发表机构
Center for Theoretical Physics, Faculty of Science, Parahyangan Catholic University（帕拉希扬天主教大学理学院理论物理中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文通过计算氢离子 Stark 位移验证了八分量相对论波动方程的不定度规形式体系，证明其与 Dirac 结果一致，并揭示了自旋-场耦合和负范数态的必要性。
AI 中文摘要
Robson 和 Staudte 提出的自旋-1/2 粒子八分量相对论波动方程（FV1/2）是 Feynman–Gell-Mann 方程的 Feshbach–Villars 线性化，它再现了 Dirac 氢原子能谱，但解空间加倍并具有不定内积。其精确 Stark 位移必须等于 Dirac 结果；其不定度规微扰理论是否能给出这些结果并不确定。我们在 Schrödinger、Dirac 和 FV1/2 理论中计算了氢离子从 H 到 U^{91+} 的 n=1–3 能级的一阶和二阶 Stark 位移，并通过 Dalgarno–Lewis 方法评估了所有谱求和（包括连续谱）。FV1/2 与 Dirac 位移逐能级一致，一阶（闭式根）达到工作精度，二阶达到 45 位有效数字。两个要素都是必需的：没有显式自旋-场耦合，一阶根变为复数；没有负范数态，(Zα)^2 系数是错误的。在加倍空间中，范数符号和宇称是锁定的：提升 2s_{1/2}–2p_{1/2} 简并的势需要其自身的自旋-梯度项，而手动放置在 κ 标记态上的位移会产生虚假的虚线性 Stark 位移。相对论线性 Stark 效应在有限因子下较小，但二阶矩求和规则表明非相对论线性 Stark 强度是守恒的，缺失部分重新出现在奇异的 (Zα)^{-2} 二阶项中。对于具有精细结构伙伴的能级，二阶位移的精确壳内/正则分割给出了 (Zα)^0 系数的解析表达式，并将 (Zα)^2 系数中的 π^2 项与奇异项联系起来。氢 n=2 的 Dirac 弱/强场交叉点位于约 2.8 kV cm^{-1} 附近，由精细结构设定的微扰窗口随 Z^5 增长。
英文摘要
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 $Z^5$.
Comments21 pages, 2 figures, 8 tables
