Stark effect of hydrogenic ions as a test of the indefinite-metric formalism of the eight-component relativistic wave equation for spin-1/2 particles
Abstract
The eight-component relativistic wave equation for spin-1/2 particles of Robson and Staudte (FV1/2), 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 U91+ in the Schrödinger, Dirac and FV1/2 theories, evaluating all spectral sums, continuum included, by the Dalgarno--Lewis method. The FV1/2 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 2s1/2--2p1/2 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 π2 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 Z5.
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BibTeX
@misc{tjiang2026stark,
title = {{Stark effect of hydrogenic ions as a test of the indefinite-metric formalism of the eight-component relativistic wave equation for spin-$\frac12$ particles}},
author = {Paulus C. Tjiang and Sylvia H. Sutanto and Vincentius E. W. Tjia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02060},
eprint = {2610.02060},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02060},
}
RIS
TY - UNPB
TI - Stark effect of hydrogenic ions as a test of the indefinite-metric formalism of the eight-component relativistic wave equation for spin-1/2 particles
AU - Paulus C. Tjiang
AU - Sylvia H. Sutanto
AU - Vincentius E. W. Tjia
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02060
UR - https://arxiv.org/abs/2610.02060
AB - The eight-component relativistic wave equation for spin-1/2 particles of Robson and Staudte (FV1/2), 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 FV1/2 theories, evaluating all spectral sums, continuum included, by the Dalgarno--Lewis method. The FV1/2 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.
ER -