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理论物理通讯：英文版

A Hierarchy of Compatibility and Comeasurability Levels in Quantum Logics with Unique Conditional Probabilities

Gerd, Niestegge

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

在量机械 Hilbert 空格形式主义，概率的解释是一个以后特定的增强程序，差不多由试验性的证据强制了，然而并非自己由数学模型激发了。从开始包含清楚的概率的解释的一个模型被量逻辑提供唯一的有条件的可能性。它在 von Neumann 代数学包括设计格子，这里，概率 conditionalization 与 L 的州的转变变得相同 ?????????? 牂敯 ? 態灵 ?????????????? € 敂獳汥 吗？？

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在量机械 Hilbert 空格形式主义，概率的解释是一个以后特定的增强程序，差不多由试验性的证据强制了，然而并非自己由数学模型激发了。从开始包含清楚的概率的解释的一个模型被量逻辑提供唯一的有条件的可能性。它在 von Neumann 代数学包括设计格子，这里，概率 conditionalization 与 L 的州的转变变得相同 ?????????? 牂敯 ? 態灵 ?????????????? € 敂獳汥 吗？？
In the quantum mechanical Hilbert space formalism, the probabilistic interpretation is a later ad-hoc add-on, more or less enforced by the experimental evidence, but not motivated by the mathematical model itself. A model involving a clear probabilistic interpretation from the very beginning is provided by the quantum logics with unique conditional probabilities. It includes the projection lattices in von Neumann algebras and here probability conditionalization becomes identical with the state transition of the Lüders-von Neumann measurement process. This motivates the definition of a hierarchy of five compatibility and comeasurability levels in the abstract setting of the quantum logics with unique conditional probabilities. Their meanings are: the absence of quantum interference or influence, the existence of a joint distribution, simultaneous measurability, and the independence of the final state after two successive measurements from the sequential order of these two measurements. A further level means that two elements of the quantum logic (events) belong to the same Boolean subalgebra. In the general case, the five compatibility and comeasurability levels appear to differ, but they all coincide in the common Hilbert space formalism of quantum mechanics, in von Neumann algebras, and in some other cases.
In 1933, Kolmogorov synthesized the basic concepts of probability that were in general use at the time into concepts and deductions from a simple set of axioms that said probability was a σ-additive function from a boolean algebra of events into [0, 1]. In 1932, von Neumann realized that the use of probability in quantum mechanics required a different concept that he formulated as a σ-additive function from the closed subspaces of a Hilbert space onto [0,1]. In 1935, Birkhoff & von Neumann replaced Hilbert space with an algebraic generalization. Today, a slight modification of the Birkhoff-von Neumann generalization is called “quantum logic”. A central problem in the philosophy of probability is the justification of the definition of probability used in a given application. This is usually done by arguing for the rationality of that approach to the situation under consideration. A version of the Dutch book argument given by de Finetti in 1972 is often used to justify the Kolmogorov theory, especially in scientific applications. As von Neumann in 1955 noted, and his criticisms still hold, there is no acceptable foundation for quantum logic. While it is not argued here that a rational approach has been carried out for quantum physics, it is argued that (1) for many important situations found in behavioral science that quantum probability theory is a reasonable choice, and (2) that it has an arguably rational foundation to certain areas of behavioral science, for example, the behavioral paradigm of Between Subjects experiments.