NONCOMMUTATIVE LATTICES AS FINITE APPROXIMATIONS

Lattice discretizations of continuous manifolds are common tools used in a variety of physical contexts. Conventional discrete approximation s, however, cannot capture all aspects of the original manifold, notab ly its topology. In this paper we discuss an approximation scheme due to Sorkin (1991) which correctly reproduces important topological aspe cts of continuum physics. The approximating topological spaces are par tially ordered sets (posets), the partial order encoding the topology, Now, the topology of a manifold M can be reconstructed from the commu tative C-algebra C(M) of continuous functions defined on it. In turn, this algebra is generated by continuous probability densities in ordi nary quantum physics on M. The latter also serves to specify the domai ns of observables like the Hamiltonian. For a poset, the role of this algebra is assumed by a noncommutative C-algebra A. This fact makes a ny poset a genuine 'noncommutative' ('quantum') space, in the sense th at the algebra of its 'continuous functions' is a noncommutative C-al gebra. We therefore also have a remarkable connection between finite a pproximations to quantum physics and noncommutative geometries. We use this connection to develop various approximation methods for doing qu antum physics using A.