The aim of this module is to extend knowledge of quantum mechanics from undergraduate courses by emphasizing its overall mathematical structure, as well as its applications to a wide range of topics in mathematical physics. We will introduce an abstract framework for quantum theory, based on tools from the theory of operator algebras, which is general enough to describe all its subareas: quantum mechanics, quantum statistical mechanics (including thermodynamics in infinite volume), quantum field theory on Minkowski space and on curved spacetimes, as well as including classical probability theory as a special case.

Spring 2021 (Monday, January 25 to Friday, March 19; Monday, April 26 to Friday, May 7)

Students should have attended either a first course in quantum mechanics, or have some knowledge of operator theory (on infinite dimensional Hilbert spaces); ideally both, but participants with experience in either area are welcome.

For some lectures, familiarity with other topics in Theoretical/Mathematical Physics will be helpful, such as Genera Relativity / differential geometry.

The presentation style is expository, i.e., each lecture will give an overview of a subject area, rather than working out the technical details. A tentative list of topics is:

* Probability theory

* Basics of quantum theory

* Algebras, states, GNS representation

* Thermodynamics and the KMS condition

* Free quantum fields on Minkowski space

* Linear quantum fields on curved spacetimes

* Frameworks for interacting quantum field theories