Constructive aspects of string-localized quantum field theory

Abstract

String-localized quantum field theory (SLFT) as an alternative to gauge theory has been under investigation for the last decade because of its conceptual benefits. This thesis is dedicated to the development and investigation of constructive tools for both perturbative and non-perturbative aspects of SLFT. On the non-perturbative side, we provide explicit formulas for the two-point function and the propagator of the so-called “escort field” of the string-localized photon potential. These are needed to describe the dressing of Dirac fields in quantum electrodynamics (QED) with photon clouds and have a potential application in the scattering theory of these dressed Dirac fields. We continue to derive the two-point functions and propagators of string-localized potentials starting from the two-point function of the photon escort field. We also prove no-go results for similar non-perturbative dressing constructions in models with massless self-interacting fields. In QED, these constructions have been implemented by Mund, Rehren and Schroer and work well. We show that such a Mund-Rehren-Schroer construction already fails at an early stage for massless Yang-Mills theory and the graviton self-coupling. The main focus of this thesis is on perturbation theory with string-localized fields. We propose a setup for string-localized perturbation theory and outline a new method to define the time-ordering operation if string-localized fields are involved. We determine the wavefront set of string-localized propagators and show that the divergences in loop graph contributions stay pure ultraviolet divergences: the new singularities arising from the string-localization are harmless in the sense of distributions. Furthermore, we investigate and compare different methods to reduce the renormalization freedom in SLFT. These considerations can be seen as a first step towards a full axiomatic construction of the string-localized S-matrix in a Bogoliubov-Epstein-Glaser spirit. Finally, we discuss examples of perturbative constructions in SLFT in low orders of perturbation theory.