A New Approach to Quantising Space-Time: I. Quantising on a General Category

TL;DR

This paper introduces a novel method for quantising space-time models by treating entities as objects in a category and constructing a corresponding quantisation monoid, extending canonical quantisation concepts to categorical structures.

Contribution

It develops a general scheme for quantising systems with configuration spaces modeled as categories, introducing the category quantisation monoid and representation framework.

Findings

Defined the category quantisation monoid from arrow fields

Constructed Hilbert space bundles over category objects

Established a foundation for quantising categorical space-time models

Abstract

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 $Q\simeq G/H$ where G and H are Lie groups. In particular, we choose as the analogue of G the monoid of `arrow fields' on 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.