非对易Landau问题中的Kohn条件和奇异Newton-Hooke对称性
Kohn condition and exotic Newton-Hooke symmetry in the non-commutative Landau problem
摘要：
$N$“外来”[别名非对易]粒子，具有质量$m_a$，电荷$e_a$和非对易参数$θ_a$，在均匀磁场$B$中运动，如果Kohn条件$e_a/m_a=\const$补充有$e_aθ_a=\const.$，则质心表现为携带系统总质量和电荷的单个外来粒子，$M$和$e$，以及适当定义的非交换参数$Θ$。对于电场消失的临界情况$eΘB\neq1$，粒子进行通常的回旋运动，修改，但相同的频率。该系统在适当的时间相关平移下是对称的，该平移跨越“奇异”的（4+2）参数中心扩展子群[即，两参数中心扩展] Newton-Hooke群。在临界情况下（$B=B_c=(eΘ)^{-1}$），系统冻结为静态“晶体”配置。加入一个恒定的电场，当$eΘB\neq1$。对于$B=B_c$，回旋运动被消除，所有粒子共同运动，遵循霍尔定律。我们的含时对称性被简化为中心扩展平移的（2+1）参数海森堡群。
英文摘要：
$N$ "exotic" [alias non-commutative] particles with masses $m_a$, charges $e_a$ and non-commutative parameters $θ_a$, moving in a uniform magnetic field $B$, separate into center-of-mass and internal motions if Kohn's condition $e_a/m_a=\const$ is supplemented with $e_aθ_a=\const.$ Then the center-of-mass behaves as a single exotic particle carrying the total mass and charge of the system, $M$ and $e$, and a suitably defined non-commutative parameter $Θ$. For vanishing electric field off the critical case $eΘB\neq1$, the particles perform the usual cyclotronic motion with modified but equal frequency. The system is symmetric under suitable time-dependent translations which span a (4+2)- parameter centrally extended subgroup of the "exotic" [i.e., two-parameter centrally extended] Newton-Hooke group. In the critical case $B=B_c=(eΘ)^{-1}$ the system is frozen into a static "crystal" configuration. Adding a constant electric field, all particles perform, collectively, a cyclotronic motion combined with a drift perpendicular to the electric field when $eΘB\neq1$. For $B=B_c$ the cyclotronic motion is eliminated and all particles move, collectively, following the Hall law. Our time-dependent symmetries are reduced to the (2+1)-parameter Heisenberg group of centrally-extended translations.
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