Abstract
Several attempts have been made to provide an axiomatic basis for the statistical transformation theory in quantum physics in the form of simple general principles. Thus, in his well-known book on Quantum Mechanics Dirac uses the superposition principle as a fundamental principle. This principle permits indeed to determine many characteristic features of the mathematical formalism. It does not, however, suffice to determine even the algebra of the state calculus, as Dirac has noticed himself. From the present point of view the superposition principle may be looked upon as an ingenious but rather artificial formulation of complementarity.
Translated from ‘Zur Begründung der Statistischen Transformationstheorie der Quantenphysik’, Sitz. Ber. Berl. Akad. Wiss., Phys.-Math. Kl. 27 (1936), 90–113.
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Notes
J. von Neumann, ‘Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik’, Goett. Nachr. (1928), 245;
Mathematische Grundlagen der Quantenmechanik, Berlin 1932, Kap. IV.
See also M. Born und P. Jordan, Elementare Quantenmechanik, Berlin 1930, 6. Kap.
[The true nature of this connection has only emerged much later in the study of the intertheory relations between quantum mechanics and classical Hamiltonian mechanics. The upshot is this: the Principle of Correspondence allows us to consider as meaningful statements of the form ‘The value of a quantity Q of a physical system S lies within the interval (q 1, q 2)’; statements of this form referring to complementary quantities are then to be treated as complementary in the sense of complementarity logic, i.e., inconnectible, as shown in this paper. On the other hand, we may use instead statements of the form ‘The physical system S is in a state where the quantity Q has a value between q 1 and q 2 ’, in this case statements referring to complementary quantities can be treated as contradictory (their conjunction would be allowed as meaningful but untrue) and the need for complementarity logic — or any other ‘logic of quantum theory’ — does not arise. The essential difference between the two statements emerges when their negations are considered: the negation of the first form would read ‘The value of quantity Q of S lies outside the interval (q1, q2)’ — this is indeed the proper negation for classical and the (improper) negation for quantum mechanics — but this is not equivalent to the (proper) negation of the second form which includes all states where the quantity Q has no value in any finite interval. The notion ‘proper negation’ as used here is a semantical one. A semantically adequate syntactic characterization of ‘proper negation’ for arbitrary languages has been attempted (e.g. by Carnap, Logical Syntax of Language, London 1937), but most attempts can be shown to be inadequate. In the present paper measurement statements are supposed to have the first form mentioned above.]
H. Reichenbach, ‘Axiomatik der Wahrscheinlichkeitsrechnung’, Math. Z. 34 (1932), 568;
H. Reichenbach, Wahrscheinlichkeitslehre, Leiden 1935.
A. Kolmogoroff, ‘Grundbegriffe der Wahrscheinlichkeitsrechnung’, Erg. Math. II/3 (1933).
A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47 (1935), 777.
E. Schroedinger, Naturwiss. 23 (1935), 807,
E. Schroedinger, Naturwiss. 23 (1935), 823,
E. Schroedinger, Naturwiss. 23 (1935), 844.
[If we take the span instead of the (set theoretical) sum we have the same situation as with the section: the span of two linear subsets is always again a linear subset (subspace). The set of subspaces then forms an orthocomplemented lattice in which — contrary to the Boolean lattice — the distributive laws do not hold. This is the ‘Quantum Logic’ advocated by Birkhoff and von Neumann. The negation in this logic has the same meaning as in our complementarity logic, i.e., it too is a nonproper negation, referring as it does to the orthogonal complement.]
P. Suppes, ‘Probability Concepts in Quantum Mechanics’, Phil. of Sc. 28 (1961), 378–389;
P. Suppes, ‘Logics Appropriate to Empirical Theories’, in The Theory of Models (ed. by J. W. Addison, L. Henkin and A. Tarski), Amsterdam 1965, p. 364–375;
P. Suppes, ‘Une logique nonclassique de la méchanique quantique’, Synthese 10 (1966), 74–85.
F. Kamber, ‘Die Struktur des Aussagenkalkuels in einer physikalischen Theorie’, Nachr. Akad. Wiss. Goettingen 10 (1964), 103–124.
H. Putnam, ‘Is Logic Empirical?’ in Boston Studies in the Philosophy of Science, vol. V (ed. by R. S. Cohen and M. W. Wartofsky), Dordrecht 1969, pp. 216–241.
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© 1975 D. Reidel Publishing Company, Dordrecht, Holland
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Strauss, M. (1975). The Logic of Complementarity and the Foundation of Quantum Theory. In: Hooker, C.A. (eds) The Logico-Algebraic Approach to Quantum Mechanics. The University of Western Ontario Series in Philosophy of Science, vol 5a. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1795-4_2
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