This paper surveys John von Neumann's work on the mathematical foundations of quantum theories in the light of Hilbert's Sixth Problem concerning the geometrical axiomatization of physics. We argue that in von Neumann's view geometry was so tied to logic that he ultimately developed a logical interpretation of quantum probabilities. That motivated his abandonment of Hilbert space in favor of von Neumann algebras, specifically the type II1factors, as the proper limit of quantum mechanics in infinite dimensions. Finally, we present the reasons why his axiomatic program remained an "unsolved problem" in mathematical physics. A recent unpublished result by Huzimiro Araki, proving that no algebra with a tracial state defined on it, such as the type II1factors, can support any (regular) representation of the canonical commutation relations, is also reviewed and its consequences for von Neumann's projects are discussed. © 2008 Elsevier Ltd. All rights reserved.

John von Neumann's mathematical "Utopia" in quantum theory

Valente, Giovanni
2008

Abstract

This paper surveys John von Neumann's work on the mathematical foundations of quantum theories in the light of Hilbert's Sixth Problem concerning the geometrical axiomatization of physics. We argue that in von Neumann's view geometry was so tied to logic that he ultimately developed a logical interpretation of quantum probabilities. That motivated his abandonment of Hilbert space in favor of von Neumann algebras, specifically the type II1factors, as the proper limit of quantum mechanics in infinite dimensions. Finally, we present the reasons why his axiomatic program remained an "unsolved problem" in mathematical physics. A recent unpublished result by Huzimiro Araki, proving that no algebra with a tracial state defined on it, such as the type II1factors, can support any (regular) representation of the canonical commutation relations, is also reviewed and its consequences for von Neumann's projects are discussed. © 2008 Elsevier Ltd. All rights reserved.
STUDIES IN HISTORY AND PHILOSOPHY OF MODERN PHYSICS
CCRs; Geometry; Logic; Quantum probabilities; von Neumann; History; Physics and Astronomy (all); History and Philosophy of Science
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/1043313
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