Spectral distances on doubled Moyal plane using Dirac eigen-spinors

TL;DR

This paper introduces a new method for computing spectral distances in the doubled Moyal plane using Dirac eigen-spinors, explicitly solving the Lipschitz ball condition and exploring geometric properties and a toy Higgs model.

Contribution

It presents a novel approach to spectral distance calculation in noncommutative geometry using Dirac eigen-spinors and addresses non-unital algebra challenges with projection operators.

Findings

Computed standard spectral distances and confirmed Pythagorean relations.

Resolved non-unital algebra issues with projection operators.

Demonstrated a toy Higgs field model through Dirac operator fluctuations.

Abstract

We present here a novel method of computing spectral distances in doubled Moyal plane in a noncommutative geometrical framework using Dirac eigen-spinors, while solving the Lipschitz ball condition explicitly through matrices. The standard results of longitudinal, transverse and hypotenuse distances between different pairs of pure states have been computed and Pythagorean equality between them have been re-produced. The issue of non-unital nature of Moyal plane algebra is taken care of through a sequence of projection operators constructed from Dirac eigen-spinors, which plays a crucial role throughout this paper. At the end, a toy model of "Higgs field" has been constructed by fluctuating the Dirac operator and the variation on the transverse distance has been demonstrated, through an explicit computation.