The book gives an introduction to Weyl non-regular quantization suitable for the description of
physically interesting quantum systems where the traditional Dirac-Heisenberg quantization is
not applicable. The latter implicitly assumes that the canonical variables describe observables
entailing necessarily the regularity of their exponentials (Weyl operators). However in
physically interesting cases -- typically in the presence of a gauge symmetry -- non-observable
canonical variables are introduced for the description of the states namely of the relevant
representations of the observable algebra. In general a gauge invariant ground state defines a
non-regular representation of the gauge dependent Weyl operators providing a mathematically
consistent treatment of familiar quantum systems -- such as the electron in a periodic
potential (Bloch electron) the Quantum Hall electron or the quantum particle on a circle --
where the gauge transformations are respectively the lattice translations the magnetic
translations and the rotations of 2pi. Relevant examples are also provided by quantum gauge
field theory models in particular by the temporal gauge of Quantum Electrodynamics avoiding
the conflict between the Gauss law constraint and the Dirac-Heisenberg canonical quantization.
The same applies to Quantum Chromodynamics where the non-regular quantization of the temporal
gauge provides a simple solution of the U(1) problem and a simple link between the vacuum
structure and the topology of the gauge group. Last but not least Weyl non-regular
quantization is briefly discussed from the perspective of the so-called polymer representations
proposed for Loop Quantum Gravity in connection with diffeomorphism invariant vacuum states.
