Lie group representations and quantization

In this paper we are concerned with the study of representations of connected Lie groups, related to physical problems. The representation technique used here is formulated on the basis of a group quantization formalism previously introduced. It generalizes the Konstant—Kirillov co-adjoint orbits method for connected Lie groups and the Borel—Weil—Bott representation algorithm for semisimple groups mainly in that it introduces the notion of higher-order polarizations which is of a crucial importance in the study of anomalies. We illustrate the fundamentals of the group approach with the help of the simplest nontrivial example of the affine group in one dimension, and the use of higher-order polarizations with the harmonic oscillator group and the Schrödinger group, the last one constituting the simplest example of an anomalous group. Also, examples of infinite-dimensional anomalous groups are briefly considered.