In this Thesis we develop a mathematical formalism of (Symmetry) Group-Approach to Quantization of Field Theories with constraints (gauge invariant theories), which incorporates techniques and tools of Algebra, Differential Geometry and Topology. This mathematical formalism is successfully applied to the quantization of anomalous systems which are difficult to handle by the conventional Canonical Quantization scheme. In particular, we provide a way out to the traditional "no-go" theorems of Groenwald and van Hove through a revision of the concept of "algebraic anomaly" and by allowing new anomalous internal degrees of freedom. We also study the Quantum Hall Effect, the appearance of integer an fractional quantum numbers of topological nature and the role played by the modular invariance. Moreover, we tackle the quantization of field theories in curved symmetric spaces like Anti-de Sitter and we offer a group-theoretical explanation of the Fulling-Unruh Effect (vacuum radiation in accelerated systems) as a spontaneous breakdown of the conformal symmetry. Lastly, a revision of the traditional concepts of gauge symmetry and constraints leads us to a unified quantization of the electromagnetic and Proca fields (massless and massive photon, respectively) and opens the door to alternatives to the Higgs-Kibble mechanism to provide mass to the intermediate vector bosons of the weak interaction in the Standard Model.
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