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Journal of Mathematical Physics

Quantum logic as a consequence of realistic measurements on deterministic systems

H. D. Doebner, W. Lücke

Published Jan 1, 1991• 4 citations
Publisher

Abstract

It is shown why, from the deterministic point of view, quantum logic and not classical logic is the appropriate language for describing realistic measurements.
DOI
10.1063/1.529127

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In concluding their major survey, “Quantum Logics,” Greechie and Gudder (1975) listed four outstanding directions for future research, the last of which was, “Explain the meaning of the word ‘logic’ in the title of this paper.” Broadly speaking, there are two radically different points of view on this. There is the view that, taken strictly, ‘logic’ is a misnomer: quantum logic studies certain algebraic structures naturally arising from the formal, apparatus of quantum theory, structures which may afford insights into quantum physics, on a number of levels, but which do not in any way involve giving up or replacing classical logical laws. (Cf., e.g., Jauch (1968, p. 77).) In contrast, there is the view that the non-Boolean structures arising from the Hilbert space formalism should be taken as giving rise to a genuine non-classical logic, in conflict with classical logic, but appropriate to the description of quantum mechanical reality.
Quantum logic is the logic of the high-level language in which we describe what is true and false of quantum systems. I use the term ‘highlevel’ in its computational sense. The language whose logic quantum logic is, is inexpressive. It has no machinery for describing probabilities, only truths and falsehoods and perhaps whatever is in between. Quantum logic conceals from its user the underlying details of the formalism of quantum mechanics, and the details (whatever they may be) of the underlying ‘machine’, the physical world. Quantum logic encourages a top-down view of the quantum world. Quantum mechanics, as done by the practising physicist, is bottom-up. The program of feeding back quantum logic into our metalevel discussion of realism fails. Quantum logic does not licence quantummechanical realism. It cannot override or rewrite the bottom-up view. The program itself is odd in that quantum logic is most naturally thought of as expressing quantum-mechanical antirealism, just as quantum mechanics itself is most naturally interpreted antirealistically. Thus quantum logic is consonant with nonlocality, in the sense that it does not allow one to derive the Bell inequalities. The conditional in quantum logic nicely expresses the effect of ideal measurements on quantum systems.
Heelan argues that the philosophy of complementarity, although successful in providing physicists with a common language with which to describe quantum phenomena, also contains a theory about scientific method and about human knowing which is open to criticism. Heelan here criticises the following points arising out of the philosophy of complementarity: psycho-physical parallelism; the view that quantum mechanical properties are to be defined classically; and the perturbation theory of measurement. In the course of the criticism, he elaborates the distinction between two types of concepts with different logical structures; viz., operational or observational concepts which state a similarity between things judged on the basis of appearance or utility to us, and explanatory concepts which state a similarity between things judged on the basis of a self-defining set of different relations between things. Heelan thus shows how a physical concept is definable by any appropriate measuring-process. The description of the measuring­process and, hence, the definition of the physical property involve the two classes of concepts described above, but in different ways. Thus one may regard the Indeterminacy Relations, not as stating limi­tations of our knowledge, but as describing more exactly the behaviour of individual systems.
The Bell-CHSH formalism is often misinterpreted as a direct refutation of local realism (LR). This paper corrects that view by demonstrating a strict deductive necessity: once LR is assumed as a physical premise alongside empirical data, the violation of measurement independence (MI) is no longer a choice or a "loophole," but a mandatory logical consequence. The work formalizes the transition from a symmetric disjunction to a deterministic implication, proving that under LR, ¬MI is a theorem.