Kohn定理与伽利略对称性
Kohn's theorem and Galilean symmetry
摘要：
利用Duval的Bargmann框架重新讨论了均匀磁场中带电粒子系统的可分性与伽利略对称性之间的关系。如果粒子的荷质比相同，即所有粒子的荷质比都是$e_a/m_a=ε$，那么磁系统的巴格曼空间与各向异性谐振子的巴格曼空间是等距的。假设粒子通过一个只取决于它们相对距离的势相互作用，系统分裂成一个代表质心加上一个解耦的内部部分的系统，并可以使用Niederer变换进一步映射成一个孤立系。相反地，孤立系的明显的伽利略助推对称性可以分别“导入”到振子和磁系统中，以产生吉本斯和波普用来证明可分性的对称性。对于相互作用势为零的孤立系是自由的，我们的程序赋予我们所有的系统与一个隐藏的薛定谔对称性，增强与独立的内部旋转。所有这些性质都来自于Galilei群的上同调结构，正如Souriau的“分解重心”所解释的那样。
英文摘要：
The relation between the separability of a system of charged particles in a uniform magnetic field and Galilean symmetry is revisited using Duval's "Bargmann framework". If the charge-to-mass ratios of the particles are identical, $e_a/m_a=ε$ for all particles, then the Bargmann space of the magnetic system is isometric to that of an anisotropic harmonic oscillator. Assuming that the particles interact through a potential which only depends on their relative distances, the system splits into one representing the center of mass plus a decoupled internal part, and can be mapped further into an isolated system using Niederer's transformation. Conversely, the manifest Galilean boost symmetry of the isolated system can be "imported" to the oscillator and to the magnetic systems, respectively, to yield the symmetry used by Gibbons and Pope to prove the separability. For vanishing interaction potential the isolated system is free and our procedure endows all our systems with a hidden Schroedinger symmetry, augmented with independent internal rotations. All these properties follow from the cohomological structure of the Galilei group, as explained by Souriau's "décomposition barycentrique"
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