Topos物理理论基金会：II。此在化与量子理论的解放
A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory
摘要：
本文是系列文章中的第二篇，其目标是发展一种构建物理学理论的全新方法。其动机来自于一种渴望，即解决在思考空间和时间的量子理论时出现的某些深层问题。我们的基本论点是，构建一个物理学理论等价于在一个附着于该系统的某种形式语言的主题中找到一个表示。当topos是集合的范畴时，经典物理学就产生了。其他类型的理论采用不同的主题。本文深入研究了量子理论中命题语言PL（S）的topos表示。在这样做的过程中，我们与早期将拓扑斯理论应用于量子物理的工作建立了直接的联系，并澄清了这些工作。关键的一步是一个过程，我们称之为“daseinisation”，通过这个过程，投影算子被映射到光谱预层的一个子对象-经典状态空间的拓扑量子模拟。在本文的第二部分，我们改变齿轮与更复杂的本地语言L（S）的介绍。从这一点开始，在本系列论文的其余部分，我们的注意力几乎全部集中在这种语言上。本文利用L（S）研究命题中的"真值对象"。这些是拓扑中扮演态角色的对象：这是一个必要的发展，因为谱前层没有全局元素，因此不存在经典物理意义上的微观态。因此，真理对象在我们的形式主义中起着至关重要的作用。
英文摘要：
This paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper, we study in depth the topos representation of the propositional language, PL(S), for the case of quantum theory. In doing so, we make a direct link with, and clarify, the earlier work on applying topos theory to quantum physics. The key step is a process we term `daseinisation' by which a projection operator is mapped to a sub-object of the spectral presheaf--the topos quantum analogue of a classical state space. In the second part of the paper we change gear with the introduction of the more sophisticated local language L(S). From this point forward, throughout the rest of the series of papers, our attention will be devoted almost entirely to this language. In the present paper, we use L(S) to study `truth objects' in the topos. These are objects in the topos that play the role of states: a necessary development as the spectral presheaf has no global elements, and hence there are no microstates in the sense of classical physics. Truth objects therefore play a crucial role in our formalism.
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