Projective Systems of Noncommutative Lattices

Abstract

The idea of a ‘discrete substratum’ underpinning the ‘continuum’ is somewhat spread among physicists. With particular emphasis this idea has been pushed by R. Sorkin who, in [141], assumes that the substratum be a finitary (see later) topological space which maintains some of the topological information of the continuum. It turns out that the finitary topology can be equivalently described in terms of a partial order. This partial order has been alternatively interpreted as determining the causal structure in the approach to quantum gravity of [15]. Recently, finitary topological spaces have been interpreted as noncommutative lattices and noncommutative geometry has been used to construct quantum mechanical and field theoretical models, notably lattice field theory models, on them [7, 8].

Given a suitable covering of a topological space M, by identifying any two points of M which cannot be ‘distinguished’ by the sets in the covering, one constructs a lattice with a finite (or in general a countable) number of points. Such a lattice, with the quotient topology, becomes a T ₀-space which turns out to be the structure space (or equivalently, the space of primitive ideals) of a postliminar approximately finite dimensional (AF) algebra. Therefore the lattice is truly a noncommutative space. In this Chapter we shall describe noncommutative lattices in some detail while in Chap. 11 we shall illustrate some of their applications in physics.