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Quantum Mechanics and Operational Probability Theory

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Abstract

We discuss a generalization of the standard notion of probability space and show that the emerging framework, to be called operational probability theory, can be considered as underlying quantal theories. The proposed framework makes special reference to the convex structure of states and to a family of observables which is wider than the familiar set of random variables: it appears as an alternative to the known algebraic approach to quantum probability.

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REFERENCES

  • Accardi, L., A. Frigerio and J.T. Lewis: 1992, Quantum Stochastic Processes. Publ. RIMS, Kyoto Univ. 18: 97–133.

    Google Scholar 

  • Bell, J.S.: 1964, On the Einstein-Podolsky-Rosen Paradox. Physics 1: 195–200.

    Google Scholar 

  • Beltrametti, E.G. and S. Bugajski: 1995, A Classical Extension of Quantum Mechanics. J. Phys. A: Math. Gen. 28: 3329–3343.

    Google Scholar 

  • Beltrametti, E.G. and S. Bugajski: 1996, The Bell Phenomenon in Classical Frameworks. J. Phys. A: Math. Gen. 29: 247–261.

    Google Scholar 

  • Beltrametti, E.G. and S. Bugajski: 1997, Effect Algebras and Statistical Physical Theories. J. Math. Phys. 38: 3020–3030.

    Google Scholar 

  • Bohm, D.: 1951, Quantum Theory. Prentice-Hall, Englewood Cliffs, N.J.

    Google Scholar 

  • Bugajski, S.: 1995, Classical and Quantal in One, or How to DescribeMesoscopic Systems, Molecular Physics Reports 11: 161–171.

    Google Scholar 

  • Bugajski, S.: 1996, Fundamentals of Fuzzy Probability Theory. Int. J. Theor. Phys. 35: 2229–2244.

    Google Scholar 

  • Bugajski, S.: 1997, Fuzzy Dynamics in Terms of Fuzzy Probability Theory. In M. Mares, R. Mesiar, V. Novak, J. Ramik and A. Stupnanova (eds.), IFSA' 97 Proceedings, Vol IV. Academia, Prague, pp. 255–260.

    Google Scholar 

  • Bugajski, S.: 1998, Net Entropies of Fuzzy Stochastic Processes. Open Systems and Information Dynamics 5: 187–200.

    Google Scholar 

  • Bugajski, S.: 1998, Fuzzy Stochastic Processes. Open Systems and Information Dynamics 5: 169–185.

    Google Scholar 

  • Bugajski, S., K.E. Hellwig and W. Stulpe: 1998, On Fuzzy Random Variables and Statistical Maps. Rep. Math. Phys. 41: 1–11.

    Google Scholar 

  • Busch, P., M. Grabowski and P.J. Lahti: 1995, Operational Quantum Physics. Springer-Verlag, Berlin.

    Google Scholar 

  • Cassinelli, G. and P.J. Lahti: 1993, Spectral Properties of Observables and Convex Mappings in Quantum Mechanics. J. Math. Phys. 34: 5468–5475.

    Google Scholar 

  • Davies, E.B. and J.T. Lewis: 1970, An Operational Approach to Quantum Probability. Comm. Math. Phys. 17: 239–260.

    Google Scholar 

  • Gudder, S.: 1998, Fuzzy Probability Theory. Demonstratio Mathematica 31: 235–254.

    Google Scholar 

  • Misra, B.: 1974, On a New Definition of Quantal States. In C.P. Enz and J. Mehra (eds.), Physical Reality and Mathematical Description. D. Reidel Publ. Co., Dordrecht, pp. 455–476.

    Google Scholar 

  • Ohya, M. and D. Petz: 1993, Quantum Entropy and Its Use. Springer-Verlag, Berlin.

    Google Scholar 

  • Stulpe, W.: 1988, Conditional Expectations, Conditional Distributions, and a Posteriori Ensembles in Generalised Probability Theory. Int. J. Theor. Phys. 27: 587.

    Google Scholar 

  • Zadeh, L.A.: 1965, Information and Control 8: 338–353.

    Google Scholar 

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Beltrametti, E., Bugajski, S. Quantum Mechanics and Operational Probability Theory. Foundations of Science 7, 197–212 (2002). https://doi.org/10.1023/A:1016007827863

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