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Valuations on Functionally Closed Sets of Quantum Mechanical Observables and Von Neumann’s ‘No-Hidden-Variables’ Theorem

Jason Zimba, Rob Clifton

The Western Ontario series in philosophy of scienceJan 1, 19989 citations

How to reconcile modal interpretations of quantum mechanics with relativity

Joseph Berkovitz, Meir Hemmo

Modal interpretations 4 are hidden-variable, no-collapse (typically) indeterministic interpretations of quantum mechanics that were designed to solve the measurement problem and to reconcile quantum mechanics with relativity. But, as recent no-go theorems by Dickson and Clifton (1998), Arntzenius

Jan 1, 20032 citations

Two No-Go Theorems for Modal Interpretations of Quantum Mechanics

Pieter E. Vermaas

Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern PhysicsSep 1, 19998 citations

What is wrong with von Neumann's theorem on "no hidden variables"

Elemér Elad Rosinger

It is shown that, although correct mathematically, the celebrated 1932 theorem of von Neumann which is often interpreted as proving the impossibility of the existence of "hidden variables" in Quantum Mechanics, is in fact based on an assumption which is physically not reasonable. Apart from that, the alleged conclusion of von Neumann proving the impossibility of the existence of "hidden variables" was already set aside in 1952 by the counterexample of the possibility of a physical theory, such as given by what is usually called the "Bohmian Mechanics". Similar arguments apply to other two well known mathematical theorems, namely, of Gleason, and of Kochen and Specker, which have often been seen as equally proving the impossibility of the existence of "hidden variables" in Quantum Mechanics.It is shown that, although correct mathematically, the celebrated 1932 theorem of von Neumann which is often interpreted as proving the impossibility of the existence of "hidden variables" in Quantum Mechanics, is in fact based on an assumption which is physically not reasonable. Apart from that, the alleged conclusion of von Neumann proving the impossibility of the existence of "hidden variables" was already set aside in 1952 by the counterexample of the possibility of a physical theory, such as given by what is usually called the "Bohmian Mechanics". Similar arguments apply to other two well known mathematical theorems, namely, of Gleason, and of Kochen and Specker, which have often been seen as equally proving the impossibility of the existence of "hidden variables" in Quantum Mechanics.

arXiv:quant-ph/0408191Aug 31, 20047 citations

Logical Characterization of Contextual Hidden-Variable Theories based on Quantum Set Theory

Masanao Ozawa

While non-contextual hidden-variable theories are proved to be impossible, contextual ones are possible.In a contextual hidden-variable theory, an observable is called a beable if the hidden-variable assigns its value in a given measurement context specified by a state and a preferred observable.Halvorson and Clifton characterized the algebraic structure of beables as a von Neumann subalgebra, called a beable subalgebra, of the full observable algebra such that the probability distribution of every observable affiliated therewith admits the ignorance interpretation.On the other hand, we have shown that for every von Neumann algebra there is a unique set theoretical universe such that the internal "real numbers" bijectively correspond to the observables affiliated with the given von Neumann algebra.Here, we show that a set theoretical universe is associated with a beable subalgebra if and only if it is ZFC-satisfiable, namely, every theorem of ZFC set theory holds with probability equal to unity.Moreover, we show that there is a unique maximal ZFC-satisfiable subuniverse "implicitly definable", in the sense of Malament and others, by the given measurement context.The set theoretical language for the ZFC-satisfiable universe, characterized by the present work, rigorously reconstructs Bohr's notion of the "classical language" to describe the beables in a given measurement context.While non-contextual hidden-variable theories are proved to be impossible, contextual ones are possible.In a contextual hidden-variable theory, an observable is called a beable if the hidden-variable assigns its value in a given measurement context specified by a state and a preferred observable.Halvorson and Clifton characterized the algebraic structure of beables as a von Neumann subalgebra, called a beable subalgebra, of the full observable algebra such that the probability distribution of every observable affiliated therewith admits the ignorance interpretation.On the other hand, we have shown that for every von Neumann algebra there is a unique set theoretical universe such that the internal "real numbers" bijectively correspond to the observables affiliated with the given von Neumann algebra.Here, we show that a set theoretical universe is associated with a beable subalgebra if and only if it is ZFC-satisfiable, namely, every theorem of ZFC set theory holds with probability equal to unity.Moreover, we show that there is a unique maximal ZFC-satisfiable subuniverse "implicitly definable", in the sense of Malament and others, by the given measurement context.The set theoretical language for the ZFC-satisfiable universe, characterized by the present work, rigorously reconstructs Bohr's notion of the "classical language" to describe the beables in a given measurement context.

arXiv:2311.09268Electronic Proceedings in Theoretical Computer ScienceNov 14, 20232 citations

Hilbert-style axiomatic completion: On von Neumann and hidden variables in quantum mechanics

Chris Mitsch

Studies in History and Philosophy of Science Part AAug 19, 202221 citations

Valuations on Functionally Closed Sets of Quantum Mechanical Observables and Von Neumann's 'No-Hidden-Variables' Theorem

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Valuations on Functionally Closed Sets of Quantum Mechanical Observables and Von Neumann's 'No-Hidden-Variables' Theorem

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