Wintersemester 2021/22

Geometric Quantization

Prof. Dr. Martin Schottenloher

Vorlesung mit Übungen (2 + 1 std.)
Vorlesung, Do 12-14. HS B 252. Beginn: 21.10.21
Übungen: Di 16-17, HS B 004; und per ZOOM.

Here are the Lecture Notes in development. They will be updated frequently. The changes from version to version are reported in Status and Changes.
The Notes will be an expanded version of the handwritten manuscript: Geometric Quantization 09/10.
The folder of Exercises.
Suggestions for Exercices

Registration: See below.

The course will be delivered in English. The planned lecture notes will be written in English.

Inhalt

One of the main objectives of Geometric Quantization is to make "Canonical Quantization" precise. In the case of a simple system of Classical Mechanics with phase space ${\mathrm{ℝ}}^{\mathrm{2n}}$ the procedure of Canonical Quantization associates with a classicle observable - represented by a function
            $f\left(p_k,q^m\right)$
with canonical variables $\left(p_k,q^m\right)$ - the quantum mechanical observable represented by the operator
            $f(-i\mathrm{ħ}\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^k},q^m)$.
This association causes problems with the ordering of the operators (consider, for example, the operator associated with the observable $p_1{q}^1$ ), it is not independent of the initial coordinates and it is not invariant under canonical transformations. Furthermore, there is no evidence how to treat constraints or general phase spaces like symplectic manifolds.

Geometric quantization is an attempt to use the differential-geometric properties of a classical phase space assumed to be a symplectic manifold M in order to define a corresponding quantum theory. One constructs the Hilbert space from the square integrable sections of a complex line bundle over M in a way which makes it completely transparent which choices or assumptions are made in the various stages of the construction.

The resulting procedure provides a way of looking at quantum theory that is distinct from conventional approaches to the subject. In particular, it can be used to construct such familiar quantum systems as e.g. the the canonical quantization of position and momentum, the Bargmann quantization, the Schrödinger equation, the quantization of spin, and the quantization of Chern-Simons-theory.
However, Geometric Quantization has its limits which are made precise through no go theorems.

This course provides the necessary differential geometric background to formulate Geometric Quantization and gives a first introduction into the subject.

The procedure of Geometric Quantization uses quite a few of interesting mathematical concepts such as e.g. connections, curvature, characteristic classes, cohomology, central extensions of Lie groups and algebras, projective and induced representations, the orbit method, etc. which are introduced and discussed in the course. In many cases these structures are not needed in full generality - e.g. one only needs the geometry of line bundles and not the general geometry of vector bundles -, so that this introduction to Geometric Quantization also can be viewed as to be a playground for learning to apply differential-geometric mathods, representation theory and other mathematical structures to physics.

Plan of the course:

• Hamiltonian Mechanics on the basis of symplectic manifolds
• Connections on line bundles
• Prequantizaion and integrality conditions
• Parallel transport and curvature
• Cohomology
• The holomorphic case and Kähler manifolds
• Geometry of polarisations and quantization
• Metaplectic correction
• Quantization of Chern-Simons-theory

Bibliography:

• S. Bates and A. Weinstein, Lectures on the geometry of quantization, Berkeley Mathematics Lecture Notes 8, AMS (1997).
• J.-L. Brylinski, Loop Spaces, characteristic classes, and geometric quantization (1993), Birkhäuser-Verlag.
• N.E. Hurt, Geometric Quantization in Action (1982), Reidel Company.
• Bertram Kostant, Quantization and unitary representations. I. Prequantization, Lecture Notes in Math. 170, p. 87-208, 1970.
• Paulette Libermann and Charles-Michel Marle, Symplectic geometry and analytical mechanics (1987), Reidel Company.
• Barrett O'Neill, Semi-Riemannian geometry - With applications to relativity (1983), Academic Press.
• M. Puta, Hamiltonian Mechanical Systems and Geometric Quantization (1993), Kluwer Publications.
• J.H. Rawnsley, A nonunitary pairing of polarizations for the Kepler problem, Trans. Amer. Math. Soc., 250 p. 167-180, 1979.
• J. Sniaticky, Geometric Quantization and Quantum Mechanics (1980) Springer-Verlag.
• Nicholas Woodhouse, Geometric quantization (1980) Oxford University Press.

Es ist beabsichtigt, zur Vorlesung Lecture Notes (in Latex) zu schreiben, die Schritt für Schritt bereitgestellt und verbessert werden soll, und über die auch in der Vorlesung diskutiert werden kann. Die Lecture Notes enthält auch Ergänzungen und Beispiele, die in der Vorlesung nicht behandelt werden. Die Notes könnzen zudem durch einige der in den Übungen eingereichten Ausarbeitungen (siehe unten) erweitert werden.
Als Ausgangspunkt dient ein handschriftliches Manuskript zu der früheren Vorlesung "Geometric Quantization 09/10" .

Übungen

Die Teilnehmer reichen Ausarbeitungen zu selbstgewählten Problemen, Themen oder Beispielen ein, die gegebenenfalls in das Skript zur Vorlesung integriert werden. In der Vorlesung und in der Übungsstunde werden Vorschläge zu möglichen Themen gemacht. In der Übungsstunde werden die Ausarbeitungen diskutiert und gelegentlich auch vorgetragen.
Eine solche Ausarbeitung als Übungsleistung soll neben der ausführlichen Erklärung der Problemstellung mit einer Motivation für das Problem beginnen und erläutern, wie das Problem in die Vorlesung oder zu anderen Entwicklungen passt. Sie sollte mit einem Ausblick enden.

Für:

Interested students of physics or of mathematics.
If you plan to participate, please contact me (martin@schottenloher.de). We need to know the number of participants. Furthermore, you might outline your main interests in the course and your prerequisites as well as your language preference. A registration will become necessary in case the course has to be delivered online.