Axiomatic formulations of nonlocal and noncommutative field theories

TL;DR

This paper explores the mathematical foundations of nonlocal and noncommutative quantum field theories, clarifying conditions for CPT symmetry, spin-statistics, and scattering theory using functional analysis and analytic functionals.

Contribution

It provides a comprehensive analysis of axiomatic formulations, linking asymptotic commutativity to Green's function regularity, and extends Wightman axioms to noncommutative settings.

Findings

Clarified the relation between asymptotic commutativity and Green's functions.

Extended Wightman axioms to noncommutative field theories.

Discussed CPT and spin-statistics theorems in noncommutative frameworks.

Abstract

We analyze functional analytic aspects of axiomatic formulations of nonlocal and noncommutative quantum field theories. In particular, we completely clarify the relation between the asymptotic commutativity condition, which ensures the CPT symmetry and the standard spin-statistics relation for nonlocal fields, and the regularity properties of the retarded Green's functions in momentum space that are required for constructing a scattering theory and deriving reduction formulas. This result is based on a relevant Paley-Wiener-Schwartz-type theorem for analytic functionals. We also discuss the possibility of using analytic test functions to extend the Wightman axioms to noncommutative field theory, where the causal structure with the light cone is replaced by that with the light wedge. We explain some essential peculiarities of deriving the CPT and spin-statistics theorems in this enlarged framework.