Linear representations of probabilistic transformations induced by context transitions

By using straightforward frequency arguments we classify transformations of probabilities which can be generated by transition from one preparation procedure (context) to another. There are three classes of transformations corresponding to statistical deviations of different magnitudes: (a) trigonometric; (b) hyperbolic; (c) hyper-trigonometric. It is shown that not only quantum preparation procedures can have trigonometric probabilistic behaviour. We propose generalizations of C-linear space probabilistic calculus to describe non-quantum (trigonometric and hyperbolic) probabilistic transformations. We also analyse the superposition principle in this framework.

[1]  R. Morrow,et al.  Foundations of Quantum Mechanics , 1968 .

[2]  W. M. de Muynck,et al.  Interpretations of quantum mechanics, joint measurement of incompatible observables, and counterfactual definiteness , 1994 .

[3]  K. Wan,et al.  Superconducting rings, superselection rules and quantum measurement problems , 1993 .

[4]  Luigi Accardi,et al.  The Probabilistic Roots of the Quantum Mechanical Paradoxes , 1984 .

[5]  Marian Grabowski,et al.  Operational Quantum Physics , 2001 .

[6]  L. E. Ballentine.  Interpretations of Probability and Quantum Theory , 2001 .

[7]  Andrei Khrennikov.  Quantum statistics via perturbation effects of preparation procedures , 2001 .

[8]  L. Ballentine,et al.  Quantum Theory: Concepts and Methods , 1994 .

[9]  Harald Bergstriim.  Mathematical Theory of Probability and Statistics , 1966 .

[10]  Jean-Luc Ville.  Étude critique de la notion de collectif , 1939 .

[11]  A. Kolmogoroff.  Grundbegriffe der Wahrscheinlichkeitsrechnung , 1933 .

[12]  L. Ballentine,et al.  Probabilistic and Statistical Aspects of Quantum Theory , 1982 .

[13]  Itamar Pitowsky,et al.  Deterministic model of spin and statistics , 1983 .

[14]  P. Dirac.  Principles of Quantum Mechanics , 1982 .

[15]  Abner Shimony,et al.  The logic of quantum mechanics , 1981 .

[16]  S. Gudder.  Review: A. S. Holevo, Probabilistic and statistical aspects of quantum theory , 1985 .

[17]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[18]  Andrei Khrennikov.  Hyperbolic quantum mechanics , 2000 .

[19]  Andrei Khrennikov,et al.  A perturbation of CHSH inequality induced by fluctuations of ensemble distributions , 2000 .

[20]  Paul Adrien Maurice Dirac,et al.  Bakerian Lecture - The physical interpretation of quantum mechanics , 1942, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[21]  W. M. de Muynck,et al.  On the Significance of the Bell Inequalities for the Locality Problem in Different Realistic Interpretations of Quantum Mechanics , 1988 .

[22]  Leslie E Ballentine,et al.  The statistical interpretation of quantum mechanics , 1970 .

[23]  W. Heisenberg.  The Physical Principles of the Quantum Theory , 1930 .

[24]  B. D'espagnat.  Veiled Reality: An Analysis Of Present-day Quantum Mechanical Concepts , 1995 .

[25]  K. Wan,et al.  Classical Systems, Standard Quantum Systems, and Mixed Quantum Systems in Hilbert Space , 1998 .

[26]  M. Sentís.  Quantum theory of open systems , 2002 .

[27]  C. Ross.  Found , 1869, The Dental register.

[28]  E. Wigner.  Quantum-Mechanical Distribution Functions Revisited , 1997 .

[29]  Andrei Khrennikov,et al.  Ensemble fluctuations and the origin of quantum probabilistic rule , 2002 .

[30]  N. Bohr.  II - Can Quantum-Mechanical Description of Physical Reality be Considered Complete? , 1935 .

[31]  J. Neumann.  Mathematical Foundations of Quantum Mechanics , 1955 .

[32]  Günther Ludwig.  Foundations of quantum mechanics , 1983 .

[33]  Andrei Khrennikov,et al.  Interpretations of Probability , 1999 .

[34]  R. Feynman,et al.  Quantum Mechanics and Path Integrals , 1965 .

[35]  Andrei Khrennikov.  Contextualist viewpoint to Greenberger-Horne-Zeilinger paradox , 2001 .

[36]  J. Linnett,et al.  Quantum mechanics , 1975, Nature.