: Reconstruction of quantum theory. Br. J. Philos. Sci. 58, 387–408 (2007)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Hardy, L.: Quantum theory from five reasonable axioms (2000). arXiv:quant-ph/00101012
  18. 18.
    Hilbert, D., von Neumann, J., Nordheim, L.: Über die Grundlagen der Quantenmechanik. Math. Ann. 98, 1–30 (1927) (Reprinted in J. von Neumann, Collected Works, Pergamon Press, Oxford, 1961, Vol. I, pp. 104–133)Google Scholar
  19. 19.
    Jammer, M.: The Philosophy of Quantum Mechanics. Wiley, New York (1974)Google Scholar
  20. 20.
    Khrennikov, A., Schumann, A.: Quantum non-objectivity from performativity of quantum phenomena. Phys. Scr. T163, 014020 (2014). arXiv:1404.7077 CrossRefADSGoogle Scholar
  21. 21.
    Landsman, N.: Mathematical Topics Between Classical and Quantum Mechanics. Spinger, New York (1998)CrossRefGoogle Scholar
  22. 22.
    Mackey, G.: Quantum mechanics and Hilbert space. Am. Math. Mon. 64, 45–57 (1957)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Manin, Y.: Complexity vs. energy: theory of computation and theoretical physics. J. Phys. 532, 012018 (2014). arXiv:1302.6695 Google Scholar
  24. 24.
    Manin, Y., Marcolli, M.: Errorcorrecting codes and phase transitions. Math. Comput. Sci. 5, pp. 155–179 (2011). arXiv:0910.5135
  25. 25.
    Manin, Y., Marcolli, M.: Kolmogorov complexity and the asymptotic bound for errorcorrecting codes. J. Differ. Geom. 97(1), 91–108 (2014). arXiv:1203.0653
  26. 26.
    Masanes, L., Müller, M.: A derivation of quantum theory from physical requirements. New J. Phys. 13, 063001 (2011)CrossRefADSGoogle Scholar
  27. 27.
    Nawareg, M., et al.: Bounding quantum theory with the exclusivity principle in a two-city experiment (2013). arXiv:1311.3495
  28. 28.
    Von Neumann, J.: Selected Letters. American Mathematical Society, London Mathematical Society, London (2005)MATHGoogle Scholar
  29. 29.
    Piron, C.: Axiomatique quantique. Helv. Phys. Acta 36, 439–468 (1964)MathSciNetGoogle Scholar
  30. 30.
    Polyakov, A.M.: Conformal symmetry of critical fluctuations. JETP Lett. 12, 381–383 (1970)ADSGoogle Scholar
  31. 31.
    Popescu, S.: Nonlocality beyond quantum mechanics. Nat. Phys. 10, 264–270 (2014)CrossRefGoogle Scholar
  32. 32.
    Popescu, S., Rohrlich, D.: Nonlocality as an axiom for quantum theory. Foundations of Physics 24, 379 (1994). arXiv:quant-ph/9508009
  33. 33.
    Ryckman, T.: The Reign of Relativity. Oxford University Press, Oxford (2005)CrossRefGoogle Scholar
  34. 34.
    Spekkens, R.: Evidence for the epistemic view of quantum states: a toy theory. Phys. Rev. A 75, 032110 (2007). arXiv:quant-ph/0401052
  35. 35.
    von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932)MATHGoogle Scholar
  36. 36.
    Weyl, H.: Raum-Zeit-Materie. Springer, Berlin (1918)Google Scholar
  37. 37.
    Wigner, E.: The unreasonable effectiveness of mathematics in the natural sciences. Commun. Pur. Appl. Math. 13(1), 1–14 (1960)MATHCrossRefADSGoogle Scholar
  38. 38.
    Wittgenstein, L.: Tractatus Logico-Philosophicus. Dover Publications, Mineola (1921/1998)Google Scholar
  39. 39.
    Zieler, N.: Axioms for non-relativistic quantum mechanics. Pac. J. Math. 11, 1151–1169 (1961)CrossRefGoogle Scholar
  40. 40.
    Zurek, W.: Algorithmic randomness and physical entropy. Phys. Rev. A 40, 4731–4751 (1989a)MathSciNetCrossRefADSGoogle Scholar
  41. 41.
    Zurek, W.: Thermodynamic cost of computation, algorithmic complexity and the information metric. Nature 341, 119–124 (1989b)CrossRefADSGoogle Scholar
  42. 42.
    Zurek, W.: Decoherence, chaos, quantum-classical correspondence, and the algorithmic arrow of time. Phys. Scr. T76, 186–198 (1998)MathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.CEA-Saclay/IRFU/LARSIMGif-sur-YvetteFrance

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