# Henning Bostelmann

# A Brief Introduction to Axiomatic Quantum Field Theory

** Abstract: **

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.