Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

ФИЗИКА ЭЛЕМЕНТАРНЫХ ЧАСТИЦ И АТОМНОГО ЯДРА

Алгебраические и статистические методы в квантовой и гравитационной физике

D. A. Slavnov

Published Jun 18, 2024• 0 citations
OpenAlex

Abstract

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.

Related papers

Algebraic and Statistical Methods in Quantum and Gravitational Physics

D. A. Slavnov

Abstract A scheme for constructing quantum mechanics is described, in which the Hilbert space and linear operators are not the primary notions of the theory. Instead, a variant of the algebraic approach is considered. The noncommutative algebra elements (observables) and the functionals on this algebra (elementary states), which are associated with the results of individual measurements, are used as primary notions. This scheme makes it possible, on the one hand, to use the apparatus of the classical (Kolmogorov) probability theory, and on the other hand, to reproduce the standard mathematical apparatus of quantum mechanics and determine the limits to its applicability.

Physics of Particles and Nuclei•Jun 1, 2022•3 citations

Measurements and mathematical formalism of quantum mechanics

D. A. Slavnov

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.

Physics of Particles and Nuclei•Mar 1, 2007•17 citations

Algebraic-statistical approach to quantum mechanics

D. A. Slavnov

It is proposed the scheme of quantum mechanics, in which a Hilbert space and the linear operators are not primary elements of the theory. Instead of it certain variant of the algebraic approach is considered. The elements of noncommutative algebra (observables) and the nonlinear functionals on this algebra (physical states) are used as the primary constituents. The functionals associate with results of a particular measurement. It is suggested to consider certain ensembles of the physical states as quantum states of the standart quantum mechanics. It is shown that in such scheme the mathematical formalism of the standart quantum mechanics can be reproduced completely.

arXiv:quant-ph/0101139•Jan 30, 2001•1 citation

Quantum mechanics without quantum logic

D. A. Slavnov

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.

arXiv:quant-ph/0404045•Apr 7, 2004•0 citations

Postulates of quantum mechanics and phenomenology

D. A. Slavnov

We describe a system of axioms that, on one hand, is sufficient for constructing the standard mathematical formalism of quantum mechanics and, on the other hand, is necessary from the phenomenological standpoint. In the proposed scheme, the Hilbert space and linear operators are only secondary structures of the theory, while the primary structures are the elements of a noncommutative algebra (observables) and the functionals on this algebra, associated with the results of a single observation.

arXiv:quant-ph/0503008•Mar 1, 2005•0 citations

Individual events and mathematical formalism of quantum mechanics

D. A. Slavnov

We describe a scheme for constructing quantum mechanics in which the Hilbert space and linear operators are only secondary structures of the theory, while the primary structures are the elements of a non-commutative algebra (observables) and the functionals on this algebra, associated with the results of a single observation.

arXiv:0810.2871•Physics of Particles and Nuclei Letters•Nov 19, 2009•0 citations
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.
Abstract A scheme for constructing quantum mechanics is described, in which the Hilbert space and linear operators are not the primary notions of the theory. Instead, a variant of the algebraic approach is considered. The noncommutative algebra elements (observables) and the functionals on this algebra (elementary states), which are associated with the results of individual measurements, are used as primary notions. This scheme makes it possible, on the one hand, to use the apparatus of the classical (Kolmogorov) probability theory, and on the other hand, to reproduce the standard mathematical apparatus of quantum mechanics and determine the limits to its applicability.
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.
It is proposed the scheme of quantum mechanics, in which a Hilbert space and the linear operators are not primary elements of the theory. Instead of it certain variant of the algebraic approach is considered. The elements of noncommutative algebra (observables) and the nonlinear functionals on this algebra (physical states) are used as the primary constituents. The functionals associate with results of a particular measurement. It is suggested to consider certain ensembles of the physical states as quantum states of the standart quantum mechanics. It is shown that in such scheme the mathematical formalism of the standart quantum mechanics can be reproduced completely.
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.