用上下文选择函子约化Topos量子理论
Topos Quantum Theory Reduced by Context-Selection Functors
摘要：
本文从两个角度研究Hilbert空间上有界算子的交换vonNeumann代数构成的背景范畴上的预层和层的量子理论。一种是通过层化来约化基于预层的拓扑量子理论，另一种是将量子概率引入到约化层的量子理论中。第一个是通过一个函子来完成的，它选择了一些方便的上下文。它定义了由所有上下文组成的范畴上的Grothendieck拓扑，从而导出了一个层拓扑，我们在这个层拓扑上构造了一个缩小的量子理论。此外，我们表明，层量子理论可以取代一个等价的，更易于管理的预层量子理论。量子概率通过Grothendieck拓扑导入，该拓扑定义在由概率组成的类别上，并能够将它们视为直觉真值。从这些拓扑中，我们构造了另一个Grothendieck拓扑，它定义在上下文范畴和概率范畴的乘积上，反映了上下文的选择和概率与真值的识别。我们在Grothendieck拓扑诱导的层拓扑上构造了一个以量子概率为真值的量子理论。
英文摘要：
In this paper, we deal with quantum theories on presheaves and sheaves on context categories consisting of commutative von Neumann algebras of bounded operators on a Hilbert space, from two viewpoints. One is to reduce presheaf-based topos quantum theory via sheafification, and the other is to import quantum probabilities to the reduced sheaf quantum theory. The first is done by means of a functor that selects some expedient contexts. It defines a Grothendieck topology on the category consisting of all contexts, hence, induces a sheaf topos on which we construct a downsized quantum theory. Also, we show that the sheaf quantum theory can be replaced by an equivalent, more manageable presheaf quantum theory. Quantum probabilities are imported by means of a Grothendieck topology that is defined on a category consisting of probabilities and enables to regard them as intuitionistic truth-values. From these topologies, we construct another Grothendieck topology that is defined on the product of the context category and the probability category and reflects the selection of contexts and the identification of probabilities with truth-values. We construct a quantum theory equipped with quantum probabilities as truth-values on the sheaf topos induced by the Grothendieck topology.
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