Projective Systems of Noncommutative Lattices
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 , 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 . 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.
KeywordsHilbert Space Topological Space Partial Order Compact Operator Projective System
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