Gauge theories in local causal perturbation theory

TL;DR

This thesis develops a rigorous framework for quantum gauge theories using local causal perturbation theory, emphasizing gauge invariance via BRS formalism, and constructs local observables and interacting fields systematically.

Contribution

It introduces a consistent method to define and normalize time ordered products with derivatives, and formulates a cohomological approach to local observables in gauge theories.

Findings

Constructed local interacting field operators perturbatively.

Established a cohomological algebra of local observables.

Compared results with existing literature on QED and Yang-Mills.

Abstract

In this thesis quantum gauge theories are considered in the framework of local, causal perturbation theory. Gauge invariance is described in terms of the BRS formalism. Local interacting field operators are constructed perturbatively and field equations are established. A nilpotent BRS transformation is defined on the local algebra of fields. It allows the definition of the algebra of local observables as an operator cohomology. This algebra of local observables can be represented in a Hilbert space. The interacting field operators are defined in terms of time ordered products of free field operators. For the results above to hold the time ordered products must satisfy certain normalization conditions. To formulate these conditions also for field operators that contain a spacetime derivative a suitable mathematical description of time ordered products is developed. Among the normalization conditions are Ward identities for the ghost current and the BRS current. The latter are generalizations of a normalization condition that is postulated by D"utsch, Hurth, Krahe and Scharf for Yang-Mills theory. It is not yet proven that this condition has a solution in every order. All other normalization conditions can be accomplished simultaneously. A principle for the correspondence between interacting quantum fields and interacting classical fields is established. Quantum electrodynamics and Yang-Mills theory are examined and the results are compared with the literature.