String chopping and time-ordered products of linear string-localized quantum fields

TL;DR

This paper addresses the challenge of defining time-ordered products for string-localized quantum fields by showing most string configurations can be ordered, aiding in the renormalization process in Epstein--Glaser scheme.

Contribution

It demonstrates that nearly all string configurations in quantum field theory can be dissected into parts that can be chronologically ordered, advancing the understanding of renormalization with string-localized fields.

Findings

Most string configurations can be dissected into orderable pieces.

Fixes the definition of time-ordered products outside a nullset.

Provides groundwork for extending definitions across nullsets.

Abstract

For a renormalizability proof of perturbative models in the Epstein--Glaser scheme with string-localized quantum fields, one needs to know what freedom one has in the definition of time-ordered products of the interaction Lagrangian. This paper provides a first step in that direction. The basic issue is the presence of an open set of $n$-tuples of strings which cannot be chronologically ordered. We resolve it by showing that almost all such string configurations can be dissected into finitely many pieces which can indeed be chronologically ordered. This fixes the time-ordered products of linear field factors outside a nullset of string configurations. (The extension across the nullset, as well as the definition of time-ordered products of Wick monomials, will be discussed elsewhere.)