”“什么东西？物理学基础中的拓扑理论
`What is a Thing?': Topos Theory in the Foundations of Physics
摘要：
本文的目的是总结在发展一个全新的方法来构建物理学理论的第一步。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。在这样做的过程中，我们为海德格尔的永恒问题“什么是物？”提供了一个新的答案。''.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。经典物理学使用集合的概念。其他理论涉及不同的主题。对于本文讨论的理论类型，一个关键目标是用箭头$\breve{A}_ϕ:\Si_ϕ\map\R_ϕ$表示任何物理量$A$，其中$\Si_ϕ$和$\R_ϕ$是适当拓扑$τ_ϕ$中的两个特殊对象（“状态对象”和“量值对象”）。我们讨论两种不同类型的语言，可以附加到一个系统，$S$。第一种是$\PL{S}$，是一种命题语言;第二种是$Ł{S}$，是一种高阶类型化语言。这两种语言都为演绎系统提供了直觉逻辑。在$\PL{S}$的帮助下，我们扩展和发展了一些早期的工作（由CJI和合作者完成）。关于拓扑理论和量子物理学一个关键步骤是一个过程，我们称之为“daseinisation”，通过这个过程，投影算子被映射到光谱预层$\Sig$的一个子对象-经典状态空间的拓扑量子模拟。所涉及的主题是$\SetH{}$：量子希尔伯特空间$\Hi$上有界算子代数的交换子代数范畴（偏序集）$\V{}$上的逆变集值函子范畴.
英文摘要：
The goal of this paper is to summarise the first steps in developing 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. In doing so we provide a new answer to Heidegger's timeless question ``What is a thing?''. 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 uses the topos of sets. Other theories involve a different topos. For the types of theory discussed in this paper, a key goal is to represent any physical quantity $A$ with an arrow $\breve{A}_ϕ:\Si_ϕ\map\R_ϕ$ where $\Si_ϕ$ and $\R_ϕ$ are two special objects (the `state-object' and `quantity-value object') in the appropriate topos, $τ_ϕ$. We discuss two different types of language that can be attached to a system, $S$. The first, $\PL{S}$, is a propositional language; the second, $Ł{S}$, is a higher-order, typed language. Both languages provide deductive systems with an intuitionistic logic. With the aid of $\PL{S}$ we expand and develop some of the earlier work (By CJI and collaborators.) on topos theory and quantum physics. A key step is a process we term `daseinisation' by which a projection operator is mapped to a sub-object of the spectral presheaf $\Sig$--the topos quantum analogue of a classical state space. The topos concerned is $\SetH{}$: the category of contravariant set-valued functors on the category (partially ordered set) $\V{}$ of commutative sub-algebras of the algebra of bounded operators on the quantum Hilbert space $\Hi$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
