Von Neumann algebras in field theory

Abstract

In this thesis, I show that modular flows of generic excited states in a quantum field theory can be studied by restricting attention to a special class of states whose associated modular flow can be characterised easily; this follows from a certain theorem involving the invertible group of operators in a general von Neumann algebra, namely that such operators are a dense subset of the algebra in the strong operator topology. I also develop tools to compute these flows using a novel perturbation expansion : the structure of the terms appearing in the expansion is made manifest to all orders in the expansion. Finally, I write down an effective action for a general charged fluid which has an anomalously broken symmetry; the novelty here is that this effective action is local (previous treatments gave a non-local effective action for such a fluid). This construction also gives a simple explanation for various puzzling coincidences which have been reported in the literature before.