Abstract
We investigate prepositional systems for local field theories, which reflect intrinsically the uncertainties of measurements made on the physical system, and satisfy the isotony and local commutativity postulates of Haag and Kastler. The space-time covariance can be implemented in a natural way in these propositional systems. New techniques are introduced to obtain these propositional systems: the lattice-valued logics. The decomposition of the complete orthomodular lattice-valued logics shows that these logics are more general than the usual two-valued ones and that in these logics there is enough structure to characterize the classical and quantum, nonrelativistic and relativistic local field theories in a natural way. The Hubert modules give the natural inner product “spaces” (modules) for the realization of the lattice-valued logics.
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References
Banai, M. (1978). preprint, KFKI-1978-20.
Banai, M. (1980a). “A New Algebraic Approach to Nonrelativistic Quantum Theory,” submitted toJ. Math. Phys.
Banai, M. (1980b). “Propositional Systems in Field Theories and Lattice Valued Quantum Logics,” inCurrent Issues in Quantum Logic, E. Beltrametti ed. Plenum Press, New York.
Banai, M. (1980c). Thesis, University Loránd Eötvös, Budapest.
Birkhoff, G., and von Neumann, J. (1936). “The Logic of Quantum Mechanics, Annals of Mathematics,37, 823–843.
Emch, G. G. (1972).Algebraic Methods in Statistical Mechanics and Quantum Field Theory. Wiley-Interscience, New York.
Gudder, S. (1970). “Axiomatic Quantum Mechanics and Generalized Probability Theory,” inProbabilistic Methods in Applied Mathematics, Vol. 2, A. T. Bharucha-Reid ed. Academic Press, New York, pp. 53–129.
Haag, R., and Kastler, D. (1964). “An Algebraic Approach to Quantum Field Theory,”Journal of Mathematical Physics,5(7), 848–861.
Jauch, J. (1968).Foundations of Quantum Mechanics. Addison-Wesley, Reading, Massachusetts.
Jech, Th. (1971).Lectures in Set Theory with Particular Emphasis on the Method of Forcing, Springer, Berlin.
Kaplansky, I. (1953). “Modules over Operator Algebras,”American Journal of Mathematics,75, 839–858.
Mackey, G. (1963).The Mathematical Foundations of Quantum Mechanics. Benjamin, New York.
Piron, C. (1976).Foundation of Quantum Physics. Benjamin, New York.
Takeuti, G., and Zaring, W. M. (1973).Axiomatic Set Theory. Springer, New York.
Takeuti, G. (1980). “Quantum Set Theory,” inCurrent Issues in Quantum Logic, E. Beltrametti ed. Plenum Press, New York.
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The author is also with the Publishing House of the Hungarian Academy of Sciences, H-1363, Budapest, P.O.B. 24.
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Banai, M. Propositional systems in local field theories. Int J Theor Phys 20, 147–169 (1981). https://doi.org/10.1007/BF00669793
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DOI: https://doi.org/10.1007/BF00669793