Euclidean Epstein-Glaser renormalization

Journal Article · Thu Oct 15 04:00:00 EDT 2009 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3202415· OSTI ID:21294396

Keller, Kai J ^[1]

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1. II. Institute for Theoretical Physics, Hamburg University, Luruper Chaussee 149, 22761 Hamburg (Germany)

In the framework of perturbative algebraic quantum field theory recently developed by Brunetti, Duetsch, and Fredenhagen (http://arxiv.org/abs/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 [Epstein and Glaser, Ann. Inst. Henri Poincare 19, 211 (1973)]. 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 (-{delta}+m{sup 2}) of Euclidean quantum field theory 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.

• OSTI ID: 21294396
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 10 Vol. 50; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
ALGEBRA
AXIOMATIC FIELD THEORY
CAUSALITY
CONSTRUCTIVE FIELD THEORY
EUCLIDEAN SPACE
FUNCTIONALS
MATHEMATICAL SOLUTIONS
MINKOWSKI SPACE
PERTURBATION THEORY
RENORMALIZATION
SCHWINGER SOURCE THEORY
SINGULARITY
SPACE-TIME