Towards an Axiomatic Formulation of Noncommutative Quantum Field Theory

TL;DR

This paper develops an axiomatic framework for noncommutative quantum field theory by defining new Wightman functions compatible with twisted Poincaré symmetry, and proves CPT and spin-statistics theorems in this setting.

Contribution

It introduces a new axiomatic approach using Wightman functions with the star-product, extending the foundational structure of NC QFT and proving key theorems.

Findings

Proved CPT theorem for space-space noncommutative spacetime.

Established spin-statistics relation for scalar fields in NC QFT.

Formulated Wightman functions compatible with twisted Poincaré symmetry.

Abstract

We propose new Wightman functions as vacuum expectation values of products of field operators in the noncommutative space-time. These Wightman functions involve the $\star$-product among the fields, compatible with the twisted Poincar\'e symmetry of the noncommutative quantum field theory (NC QFT). In the case of only space-space noncommutativity ($\theta_{0i}=0$), we prove the CPT theorem using the noncommutative form of the Wightman functions. We also show that the spin-statistics theorem, demonstrated for the simplest case of a scalar field, holds in NC QFT within this formalism.