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.