Pauli Hamiltonian for a spin one-half particle carrying a non-Abelian charge in the presence of non-Abelian external fields

Abstract

We study the solution of non-relativistic approximation of Dirac equation in the presence of non-Abelian external fields. The dynamics of the system is described by the Pauli Hamiltonian for a spin one-half particle carrying a non-Abelian charge. This non-Abelian charge, at the quantum level, is proportional to the generators of the U(2) gauge group and reveals the effect of a spin-orbit coupling across the non-Abelian gauge field. By choosing a suitable non-Abelian gauge field, the total angular momentum is the conserved quantity of the system as well as the corresponding supercharge. The Hamiltonian of the system is then invariant under the transformations generated by the supercharge. This allows us to apply the algebraic method. The Landau levels obtained are deformed by the non-Abelian contribution of the gauge potential. Each of these energy levels are doubly degenerate except the lowest Landau level whose eigenvalue is zero. In the Abelian limit, the Landau level energies are recovered with a fourfold degeneracy. These results correspond to the supersymmetric extension of the ordinary Landau levels.