Abstract
We presented a contextual statistical model of the probabilistic description of physical reality. Here contexts (complexes of physical conditions) are considered as basic elements of reality. There is discussed the relation with QM. We propose a realistic analogue of Bohr’s principle of complementarity. In the opposite to the Bohr’s principle, our principle has no direct relation with mutual exclusivity for observables. To distinguish our principle from the Bohr’s principle and to give better characterization, we change the terminology and speak about supplementarity, instead of complementarity. Supplementarity is based on the interference of probabilities. It has quantitative expression trough a coefficient which can be easily calculated from experimental statistical data. We need not appeal to the Hilbert space formalism and noncommutativity of operators representing observables. Moreover, in our model there exists pairs of supplementary observables which cannot be represented in the complex Hilbert space. There are discussed applications of the principle of supplementarity outside quantum physics.
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N. Bohr (1987) The Philosophical Writings of Niels Bohr NumberInSeries3 vols. Ox Bow Press Woodbridge, Conn
D. Hilbert J. Neumann Particlevon L. Nordheim (1927) Math. Ann. 98 1 Occurrence Handle1512390
P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford Univ. Press, 1930).
W. Heisenberg (1930) Physical Principles of Quantum Theory Chicago Univ. Press Chicago
J. Neumann Particlevon (1932) Mathematische Grundlagen der Quantenmechanik Springer Berlin
J. Neumann Particlevon (1955) Mathematical foundations of Quantum Mechanics Princeton Univ. Press Princeton, NJ
A. Einstein B. Podolsky N. Rosen (1935) Phys. Rev. 47 777 Occurrence Handle10.1103/PhysRev.47.777
N. Bohr (1935) Phys. Rev. 48 696 Occurrence Handle10.1103/PhysRev.48.696 Occurrence Handle0012.42701
A.S. Wightman (1976) Proc. Symp. Pure Math. 28 147 Occurrence Handle0339.46051 Occurrence Handle55 #9739
G.W. Mackey (1963) Mathematical Foundations of Quantum Mechanics W. A. Benjamin Inc. New York
A. Landé, Foundations of Quantum Theory (Yale Univ. Press, 1955); New foundations of quantum mechanics (Cambridge Univ. Press, Cambridge, 1968).
L. De Broglie, The Current Interpretation of Wave Mechanics. A Critical Study (Elsevier, London, 1964). D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1951).
S. P. Gudder, Trans. AMS 119, 428 (1965); Axiomatic Quantum Mechanics and Generalized Probability Theory (Academic Press, New York, 1970); “An Approach to Quantum Probability,” in Foundations of Probability and Physics, A. Yu. Khrennikov, ed., Quantum Prob. White Noise Anal. 13, 147 (2001).
L. E. Ballentine, Rev. Mod. Phys. 42, 358 (1970); Quantum Mechanics (Englewood Cliffs, New Jersey, 1989); “Interpretations of probability and quantum theory,” in Foundations of Probability and Physics, A. Yu. Khrennikov, ed., Quantum Prob. White Noise Anal. 13, 71 (2001); Quantum Mechanics (WSP, Singapore, 1998).
L. Accardi, Phys. Rep. 77, 169 (1981); “The probabilistic Roots of the Quantum Mechanical Paradoxes,” in The Wave–Particle Dualism: A Tribute to Louis de Broglie on his 90th Birthday, S. Diner, D. Fargue, G. Lochak, and F. Selleri, eds. (D. Reidel Publ. Company, Dordrecht, 1984), pp. 47–55; Urne e Camaleoni: Dialogo sulla realta, le leggi del caso e la teoria quantistica (Il Saggiatore, Rome, 1997); “Locality and Bell’s Inequality”, in Foundations of Probability and Physics, A. Yu. Khrennikov, ed., Q. Prob. White Noise Anal. 13, 1 (2001).
