Finite Approximations to Quantum Physics: Quantum Points and their Bundles

TL;DR

This paper explores how finite topological spaces called posets can effectively approximate manifolds for quantum physics, capturing key topological features with minimal points.

Contribution

It develops tools for formulating quantum physics on posets, enabling the study of topological phenomena in a simplified, finite setting.

Findings

Posets can reproduce topological features like winding numbers and fractional statistics.

Application to covering space quantization and soliton physics demonstrates practical utility.

Simple examples illustrate the approach's effectiveness in modeling quantum systems.

Abstract

There exists a physically well motivated method for approximating manifolds by certain topological spaces with a finite or a countable set of points. These spaces, which are partially ordered sets (posets) have 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 essential tools for doing quantum physics on posets. The poset approach to covering space quantization, soliton physics, gauge theories and the Dirac equation are discussed with emphasis on physically important topological aspects. These ideas are illustrated by simple examples like the covering space quantization of a particle on a circle, and the sine-Gordon solitons.