量子拓扑中量子化诱导的层列
Sheaves in Quantum Topos Induced by Quantization
摘要：
本文表明，量子化诱导Lawvere-Tierney拓扑（因此，层拓扑）的量子拓扑。我们表明，从经典可观的自伴算子的希尔伯特空间上的量子映射自然诱导几何态射从预层拓扑相关的经典系统到一个预层拓扑，称为量子拓扑，在上下文范畴的交换冯诺依曼代数的有界算子的希尔伯特空间。借助于几何态射，我们定义了量子拓扑上的Lawvere-Tierney拓扑（以及它们在上下文范畴上的等价Grothendieck拓扑）。我们表明，其中，我们可以唯一地选择最翔实的，我们称之为量化拓扑。此外，我们还构造了量子化拓扑诱导的层。这可以以基本和自包含的方式完成，因为量化拓扑具有相当简单的表达式。
英文摘要：
This paper shows that quantization induces a Lawvere-Tierney topology on (hence, a sheaf topos in) the quantum topos. We show that a quantization map from classical observables to self-adjoint operators on a Hilbert space naturally induces geometric morphisms from presheaf topoi related to the classical system into a presheaf topos, called the quantum topos, on the context category consisting of commutative von Neumann algebras of bounded operators on the Hilbert space. By means of the geometric morphisms, we define Lawvere-Tierney topologies on the quantum topos (and their equivalent Grothendieck topologies on the context category). We show that, among them, we can uniquely select the most informative one, which we call the quantization topology. We furthermore construct sheaves induced by the quantization topology. This can be done in an elementary and self-contained way, because the quantization topology has a quite simple expression.
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