Renormalization of quantum field theory on curved space-times, a causal approach

TL;DR

This thesis rigorously develops a perturbative quantum field theory on curved space-times using a causal approach, extending previous methods to clarify mathematical details and renormalization ambiguities without gauge invariance.

Contribution

It provides a complete, rigorous formulation of perturbative quantum field theory on curved space-times, addressing unresolved mathematical issues and detailing renormalization counterterms and ambiguities.

Findings

Clarified the geometry of wave front sets of n-point functions

Resolved mathematical questions in the renormalization process

Provided detailed analysis of counterterms and ambiguities

Abstract

The subject of the thesis is the construction of a perturbative quantum theory of interacting fields on a curved space-time, following a point of view pioneered by Stueckelberg and Bogoliubov and developed by Epstein-Glaser on the flat Minkowski space-time. In 2000, a breakthrough was done by Brunetti and Fredenhagen who were able to extend the Epstein-Glaser theory by exploiting the point of view developed by Radzikowski to define quantum states on a curved space-time in terms of wave-front sets. These results were further extended by Fredenhagen, Brunetti, Hollands, Wald, Rejzner, etc. to Yang-Mills fields and the gravitation. However, even for theories without gauge invariance, many mathematical details were left unexplored and unquestioned. Our task was precisely to derive fully rigorously this theory in the case there is no gauge invariance. We propose in our work a complete review of the result, solving numerous questions, adding many new results around this program and, eventually, giving more precise details on the counterterms and ambiguities in the renormalization process, and a deeper understanding of the geometry of the wave front set of the n-point functions.