Von Neumann, Gödel and Quantum Incompleteness

Thomas Breuer

Claudio Garola

Susanna Tall, Arvi Hakanen

Kurt Gödel’s incompleteness theorem refers to incompleteness in the sense that there are always statements which can be formulated within a formal system, but which re-main unproved within the system. Gödel’s theorem was first connected with the theory of computability by Alan Turing. Until recently, the inconsistency that Gödel’s theo-rem tackle has not been directly shown to correspond to any physical phenomenon. A recent article showed how to apply Gödel’s theorem in the context of many-body quan-tum mechanical calculations. Even so, it is still not known whether it can be applied to the most primitive quantum physical phenomena. In the present study, we used Gödel’s incompleteness theorem in the case of the quantum double-slit experiment by applying a proof of Turing machine’s halting problem. In addition, we made an analogy between Gödel’s theorem (and Church - Turing theses of computability), and quantum compu-ting by extrapolating the term “computation” as low as to the single-quantum level.

Euan J. Squires

M. Dalla Chiara, Roberto Giuntini, Richard J. Greechie

Robert Golub, S. K. Lamoreaux

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 (Gödel'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 an ``uncashable 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.