Mathematisches Seminar: Functional integration

Time / location:      tba / online via zoom

First meeting: Mon 09/11/2020 at 14:15

(Discussion of topics and assignment of talks)

If you plan to attend, please register by email until 04/11/2020.

Talks can be given in English or German!


Synopsis

Mark Kac, who was inspired by Richard Feynman and Norbert Wiener, constructed 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 theory 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 ground-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 in the programmes MSc Mathematics and TMP

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.