Landau Levels on a Torus

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 Quantiza-tion. 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 ] ∝ iℏB, and, at the same time, live in a Landau level of finite dimension B L 1 L 2 /(hc/e), which is impossible from Wintner's theorem. The paper shows the way out of this conundrum.