Title: Algebraic Quantum Field Theory: Introduction and Applications of the Lax-Phillips Theorem
Abstract: Algebraic quantum field theory is one of the main attempts to provide a formal mathematical framework for quantum field theory. We give an introduction to algebraic quantum field theory, whose core idea is to assign to every space-time region an algebra of local observables. These algebras, who are assumed to be von Neumann algebras, should obey the Haag-Kastler axioms, describing how the algebras assigned to different space-time regions should interact with each other. We show how to translate the Haag-Kastler axioms from von Neumann algebras to the simpler concept of standard subspaces. This leads us to the investigation of one-parameter semigroups of unitary endomorphisms of standard subspaces, which we analyze through the lens of a real version of the classical Lax-Phillips Theorem, originally developed in the context of scattering theory.