Learning Algebraic Quantum Field Theory

Lecture Operator-Algebraic Quantum Field Theory

In the summer term of 2025, I am giving a lecture on an introduction to algebraic quantum field theory (10 ECTS, 4 hours of lectures + 2 hours of tutorials per week).

Preprequisites: The target audience consists of mathematics and physics students alike, but the focus is on the mathematics students that already know about functional analysis (in particular, Hilbert space theory including measurable functional calculus for unbounded operators, and the elementary theory of C*- and von Neumann algebras), but not so much about quantum physics.

Planned Contents: The precise contents are not completely fixed yet, but a rough outline of the first chapters is the following:

• Chapter 1: The general operator-algebraic framework of quantum physics. C*-algebras and their states, probability interpretation. Weyl algebras. Symmetries and symmetry groups. C*-dynamical systems and ground states.
• Chapter 2: The operator-algebraic framework of quantum field theory. Minkowski  spacetime and Poincaré symmetry. Vacuum states and their energy-momentum spectrum. Locality and the net structure of the algebra of observables. The Reeh-Schlieder Theorem and its consequences.
• Chapter 3: Free quantum field theories. Free Klein-Gordon and Dirac field. Fock representations of Weyl and CAR algebras. Comparison with the Wightman formalism.
• Chapter 4: Main structural results of AQFT.
• Chapter 5: Scattering Theory.
• Chapter 6: Modular Theory in QFT.

Lecture notes, exercises and solutions will be provided on StudOn.

While these lectures are primarily for FAU students, it is also possible to follow them as an external student. If you are interested to follow the lectures as an external student, please get in touch via email.