Connes' Distance of One-Dimensional Lattices: General Cases

TL;DR

This paper applies Connes' distance formula to one-dimensional lattices with various topologies, generalizing the lattice Dirac operator to include non-unitary link-variables, and interprets these geometrically as lattice spacing and parallel transport.

Contribution

It introduces a generalized lattice Dirac operator with non-unitary link-variables and explores their geometric interpretation in 1D lattices with different topologies.

Findings

Connes' distance formula effectively measures geometry on 1D lattices.

Generalization of Dirac operator includes non-unitary link-variables.

Geometric interpretation links link-variables to lattice spacing and transport.

Abstract

Connes' distance formula is applied to endow linear metric to three 1D lattices of different topology, with a generalization of lattice Dirac operator written down by Dimakis et al to contain a non-unitary link-variable. Geometric interpretation of this link-variable is lattice spacing and parallel transport.