Quantization on a Lie Group: Higher-Order Polarizations

Much effort has been devoted to the geometrization of quantum mechanics during the second half of this century in an attempt to emulate classical mechanics and classical gravity at mathematical beauty and, why not, to better understand quantum gravity. We wish to report on one particular line of this task, which lies mostly on symmetry grounds and has been developed in the last years trying to accomodate modern aspects of quantum mechanics such as global quantization of systems with non-trivial topology, in particular systems suffering from topological anomalies, and accounting for more general obstructions to the basic rules of local quantization, to be referred to as algebraic anomalies, directly attached to the well-known no-go theorems [1] of the original, standard quantum mechanics. This group approach to quantization (GAQ) [2], in some respect generalizes geometric quantization (GQ), originally developed by Kirillov, Kostant and Souriau [3–5] both as a method of quantization and as a group representation technique, and the more specific representation algorithm of Borel-Weyl-Bott (B-W-B) [6], which essentially applies to finite-dimensional semisimple groups.