Euclidean Epstein-Glaser Renormalization

TL;DR

This paper extends Epstein-Glaser renormalization to Euclidean quantum field theories by developing a recursive construction of Euclidean time-ordered products, adapting the causal approach to the Euclidean setting.

Contribution

It generalizes the Epstein-Glaser recursive construction for Minkowski space to Euclidean space, introducing Euclidean causality and handling the lack of causal structure.

Findings

Constructed Euclidean time-ordered products for scalar QFTs.

Modified Epstein-Glaser method for Euclidean signature.

Established a Euclidean causality condition.

Abstract

In the framework of perturbative Algebraic Quantum Field Theory (pAQFT), recently introduced by Brunetti, Duetsch and Fredenhagen, I give a general construction of so-called "Euclidean time-ordered products", i.e. algebraic versions of the Schwinger functions, for scalar quantum field theories on spaces of Euclidean signature. This is done by generalizing the recursive construction of time-ordered products by Epstein and Glaser, originally formulated for quantum field theories on Minkowski space (MQFT). An essential input of Epstein-Glaser renormalization is the causal structure of Minkowski space. The absence of this causal structure in the Euclidean framework makes it necessary to modify the original construction of Epstein and Glaser at two points. First, the whole construction has to be performed with an only partially defined product on (interaction-) functionals. This is due to the fact that the fundamental solutions of the Helmholtz operator of EQFT have a unique singularity structure, i.e. they are unique up to a smooth part. Second, one needs to (re-)introduce a (rather natural) "Euclidean causality" condition for the recursion of Epstein and Glaser to be applicable.