dorsal/arxiv
View SchemaQuantum Probability Theory
| Authors | Miklos Redei, Stephen J. Summers |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0601158 |
| URL | https://arxiv.org/abs/quant-ph/0601158 |
Abstract
The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient. These illustrate the physical significance of the previously mentioned differences.
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"abstract": "The mathematics of classical probability theory was subsumed into classical\nmeasure theory by Kolmogorov in 1933. Quantum theory as nonclassical\nprobability theory was incorporated into the beginnings of noncommutative\nmeasure theory by von Neumann in the early thirties, as well. To precisely this\nend, von Neumann initiated the study of what are now called von Neumann\nalgebras and, with Murray, made a first classification of such algebras into\nthree types. The nonrelativistic quantum theory of systems with finitely many\ndegrees of freedom deals exclusively with type I algebras. However, for the\ndescription of further quantum systems, the other types of von Neumann algebras\nare indispensable. The paper reviews quantum probability theory in terms of\ngeneral von Neumann algebras, stressing the similarity of the conceptual\nstructure of classical and noncommutative probability theories and emphasizing\nthe correspondence between the classical and quantum concepts, though also\nindicating the nonclassical nature of quantum probabilistic predictions. In\naddition, differences between the probability theories in the type I, II and\nIII settings are explained. A brief description is given of quantum systems for\nwhich probability theory based on type I algebras is known to be insufficient.\nThese illustrate the physical significance of the previously mentioned\ndifferences.",
"arxiv_id": "quant-ph/0601158",
"authors": [
"Miklos Redei",
"Stephen J. Summers"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"title": "Quantum Probability Theory",
"url": "https://arxiv.org/abs/quant-ph/0601158"
},
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