# Geometric Quantization

## " 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

- Symplectic manifolds & Hamiltonian formalism
- Hermitian line bundles and their characterization via the first chern class
- Integrable distributions (in particular Lagrangian distributions)
- Metaplectic group & metaplectic structures
- Theory of unitary representations

The description of the module can be found here and the detailed list of topics and references can be found here .

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:

- Get acquainted with the general idea of the seminar by reading the summary of all the talks and the introduction of [Baykara] .
- Read carefully the guidelines of your topic.
- Ask us if you need help, more references, copies of books, etc.
- Take appointment with the person tutoring your topic to schedule a meeting 1-2 weeks before your talk. During the meeting you will have the opportunity to ask questions about the content and the structure of the talk.
- One day before the meeting send us a short schematic description of the results you will present in the talk saying how much detail you will give.
- The day before your talk send us the slides of the talk or your notes so that we can distribute them to the audience in advance.
- Give your talk by writing on a tablet in real time (preferred option) or prepare slides using Latex beamer - if you need help with this, ask us. We will also reserve a room in the Mathematikon if the speaker wants to give the talk from there.

In preparing the talk two things are extremely important:

- Be sure that you understand the content and ask us if the math is not clear
- Structure the presentation in such a way that the other students, who will be new to the content, can learn something from it. Suggestions (in German) on how to give math presentations can be found here .

Evaluation and registration

- To pass this module you will need to give a 90-minute talk and to participate in the talks of others.
- Please register to the seminar on MÜSLI and on MaMpf where you will find the notes of the talks.
- On MaMpf you can ask questions to the material of the talks leaving a comment in the corresponding section. If you want to ask a question that is not related to a talk in particular or you want to start a discussion with the other members of the seminar on some topic you are encouraged to write a post in the forum.

- 15.04.2021 (Daniel): Mathematical Model of Classical Mechanics
- 22.04.2021 (Eugen): Symmetries in Classical Mechanics
- 29.04.2021 (Julian): Mathematical Model of Quantum Mechanics
- 06.05.2021 (Erik/ David): Prequantization and Hermitian Line Bundles 1
- 20.05.2021 (Erik/ David): Prequantization and Hermitian Line Bundles 2
- 27.05.2021 (Simon): Lagrangian Distributions and Polarization
- 10.06.2021 (Michael): Half-Form Correction for Real Polarizations
- 17.06.2021 (Gabriele): Half-Form Correction for Kähler Polarizations and Metalinear Structures
- 24.06.2021 (Henri): Pairing of Polarizations and the BKS Construction
- 01.07.2021 (Jonas): Groenewold-Van Hove Problem
- 08.07.2021 (Jan): Unitary Representations via the Orbit Method
- 15.07.2021 (Steffen/Johanna): Conformal Quantum Mechanics

- [1] B. C. Hall; Quantum Theory for Mathematicians
- [2] N. M. J. Woodhouse; Geometric quantization.
- [3] A. Echeverrıa-Enrıquez, M. C. Munoz Lecanda, N. Roman-Roy, and C. Victoria-Monge; Mathematical Foundations of Geometric Quantization
- [4] S. Bates and A. Weinstein; Lectures on the Geometry of Quantization
- [5] E. Lerman; Geometric quantization, a crash course.