n能级系统的直觉量子逻辑
Intuitionistic quantum logic of an n-level system
摘要：
十年前，Isham和Butterfield提出了量子力学的拓扑理论方法，Doering和Isham扩展了该方法，为所有物理学提供了新的数学基础。去年，本书的三位作者通过将量子理论的C*-代数方法与拓扑理论的所谓内部语言相结合，重新发展和完善了这些思想（见arXiv：0709.4364）。本文的目的是通过复n乘n矩阵的C*-代数的具体例子来说明我们的抽象设置。这导致了一个明确的表示的pointfree量子相空间和相关的逻辑结构和Gelfand变换的n能级系统。我们还确定了相关的非probabilisitic状态命题配对（或估值），并给出了一个非常自然的拓扑理论的Kochen-Specker定理。关键的一点是，量子n能级系统的逻辑结构是直观的，这意味着它是分配的，但不满足排中律（两者都与通常的量子逻辑相反）。
英文摘要：
A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Doering and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (see arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*-algebra of complex n by n matrices. This leads to an explicit expression for the pointfree quantum phase space and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen--Specker Theorem. The essential point is that the logical structure of a quantum n-level system turns out to be intuitionistic, which means that it is distributive but fails to satisfy the law of the excluded middle (both in opposition to the usual quantum logic).
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