Distances on a Lattice from Non-commutative Geometry

TL;DR

This paper uses noncommutative geometry to analyze lattice distances, revealing that discretized Dirac operators produce non-intuitive metric properties, which impacts how lattice discretizations should be chosen.

Contribution

It introduces a noncommutative geometric approach to evaluate lattice distances, providing a new criterion for discretizing the Dirac operator.

Findings

Lattice distances differ from continuum expectations.

Discretized Dirac operators affect the metric structure.

Provides a new perspective on lattice discretization choices.

Abstract

Using the tools of noncommutative geometry we calculate the distances between the points of a lattice on which the usual discretized Dirac operator has been defined. We find that these distances do not have the expected behaviour, revealing that from the metric point of view the lattice does not look at all as a set of points sitting on the continuum manifold. We thus have an additional criterion for the choice of the discretization of the Dirac operator.