  We report briefly on an approach to quantum theory entirely based on symmetry
grounds which improves Geometric Quantization in some respects and provides an
alternative to the canonical framework. The present scheme, being typically
non-perturbative, is primarily intended for non-linear systems, although
needless to say that finding the basic symmetry associated with a given
(quantum) physical problem is in general a difficult task, which many times
nearly emulates the complexity of finding the actual (classical) solutions.
Apart from some interesting examples related to the electromagnetic and
gravitational particle interactions, where an algebraic version of the
equivalence principle naturally arises, we attempt to the quantum description
of non-linear sigma models. In particular, we present the actual quantization
of the partial-trace non-linear SU(2) sigma model as a representative case of
non-linear quantum field theory.
