Dirac Hierarchy Research Articles

Integrable couplings of the multi-component Dirac hierarchy and its Hamiltonian structure

Integrable couplings of the multi-component Dirac hierarchy is obtained by use of the vector loop algebra G ∼ M , then the Hamiltonian structure of the above system is given by the quadratic-form identity.

Hierarchy of Dirac, Pauli, and Klein–Gordon conserved operators in Taub-NUT background

The algebra of conserved observables of the SO(4,1) gauge-invariant theory of the Dirac fermions in the external field of the Kaluza–Klein monopole is investigated. It is shown that the Dirac conserved operators have physical parts associated with Pauli operators that are also conserved in the sense of the Klein–Gordon theory. In this way one gets simpler methods of analyzing the properties of the conserved Dirac operators and their main algebraic structures including the representations of dynamical algebras governing the Dirac quantum modes.

On different integrable systems sharing the same nondynamical r-matrix

In a recent paper [Zhijun Qiao and Ruguang Zhou, Phys. Lett. A 235, 35 (1997)], the amazing fact was reported that a discrete and a continuous integrable system share the same r-matrix with the interesting property of being nondynamical. Now, we present three further pairs of different continuous integrable systems sharing the same r-matrix again being nondynamical. The first pair is the finite-dimensional constrained system (FDCS) of the famous AKNS hierarchy and the Dirac hierarchy; the second pair is the FDCS of the well-known geodesic flows on the ellipsoid and the Heisenberg spin chain hierarchy; and the third pair is the FDCS of one hierarchy studied by Xianguo Geng [Phys. Lett. A 162, 375 (1992)] and another hierarchy proposed by Zhijun Qiao [Phys. Lett. A 192, 316 (1994)]. All those FDCSs possess Lax representations and from the viewpoint of r-matrix can be shown to be completely integrable in Liouville’s sense.