Mathematisches Seminar: Geometric Quantization
Prof. Dr. Martin Schottenloher
Di 12-14, HS B 251, Beginn: 16.04.24 (Vorbesprechung)
Vorträge können in Deutsch oder in Englisch gehalten werden.
Schedule
- 16.04.24 Vorbesprechung
- 07.05.24 Prequantization (Benjamin Fox)
- 04.06.24 Polarization (Navid Emami)
- 11.06.24 Integrality (Aaron Willen)
- Kähler Quantization (Luca Nahs)
- Densities and Their Geometry (Afonso Paulouro)
- Half-Form Quantization (Giulia Tazzoli)
Inhalt
Geometric quantization is a general framework for constructing quantum systems from their classical counterparts, - based on ideas of canonical quantization und using the geometry of the classical system. Such a classical system consists first of all of a symplectic manifold M together with a set R of classical observables. In order to apply Geometric Quantization one has to select in addition some geometric data like a suitable connection an a complex line bundle on M, a polarization, a measure, and possibly more. These geometric data are used to define the Hilbert space H of "wave functions" and to construct a unique correspondence assigning to each f in R a quantum operator Q(f) on H.
Geometric Quantization does help to understand canonical quantization in several instances. But it seems to become too difficult to generate new quantum models for highly complex classical systems. However, the essential properties of Geometric Quantization have applications in theoretical considerations of several domains of mathematical physics.
The objective of the seminar is to present an introduction to Geometric Quantization. A detailed exposition can be found in my (incomplete) notes: Lecture Notes on Geometric Quantization. Based on the Course Given in 2021/22.
Examples for topics:
- Line bundles (Geradenbündel)
- Connections on line bundles
- Parallel transport and curvature on line bundles
- Prequantization
- Integrality as the quantum condition
- Geometry of polarisations
- Representation space (first step of geometric quantization) More advanced:
- Kähler quantization and Kähler geometry
- Half-density quantization
- Half-form Quantization
- Metaplectic correction
- Coadjoint orbits
- Quantization of Chern-Simons-theory
- Reduction and Quantization
- S. Bates and A. Weinstein, Lectures on the geometry of quantization, Berkeley Mathematics Lecture Notes 8, AMS (1997).
- B.C. Hall Quantum Theory for Mathematicians. Springer-Verlag, 2013.
- N.E. Hurt, Geometric Quantization in Action (1982), Reidel Company.
- M. Puta, Hamiltonian Mechanical Systems and Geometric Quantization (1993), Kluwer Publications.
- G.M. Tuynman The metaplectic correction in geometric quantization. Journal of Geometry and Physics 106 (2016), 401 - 426.
- J. Sniaticky, Geometric Quantization and Quantum Mechanics (1980) Springer-Verlag.
- Nicholas Woodhouse, Geometric quantization (1980) Oxford University Press. 2nd ed. 1991.
Für:
Interested students of physics or of mathematics.
Voraussetzungen:
Basic knowledge of Classical Mechanics, Quantum Mechanics, differentiable manifolds. For example as presented in the above mentioned notes.
Anmeldung:
By mail: martin@schottenloher.de