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.

中文翻译：

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量子和引力物理学中的代数和统计方法

摘要

描述了构建量子力学的方案，其中希尔伯特空间和线性算子不是该理论的主要概念。相反，考虑了代数方法的一种变体。与单个测量结果相关联的非交换代数元素（可观测量）和该代数上的泛函（基本状态）被用作主要概念。该方案一方面可以使用经典（Kolmogorov）概率论的装置，另一方面可以重现量子力学的标准数学装置并确定其适用性的限制。