ALGEBRAIC QUANTIZATION, GOOD OPERATORS AND FRACTIONAL QUANTUM NUMBERS

The problems arising when quantizing systems with periodic boundary co nditions are analysed, in an algebraic (group-) quantization scheme, a nd the ''failure'' of the Ehrenfest theorem is clarified in terms of t he already defined notion of good (and bad) operators. The analysis of ''constrained'' Heisenberg-Weyl groups according to this quantization scheme reveals the possibility for quantum operators without classica l analogue and for new quantum (fractional) numbers extending those al lowed for Chern classes in traditional Geometric Quantization. This st udy is illustrated with the examples of the free particle on the circu mference and the charged particle in a homogeneous magnetic field on t he torus, both examples featuring ''anomalous'' operators, non-equival ent quantization and the latter, fractional quantum numbers. These pro vide the rationale behind flux quantization in superconducting rings a nd Fractional Quantum Hall Effect, respectively.