Topos物理理论基金会：III。用箭头表示物理量
A Topos Foundation for Theories of Physics: III. The Representation of Physical Quantities With Arrows
摘要：
本文是系列文章中的第三篇，其目标是发展一种全新的看待物理学理论的方式。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。在论文II中，我们研究了量子理论情况下命题语言PL（S）的topos表示，在本文中，我们对更广泛的局部语言L（S）做了同样的事情。主要成果之一是找到了自伴算子的拓扑表示。这涉及到表明，对于任何物理量A，都有一个箭头$\breveδ^o(A):\Sig\map\SR$，其中$\SR$是这个理论的量值对象。$\breveδ^o(A)$的构造是在论文II中讨论的投影算子的daseinisation的扩展。对象$\SR$是一个monoid对象，只有在拓扑结构，$τ_ϕ$，理论，并提高适用性的形式主义，我们适用于$\SR$的拓扑结构模拟的Grothendieck扩展的一个群。结果对象$\kSR$是$τ_ϕ$中的阿贝尔群对象。我们还讨论了数量-值对象的另一个候选对象$\PR{\mathR}$。在这个前层中，内外的此在化都以对称的方式使用。最后，简要讨论了量子拓扑方案中幺正算符的作用。
英文摘要：
This paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics. 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. In paper II, we studied the topos representations of the propositional language PL(S) for the case of quantum theory, and in the present paper we do the same thing for the, more extensive, local language L(S). One of the main achievements is to find a topos representation for self-adjoint operators. This involves showing that, for any physical quantity A, there is an arrow $\breveδ^o(A):\Sig\map\SR$, where $\SR$ is the quantity-value object for this theory. The construction of $\breveδ^o(A)$ is an extension of the daseinisation of projection operators that was discussed in paper II. The object $\SR$ is a monoid-object only in the topos, $τ_ϕ$, of the theory, and to enhance the applicability of the formalism, we apply to $\SR$ a topos analogue of the Grothendieck extension of a monoid to a group. The resulting object, $\kSR$, is an abelian group-object in $τ_ϕ$. We also discuss another candidate, $\PR{\mathR}$, for the quantity-value object. In this presheaf, both inner and outer daseinisation are used in a symmetric way. Finally, there is a brief discussion of the role of unitary operators in the quantum topos scheme.
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