Distances in Finite Spaces from Noncommutative Geometry

TL;DR

This paper explores how noncommutative geometry can define and analyze metrics on finite spaces, generalizing classical Riemannian metrics and providing computational examples in both commutative and noncommutative contexts.

Contribution

It introduces a framework for defining metrics on finite spaces using noncommutative geometry principles, extending classical concepts and providing explicit examples.

Findings

The metric generalizes the classical Riemannian metric.

Properties of the noncommutative metric in finite spaces are characterized.

Examples demonstrate the application in both commutative and noncommutative cases.

Abstract

Following the general principles of noncommutative geometry, it is possible to define a metric on the space of pure states of the noncommutative algebra generated by the coordinates. This metric generalizes the usual Riemannian one. We investigate some general properties of this metric in the finite commutative case which corresponds to a metric on a finite set, and also give some examples of computations in both commutative and noncommutative cases.