Dimensional singularity analysis of relativistic equations.

Abstract

For a nonrelativistic hydrogenic atom, the dimension dependence of the energy levels is nonsingular except for a second-order pole at ${\mathit{D}}^{\mathrm{*}}$=3-2n, where n is the principal quantum number. For the relativistic Klein-Gordon and Dirac equations, the dimension dependence has a much more complicated singularity structure, involving branch points. For all eigenstates there are branch points at ${\mathit{D}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$\ifmmode\pm\else\textpm\fi{}2Z\ensuremath{\alpha}, where Z is the nuclear charge, \ensuremath{\alpha} is the fine-structure constant, and ${\mathit{D}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ is independent of n but varies linearly with orbital angular momentum. For most states there is in addition a pair of branch points near ${\mathit{D}}^{\mathrm{*}}$ but slightly off the real axis. The customary perturbation expansion in terms of Z\ensuremath{\alpha} gives qualitatively incorrect dimension dependence; it predicts only poles located on the real axis at ${\mathit{D}}^{\mathrm{*}}$ and ${\mathit{D}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$, no matter how high the order of the expansion. The dimensional singularities result from the behavior at r=0. The qualitatively incorrect results occur because the perturbing potential, proportional to ${\mathrm{\ensuremath{\alpha}}}^{2}$/${\mathit{r}}^{2}$, overwhelms the unperturbed 1/r potential at small r. Because of the complexity of the dimensional singularity structure, the popular ``shifted expansion'' method for summing the 1/D expansion does not work well for these equations. We demonstrate a general method for identifying the dimensional singularities that leads to an exact summation of the 1/D expansion for all eigenstates of both the Klein-Gordon and the Dirac equations for a particle in a Coulomb potential.