Rigorous Definition of Quantum Field Operators in Noncommutative Quantum Field Theory

TL;DR

This paper establishes a rigorous framework for defining quantum field operators in noncommutative quantum field theory using a star-product, facilitating the derivation of the Wightman reconstruction theorem.

Contribution

It provides a rigorous mathematical construction of quantum fields in NC QFT with star-products, enabling foundational theorems to be extended to noncommutative settings.

Findings

Constructed the space of quantum fields with star-product multiplication.

Extended the Wightman reconstruction theorem to NC QFT.

Provided a rigorous foundation for noncommutative quantum field operators.

Abstract

The space, on which quantum field operators are given, is constructed in any theory, in which the usual product between test functions is substituted by the $\star$-product (the Moyal-type product). The important example of such a theory is noncommutative quantum field theory (NC QFT). This construction is the key point in the derivation of the Wightman reconstruction theorem.