Abstract
The basic rules of quantum mechanics are reformulated. They deal primarily with individual systems and do not assume that every ket may represent a physical state. The customary kinematic and dynamic rules then allow to construct consistent Boolean logics describing the history of a system, following essentially Griffiths' proposal. Logical implication is defined within these logics, the multiplicity of which reflects the complementary principle. Only one interpretative rule of quantum mechanics is necessary in such a framework. It states that these logics providebona fide foundations for the description of a quantum system and for reasoning about it. One attempts to build up classical physics, including classical logic, on these quantum foundations. The resulting theory of measurement needs not to statea priori that the eigenvalues of an observable have to be the results of individual measurements nor to assume wave packet reduction. Both these properties can be obtained as consequences of the basic rules. One also needs not to postulate that every observable is measurable, even in principle. A proposition calculus is obtained, allowing in principle the replacement of the discussion of problems concerned with the practical interpretation of experiments by due calculations.
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References
J. Von Neumann,Mathematical Foundations of Quantum Mechanics (Princeton University Press, 1955).
R. Griffiths,Stat. Phys. 36:219 (1984).
R. Griffiths,Am. J. Phys. 55:11 (1987).
R. Omnès,C. R. Acad. Sci. 304:1039 (1987).
R. Omnès,Phys. Lett. A 125:169 (1987).
P. A. M. Dirac,The Principles of Quantum Mechanics, 4th ed. (Clarendon Press, Oxford, 1956).
B. d'Espagnat,Conceptual Foundations of Quantum Mechanics, 2nd ed. (W. A. Benjamin, Reading, Massachusetts, 1976).
H. Primas,Chemistry, Quantum Mechanics and Reductionism (Springer-Verlag, Berlin, 1981).
M. Reed and B. Simon,Functional Analysis, Vol. I (Academic Press, New York, 1972).
A. Einstein, B. Podolsky, and N. Rosen,Phys. Rev. 47:777 (1935).
W. V. Quine,Mathematical Logic (Harvard University Press, Cambridge, Massachusetts, 1965).
R. P. Feynman,Rev. Mod. Phys. 20:367 (1948).
V. Aharonov, P. G. Bergman, and J. L. Lebowitz,Phys. Rev. 134B:1410 (1964).
A. Renyi,Foundations of Probability (Holden-Day, San Francisco, 1970).
B. d'Espagnat,Phys. Lett. A 124:204 (1987).
M. Jammer,The Philosophy of Quantum Mechanics (Wiley, New York, 1974);The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966).
H. Weyl,Bull. Am. Math. Soc. 56:115 (1950).
C. L. Fefferman,Bull. Am. Math. Soc 9:129 (1983).
L. Hörmander,The Analysis of Linear Partial Differential Operators (Springer-Verlag, Berlin, 1985).
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer,Ann. Phys. (N.Y.) 111:61, 111 (1978).
I. Daubechies, unpublished.
I. Daubechies, A. Grossmann, and Y. Meyer,J. Math. Phys. 27:1271 (1986).
I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai,Ergodic Theory (Springer-Verlag, Berlin, 1982).
E. P. Wigner,Am. J. Phys. 31:6 (1963).
W. Heisenberg,Z. Phys. 43:172 (1927).
W. Pauli,Die Allgemeinen Prinzipien der Wellenmechanik, in Handbuch der Physik (Springer-Verlag, Berlin, 1933).
L. Van Hove, private communication.
H. Margenau, inQuantum Theory of Atoms, Molecules and the Solid State. A Tribute to John G. Slater, P. O. Löwdin, ed. (Academic Press, New York, 1966).
N. Bohr,Atomic Physics and Human Knowledge (Wiley, New York, 1963).
E. Schrödinger,Naturwissenschaften 23:807, 823, 844 (1935).
D. Bohm,Quantum Theory (Constable, London, 1954).
F. London and E. Bauer,La théorie de l'observation en mécanique quantique (Hermann, Paris, 1939).
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Omnès, R. Logical reformulation of quantum mechanics. I. Foundations. J Stat Phys 53, 893–932 (1988). https://doi.org/10.1007/BF01014230
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DOI: https://doi.org/10.1007/BF01014230