环面上粒子在恒定磁场中的离散偶然对称性
Discrete Accidental Symmetry for a Particle in a Constant Magnetic Field on a Torus
摘要：
经典粒子在恒定磁场中沿圆形轨道作回旋运动。在量子水平上，所有经典轨道都是闭合的这一事实引起了光谱的简并。众所周知，在恒定磁场中的带电粒子的能谱由无限简并的朗道能级组成。正如对于$1/r$和$r^2$势一样，人们因此期待着某种隐藏的偶然对称性，在这种情况下是无限维表示的。事实上，回旋加速器圆的中心位置起着龙格-伦茨矢量的作用。在确定了相应的偶然对称代数之后，我们在有限周期体积中重新分析了系统。有趣的是，类似于非阿贝尔规范理论中由于$θ$ -真空角导致的CP不变性的量子力学破缺，由于两个自伴扩展参数$θ_x$和$θ_y$导致的量子效应明确地打破了经典理论的连续平移不变性。这将对称性降低到离散的磁平移群，并导致有限的简并。类似于在圆锥上运动的粒子，在恒定磁场中的粒子显示出量子力学中非常特殊的偶然对称性。
英文摘要：
A classical particle in a constant magnetic field undergoes cyclotron motion on a circular orbit. At the quantum level, the fact that all classical orbits are closed gives rise to degeneracies in the spectrum. It is well-known that the spectrum of a charged particle in a constant magnetic field consists of infinitely degenerate Landau levels. Just as for the $1/r$ and $r^2$ potentials, one thus expects some hidden accidental symmetry, in this case with infinite-dimensional representations. Indeed, the position of the center of the cyclotron circle plays the role of a Runge-Lenz vector. After identifying the corresponding accidental symmetry algebra, we re-analyze the system in a finite periodic volume. Interestingly, similar to the quantum mechanical breaking of CP invariance due to the $θ$-vacuum angle in non-Abelian gauge theories, quantum effects due to two self-adjoint extension parameters $θ_x$ and $θ_y$ explicitly break the continuous translation invariance of the classical theory. This reduces the symmetry to a discrete magnetic translation group and leads to finite degeneracy. Similar to a particle moving on a cone, a particle in a constant magnetic field shows a very peculiar realization of accidental symmetry in quantum mechanics.
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