A New Approach to Quantising Space-Time: I. Quantising on a General Category
Christopher John Isham · Advances in Theoretical and Mathematical Physics · 2003
A new approach i s 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 `conguration space' (or history-theory analogue) can be regarded as the set of objects Ob(Q) in a category Q.In this rst 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 conguration 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 e l d s ' o n Q. Physically, this means that an arrow b e t ween two objects in the category is viewed as some sort of analogue of momentum.After nding the `category quantisation monoid', we s h o w h o w suitable representations can be constructed using a bundle of Hilbert spaces over Ob(Q).