Self-duality, four-forms, and the eight-dimensional Yang-Mills/Dittmann-Bures field over the three-level quantum systems
Abstract
Description
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.
eleven pages, LaTeX, ten new figures, we substantially revise the paper, adding new analyses, while omitting much of the previous detail concerning the implementation of the (Euclidean) CDFN equations. This detail turned out to be largely irrelevant to the main (non-Euclidean) results of the paper
eleven pages, LaTeX, ten new figures, we substantially revise the paper, adding new analyses, while omitting much of the previous detail concerning the implementation of the (Euclidean) CDFN equations. This detail turned out to be largely irrelevant to the main (non-Euclidean) results of the paper