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The quantization of a Hamiltonian mechanical system has proved to be a delicate as well as a difficult problem. In the past decade, however, considerable progress has been made by returning to an examination of the mathematical foundations of classical physics and noting that they can be simply and elegantly phrased in terms of symplectic geometry. The resulting quantization theory, geometric quantization, is an outgrowth of independent work by Kostant  and Souriau .
In this chapter we present the first step of geometric quantization, the so called geometric prequantization. The second step, this means the construction of polarizations, half-densities and half-forms will make the object of the following chapter.
KeywordsPoisson Bracket Symplectic Manifold Cotangent Bundle Geometric Quantization Complex Line Bundle
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