Final Report Summary - QC&C (Quantum fields and Curvature--Novel Constructive Approach via Operator Product Expansion)
The purpose of this project was to enhance the theoretical understanding of quantum field theories. Such theories were originally devised to describe elementary particles, but many of their underlying ideas have penetrated deeply into other fields of physics, or mathematics. We set as our goal to investigate features in such theories which broadly speaking focus on their „algebraic backbone,“ and which manifest themselves for instance in the „operator product expansion“, as well as related structures. Such expansions are not only useful in practice as an approximation method, but have a wider conceptual significance as a possible mathematical foundation of such theories. A key focus was on detailed investigations of this expansion as well as related concepts from various viewpoints. These, technically highly complex, investigations have led, for instance, to:
(i) a detailed understanding of the convergence and associativity properties of such expansions,
(ii) a self-consistent procedure for determining the coefficients in them and how they can serve as a constructive tool for quantum field theories,
(iii) a detailed understanding of the relation of such expansions to renormalization group flows,
(iv) an understanding of the interplay of such expansion with gauge invariance,
(v) an understanding of the significance of algebraic structures in field theory from a geometric viewpoint.
(i) a detailed understanding of the convergence and associativity properties of such expansions,
(ii) a self-consistent procedure for determining the coefficients in them and how they can serve as a constructive tool for quantum field theories,
(iii) a detailed understanding of the relation of such expansions to renormalization group flows,
(iv) an understanding of the interplay of such expansion with gauge invariance,
(v) an understanding of the significance of algebraic structures in field theory from a geometric viewpoint.