Stark effect of hydrogenic ions as a test of the indefinite-metric formalism of the eight-component relativistic wave equation for spin-$\frac{1}{2}$ particles

Abstract

The eight-component relativistic wave equation for spin-$\frac{1}{2}$ particles of Robson and Staudte (FV$\frac{1}{2}$), a Feshbach--Villars linearization of the Feynman--Gell-Mann equation, reproduces the Dirac hydrogenic spectrum with a doubled solution space and an indefinite inner product. Its exact Stark shifts must equal the Dirac ones; whether its indefinite-metric perturbation theory delivers them is not guaranteed. We compute the first- and second-order Stark shifts of the $n=1$--3 levels of hydrogenic ions from H to U$^{91+}$ in the Schrödinger, Dirac and FV$\frac{1}{2}$ theories, evaluating all spectral sums, continuum included, by the Dalgarno--Lewis method. The FV$\frac{1}{2}$ and Dirac shifts coincide level by level, to working precision at first order (closed-form roots) and to 45 significant figures at second order. Both ingredients are needed: without the explicit spin--field coupling the first-order roots become complex, and without the negative-norm states the $(Zα)^2$ coefficient is wrong. Norm sign and parity are locked in the doubled space: a potential lifting the $2s_{1/2}$--$2p_{1/2}$ degeneracy needs its own spin--gradient term, and shifts placed by hand on the $κ$-labeled states give a spurious imaginary linear Stark shift. The relativistic linear Stark effect is smaller by finite factors, but a second-moment sum rule shows the non-relativistic linear Stark strength is conserved, the missing part reappearing in singular $(Zα)^{-2}$ second-order terms. For levels with a fine-structure partner, an exact intra-shell/regular split of the second-order shift gives the $(Zα)^0$ coefficients analytically and relates the $π^2$ terms of the $(Zα)^2$ coefficients to the singular ones. The Dirac weak/strong-field crossover for $n=2$ of hydrogen lies near 2.8~kV cm$^{-1}$, and the perturbative window set by the fine structure grows as $Z^5$.