Measurements and Mathematical Formalism of Quantum Mechanics

TL;DR

This paper introduces an algebraic framework for quantum mechanics that does not rely on Hilbert spaces, enabling a unified approach to quantum measurement and classical probability, and addressing quantum paradoxes.

Contribution

It presents a novel algebraic formalism for quantum mechanics that integrates classical probability theory and avoids quantum paradoxes.

Findings

Reproduces standard quantum formalism within an algebraic scheme

Allows the use of classical probability in quantum measurement

Addresses and avoids quantum paradoxes

Abstract

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.