Advances in Theoretical and Mathematical Physics

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

C.J. Isham

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


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