On the Borchers Class of a Non-Commutative Field

TL;DR

This paper extends Borchers' class of local quantum fields to non-commutative scalar fields by introducing a new equivalence relation based on asymptotic commutativity, generalizing the concept of relative locality.

Contribution

It proposes a novel equivalence class framework for non-commutative quantum fields using asymptotic commutativity, expanding Borchers' class concept to nonlocal quantum field theories.

Findings

Defined a new equivalence relation for non-commutative fields.

Generalized Borchers' class to nonlocal quantum field theories.

Focused on scalar fields with space-space non-commutativity.

Abstract

In this paper, we arrive at the notion of equivalence classes of a non-commutative field exploring some ideas by Soloviev to nonlocal quantum fields. Specifically, an equivalence relation between non-commutative fields is formulated by replacing the weak relative locality condition by a weak relative asymptotic commutativity property, generalizing the notion of relative locality proved by Borchers in the framework of local QFT. We restrict ourselves to the simplest case of a scalar field theory with space-space non-commutativity.