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Landau levels on a torus. (English) Zbl 0984.81200
Summary: Landau levels have represented a very rich field of research, which has gained widespread attention after their application to the 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 $$[{\mathfrak p}_x,{\mathfrak p}_y] \approx i\hbar B$$, and, at the same time, live in a Landau level of finite dimension $$BL_1L_2/(hc/e)$$, which is impossible from Wintner’s theorem. The paper shows the way out of this conundrum.

##### MSC:
 81V70 Many-body theory; quantum Hall effect 81S10 Geometry and quantization, symplectic methods 81T13 Yang-Mills and other gauge theories in quantum field theory 81T70 Quantization in field theory; cohomological methods
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