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arXiv:quant-ph/0205125

Quantum mechanics as the objective local theory

D. A. Slavnov

Published May 20, 2002• 0 citations
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Abstract

In the work it is shown that the principles "the objective local theory" and corollaries of the standard quantum mechanics are not in such antagonistic inconsistency as it is usually supposed. In the framework of algebraic approach, the postulates are formulated which allow constructing the updated mathematical scheme of quantum mechanics. This scheme incorporates the standard mathematical apparatus of quantum mechanics. Simultaneously, in it there is a mathematical object, which adequately describes individual experiment.

DOI
10.48550/arxiv.quant-ph/0205125

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In the work it is shown that the principles "the objective local theory" and corollaries of the standard quantum mechanics are not in such antagonistic inconsistency as it is usually supposed. In the framework of algebraic approach, the postulates are formulated which allow constructing the updated mathematical scheme of quantum mechanics. This scheme incorporates the standard mathematical apparatus of quantum mechanics. Simultaneously, in it there is a mathematical object, which adequately describes individual experiment.
We describe a scheme of quantum mechanics in which the Hilbert space and linear operators are only secondary structures of the theory. As primary structures we consider observables, elements of noncommutative algebra, and the physical states, the nonlinear functionals on this algebra, which associate with results of single measurement. We show that in such scheme the mathematical apparatus of the standard quantum mechanics does not contradict a hypothesis on existence of an objective local reality, a principle of a causality and Kolmogorovian probability theory.
We describe the scheme of quantum mechanics, in which the Hilbert space and linear operators are not primary notions of the theory. Instead of this we consider some variant of the algebraic approach. As primary notions we use elements of thenoncommutative algebra (observable) and functionals on this algebra (elementary states), which we associate with results of individual measurements. Such a scheme allows us to use, on the one hand, the apparatus of classical (Kolmogorov’s) probability theory, and on the other hand, to reproduce the standard mathematical apparatus of quantum mechanics and to specify borders of its applicability.
Interest in quantum mechanics is growing in connection with new experiments with quantum particles and a deeper knowledge of nano- and subatomic principles. The practical use of quantum mechanic methods goes in parallel with the constant rethinking of its foundations, its ideological and epistemological role. A rigorous exposition of the mathematical foundations of quantum mechanics in abstract-Hilbert spaces was given in the book of J. Von Neumann in 1933. It is the first and only experiment that has been brought to the end, the presentation of the apparatus of quantum mechanics with the sequence and rigor which usually presented in a purely mathematical theory. Later in 1949 the group of French mathematicians under the pseudonym N. Bourbaki introduced the concept of "mathematical structure" into mathematics. To date, there is no narration of quantum mechanics, where mathematical rigor is combined with the concept of "mathematical structure". In the framework of the approach formulated by the group of N. Burbaki, quantum mechanics is a composite structure consisting of three simple mathematical structures: the Hilbert space of complex-valued vectors, the space of linear self-adjoint operators and the structure of classical mechanics are showed in this article.
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.