P. Mittelstaedt and E. W. Stachow, Int. J. Theor. Phys. 22, 517 (1983); P. Mittelstaedt, The Interpretation of Quantum Mechanics and the Measurement Process (Cambridge Univ. Press, Cambridge, 1997); Ann. Phys. 7, 710 (1998); “Quantum Mechanics without Probabilities”, in John von Neumann and the Foundations of Quantum Physics, M. Rdei and Stltzner, eds. (Kluwer Academic, Dordrecht, Holland, 2001), pp. 121–134; “Universality and Consistency in Quantum Mechanics—New Problems of an Old Theory”, in Fsica cuntica y realidad. Quantum Physics and Reality (Madrid, 2000), C. Mataix and A. Rivadulla eds. (Editorial Complutense, Madrid, 2002), pp. 197–213.
W. M. De Muynck, “Interpretations of Quantum Mechanics, and Interpretations of Violations of Bell’s Inequality”, in Foundations of Probability and Physics, A. Yu. Khrennikov, ed., Q. Prob. White Noise Anal. 13, 95 (2001).
W.M. De Muynck (2002) Foundations of Quantum Mechanics, An Empiricists Approach Kluwer Dordrecht
A. Yu. Khrennikov, ed., “Foundations of Probability and Physics”, Q. Prob. White Noise Anal. 13 (WSP, Singapore, 2001).
A. Yu. Khrennikov, ed., “Quantum Theory: Reconsideration of Foundations,” Ser. Math. Modeling Vol. 2 (Växjö Univ. Press, Växjö, 2002).
A. Yu. Khrennikov, ed., “Foundations of Probability and Physics −2,” Ser. Math. Modeling Vol. 5 (Växjö Univ. Press, Växjö, 2003).
A. Yu. Khrennikov, ed., “Quantum Theory: Reconsideration of Foundations −2,” Ser. Math. Modeling Vol. 10 (Växjö Univ. Press, Växjö, 2004).
A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen, in Foundations of Probability and Physics-3, A. Yu. Khrennikov, ed., AIP Conference Proceedings, 2005, pp. 16–24.
O. Nánásiová (2003) Int. J. Theor. Phys. 42 1889 Occurrence Handle1053.81006
A. Plotnitsky, “Quantum Atomicity and Quantum Information: Bohr, Heisenberg, and Quantum Mechanics as an Information Theory”, in Quantum Theory: Reconsideration of Foundations, A. Yu. Khrennikov, ed., (Växjö Univ. Press, 2002), pp. 309–343.
A. Plotnitsky (2001) Reading Bohr: Complementarity, Epistemology, Entanglement, and Decoherence A. Gonis P. Turchi (Eds) NATO Workshop Decoherence and its Implications for Quantum Computations IOS Press Amsterdam 3–37
A. Plotnitsky, The Knowable and Unknowable (Modern Science, Nonclassical Thought, and the “Two Cultures” (Univ. Michigan Press, 2002); Found. Phys. 33, 1649 (2003).
S. Gudder S. Pulmannova (1998) Comm. Math. Univ. Carolinae 39 645 Occurrence Handle2000i:81009
A. Yu. Khrennikov, “On Foundations of Quantum Theory,” in Quantum Theory: Reconsideration of Foundations, A. Yu. Khrennikov, ed., Ser. Math. Modeling, Vol. 2 (Växjö University Press, 2002), pp. 163–173.
A. Yu. Khrennikov, Annalen der Physik 12, 575 (2003); “On the classical limit for the hyperbolic quantum mechanics,” quant-ph/0401035; J. Phys. A: Math. Gen. 34, 9965 (2001); Il Nuovo Cimento B 117, 267 (2002); J. Math. Phys. 43 789 (2002); Ibid 45, 902 (2004).
K. Svozil, “Counterfactuals and Contextuality,” in Quantum Theory: Reconsideration of Foundations-3, A. Yu. Khrennikov, ed., AIP Conference Proceedings, 2005, pp. 425–437.
