## Advances in Theoretical and Mathematical Physics

- Adv. Theor. Math. Phys.
- Volume 7, Number 2 (2003), 331-367.

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

#### 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
Ob(*Q*) in a category *Q*. 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* is isomorphic to *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 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 Ob(*Q*).

#### Article information

**Source**

Adv. Theor. Math. Phys., Volume 7, Number 2 (2003), 331-367.

**Dates**

First available in Project Euclid: 4 April 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.atmp/1112627636

**Mathematical Reviews number (MathSciNet)**

MR2015168

**Zentralblatt MATH identifier**

1093.81058

#### Citation

Isham, C.J. A New Approach to Quantising Space-Time: I. Quantising on a General Category. Adv. Theor. Math. Phys. 7 (2003), no. 2, 331--367. https://projecteuclid.org/euclid.atmp/1112627636