代数量子理论的一个论题
A topos for algebraic quantum theory
摘要：
本文的目的是将代数量子力学与拓扑理论联系起来，为量子逻辑和量子空间建立新的基础。玻尔的想法，量子物理学的经验内容只能通过经典物理学的启发，我们展示了如何观察A的C*-代数诱导拓扑T（A），其中所有的交换子代数的合并包括一个单一的交换C*-代数。根据Banaschewski和Mulvey的构造性Gelfand对偶定理，后者在T（A）中有一个内谱S（A），在我们的方法中，它扮演着系统的量子相空间的角色。因此，我们将locale（空间的拓扑理论概念，本质上带有Heyting代数的直觉主义逻辑结构）与C*-代数（空间的非交换概念）联系起来。在这种情况下，A上的状态成为S（A）上的概率测度（更准确地说，赋值），A的自伴元素定义了从S（A）到斯科特区间域的连续函数（更准确地说，区域映射）。注意到S（A）的开子集对应于关于系统的命题，将（广义）真值分配给状态和命题的配对映射假定为极其简单的范畴形式。以这种方式表述，由A定义的量子理论实质上变成了一个经典理论，内在于拓扑T（A）。
英文摘要：
The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr's idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra of observables A induces a topos T(A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum S(A) in T(A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on S(A), and self-adjoint elements of A define continuous functions (more precisely, locale maps) from S(A) to Scott's interval domain. Noting that open subsets of S(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T(A).
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