标题： 二阶量化用于开普勒问题 显示英文标题

标题： Second Quantization for the Kepler Problem

摘要： 开普勒问题涉及一个在吸引力平方反比力作用下的点粒子。 在简要回顾了这个问题的经典和量子版本，重点在于它们的隐藏$\text{SU}(2) \times \text{SU}(2)$对称性之后，我们讨论自旋为$\frac{1}{2}$的量子开普勒问题。 我们表明，该问题的束缚态希尔伯特空间$\mathcal{H}$作为$\text{SU}(2) \times \text{SU}(2)$的表示，与时空$\mathbb{R} \times S^3$上威耳方程解的希尔伯特空间是单元等价的。 这个方程描述了一个无质量的左旋自旋为$\frac{1}{2}$的粒子。 然后，我们在 $\mathcal{H}$ 上构建费米子福克空间，并证明它酉等价于 $\mathbb{R} \times S^3$ 上无质量左手自旋$\frac{1}{2}$自由量子场的希尔伯特空间，同样以 $\text{SU}(2) \times \text{SU}(2)$ 的表示形式呈现。 通过修改该自由场理论的哈密顿量，我们得到了著名的“马德隆规则”。 这些规则合理地近似了我们考虑电子数越来越多的元素时观察到的子壳层填充情况，并与元素周期表的大致整体结构相匹配。 显示英文摘要

摘要： The Kepler problem concerns a point particle in an attractive inverse square force. After a brief review of the classical and quantum versions of this problem, focused on their hidden $\text{SU}(2) \times \text{SU}(2)$ symmetry, we discuss the quantum Kepler problem for a spin-$\frac{1}{2}$ particle. We show that the Hilbert space $\mathcal{H}$ of bound states for this problem is unitarily equivalent, as a representation of $\text{SU}(2) \times \text{SU}(2)$, to the Hilbert space of solutions of the Weyl equation on the spacetime $\mathbb{R} \times S^3$. This equation describes a massless left-handed spin-$\frac{1}{2}$ particle. We then form the fermionic Fock space on $\mathcal{H}$ and show this is unitarily equivalent to the Hilbert space of a massless left-handed spin-$\frac{1}{2}$ free quantum field on $\mathbb{R} \times S^3$, again as representations of $\text{SU}(2) \times \text{SU}(2)$. By modifying the Hamiltonian of this free field theory, we obtain the well-known "Madelung rules". These give a reasonable approximation to the observed filling of subshells as we consider elements with more and more electrons, and match the rough overall structure of the periodic table.