Quantum field theory and renormalization a la Stuckelberg-Petermann-Epstein-Glaser

Virginia Gali

Published 2016-09-29Version 1

The objective of this thesis is to analyze certain results presented by Nguyen Viet Dang in his article on the extension of distributions on Riemannian manifolds (cf. [5]). Some of his proofs were thoroughly revised. In addition, corrections or more detailed descriptions and explanations were added to them. In the present work, we study the renormalization of perturbative quantum field theory (pQFT) as a problem of extension of distributions originally defined on the complement of a closed set in a manifold. Our approach is simpler in the case of locally Euclidean quantum field theories (QFT). There- fore, we choose to work with spacetimes which are d-dimensional Riemannian manifolds (instead of pseudo-Riemannian manifolds). This has the advantage that we deal only with the Green functions and there are no time ordered products of fields. The geometric view also favors the study of renormalization in coordinate space. This is crucial for theories with curved spacetime backgrounds; in such scenarios, translation invariance is lost and the study of renormalization in momentum space is not possible. In addition, we do not specify the theory to which the Green functions belong to, so our study separates the problem from particular models of pQFT.