  The problems arising when quantizing systems with periodic boundary
conditions are analysed, in an algebraic (group-) quantization scheme, and the
``failure" of the Ehrenfest theorem is clarified in terms of the already
defined notion of {\it good} (and {\it bad}) operators. The analysis of
``constrained" Heisenberg-Weyl groups according to this quantization scheme
reveals the possibility for new quantum (fractional) numbers extending those
allowed for Chern classes in traditional Geometric Quantization. This study is
illustrated with the examples of the free particle on the circumference and the
charged particle in a homogeneous magnetic field on the torus, both examples
featuring ``anomalous" operators, non-equivalent quantization and the latter,
fractional quantum numbers. These provide the rationale behind flux
quantization in superconducting rings and Fractional Quantum Hall Effect,
respectively.
