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 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.