Wintersemester 2017/18: Seminar Geometrische Quantisierung

Prof. Dr. Katrin Wendland
Dr. Santosh Kandel

Wann und wo: Di 14 - 16, SR 404, Eckerstr. 1

Vorbesprechung: Do 13.07.2017, 14:00, SR 403, Eckerstrasse 1.
Um teilzunehmen, kommen Sie bitte in die Vorbesprechung des Seminares; eine Teilnehmerliste wird nicht vorab ausliegen.

Topic:
Classical physics does not predict the behaviour of atoms and molecules correctly. Indeed, classically, Coulomb's law implies that the electron of the hydrogen atom should orbit around the proton, and thus the electron continuously radiates energy and causes the hydrogen atom to collapse. This contradicts the observed stability of the hydrogen atom. One of the major triumphs of quantum mechanics is its explanation for the stability of atoms.
Mathematically, a classical mechanical system can be described by a so-called symplectic manifold M called the state space, and the observables are functions on M. A quantum mechanical system, on the other hand, is described by a Hilbert space, and the observables are "operators" on this Hilbert space. A process which roughly associates to a classical theory a quantum theory is called "quantization". Ideally, one would like to associate to each classical observable a quantum observable, but it is impossible to achieve this: there are no go theorems. In practice, one has to lower one's expectation so that a reasonable quantization process can be constructed.
The goal of this seminar is to study one particular method of quantization called geometric quantization. Position space quantization, momentum space quantization and holomorphic quantization are particular instances of geometric quantization. In geometric quantization, one constructs the Hilbert space from the square integrable sections of a so-called complex line bundle over M.
Within the seminar, we will motivate and introduce the mathematical notions that are needed for geometric quantization, starting from Newtonian mechanics. Background knowledge from physics is helpful but is not required.

Literatur:
Die Links führen auf Webseiten, von denen aus dem Universitätsnetz die jeweiligen Referenzen zugänglich sind. Falls kein Link gesetzt ist, finden Sie die Referenz in der Bibliothek des Mathematischen Institutes Freiburg.

• S. Bates and A. Weinstein, Lectures on the geometry of quantization , Berkeley Mathematics Lecture Notes 8, AMS (1997)
• Brian C. Hall, Quantum theory for mathematicians , volume 267 of Graduate Texts in Mathematics, Springer, New York, 2013
• William D. Kirwin and Siye Wu, Geometric quantization, parallel transport and the Fourier transform , Comm. Math. Phys. 266(3):577-594, 2006.
• Bertram Kostant, Quantization and unitary representations. I. Prequantization , pages 87-208. Lecture Notes in Math., Vol. 170, 1970.
• John M. Lee, Introduction to smooth manifolds , volume 218 of Graduate Texts in Mathematics. Springer, New York, second edition, 2013.
• Paulette Libermann and Charles-Michel Marle, Symplectic geometry and analytical mechanics , volume 35 Mathematics and its Applications. 1987.
• Barrett O'Neill, Semi-Riemannian geometry - With applications to relativity, volume 103 of Pure and Applied Mathematics. Academic Press, Inc. Harcourt Brace Jovanovich, Publishers, New York, 1983.
• J.H. Rawnsley, A nonunitary pairing of polarizations for the Kepler problem , Trans. Amer. Math. Soc., 250:167-180, 1979.
• Ana Cannas da Silva, Lectures on symplectic geometry , volume 1764 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2001.
• Nicholas Woodhouse, Geometric quantization, The Clarendon Press, Oxford University Press, New York, 1980, Oxford Mathematical Monographs
• Siye Wu, Projective flatness in the quantisation of bosons and fermions , J. Math. Phys., 56(7), 2015.

Vortragsprogamm: hier

Tutorium: Santosh Kandel