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 appro ximation scheme due to Sorkin which reproduces physically important as pects of manifold topology with striking fidelity. The approximating t opological spaces in this scheme are partially ordered sets (posets). Now, in ordinary quantum physics on a manifold M, continuous probabili ty densities generate the commutative C-algebra C(M) of continuous fu nctions on M. It has a fundamental physical significance, containing t he information to reconstruct the topology of M, and serving to specif y the domains of observables like the Hamiltonian. For a poset, the ro le of this algebra is assumed by a noncommutative Cs-algebra A. As non commutative geometries are based on noncommutative C-algebras, we the refore have a remarkable connection between finite approximations to q uantum physics and noncommutative geometries. Various methods for doin g quantum physics using A are explored. Particular attention is paid t o developing numerically viable approximation schemes which at the sam e time preserve important topological features of continuum physics.