Measurements and mathematical formalism of quantum mechanics

A scheme for constructing quantum mechanics is given that does not have Hilbert space and linear operators as its basic elements. Instead, a version of algebraic approach is considered. Elements of a noncommutative algebra (observables) and functionals on this algebra (elementary states) associated...

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Bibliographic Details
Published inPhysics of particles and nuclei Vol. 38; no. 2; pp. 147 - 176
Main Author Slavnov, D. A.
Format Journal Article
LanguageEnglish
Published United States 01.03.2007
Subjects
ALGEBRA
COMMUTATION RELATIONS
FIELD OPERATORS
HILBERT SPACE
PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
QUANTUM FIELD THEORY
QUANTUM MECHANICS
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Summary:A scheme for constructing quantum mechanics is given that does not have Hilbert space and linear operators as its basic elements. Instead, a version of algebraic approach is considered. Elements of a noncommutative algebra (observables) and functionals on this algebra (elementary states) associated with results of single measurements are used as primary components of the scheme. On the one hand, it is possible to use within the scheme the formalism of the standard (Kolmogorov) probability theory, and, on the other hand, it is possible to reproduce the mathematical formalism of standard quantum mechanics, and to study the limits of its applicability. A short outline is given of the necessary material from the theory of algebras and probability theory. It is described how the mathematical scheme of the paper agrees with the theory of quantum measurements, and avoids quantum paradoxes.
ISSN:1063-7796
1531-8559
DOI:10.1134/S1063779607020013