Quantization of the Harmonic Oscillator – Ariadne’s Thread in Quantization

In this chapter, we will study the following quantization methods:

• • Heisenberg quantization (matrix mechanics; creation and annihilation operators),

• • Schrödinger quantization (wave mechanics; the Schrödinger partial differential equation),

• • Feynman quantization (integral representation of the wave function by means of the propagator kernel, the formal Feynman path integral, the rigorous infinite-dimensional Gaussian integral, and the rigorous Wiener path integral),

• • Weyl quantization (deformation of Poisson structures),

• • Weyl quantization functor from symplectic linear spaces to C^*-algebras,

• • Bargmann quantization (holomorphic quantization),

• • supersymmetric quantization (fermions and bosons).