Polarization-Free Generators and the S-Matrix

TL;DR

This paper investigates polarization-free generators in quantum field theories, showing they exist in all dimensions but are mainly useful in two-dimensional models where they do not allow particle production.

Contribution

It demonstrates the existence and properties of polarization-free generators across dimensions and clarifies their limitations in higher-dimensional interacting theories.

Findings

Existence of polarization-free generators in all spacetime dimensions.

In higher dimensions, these generators have delicate domain properties.

In two dimensions, they support non-trivial interactions without particle production.

Abstract

Polarization-free generators, i.e. ``interacting'' Heisenberg operators which are localized in wedge-shaped regions of Minkowski space and generate single particle states from the vacuum, are a novel tool in the analysis and synthesis of two-dimensional integrable quantum field theories. In the present article, the status of these generators is analyzed in a general setting. It is shown that such operators exist in any theory and in any number of spacetime dimensions. But in more than two dimensions they have rather delicate domain properties in the presence of interaction. If, for example, they are defined and temperate on a translation-invariant, dense domain, then the underlying theory yields only trivial scattering. In two-dimensional theories, these domain properties are consistent with non-trivial interaction, but they exclude particle production. Thus the range of applications of polarization-free generators seems to be limited to the realm of two-dimensional theories.