Self-duality, four-forms, and the eight-dimensional Yang-Mills/Dittmann-Bures field over the three-level quantum systems

TL;DR

This paper explores the properties of self-dual and anti-self-dual four-forms related to the Yang-Mills field over three-level quantum systems, revealing eigenvalue structures and behaviors within a non-Euclidean Bures metric context.

Contribution

It extends the understanding of self-duality and four-form properties in the non-Euclidean Bures metric setting for three-level quantum systems, including solving eigenvalue problems.

Findings

Eigenvalues consist of four singlets and three octets.

Four-forms exhibit simple behaviors in the Bures metric context.

Solutions to the Hodge * operator eigenproblem are obtained for specific density matrices.

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. Parallelling the decomposition of eight-dimensional Euclidean fields by Corrigan, Devchand, Fairlie and Nuyts, as well as Figueoroa-O'Farrill and others, we investigate the properties of self-dual and anti-self-dual four-forms corresponding specifically to our Bures/non-Euclidean context. For any of a number of (nondegenerate) 3 x 3 density matrices, we are able to solve the eigenequation of the associated Hodge * operator with respect to the Bures metric. We obtain sets of (traceless) twenty-eight real eigenvalues, consisting of four singlets and three octets. The associated four-forms are found to exhibit quite simple behaviors, though we are not able to derive them in full generality.