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International Journal of Theoretical Physics

The significance of a hidden variable proof and the logical interpretation of quantum mechanics

Donald Hockney

Published Sep 1, 1978• 1 citation
Publisher
DOI
10.1007/bf00669974

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The interpretation of quantum mechanics has been a problem since its founding days. A large contribution to the discussion of possible interpretations of quantum mechanics is given by the so-called impossibility proofs for hidden variable models; models that allow a realist interpretation. In this thesis some of these proofs are discussed, like von Neumann's Theorem, the Kochen-Specker Theorem and the Bell-inequalities. Some more recent developments are also investigated, like Meyer's nullification of the Kochen-Specker Theorem, the MKC-models and Conway and Kochen's Free Will Theorem. This last one is taken to suggest that the problems that arise for certain interpretations of quantum mechanics are not limited to realist interpretations only, but also affect certain instrumentalist interpretations. It is argued that one may arrive at a more satisfying interpretation of quantum mechanics if one adopts a logic that seems more compatible with the instrumentalist viewpoint namely, intuitionistic logic. The motivations for adopting this form of logic rather than classical logic or quantum logic are linked to some of the philosophical ideas of Bohr. In particular a new interpretation of Bohr's notion of complementarity is proposed. Finally some possibilities are explored for linking the intuitionistic interpretation of quantum mechanics to the mathematical formalism of the theory.

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Problem- Many attempts have been made, starting from Quantum mechanics (QM), to prove the non-viability of hidden-variable (HV) solutions, and to justify the unnaturalness of QM explanations over physical realism. In particular, Colbeck & Renner (C&R) (2011) claimed to prove that no extension of quantum theory can exist with better predictive power than quantum mechanics itself. This implies that QM is an ideal theory, and that physical realism should be abandoned. Purpose- This paper critiques this proof. Approach- Logical considerations are used to examine the premises in the proof. Findings- The C&R proof is show to be capable of entirely the opposite interpretation that its authors intended, namely that a new deeper physics may exist based on physical realism, but that it will definitely not be quantum theory or even an extension thereof. Also, the proof is falsified by presenting a viable theory of internal structures (hidden-variable solution) with demonstrably better ontological explanatory power than QM. Implications- QM is interpreted as an approximate stochastic representation of a deeper dynamic behaviour at the finer scale. Originality- This work predicts that physical realism still applies, and that hidden-variable solutions still have vitality as the way forward for fundamental physics.

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The interpretation of quantum mechanics has been a problem since its founding days. A large contribution to the discussion of possible interpretations of quantum mechanics is given by the so-called impossibility proofs for hidden variable models; models that allow a realist interpretation. In this thesis some of these proofs are discussed, like von Neumann's Theorem, the Kochen-Specker Theorem and the Bell-inequalities. Some more recent developments are also investigated, like Meyer's nullification of the Kochen-Specker Theorem, the MKC-models and Conway and Kochen's Free Will Theorem. This last one is taken to suggest that the problems that arise for certain interpretations of quantum mechanics are not limited to realist interpretations only, but also affect certain instrumentalist interpretations. It is argued that one may arrive at a more satisfying interpretation of quantum mechanics if one adopts a logic that seems more compatible with the instrumentalist viewpoint namely, intuitionistic logic. The motivations for adopting this form of logic rather than classical logic or quantum logic are linked to some of the philosophical ideas of Bohr. In particular a new interpretation of Bohr's notion of complementarity is proposed. Finally some possibilities are explored for linking the intuitionistic interpretation of quantum mechanics to the mathematical formalism of the theory.
In this paper I provide a detailed history of von Neumann’s “No Hidden Variables” theorem, and I argue it is a demonstration that his axiomatization mathematically captures a salient feature of the statistical interpretation (namely, that hidden variables are incompatible). I show that this reading of von Neumann’s theorem is obvious once one recalls several contextual factors of his work. First, his axiomatization was what I call a Hilbert-style axiomatic completion; indeed, it developed from work initiated by Hilbert (and Nordheim). Second, it was responsive to specific mathematical and theoretical problems faced by Dirac and Jordan’s statistical transformation theory (then called ‘quantum mechanics’). Third, the axiomatization was essentially completed already in his 1927 papers, at least concerning the status of hidden variables, and this would have been obvious to the audience for those papers. Thus, the theorem’s statement and proof were only necessary when the material was presented for a general mathematical audience, i.e., in his 1932 Mathematical Foundations of Quantum Mechanics . With this reading in mind, his claim that quantum mechanics was in “compelling logical contradiction with causality” appears as a straightforward consequence of his theorem. I conclude by reassessing the theorem’s broader historical and scientific significance.
Problem- Many attempts have been made, starting from Quantum mechanics (QM), to prove the non-viability of hidden-variable (HV) solutions, and to justify the unnaturalness of QM explanations over physical realism. In particular, Colbeck & Renner (C&R) (2011) claimed to prove that no extension of quantum theory can exist with better predictive power than quantum mechanics itself. This implies that QM is an ideal theory, and that physical realism should be abandoned. Purpose- This paper critiques this proof. Approach- Logical considerations are used to examine the premises in the proof. Findings- The C&R proof is show to be capable of entirely the opposite interpretation that its authors intended, namely that a new deeper physics may exist based on physical realism, but that it will definitely not be quantum theory or even an extension thereof. Also, the proof is falsified by presenting a viable theory of internal structures (hidden-variable solution) with demonstrably better ontological explanatory power than QM. Implications- QM is interpreted as an approximate stochastic representation of a deeper dynamic behaviour at the finer scale. Originality- This work predicts that physical realism still applies, and that hidden-variable solutions still have vitality as the way forward for fundamental physics.