Analyses of a Yang-Mills field over the three-level quantum systems

TL;DR

This paper investigates the geometric and analytical properties of Yang-Mills fields over three-level quantum systems using numerical methods, decomposing the fields into self-dual and anti-self-dual parts and approximating key functionals.

Contribution

It introduces a numerical approach to analyze Yang-Mills fields on the convex set of three-level quantum systems, decomposing fields into self-dual and anti-self-dual components and approximating relevant functionals.

Findings

Approximate Yang-Mills functional values for the fields.

Decomposition into self-dual and anti-self-dual components.

Analysis of quantities related to self-dual Yang-Mills fields in eight dimensions.

Abstract

Utilizing a number of results of Dittmann, we investigate the nature of the Yang-Mills field over the eight-dimensional convex set, endowed with the Bures metric, of three-level quantum systems. Adopting a numerical strategy, we first decompose the field into self-dual and anti-self-dual components, by implementing the octonionic equations of Corrigan, Devchand, Fairlie and Nuyts. For each of these three fields, we obtain approximations to: (1) the Yang-Mills functional; (2) certain quantities studied by Bilge, Dereli, and Kocak in their analysis of self-dual Yang-Mills fields in eight dimensions; and (3) other measures of interest.