Pythagoras' Theorem on a 2D-Lattice from a "Natural" Dirac Operator and Connes' Distance Formula

TL;DR

This paper extends a natural Dirac operator from 1D to 2D lattices within noncommutative geometry, demonstrating it induces Euclidean distance and can be generalized to higher dimensions.

Contribution

It generalizes a Dirac operator to 2D lattices that recovers Euclidean distance, advancing the understanding of metric structures in noncommutative geometry.

Findings

Dirac operator induces Euclidean distance on 2D lattice

Generalizable to higher-dimensional lattices

Maintains local eigenvalue property

Abstract

One of the key ingredients of A. Connes' noncommutative geometry is a generalized Dirac operator which induces a metric(Connes' distance) on the state space. We generalize such a Dirac operator devised by A. Dimakis et al, whose Connes' distance recovers the linear distance on a 1D lattice, into 2D lattice. This Dirac operator being "naturally" defined has the so-called "local eigenvalue property" and induces Euclidean distance on this 2D lattice. This kind of Dirac operator can be generalized into any higher dimensional lattices.