On-shell extension of distributions

Abstract

We consider distributions on ${\mathbb{R}}^n\setminus 0$ which satisfy a given set of partial differential equations and provide criteria for the existence of extensions to ${\mathbb{R}}^n$ that satisfy the same set of equations on ${\mathbb{R}}^n$. We use the results to construct distributions satisfying specific renormalisation conditions in the Epstein and Glaser approach to perturbative quantum field theory. Contrary to other approaches, we provide a unified apporach to treat Lorentz covariance, invariance under global gauge group and almost homogeneity, as well as discrete symmetries. We show that all such symmetries can be recovered by applying a linear map defined for all degrees of divergence. Using similar techniques, we find a relation between on-shell and off-shell time-ordered products involving higher derivatives of the fields.

23 p, Lemma 3.5 corrected, published version

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• Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity
• The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory
• On the assertion that PCT violation implies Lorentz non-invariance
• Improved Epstein-Glaser Renormalization II. Lorentz invariant framework
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Cited by in corpus (2)

• Stora's fine notion of divergent amplitudes
• Local operators in the Sine-Gordon model: ${\partial }_{\mu }\varphi {\partial }_{\nu }\varphi$ and the stress tensor