LANDAU-LEVEL GROUND-STATE DEGENERACY AND ITS RELEVANCE FOR A GENERAL QUANTIZATION PROCEDURE

The quantum dynamics of a two-dimensional charged spin-1/2 particle is studied for general, symmetry-free curved surfaces and general, nonun iform magnetic fields that are, when different from zero, orthogonal t o the defining two surface. Although higher Landau levels generally lo se their degeneracy under such general conditions, the lowest Landau l evel, the ground state, remains degenerate. Previous discussions of th is problem have had less generality and/or used supersymmetry, or else have appealed to very general mathematical theorems from differential geometry. In contrast our discussion relies on simple and standard qu antum-mechanical concepts. The mathematical similarity of the physical problem at hand and that of a phase-space path-integral quantization scheme of a general classical system is emphasized. Adopting this anal ogy in the general case leads to a general quantization procedure that is invariant under general coordinate transformations-completely unli ke any of the conventional quantization prescriptions-and therefore ge neralizes the concept of quantization to hitherto inaccessible situati ons. In a complementary fashion, the so-obtained picture of general qu antization helps to derive useful semiclassical formulas for a Hall cu rrent in the case of a filling factor equal to one for a general surfa ce and magnetic field.