环面上的朗道能级
Landau levels on a torus
摘要：
朗道能级代表了一个非常丰富的研究领域，在应用于量子霍尔效应后，受到了广泛的关注。在一个特殊的规范，全纯规范，他们给出了巴格曼的希尔伯特空间的物理实现的整个功能。它们也被认为是费曼路径积分和几何量子化之间的天然桥梁。本文讨论了当人们试图研究具有均匀磁场和周期边界条件的有限条L_1、L_2上的量子力学时，在问题的表述中所涉及的一些数学上的微妙之处。这里存在一个明显的悖论：无穷小平移应该与正则算子[p_x，p_y] \propto i \hslash B相联系，同时又存在于有限维的Landau水平BL_1L_2/（hc/e）中，这在Wintner定理中是不可能的。这篇论文指出了解决这一难题的方法。
英文摘要：
Landau levels have represented a very rich field of research, which has gained widespread attention after their application to quantum Hall effect. In a particular gauge, the holomorphic gauge, they give a physical implementation of Bargmann's Hilbert space of entire functions. They have also been recognized as a natural bridge between Feynman's path integral and Geometric Quantization. We discuss here some mathematical subtleties involved in the formulation of the problem when one tries to study quantum mechanics on a finite strip of sides L_1, L_2 with a uniform magnetic field and periodic boundary conditions. There is an apparent paradox here: infinitesimal translations should be associated to canonical operators [p_x,p_y] \propto i\hslash B, and, at the same time, live in a Landau level of finite dimension B L_1L_2/(hc/e), which is impossible from Wintner's theorem. The paper shows the way out of this conundrum.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