H. Atmanspacher and H. Primas, “Epistemic and Ontic Quantum Realities”, in Foundations of Probability and Physics-3, A. Yu. Khrennikov, ed., AIP Conference Proceedings, 2005, pp. 10–25.
E. Beltrametti G. Cassinelli (1981) The Logic of Quantum Mechanics Addison-Wesley Publ. Com. London-Amsterdam
P. Busch M. Grabowski P. Lahti (1995) Operational Quantum Physics Springer Verlag Berlin
E.B. Davies (1976) Quantum Theory of Open Systems Academic Press London
A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982).
S. P. Gudder, J. Math. Phys. 11, 431 (1970); Quantum Probability (Academic Press, San Diego, 1988).
W. Muynck Particlede W. De Baere H. Martens (1994) Found. Physics 24 1589
A.N. Shiryayev (1984) Probability Springer New York–Berlin–Heidelberg
A.Yu. Khrennikov (2004) ArticleTitleOn Quantum-like Probabilistic Structure of Mental Information Open Syst. Inform. Dynam. 11 IssueID3 267 Occurrence Handle1059.82033 Occurrence Handle2102381
E. Conte, O. Todarello, A. Federici, F. Vitiello, M. Lopane, and A. Yu. Khrennikov, “A Preliminary Evidence of Quantum-like Behavior in Measurements of Mental States,” in Quantum Theory: Reconsideration of Foundations-2, A. Yu. Khrennikov, ed., Vol. 10 (Växjö Univ. Press, Växjö, 2004), pp. 679–702; http://xxx.lanl.gov/abs/quant-ph/0307201; A. A. Grib, A. Yu. Khrennikov, and K. Starkov, “Probability Amplitude in Quantum-like Games”, in Quantum Theory: Reconsideration of Foundations-2, A. Yu. Khrennikov, ed., Vol. 10 (Växjö Univ. Press, Växjö, 2004), pp. 703–722.
A.Yu. Khrennikov (2004) Information Dynamics in Cognitive, Psychological and Anomalous Phenomena Kluwer Dordrecht
R. Mises Particlevon (1964) The Mathematical Theory of Probability and Statistics Academic London
A.Yu. Khrennikov (1999) Interpretations of Probability VSP Int. Sc. Publ. Utrecht
A. N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer Verlag, Berlin, 1933); reprinted: Foundations of the Probability Theory (Chelsea Publ. Comp., New York, 1956).
R.T. Cox (1961) The Algebra of Probable Inference J. Hopkins Univ. Press Baltimore, MD
R. Ashby, Design of Mind (Chapman–Hall, London, 1952); D. Amit. Modeling Brain Function (Cambridge Univ Press, Cambridge, 1989).
R. Penrose, The Emperor’s New Mind (Oxford Univ. Press, New-York, 1989); Shadows of the Mind (Oxford Univ. Press, Oxford, 1994).
A. Yu. Khrennikov, Found. Phys. 32, 1159 (2002); Phys. Lett. A. 278, 307 (2001); Il Nuovo Cimento 119 (2004).
A.Yu. Khrennikov (2004) Found. Phys. Lett. 17 691 Occurrence Handle10.1007/s10702-004-1122-z Occurrence Handle1063.81020 Occurrence Handle2005h:81025
W. De Baere, Lett. Nuovo Cimento 39, 234 (1984); 40, 488 (1984); Advances in Electron. Electron Phys. 68, 245 (1986).
K. Hess and W. Philipp, Proc. Nat. Acad. Sc. 98, 14224 (2001); 98, 14227 (2001); 101, 1799 (2004); Europhys. Lett. 57, 775 (2002).
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Khrennikov, A. The Principle of Supplementarity: A Contextual Probabilistic Viewpoint to Complementarity, the Interference of Probabilities and Incompatibility of Variables in Quantum Mechanics. Found Phys 35, 1655–1693 (2005). https://doi.org/10.1007/s10701-005-6511-z
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DOI: https://doi.org/10.1007/s10701-005-6511-z