Probabilities, Laws, and Structures pp 497-506 | Cite as

# Some Historical and Philosophical Aspects of Quantum Probability Theory and its Interpretation

## Abstract

This paper argues that von Neumann’s work on the theory of ‘rings of operators’ has the same role and significance for quantum probability theory that Kolmogorov and his work represents for classical probability theory: Kolmogorov established classical probability theory as part of classical measure theory (Kolmogorov 1933); von Neumann established quantum probability theory as part of non-classical (non-commutative)measure theory based on von Neumann algebras (1935–1940). Since the quantum probability theory based on general von Neumann algebras contains as a special case the classical probability theory (Sect. 36.2), there is a very tight conceptual-structural *similarity* between classical and quantum probability theory. But there is a major interpretational *dissimilarity* between classical and quantum probability: a straightforward frequency interpretation of non-classical probability is not possible (Sect. 36.3). A possible way of making room for a frequency interpretation of quantum probability theory is to accept the so-called Kolmogorovian Censorship Hypothesis, which can be shown to hold for quantum probability theories based on the theory of von Neumann algebras (Sect. 36.4), which however has both technical weaknesses and philosophical ramifications that are unattractive, as will be seen in Sect. 36.4.

## Keywords

Boolean Algebra Selfadjoint Operator Tracial State Quantum Probability Dimensional Hilbert Space## Preview

Unable to display preview. Download preview PDF.

## References

- G. Bana and T. Durt. Proof of Kolmogorovian censorship.
*Foundations of Physics*, 27:1355–1373, 1997.CrossRefGoogle Scholar - A. Döring. Kochen-Specker theorem for general von Neumann algeberas.
*International Journal of Theoretical Physics*, 44:139–160, 2005.Google Scholar - R. V. Kadison and J. R. Ringrose.
*Fundamentals of the Theory of Operator Algebras*, volume I. and II. Academic Press, Orlando, 1986.Google Scholar - A. N. Kolmogorov.
*Grundbegriffe der Wahrscheinlichkeitsrechnung*. Springer, Berlin, 1933. English translation:*Foundations of the Theory of Probability*, (Chelsea, New York, 1956).Google Scholar - F. J. Murray and J. von Neumann. On rings of operators.
*Annals of Mathematics*, 37:116–229, 1936. Reprinted in Taub (1961) No. 2.Google Scholar - F. J. Murray and J. von Neumann. On rings of operators, II.
*American Mathematical Society Transactions*, 41:208–248, 1937. Reprinted in Taub (1961) No. 3.Google Scholar - F. J. Murray and J. von Neumann. On rings of operators, IV.
*Annals of Mathematics*, 44:716–808, 1943. Reprinted in Taub (1961) No. 5.Google Scholar - D. Petz and J. Zemanek. Characterizations of the trace.
*Linear Algebra and its Applications*, 111:43–52, 1988.CrossRefGoogle Scholar - M. Rédei. Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead).
*Studies in the History and Philosophy of Modern Physics*, 27:1309–1321, 1996.Google Scholar *Theories of Physics*. Kluwer Academic Publisher, 1998.Google Scholar- M. Rédei. ‘Unsolved problems in mathematics’ J. von Neumann’s address to the International Congress of Mathematicians Amsterdam, September 2-9, 1954.Google Scholar
*The Mathematical Intelligencer*, 21:7–12, 1999.Google Scholar- M. Rédei. Von Neumann’s concept of quantum logic and quantum probability. In M. Rédei and M. St¨oltzner, editors,
*John von Neumann and the Foundations of Quantum Physics*, Institute Vienna Circle Yearbook, pages 153–172. Kluwer Academic Publishers, Dordrecht, 2001.Google Scholar - M. Rédei, editor.
*John von Neumann: Selected Letters*, volume 27 of*History of Mathematics*, Rhode Island, 2005. American Mathematical Society and London Mathematical Society.Google Scholar - M. Rédei. The birth of quantum logic.
*History and Philosophy of Logic*, 28:107–122, May 2007.Google Scholar - M. Rédei. Kolmogorovian Censorship Hypothesis for general quantum probability theories.
*Manuscrito - Revista Internacional de Filosofia*, 33:365–380, 2010.Google Scholar - M. Rédei and S. J. Summers. Quantum probability theory.
*Studies in the History and Philosophy of Modern Physics*, 38:390–417, 2007.Google Scholar - L. E. Szabó. Critical reflections on quantum probability theory. In M. Rédei and M. St¨oltzner, editors,
*John von Neumann and the Foundations of Quantum Physics*, Institute Vienna Circle Yearbook, pages 201–219. Kluwer Academic Publishers, Dordrecht, 2001.Google Scholar - M. Takesaki.
*Theory of Operator Algebras*, volume I. Springer Verlag, New York, 1979.Google Scholar - A. H. Taub, editor.
*John von Neumann: Collected Works*, volume III. Rings of Operators, New York and Oxford, 1961. Pergamon Press.Google Scholar - R. von Mises. Grundlagen der Wahrscheinlichkeitsrechnung.
*Mathematische Zeitschrift*, 5:52–99, 1919.CrossRefGoogle Scholar - R. von Mises.
*Probability, Statistics and Truth*. Dover Publications, New York, 2nd edition, 1981. Originally published as ‘Wahrscheinlichkeit, Statistik und Wahrheit’ (Springer, 1928).Google Scholar - J. von Neumann. On rings of operators, III.
*Annals of Mathematics*, 41:94–161, 1940. Reprinted in Taub (1961) No. 4.Google Scholar