ICTP: IC/94/38 Napoli: DSF-T-2/94 Syracuse: SU-4240-567 hep-th/9403067 March 1994 FINITE QUANTUM PHYSICS AND NONCOMMUTATIVE GEOMETRY

Conventional discrete approximations of a manifold do not preserve its nontrivial topological features. In this article we describe an approximation scheme due to Sorkin which reproduces physically important aspects of manifold topology with striking delity. The approximating topological spaces in this scheme are partially ordered sets (posets). Now, in ordinary quantum physics on a manifold M , continuous probability densities generate the commutative C*-algebra C(M) of continuous functions on M . It has a fundamental physical signi cance, containing the information to reconstruct the topology of M , and serving to specify the domains of observables like the Hamiltonian. For a poset, the role of this algebra is assumed by a noncommutative C*-algebra A. As noncommutative geometries are based on noncommutative C*-algebras, we therefore have a remarkable connection between nite approximations to quantum physics and noncommutative geometries. Various methods for doing quantum physics using A are explored. Particular attention is Fellow of the Italian National Council of Research (CNR) under Grant No. 203.01.60. paid to developing numerically viable approximation schemes which at the same time preserve important topological features of continuum physics.