具有de Sitter时空对称性的狭义相对论中的单电子原子
One Electron Atom in Special Relativity with de Sitter Space-Time Symmetry
摘要：
德西特不变量狭义相对论（deSitter invariant，dS-SR）是常曲率的狭义相对论，是通常爱因斯坦狭义相对论（Einstein SR，E-SR）的自然推广。本文用绝热近似和量子力学的准定态微扰计算方法求解了氢原子的dS-SR Dirac方程。氢原子位于宇宙的光锥上。利用FRW度规和$Λ$ CDM宇宙学模型讨论了这个问题。对于原子，考虑了Beltrami度规描述的de Sitter时空几何的影响。dS-SR Dirac方程是一个含时量子哈密顿系统。结果表明：1、在dS-SR QM框架下，基本物理常数$m_e,\;\hbar,\;e$随宇宙学时间的变化而变化。2、由于dS-SR QM效应引起的$α\equiv e^2/(\hbar c)$-分裂：利用微扰理论，解析计算了$ΔE(z)$的分裂，属于dS-SR QM的$\mathcal{O}(1/R^2)$ -物理。在数值上，我们发现当$|R|\simeq \{10^3 Gly,\;10^4 Gly,\;10^5 Gly\;\}$和$z\simeq \{1,\;{\rm or}\;2\}$，$ΔE(z)>> 1{\rm (Lamb\; shift)}$。这表明在这些情况下，QED引起的超精细结构效应可以忽略，dS-SR精细结构效应占主导地位。这种效应可以用来确定dS-SR中的普适常数$R$，被认为是超越E-SR的新物理。
英文摘要：
The de Sitter invariant Special Relativity (dS-SR) is a SR with constant curvature, and a natural extension of usual Einstein SR (E-SR). In this paper, we solved the dS-SR Dirac equation of Hydrogen by means of the adiabatic approach and the quasi-stationary perturbation calculations of QM. Hydrogen atoms are located on the light cone of the Universe. FRW metric and $Λ$CDM cosmological model are used to discuss this issue. To the atom, effects of de Sitter space-time geometry described by Beltrami metric are taken into account. The dS-SR Dirac equation turns out to be a time dependent quantum Hamiltonian system. We revealed that: 1,The fundamental physics constants $m_e,\;\hbar,\;e$ variate adiabatically along with cosmologic time in dS-SR QM framework. But the fine-structure constant $α\equiv e^2/(\hbar c)$ keeps to be invariant; 2,$(2s^{1/2}-2p^{1/2})$-splitting due to dS-SR QM effects: By means of perturbation theory, that splitting $ΔE(z)$ were calculated analytically, which belongs to $\mathcal{O}(1/R^2)$-physics of dS-SR QM. Numerically, we found that when $|R|\simeq \{10^3 Gly,\;10^4 Gly,\;10^5 Gly\;\}$, and $z\simeq \{1,\;{\rm or}\;2\}$, the $ΔE(z)>> 1{\rm (Lamb\; shift)}$. This indicate that for these cases the hyperfine structure effects due to QED could be ignored, and the dS-SR fine structure effects are dominant. This effect could be used to determine the universal constant $R$ in dS-SR, and be thought as a new physics beyond E-SR.
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