Nonlocal Extension of the Borchers Classes of Quantum Fields

TL;DR

This paper extends the concept of Borchers classes to nonlocal quantum fields, demonstrating that fields within the same class share the same S-matrix and establishing foundational properties for nonlocal quantum field theories.

Contribution

It introduces a generalized equivalence relation for nonlocal quantum fields, extending Borchers classes to include fields with singular ultraviolet behavior, and proves key properties like transitivity and S-matrix invariance.

Findings

Extended Borchers classes accommodate nonlocal fields with singular ultraviolet behavior.

All fields in the same extended class have identical S-matrices.

The asymptotic commutativity condition ensures key symmetries in nonlocal QFT.

Abstract

We formulate an equivalence relation between nonlocal quantum fields, generalizing the relative locality which was studied by Borchers in the framework of local QFT. The Borchers classes are shown to allow a natural extension involving nonlocal fields with arbitrarily singular ultraviolet behavior. Our consideration is based on the systematic employment of the asymptotic commutativity condition which, as established previously, ensures the normal spin and statistics connection as well as the existence of PCT symmetry in nonlocal field theory. We prove the transitivity of the weak relative asymptotic commutativity property generalizing Jost-Dyson's weak relative locality and show that all fields in the same extended Borchers class have the same S-matrix.