Pseudospin, Spin, and Coulomb Dirac-symmetries: Doublet Structure and Supersymmetric Patterns

The Dirac equation serves as the basis for the relativistic description of atoms, nuclei and hadrons. In atoms the relevant potentials felt by the electron are Coulombic vector potentials. A Dirac Hamiltonian with a Coulomb potential exhibits a finestructure spectrum with characteristic two-fold degeneracy. Relativistic mean fields in nuclei generated by meson exchanges , and quark confinement in hadrons 2 necessitate a mixture of Lorentz vector and scalar potentials. Recently symmetries of Dirac Hamiltonians with such Lorentz structure have been shown to be relevant for explaining the observed degeneracies of certain shell-model orbitals in nuclei (“pseudospin doublets”) , and the absence of quark spin-orbit splitting (“spin doublets”) , as observed in heavy-light quark mesons. The degenerate doublets associated with the relativistic Coulomb, pseudospin, and spin symmetries are shown in Table 1. In the current contribution we show 5 that the degeneracy patterns and relations between wave functions implied by such relativistic symmetries resemble