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

arXiv:quant-ph/0503008

Postulates of quantum mechanics and phenomenology

D. A. Slavnov

Published Mar 1, 2005• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0503008

Related papers

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

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

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

Алгебраические и статистические методы в квантовой и гравитационной физике

D. A. Slavnov

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.

ФИЗИКА ЭЛЕМЕНТАРНЫХ ЧАСТИЦ И АТОМНОГО ЯДРА•Jun 18, 2024•0 citations

Elements of quantum theory

Giuseppe Di Molfetta

In this chapter we will expose the fundamental principles of quantum mechanics alongside the mathematical formalism on which they are grounded, which is that of Hilbert spaces. In particular we will briefly present the necessary fundamentals of linear algebra, including the definitions of vector spaces, of scalar and tensorial products, and the definition of linear operators.

Institute of Physics eBooks•Sep 1, 2024•1 citation

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
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.
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.
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.
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.