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28 March 2024
 
  » arxiv » quant-ph/0401175

 Article overview


Quantising on a category
C J Isham ;
Date 28 Dec 2003
Subject quant-ph
AbstractWe 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 $ObQ$ 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 $Qsimeq G/H$, 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 $ObQ$. For the example of a category of finite sets, we construct an explicit representation structure of this type.
Source arXiv, quant-ph/0401175
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