" Quantization is an art, not a functor."
− general consensus
Quantization describes the process of assigning a quantum system to a given classical system. Even though there is no general recipe working in all cases, in the last fifty years a successful mathematical approach, known as Geometric Quantization, has been developed. Such an approach entails the following three steps: Prequantization, Polarization, and Metaplectic correction. Prequantization produces a natural Hilbert space and transforms Poisson brackets of functions on the classical side into commutators of operators on the quantum side. Nevertheless, the prequantum Hilbert space is generally "too large" to be physically meaningful. Therefore, the choice of a polarization and, in some cases, the introduction of a metaplectic correction are needed to get the right quantum Hilbert space. Each step will be clarified using concrete examples such as the harmonic oscillator and the spin of a particle. Following the story of geometric quantization we will learn about many fascinating and crucial mathematical concepts such as
For a nice overview on what else to expect check out this paper about geometric quantization by Z.K.Baykara.
Basic knowledge of differential geometry and topology (i.e. manifolds, vector fields, differential forms, vector bundles...) and also functional analysis is needed. We will not expect you to know any physics, nevertheless some things might seem more reasonable with knowledge of some quantum mechanics.
If you are giving a talk:
In preparing the talk two things are extremely important:
Evaluation and registration