Hidden Variables: Rehabilitation of von Neumann's Analysis, and Pauli's Uncashable Check
Abstract: In his book \textit{The Mathematical Foundations of Quantum Mechanics}, published in 1932, J. von Neumann performed an analysis of the consequences of introducing hidden parameters (hidden variables) into quantum mechanics. He showed that hidden variables cannot be incorporated into the existing theory of quantum mechanics without major modifications, and concluded that if they did exist, the theory would have already failed in situations where it has been successfully applied. von Neumann left open the possibility that the theory is not complete, and his analysis for internal consistency is the best that can be done for a self-referenced logical system (G\"odel's theorem). This analysis had been taken as an incorrect proof" against the existence of hidden variables. von Neumann's so-called proof isn't even wrong as such a proof does not exist. One of the earliest attempts at a hidden variable theory was by D. Bohm, and because there were no experimental consequences, W. Pauli referred to it as anuncashable check." To our knowledge, a successful hidden variable extension to quantum mechanics with testable consequences has not yet been produced, suggesting that von Neumann's analysis is worthy of rehabilitation, which we attempt to provide in a straightforward manner.
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- Our discussion overlaps partly with that of Jeffrey Bub presented in “von Neumann’s ‘No Hidden Variables’ Proof: A Re-appraisal,” Found. Phys. 40, 1333 (2010), arXiv:1006.0499 [quant-ph].
- Von Neumann’s proof is cast in modern notation by James Albertson in “Von Neumann’s Hidden-Parameter Proof,” Am. J Phys. 29, 478 (1961).
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- Pauli to Bohm Dec.3 1951 in Meyenn, K. von, ed., “Wolfgang Pauli - Scientific Correspondence with Bohr, Einstein, Heisenberg, a.o. Volume IV, Part I: 1950-1952,” p.436 no. 1313 (Springer 1966).
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- Pauli, W. to Fierz, M., Jan. 6, 1952, in Ref. [10], p. 499 no. 1337. See also : Pauli, W. “Remarques sur le problem des parametres cache dans la mecanique quantiqe et sur la theorie de l’onde pilote,” (Remarks on the hidden parameter problem in quantum mechanics and the theory of the pilot wave), in Louis de Broglie, Physicien et Penseur (Paris: Michel, 1953). See also CERN-ARCH-PMC-06-071 at the Pauli Archives, http://cds.cern.ch/record/96311?ln=sv.
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