On the Construction of Quantum Field Theories with Factorizing S-Matrices

TL;DR

This thesis introduces a new method for constructing interacting quantum field theories in two dimensions using prescribed factorizing S-matrices, focusing on algebraic and scattering properties of the models.

Contribution

It develops a novel construction approach for quantum field theories based on factorizing S-matrices, including explicit models like the Sinh-Gordon and Ising models, and proves their key properties.

Findings

Constructed wedge-localized quantum fields for a large class of S-matrices.

Proved the existence of local observables using the modular nuclearity condition.

Demonstrated asymptotic completeness and solved the inverse scattering problem for the models.

Abstract

The subject of this thesis is a novel construction method for interacting relativistic quantum field theories on two-dimensional Minkowski space. The input in this construction is not a classical Lagrangian, but rather a prescribed factorizing S-matrix, i.e. the inverse scattering problem for such quantum field theories is studied. For a large class of factorizing S-matrices, certain associated quantum fields, which are localized in wedge-shaped regions of Minkowski space, are constructed explicitely. With the help of these fields, the local observable content of the corresponding model is defined and analyzed by employing methods from the algebraic framework of quantum field theory. The abstract problem in this analysis amounts to the question under which conditions an algebra of wedge-localized observables can be used to generate a net of local observable algebras with the right physical properties. The answer given here uses the so-called modular nuclearity condition, which is shown to imply the existence of local observables and the Reeh-Schlieder property. In the analysis of the concrete models, this condition is proven for a large family of S-matrices, including the scattering operators of the Sinh-Gordon model and the scaling Ising model as special examples. The so constructed models are then investigated with respect to their scattering properties. They are shown to solve the inverse scattering problem for the considered S-matrices, and a proof of asymptotic completeness is given.