Finite Quantum Physics and Noncommutative Geometry

TL;DR

This paper explores a novel approach to approximating manifold topology using posets and noncommutative C*-algebras, linking finite quantum physics models with noncommutative geometry to preserve topological features.

Contribution

It introduces an approximation scheme that reproduces manifold topology with posets and noncommutative C*-algebras, bridging finite quantum physics and noncommutative geometry.

Findings

Posets can effectively approximate manifold topology.

Noncommutative C*-algebras encode topological information.

Numerical schemes can preserve continuum topological features.

Abstract

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 fidelity. 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 $\cc(M)$ of continuous functions on $M$. It has a fundamental physical significance, 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 $\ca $. As noncommutative geometries are based on noncommutative C*-algebras, we therefore have a remarkable connection between finite approximations to quantum physics and noncommutative geometries. Various methods for doing quantum physics using $\ca $ are explored. Particular attention is paid to developing numerically viable approximation schemes which at the same time preserve important topological features of continuum physics.