Thomas Mainiero Syed Asif

The aim of the geometric quantization program is to describe a quantization procedure in terms of natural geometric structures. To date, this program has succeeded in unifying various older methods of quantizing finite dimensional physical systems. The generalization to infinite-dimensional systems (for example, field theories) remains an active area of research. As such, we will restrict our attention the finite dimensional case. We will describe the basic construction procedure and the geometric structures involved, and for concreteness we will show the explicit details of the construction in the case of the n-dimensional harmonic oscillator. Despite its apparent simplicity, the harmonic oscillator is sufficiently rich a physical system to highlight the main points of the geometric quantization procedure while requiring us to grapple with some of the subtle issues which arise. In what follows we will primarily follow the exposition given by Woodhouse and Simms [1][2]. First we will provide a mathematical description of a classical physical system and define the conventions used in this paper. The basic object is a symplectic manifold: a pair (M,ω) with M a 2n dimensional manifold M equipped with a closed nondegenerate 2-form ω. We are then naturally concerned with morphisms which preserve this structure; in particular, these are the symplectomorphisms ρ : M1 → M2 for symplectic manifolds (M1, ω1) and (M2, ω2) such that ρω2 = ω1. When appropriate we will concern ourselves with M an affine symplectic manifold and restrict our attention to the linear symplectomorphisms M → M which form the group Sp(M) ∼= Sp(2n,R). Without loss of generality for the following discussions, we will blur the distinction between an affine space and its associated vector space. Via Darboux’s theorem, around any point m ∈ M a symplectic manifold M we can find a neighborhood U containing m and local coordinates pa, qa on U such that