Henning Bostelmann

A Brief Introduction to Axiomatic Quantum Field Theory


Quantum field theory aims at unifying quantum theory with the principles of (special) relativity. Its consistent mathematical description, beyond the level of formal perturbation theory, remains a challenging problem. On Minkowski space, two mathematical frameworks have emerged for the description of quantum field theory: First, the Wightman axioms, which focus on the more familiar notion of quantum fields and deal with unbounded operators; second, the more abstract Haag-Kastler setting, which takes algebras of bounded operators as its fundamental objects. This lecture gives a brief introduction to each of the two frameworks, motivates their conceptual foundations, and sketches basic techniques and results. Also, we discuss how to pass from the Wightman to the Haag-Kastler setting and vice versa.