{
  "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 $Z^5$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02060",
  "author_total": 3,
  "authors": [
    {
      "name": "Paulus C. Tjiang"
    },
    {
      "name": "Sylvia H. Sutanto"
    },
    {
      "name": "Vincentius E. W. Tjia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/38hft5y79acc2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02060",
  "fulltext_url": null,
  "id": "38hft5y79acc2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02060",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "physics.atom-ph"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "3107|Atomic and Molecular Physics"
    ]
  ],
  "openalex": "W7216244918",
  "pdf_url": "https://arxiv.org/pdf/2610.02060",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/38hft5y79acc2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2610.02060",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02060"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:physics.atom-ph",
      "subfield": "amo"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "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",
  "title_text": "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",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:03Z",
  "url": "https://research.draconitic.com/paper/38hft5y79acc2",
  "venue": "arXiv",
  "versions": []
}