Abstract
Following the idea of Zadeh, the concept of a statistical (or fuzzy)σ algebra is introduced. For two extreme cases of classical and quantum statisticalσ algebras the representation theorems are proved. The basic feature distinguishing these two cases is the possibility of producing nontrivial superpositions of pure quantum states, which is absent in the classical case.
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Bugajska, K., and Bugajski, S. (1973).Ann. Inst. Henri Poincaré,19, 333.
Bugajska, K., and Bugajski, S. (1973).Bull. Acad. Polon. Sci., Ser. Math,21, 873.
Gerelle, E. R., Greechie, R. J., and Miller, F. R. (1974). InPhysical Reality and Mathematical Description, pp. 169–192. Reidel, Dordrecht.
Gudder, S. P. (1970).J. Math. Phys., 1037.
Guz, W. (1974).Rep. Math. Phys.,6, 445.
Guz, W. (1975).Rep. Math. Phys.,7, 313.
Guz, W. (1978).Ann. Inst. Henri Poincaré 29, 357.
Guz, W. (1980).Rep. Math. Phys.,17, 385.
Guz, W. (1981a).Ann. Inst. Henri Poincaré,34, 373
Guz, W. (1981b).Fortschr. Physik,29, 345.
Jauch, J. M., and Piron, C. (1969).Helv. Phys. Acta 42, 842.
Mackey, G. W. (1963).The Mathematical Foundations of Quantum Mechanics. Benjamin, New York.
Maeda, S., and Maeda, F. (1970).Theory of Symmetric Lattices. Springer, New York.
Pool, J. C. T. (1968).Commun. Math. Phys.,9, 212.
Pulmannova, (1976),Commun. Math. Phys.,49, 47.
Randall, C. H., and Foulis, D. J. (1972). InFoundations of Quantum Mechanics and Ordered Linear Spaces, Lecture Notes in Physics, No. 29. Springer, New York.
Varadarajan, V. S. (1968).Geometry of Quantum Theory, Vol. 1, Van Nostrand, Princeton, New Jersey.
Zadeh, L. (1965).Inform. Control,8, 338.
Zierler, N. (1961).Pacific J. Math. 11, 1151.
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A part of this work has been written during the author's stay at the Mathematics Department, University of Toronto (Canada). The financial support from the NSERC research grant No. A5206 is gratefully acknowledged.
On leave of absence from the Institute of Theoretical Physics, University of Gdańsk, 80-952 Gdańsk, Poland.
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Guz, W. Fuzzy σ algebras of physics. Int J Theor Phys 24, 481–493 (1985). https://doi.org/10.1007/BF00669908
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DOI: https://doi.org/10.1007/BF00669908