Mathematisches Seminar: Functional integration in quantum physics

Time / location: Mi 8 – 10, B 039

First meeting: Fri 17/04/2015 at 09:30 in B 448

(Discussion of topics and assignment of talks)

Talks can be given in English or German!


Synopsis

Following Mark Kac, who was inspired by Richard Feynman and Norbert Wiener, we construct a Brownian-motion representation of Schrödinger semigroups. Such representations, which also go under the name functional integrals (or path integrals in physics), are a very useful technical tool in analysis and probability theory. In fact, they allow to attack spectral problems of Schrödinger operators with methods from probability and, conversely, problems in probability theory with methods from operator theory. Applications in mathematical physics are numerous and include a simple proof of the diamagnetic inequality, the existence and self-averaging of the integrated density of states for random Schrödinger operators and gound-state properties of the Fröhlich polaron.

Prerequisites

Functional analysis, basics of the theory of self-adjoint operators in Hilbert spaces and basics of probability theory

Audience

Students of the programmes TMP and M.Sc. Mathematics

Suggested reading

• B. Simon, Functional integration and quantum physics, Academic Press, New York, 1979.
• J. Lőrinczi, F. Hiroshima and V. Betz, Feynman-Kac-type theorems and Gibbs measures on path space, de Gruyter, Berlin, 2011, Part I.
• A.-S. Sznitman, Brownian motion, obstaces and random media, Springer, Berlin, 1998, Part I.