Short-distance analysis for algebraic euclidean field theory

TL;DR

This paper explores how short-distance limits in Euclidean quantum field theories relate to their Minkowski counterparts, extending the algebraic approach to include Euclidean models and their scaling behaviors.

Contribution

It establishes a connection between Euclidean and Minkowski short-distance limits, enabling analysis of Euclidean models' behavior in Minkowski space.

Findings

Proves Euclidean short-distance limits imply Minkowski limits.

Shows the correspondence between Euclidean and Minkowski theories under scaling.

Provides a framework for analyzing confinement and particle content.

Abstract

Recently D. Buchholz and R. Verch have proposed a method for implementing in algebraic quantum field theory ideas from renormalization group analysis of short-distance (high energy) behavior by passing to certain scaling limit theories. Buchholz and Verch distinguish between different types of theories where the limit is unique, degenerate, or classical, and the method allows in principle to extract the `ultraparticle' content of a given model, i.e. to identify particles (like quarks and gluons) that are not visible at finite distances due to `confinement'. It is therefore of great importance for the physical interpretation of the theory. The method has been illustrated in a simple model in with some rather surprising results. This paper will focus on the question how the short distance behavior of models defined by euclidean means is reflected in the corresponding behavior of their Minkowski counterparts. More specifically, we shall prove that if a euclidean theory has some short distance limit, then it is possible to pass from this limit theory to a theory on Minkowski space, which is a short distance limit of the Minkowski space theory corresponding to the original euclidean theory.