量化一个类别
Quantising on a category
摘要：
我们回顾的问题，找到一个一般的框架内，可以构建量子理论的非标准模型的空间，或时空。出发点是观察到这种类型的实体通常可以被视为类别中的对象，其箭头是结构保持映射。这激发了研究量化一个系统的一般问题，该系统的“配置空间”（或历史-理论模拟）是一个范畴$\Q$中的对象$\Ob\Q$的集合。我们开发了一个计划的基础上构建一个类似的组中使用的规范量子化的系统，其配置空间是一个流形$Q\simeq G/H$，其中$G$和$H$是李群。特别是，我们选择的模拟$G$的幺半群的“箭头字段”上$\Q$。从物理上讲，这意味着该类别中两个物体之间的箭头被视为动量的模拟。在找到“类量化幺半群”，我们展示了如何合适的表示可以使用一束（或者，更准确地说，预层）的希尔伯特空间在$\Ob\Q$。对于有限集范畴的例子，我们构造了这种类型的显式表示结构。
英文摘要：
We review the problem of finding a general framework within which one can construct quantum theories of non-standard models for space, or space-time. The starting point is the observation that entities of this type can typically be regarded as objects in a category whose arrows are structure-preserving maps. This motivates investigating the general problem of quantising a system whose `configuration space' (or history-theory analogue) is the set of objects $\Ob\Q$ in a category $\Q$. We develop a scheme based on constructing an analogue of the group that is used in the canonical quantisation of a system whose configuration space is a manifold $Q\simeq G/H$, where $G$ and $H$ are Lie groups. In particular, we choose as the analogue of $G$ the monoid of `arrow fields' on $\Q$. Physically, this means that an arrow between two objects in the category is viewed as an analogue of momentum. After finding the `category quantisation monoid', we show how suitable representations can be constructed using a bundle (or, more precisely, presheaf) of Hilbert spaces over $\Ob\Q$. For the example of a category of finite sets, we construct an explicit representation structure of this type.
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