An Introduction to the Orbit Method

Abstract

The Orbit Method is a method to determine all irreducible unitary representations of a Lie group. It is entangled with its physical counterpart geometric quantization, which is an extension of the canonical quantization scheme to general curved manifolds. The main ingredient of the Orbit Method is the notion of coadjoint orbits, which will be explained. Coadjoint orbits of a Lie group have the natural structure of a symplectic manifold, as does the phase space of a classical mechanical system. Naturally, geometric quantization will be treated next, since it attempts to provide a geometric interpretation of quantization within an extension of the mathematical framework of classical mechanics (symplectic geometry). In particular, the axioms imposed on a quantization will be discussed. Finally, as an application, coadjoint orbits and geometric quantization will be brought together by indicating how to determine the irreducible unitary representations of SU(2) by means of the Orbit Method.