FINITE APPROXIMATIONS TO QUANTUM PHYSICS - QUANTUM POINTS AND THEIR BUNDLES

There exists a physically well-motivated method for approximating mani folds by certain topological spaces with a finite or a countable set o f points. These spaces, which are partially ordered sets (posets), hav e the power to effectively reproduce important topological features of continuum physics like winding numbers and fractional statistics, and that too often with just a few points. In this work we develop the es sential tools for doing quantum physics on posets. The poset approach to covering space quantization, soliton physics, gauge theories and th e Dirac equation are discussed with emphasis on physically important t opological aspects. These ideas are illustrated by simple examples lik e the covering space quantization of a particle on a circle, and the s ine-Gordon solitons.