Euclidean Epstein–Glaser renormalization

Abstract

In the framework of perturbative Algebraic Quantum Field Theory (pAQFT) recently developed by Brunetti, Dütsch, and Fredenhagen (arXiv:0901.2038), 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 −∆ + m 2 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. * Electronic address: kai.johannes.keller@desy.de 24 A. The Wave Front Set of the Dirac Delta Distribution25

The algebra of Wick polynomials of a scalar field on a Riemannian manifold

“…charged Dirac fields, [Za15]. Finally it is worth mentioning that our results are complementary to those obtained by Keller in [Kel09,Kel10] in the analysis of Epstein-Glaser renormalization in an Euclidean framework. We remark that the net of algebras that we obtain at the end of our construction seems to bear similarities with factorization algebras and, in our opinion, it would be worth making a detailed comparison along the same lines of [GR17,BPS19] in the Lorentzian setting.…”

“…The key observation originates from the support factorization axiom (5) in Definition 31, which entails that the E-product of n + 1 Wick powers is completely determined on M n+1 \ Diag(M n+1 ) by its prescription on n factors. This was first observed in [Kel09,Kel10] and it is the Riemannian counterpart of the same procedure followed in causal perturbation theory on a globally hyperbolic spacetime.…”

“…Moreover we call P loc : BkgG → Vec the functor such that, for any Construction of the map R V . Our starting point is the work of [Kel09]. In this paper it is argued that the map R V should be realized as the algebraic version of the formal path integral formula…”

Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory

“…Starting from the early seventies a few works in this direction have appeared in the literature [OS73,OS75] and, despite most of the efforts went towards formulating algebraic quantum field theory on Lorentzian backgrounds, it is clear that most of the ideas, of the technique and of the structural aspects admit a well-defined Euclidean counterpart. A notable example in this direction are the recent works by Keller on the formulation of Euclidean Epstein-Glaser renormalization [Kel09,Kel10], see also [Sch98,Wa79].…”

Euclidean Configuration Space Renormalization, Residues and Dilation Anomaly

“…It is therefore not surprising that this approach attracts more attention now than half a century ago when it was originally conceived -see e.g. EGP,FHS,K,N,K10,B]. Papers like [BBK] reflect, surely, later developments in both renormalization theory (Kreimer's Hopf algebra structure -see e.g.…”