2003/03/31 by C. J. Isham
Physics and Astronomy · #gr-qc #hep-th #quant-ph
published as Adv.Theor.Math.Phys. 7 (2003) 331-367 · Latex2e
arxiv created 2003/12/24 · arxiv updated 2009/11/30
A new approach is suggested to the problem of quantising causal sets, or topologies, or other such models for space-time (or space). The starting point is the observation that entities of this type can 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) can be regarded as the set of objects in a category. In this first of a series of papers, we study this question in general and 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≃ G/H where G and H are Lie groups. In particular, we choose as the analogue of G the monoid of `arrow fields' on the category. Physically, this means that an arrow between two objects in the category is viewed as some sort of analogue of momentum. After finding the `category quantisation monoid', we show how suitable representations can be constructed using a bundle of Hilbert spaces over the set of objects.