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arXiv:math-ph/0108005

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

Paul B. Slater

Published Aug 15, 2001• 0 citations
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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.

DOI
10.48550/arxiv.math-ph/0108005

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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.
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.
Hubner's formula for the Bures (statistical distance) metric is applied to both a one-parameter and a two-parameter series (n=2,...,7) of sets of 2^n x 2^n density matrices. In the doubly-parameterized series, the sets are comprised of the n-fold tensor products --- corresponding to n independent, identical quantum systems --- of the 2 x 2 density matrices with real entries. The Gaussian curvatures of the corresponding Bures metrics are found to be constants (4/n). In the second series of 2^n x 2^n density matrices studied, the singly-parameterized sets are formed --- following a study of Krattenthaler and Slater --- by averaging with respect to a certain Gibbs distribution, the n-fold tensor products of the 2 x 2 density matrices with complex entries. For n = 100, we are also able to compute the Bures distance between two arbitrary (not necessarily neighboring) density matrices in this particular series, making use of the eigenvalue formulas of Krattenthaler and Slater, together with the knowledge that the 2^n x 2^n density matrices in this series commute.
Given a simple self-dual quantum Hamiltonian $H=KB+\ensuremath{\Gamma}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}$, where KKK and $\ensuremath{\Gamma}$ are coupling constants, and the condition that $[B,[B,[B,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}]]]=16[B,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}]$, then we construct an infinite set of conserved charges Q2n{Q}_{2n}Q2n​; [H,Q2n]=0[H,{Q}_{2n}]=0[H,Q2n​]=0. In simple models, like the two-dimensional Ising or Baxter eight-vertex, these charges appear in the associated quantum theories and are equivalent to those which result from the transfer-matrix formulation and exact quantum integrability of the system. The power of our result is that it is an operator statement and does not refer to the number of dimensions or the nature of the space-time manifold: lattice, continuum, or loop space. It is suggested how the establishment of this link between duality and integrability could be used to exploit the Kramers-Wannier-type self-duality of the four-dimensional SU(N)\mathrm{SU}(N)SU(N) gauge theory to find hidden symmetry.
An explicit parameterization for the state space of an nnn-level density matrix is given. The parameterization is based on the canonical coset decomposition of unitary matrices. We also compute, explicitly, the Bures metric tensor over the state space of two- and three-level quantum systems.