Geometric quantization and internal symmetry

Abstract

As a part of an attempt to geometrize physics, internal symmetries in the covariant classification of matter by itsT μυ type are considered in relation to phase transformations generated by complex and quaternionic structures on space-time. The Rainich theory of electromagnetism and neutrinos is compared with the theory ofU(1) ×SO(1, 3) torsional gauge fields, and extended to the quaternionic case. It is shown by the Kostant technique of geometric quantization that complex and quaternionic phase transformations for an Einstein space are associated with one-dimensional and three-dimensional harmonic oscillators.