John von Neumann's mathematical “Utopia” in quantum theory

Authors
Giovanni Valente
University of Pittsburgh
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 II1II1 factors, 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 A recent unpublished result by Huzimiro Araki, proving that no algebra with a tracial state defined on it, such as the type II1II1 factors, can support any (regular) representation of the canonical commutation relations, is also reviewed and its consequences for von Neumann's projects are discussed.
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.1016/j.shpsb.2008.06.002
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

Our Archive


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 37,153
Through your library

References found in this work BETA

Interpreting Quantum Field Theory.Laura Ruetsche - 2002 - Philosophy of Science 69 (2):348-378.
The Logic of Quantum Mechanics.Garrett Birkhoff & John von Neumann - 1937 - Journal of Symbolic Logic 2 (1):44-45.
Von Neumann’s Concept of Quantum Logic and Quantum Probability.Miklós Rédei - 2001 - Vienna Circle Institute Yearbook 8:153-172.

View all 7 references / Add more references

Citations of this work BETA

Add more citations

Similar books and articles

Von Neumann’s Concept of Quantum Logic and Quantum Probability.Miklós Rédei - 2001 - Vienna Circle Institute Yearbook 8:153-172.
What John von Neumann Thought of the Bohm Interpretation.Michael Stöltzner - 1999 - Vienna Circle Institute Yearbook 7:257-262.
Von Neumann, Gödel and Quantum Incompleteness.Thomas Breuer - 2001 - Vienna Circle Institute Yearbook 8:75-82.
Von Neumann's 'No Hidden Variables' Proof: A Re-Appraisal. [REVIEW]Jeffrey Bub - 2010 - Foundations of Physics 40 (9-10):1333-1340.
Entropy, von Neumann and the von Neumann Entropy.Dénes Petz - 2001 - Vienna Circle Institute Yearbook 8:83-96.
Mathematical Foundations of Quantum Mechanics.John von Neumann & R. T. Beyer - 1955 - British Journal for the Philosophy of Science 8 (32):343-347.
Mathematical Physics and Philosophy of Physics.Miklós Rédei - 2002 - Vienna Circle Institute Yearbook 9:239-243.
The Von Neumann Entropy: A Reply to Shenker.Leah Henderson - 2003 - British Journal for the Philosophy of Science 54 (2):291-296.
Von Neumann’s Theory of Quantum Measurement.Jeffrey Bub - 2001 - Vienna Circle Institute Yearbook 8:63-74.

Analytics

Added to PP index
2009-01-28

Total downloads
45 ( #147,891 of 2,308,776 )

Recent downloads (6 months)
1 ( #448,127 of 2,308,776 )

How can I increase my downloads?

Monthly downloads

Sorry, there are not enough data points to plot this chart.

My notes

Sign in to use this feature